11institutetext: Vanderbilt University 22institutetext: Dept. of Mathematics,
1326 Stevenson Center, Station B 407807, Nashville, TN 37240. 22email:
<EMAIL_ADDRESS>33institutetext: University of Virginia
44institutetext: Dept. of Biomedical Engineering, Thornton Hall, 351 McCormick
Road, Charlottesville, VA 22904. 44email<EMAIL_ADDRESS>55institutetext:
University of Virginia 66institutetext: Dept. of Biomedical Engineering and
Electrical and Computer Engineering, Thornton Hall, 351 McCormick Road,
Charlottesville, VA 22904. 66email<EMAIL_ADDRESS>
# Data representation with optimal transport

Rocío Díaz Martín0000-0002-3732-6296 and
Ivan Vladimir Medri0000-0003-2419-2193 and
Gustavo Kunde Rohde0000-0003-1703-9035

##  1 Introduction

Finding mathematical formulas for representing data (such as signals, images,
vectors, measures, etc.) has long interested mathematicians, engineers, and
scientists. The underlying premise is that certain mathematical
representations can facilitate solutions to various problems. To achieve this,
operators that map the original space of the objects of interest into a space
of representations are often defined and referred to as transforms.

For instance, in the field of Signal Processing Analysis111In signal
processing, the term analysis typically replaces the notion of
representation., there are many examples commonly used: (1) The Fourier
transform represents signals in terms of frequency components, allowing for
the practical solution of problems related to shift-invariant linear
operators, including the heat (diffusion) equation, convolution problems, and
numerous others stein2011fourier . (2) The Laplace transform represents
derivatives in a way that allows the conversion of initial value problems into
algebraic problems siebert1986circuits . (3) In medical imaging, if a function
describes an unknown density in a patient, we can also represent it by the
collection of all the X-ray projections taken at different angles around the
patient, and use the Radon transform for reconstruction purposes
helgason2011integral . (4) The wavelet transform analyzes signals in time by
introducing two new variables, namely scale, and resolution mallat1999wavelet
. The listed mathematical transformations (Fourier, Laplace, Radon, and
Wavelet transform) are linear operators, and thus often fail to deal with the
non-linearities present in modern data science applications. There are some
exceptions to this shortcoming. For example, the scattering transform is non-
linear and has been successfully applied to machine learning applications
mallat2012group .

In this chapter, we will describe a family of non-linear transforms rooted in
the theory of optimal transport (OT) (see also kolouri2016transport ). In
addition to obtaining new representations through these transforms, they will
allow us to create new metrics or distances to compare our original signals or
data (see, for e.g., rubaiyat2020parametric ; shifat2021radon ;
rubaiyat2024end ; basu2014detecting ; kundu2020enabling ).

The theory of optimal transport addresses matching problems between different
configurations of mass by posing a minimization problem sant2015 ;
Villani2003Topics ; Villani2009Optimal . The total cost that results from
solving the optimization problem introduces new tools for comparing signals:
the so-called Wasserstein distances. Since these distances arise from
alignment problems, they are often more ‘natural’ for comparing probability
densities or measures than typical Euclidean distances (i.e.,
$L^{2}$-distances). However, Wasserstein distances are difficult and expensive
to compute.

The transforms we will describe in this chapter serve as a trade-off between
transport theory and classical $L^{2}$-theory. We will interpret our data or
signals as measures and embed the space of measures into an $L^{2}$-space.
This embedding will be a one-to-one transform that relates to optimal
transport. These transport transforms, which we will usually denote by
$\widehat{\cdot}$, will enable us to define metrics in the original space of
measures that retain some natural properties of Wasserstein distances while
also being expressible as a simple Euclidean norm, thus allowing the
computational framework of Hilbert spaces. See Figures 1 and 5.

Figure 1: The space of measures $\mathcal{M}(\Omega)$ defined on
$\Omega\subseteq\mathbb{R}^{d}$ is embedded into a Euclidean space through a
transport transform $\widehat{\cdot}$. By pulling the $L^{2}$-norm back we
define a distance $d(\cdot,\cdot)$ in the space of measures.

Motivating tasks. Let us assume that in a physical system, a density function
(that might represent mass at every point in space) suffers deformations given
by translations and dilations without modifying the total mass. Thus, it would
take the form $f_{(\omega,\tau)}(x)=\omega f(\omega x-\tau)$, where $f$ is a
template configuration and $(\omega,\tau)$ are deformation parameters. Suppose
that we experimentally measure this quantity $f_{(\omega,\tau)}$ obtaining an
estimated density $h$. Some example questions that transport transforms can
help to solve are:

  1. 1.

(Classification.) If the initial configuration $f$ is unknown but lies within
a finite number of template options $\\{f_{k}\\}_{k=1}^{K}$, how can we
identify the class $k$ of our system?

  2. 2.

(Estimation.) If the initial template is known, how can we find the correct
deformation parameters $(\omega,\tau)$?

  3. 3.

(Reconstruction or generation.) If the template $f$ and the deformation
parameters $(\omega,\tau)$ are known, is it possible to reconstruct the
intermediate steps that changed the template $f$ to $f_{(\omega,\tau)}$ in a
‘natural’ way?

The first task is a classification problem where each class is a collection of
functions of the form $\mathbb{S}_{k}:=\\{x\mapsto\omega f_{k}(\omega
x-\tau):\,\omega\in\mathbb{R}_{>0},\tau\in\mathbb{R}\\}$ with $f_{k}$,
$k=1,\dots,K$, fixed templates. Those classes are not linearly separable as
can be seen in the left panel of Figure 2. Nevertheless, if we apply certain
transport transform, the transformed classes take the form
$\widehat{\mathbb{S}_{k}}=\\{\xi\mapsto
a\widehat{f_{k}}(\xi)+b:\,a\in\mathbb{R}_{>0},b\in\mathbb{R}\\}$ (as will be
explained in Section 3) creating half-planes, which are linearly separable in
transform space as shown in the right panel of Figure 2.

Figure 2: (a) Represents three template functions $f_{1},f_{2},f_{3}$. (b)
Represents the translations and dilations of the templates,
$\mathbb{S}_{1},\mathbb{S}_{2},\mathbb{S}_{3}$, in 3D space using three
coordinate values. (c) Represents the transforms of translations and dilations
of the templates,
$\widehat{\mathbb{S}_{1}},\widehat{\mathbb{S}_{2}},\widehat{\mathbb{S}_{3}}$,
in 3D space.

For the second point, when the template $f$ is known, we seek for
$(\omega,\tau)$ so that $f_{(\omega,\tau)}$ best matches the measured signal
$h$. A naive attempt to solve this problem would be to minimize the functional
$J_{1}(\omega,\tau)=\|f_{(\omega,\tau)}-h\|_{2}^{2}$. Nevertheless, this is a
non-convex problem (with respect to $(\omega,\tau)$) and global minima might
be hard to find. With the use of transport transforms, a different functional
$J_{2}(a,b)=\|a\widehat{f}+b-\widehat{h}\|_{2}^{2}$ with
$a=1/\omega,b=\tau/\omega$ can be utilized. Minimizing $J_{2}$ is a convex
problem, a global minimum is easily found, and the parameters $(\omega,\tau)$
can be obtained afterward from $({a},{b})$. Using the injectivity and
properties of the transform, it can be proven that the parameters found with
this method are exactly the ones that make $h=f_{(\omega,\tau)}$.

The third task is an interpolation problem as we want to generate intermediate
signals between the template $f$ and the target $f_{(\omega,\tau)}$. Since the
transport transforms are embeddings in $L^{2}$ spaces, the interpolation can
be obtained as a simple convex combination
$(1-t)\widehat{f}+t\widehat{f_{(\omega,\tau)}}$, for $0\leq t\leq 1$, and then
applying the inverse transform. Figure 3 compares such transition with the
convex combination $(1-t)f+t{f_{(\omega,\tau)}}$ taken directly in the
original signal space.

Figure 3: (Left) Interpolation in original space. A Gaussian function $f$ that
moves to another Gaussian $f_{(\omega,\tau)}$ has intermediate states given by
a combination of two Gaussians. (Right) A Gaussian function $f$ moves to
another Gaussian $f_{(\omega,\tau)}$ using interpolation in transform space.
Intermediate densities behave in accordance with the initial and final
configurations ($f$ and $f_{(\omega,\tau)}$).

The underlying concept among the above-mentioned problems is that a template
function is being altered through mass-preserving transformations. Although we
described our examples using dilations and translations, more complicated
deformations can be considered. Usually, they are associated with changes in
the domain of the functions that are hard to express in a simple form for
classification, estimation, or reconstruction. Transport theory conveniently
captures these deformations, and transport transforms will allow us to express
them in a simplified formulation that can utilize the computational power of
Euclidean space methods. The advantages illustrated above will be detailed in
this chapter. These are not exhaustive; other benefits include the possibility
of performing PCA in transform-space and optimizing costs using gradient
descent, among others.

Signals and images, point clouds, sampled data, density functions, and
measures. The subject matter of this chapter is related to the mathematical
modeling of signals, images, and data in general. We will adopt measure theory
as a unifying framework. Measures $\mu$ generalize considering signals and
images as functions $f_{\mu}$. In many cases, the physical meaning of
$f_{\mu}$ is related to the density of a certain quantity (mass, energy,
attenuation, pressure, etc.222For example, in fluorescence microscopy, the
amount of light (i.e., the number of photons) hitting a detector at location
$x$ is directly proportional to the amount of fluorescently tagged protein in
the optical path at that location. Similarly, in an image obtained by an
optical camera, $f_{\mu}(x)$ refers to the optical flux per unit area (radiant
exitance) barrett2013foundations . In 3D magnetic resonance imaging (MRI), the
intensity of the proton density image $f_{\mu}(x)$ is directly proportional to
the proton density at the location $x$, which means that the image intensity
$f_{\mu}(x)$ reflects the concentration of protons at each point in the
scanned volume. Similar interpretations can be made about numerous other
sensing and imaging modalities, including nuclear medicine (density of a
radioactive tracer), X-ray computed tomography (a 2D or 3D distribution of
attenuation coefficients), and sound intensity (pressure per unit area per
time), among other examples.). The measure $\mu$ with density $f_{\mu}$
computes the total amount of that quantity within a region
$A\subseteq\mathbb{R}^{d}$ via

$\mu(A)=\int_{A}f_{\mu}(x)\,dx.$ (1)

For digital data (e.g., discrete images333A two-dimensional grayscale image is
usually regarded as a function mapping a spatial domain
$\Omega\subseteq\mathbb{R}^{2}$ to $\mathbb{R}$. Typically, when using pixels,
$\Omega$ is a finite discrete set., point clouds444Point clouds are
collections of points representing object surfaces often produced 3D. Each
point $x$ encodes a position in Cartesian coordinates and can be viewed as a
delta measure $\delta_{x}$. , or when considering sampled data) we may have
intensity measurements $f_{\mu}(x_{k})$ only at locations
$x_{k}\in\mathbb{R}^{d},k=1,\dots,K$. Thus, we can view our data as a discrete
measure $\mu=\sum_{k=1}^{K}f_{\mu}(x_{k})\delta_{x_{k}}$ (that is,
$\mu(A)=\sum_{k=1}^{K}f_{\mu}(x_{k})\delta_{x_{k}}(A)$, where
$\delta_{x_{k}}(A)=1$ if $x_{k}\in A$ and 0 otherwise).

Measures are the natural objects when studying optimal transport theory. Thus,
as our transforms will be motivated by mass-transportation problems, using
measure theory will simplify the notation and formula as will be seen later in
this chapter.

Chapter Organization. In Section 2, we provide a brief introduction to Optimal
Transport. Section 3 starts by synthesizing the general framework of transport
transforms, and later summarizes its variants related to closed-form solutions
for Optimal Transport in 1D. Applications of these tools are given in Section
4, where we revisit the motivating tasks stated in this introduction.

##  2 Formulation of the Optimal Transport problem

Monge’s formulation. Let $\mathcal{P}(\Omega)$ be the set of probability
measures defined in a subset $\Omega\subseteq\mathbb{R}^{d}$, and consider
$\mu$, $\nu\in\mathcal{P}(\Omega)$. The Monge Optimal Transport (OT) problem
between $\mu$ and $\nu$ is to find the most economical procedure to move all
the mass distributed according to $\mu$ onto the target $\nu$ by using a map
$T$. Mathematically, it is stated as the minimization problem

$OT(\mu,\nu)=\inf_{T\in\mathrm{MP}(\mu,\nu)}\int_{\Omega}c(x,T(x))d\mu(x),$

where $c:\Omega\times\Omega\to\mathbb{R}_{\geq 0}$ represents the cost of
transporting one unit of mass from position $x$ to $y=T(x)$, and
$\mathrm{MP}(\mu,\nu)$ is the set of all mass-preserving functions, that is,
measurable functions $T:\Omega\to\Omega$ that push measure $\mu$ into $\nu$ in
the sense that

$\nu(B)=\mu(T^{-1}(B))=\mu(\\{x:\,T(x)\in B\\})\quad\text{ for every
$\nu$-measurable set $B$.}$

If $T$ satisfies this relation, it is called transport map, and we say that
$\nu$ is the pushforward of $\mu$ by $T$, denoted as $\nu=T_{\\#}\mu$
555$\nu=T_{\\#}\mu$ can also be defined through the change of variables
formula
$\int\psi(y)\,d\nu(y)=\int\psi(y)\,d(T_{\\#}\mu)(y)=\int\psi(T(x))\,d\mu(x)$
for every test function $\psi$. Also, if $X$ and $Y$ are random variables with
distributions $\mu$ and $\nu$, respectively, we have that $Y=T(X)$. . They
encode that all mass at point $x$ should be moved to $y=T(x)$. In the special
case where $\mu$ and $\nu$ have density functions $f_{\mu}$ and $f_{\nu}$
(i.e., $\mu$ is defined according to (1), and analogously for $\nu$), the
above relation can also be written as

$\int_{T^{-1}(B)}d\mu=\int_{T^{-1}(B)}f_{\mu}(x)dx=\int_{B}f_{\nu}(x)dx=\int_{B}d\nu.$

In addition, if $T$ is smooth and one-to-one, using the change of variable
formula and denoting by $DT$ the Jacobian of $T$, we obtain $\nu=T_{\\#}\mu$
if and only if

$|\mathrm{det}(DT(x))|\,f_{\nu}(T(x))=f_{\mu}(x).$

A closed-form solution for Monge’s problem can be obtained when working with
one-dimensional measures. Given $\nu\in\mathcal{P}(\mathbb{R})$ we define its
cumulative distribution function (CDF) as $F_{\nu}(x):=\nu([-\infty,x))$
together with its generalized inverse666To be more precise, when considering
$\nu\in\mathcal{P}(I)$ where $I=[a,b]$ is a real interval, we define the
generalized inverse of $F_{\nu}:[a,b]\to[0,1]$ as the function
$F_{\nu}^{\dagger}:[0,1]\to[a,b]$ given by
$F_{\nu}^{\dagger}(y):=\inf\\{x\in[a,b]:\,F_{\nu}(x)>y\\}$, where we impose
$\inf\emptyset=b$. Notice that it is similar to the quantile function the
distribution $\nu$.

$F_{\nu}^{\dagger}(y):=\inf\\{x\in\mathbb{R}:\,F_{\nu}(x)>y\\}.$

Note that when $F_{\nu}$ is invertible, the generalized inverse coincides with
the inverse. Otherwise, for non-decreasing functions, it is just the function
whose graph is a reflection through $y=x$ of the graph of $F_{\nu}$
considering that flat regions are reflected into a discontinuous jump and
vice-versa. With these definitions, in the case when $\mu$ absolutely
continuous with respect to the Lebesgue measure ($\mu\ll\mathcal{L}$) an
optimal Monge map from $\mu$ to $\nu$ can be explicitly obtained as

$T(x)=F_{\nu}^{\dagger}(F_{\mu}(x)).$ (2)

In terms of densities, that is, if $\mu$ and $\nu$ have probability density
functions (pdf) $f_{\mu}$ and $f_{\nu}$, respectively, $T$ in (2) satisfies
the following integral equation (see Figure 4):

$\underbrace{\int_{-\infty}^{x}f_{\mu}(x)\,dx}_{F_{\mu}(x)}=\int_{-\infty}^{x}f_{\nu}(T(x))T^{\prime}(x)\,dx=\underbrace{\int_{-\infty}^{T(x)}f_{\nu}(y)\,dy}_{F_{\nu}(T(x))}$
(3)

Figure 4: Illustration of mass transportation from reference $\mu$ to target
$\nu$ through a MP map $T$ in 1D. The measures $\mu$ and $\nu$ have pdfs
$f_{\mu}$ and $f_{\nu}$, and CDFs $F_{\mu}$ and $F_{\nu}$, respectively. The
gray shadowed areas coincide. They represent the accumulated mass until point
$x$ (on the left) and until point $T(x)$ (on the right). To achieve this
conservation of mass, the increasing function $T$ is obtained as (3). Such
formulation (3) is visualized in this figure: $x\mapsto F_{\mu}(x)\mapsto
F_{\nu}^{\dagger}(F_{\mu}(x))=T(x)$.

Kantorovich’s formulation. For higher dimensions, Monge’s optimal transport
problem has no closed-form solution. In addition, it is sometimes ill-posed
when working with discrete measures. For example, there is no function $T$
that pushes a single delta measure $\mu=\delta_{0}$ onto
$\nu=0.5(\delta_{-1}+\delta_{1})$ since it would require to split the mass at
$x=0$ into two separate targets $y=\pm 1$. Besides, the constraint $T:X\to Y$
such that $\nu=T_{\\#}\mu$ is non-linear. This makes the minimization
procedure rather convoluted. An alternative formulation of the optimal
transport problem that solves most of these issues was proposed by
Kantorovich. Specifically, Kantorovich’s optimal transport problem is stated
as the minimization problem

$OT(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\int_{\Omega^{2}}c(x,y)d\pi(x,y),$ (4)

where $\Pi(\mu,\nu)$ is the set of all joint probability measures in
$\Omega^{2}$ with marginals $\mu$ and $\nu$, that is,
$\pi(A\times\Omega)=\mu(A)$, $\pi(\Omega\times B)=\nu(B)$ for all measurable
sets $A,B\subseteq\Omega$. A measure $\pi\in\Pi(\mu,\nu)$ is called a
transportation plan, and the quantity $\pi(A\times B)$ represents how much
mass from $A$ should be moved to region $B$. Now, when working with the
example mentioned before777In general, for the discrete case,
$\mu=\sum_{i=1}^{n}\mu(x_{i})\delta_{x_{i}}$,
$\nu=\sum_{j=1}^{m}\nu(y_{j})\delta_{y_{j}}$ the problem (4) reads as
${\min\sum_{i=1}^{n}\sum_{j=1}^{m}c(x_{i},y_{j})\,\pi(x_{i},y_{j})}$ subject
to $\sum_{i=1}^{n}\pi(x_{i},y_{j})=\nu(y_{j})$,
$\sum_{j=1}^{m}\pi(x_{i},y_{j})=\mu(x_{i})$. Understanding the discrete plans
as matrices $\pi\in\mathbb{R}^{n\times m}$, this becomes a linear
programming., moving $\mu=\delta_{0}$ onto $\nu=0.5(\delta_{-1}+\delta_{1})$
can be done by using a plan of the form
$\pi=0.5\delta_{0,1}+0.5\delta_{0,-1}$.

The set of all transportation plans is non-empty as
$\pi=\mu\times\nu\in\Pi(\mu,\nu)$. A minimizer of (4) is called an optimal
transport plan and its existence is guaranteed when the cost function $c$ is
continuous. Nevertheless, optimal plans are not necessarily unique. For our
exposition, we will work with the Euclidean cost $c(x,y)=\|x-y\|^{2}$ which is
strictly convex. To guarantee (4) to be finite, we will restrict the space of
measures to those with finite second moment

$\mathcal{P}_{2}(\Omega):=\left\\{\nu\in\mathcal{P}(\Omega):\,\int_{\Omega}\|x\|^{2}d\nu<\infty\right\\}.$

In this case, a stronger result can be obtained that guarantees the existence,
uniqueness, and finiteness of the solution of Kantorovich and Monge’s problems
simultaneously. A space of functions that will be useful in the next theorem
and the rest of the chapter is the space of square-integrable functions that
are gradients of convex functions

$\displaystyle
L_{G}^{2}(\mu):=\\{T:\mathbb{R}^{d}\to\mathbb{R}^{d}:\,T=\nabla\varphi,\,\varphi\text{
convex and }\int\|T(x)\|^{2}d\mu(x)<\infty\\}.$

######  Theorem 2.1 (Brenier’s theorem)

Let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $c(x,y)=\|x-y\|^{2}$.
Suppose that $\mu$ is absolutely continuous with respect to the Lebesgue
measure in $\mathbb{R}^{d}$ ($\mu\ll\mathcal{L}$). Then, there exists a unique
solution to Kantorovich’s problem. It is of the form
$\pi=(\mathrm{id},T)_{\\#}\mu$, with $T\in L_{G}^{2}(\mu)$. Moreover, $T$ is
also the unique solution to Monge’s problem, and $T_{\\#}\mu$, and we will
denote it by $T=T_{\mu\to\nu}$. Conversely, let
$\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $T\in L_{G}^{2}(\mu)$, then $T$ is
optimal for the Monge Problem between the measures $\mu$.

###### Remark 1

Let $\mu,\nu$ as in Brenier’s Theorem. If $T_{1}=\nabla\varphi_{1}$ and
$T_{2}=\nabla\varphi_{2}$ for $\varphi_{1},\varphi_{2}$ convex functions, and
$(T_{1})_{\\#}\mu=\nu=(T_{2})_{\\#}\mu$, then $T_{1}=T_{2}$.

Wasserstein distance. Intuitively speaking, moving a distribution of mass
$\mu$ to $\nu$ in an optimal way should cost less than moving $\mu$ to a third
measure and then to $\nu$, which resembles a triangle inequality. In fact, in
$\mathcal{P}_{2}(\Omega)$ the square root of the transport cost
$OT(\cdot,\cdot)$ defines the so called Wasserstein metric

$W_{2}(\mu,\nu):=OT(\mu,\nu)^{1/2}.$

Dynamic formulation. The Wasserstein distance renders
$\mathcal{P}_{2}(\Omega)$ a metric space. This structure is very rich
geometrically. For simplicity and from now on, let us assume that
$\Omega\subset\mathbb{R}^{d}$ is convex and compact.
$(\mathcal{P}(\Omega),W_{2})$ is also a geodesic space: given two measures
$\mu,\nu\in\mathcal{P}(\Omega)$, there exists a curve of measures
$\rho_{t}\in\mathcal{P}(\Omega)$, for $0\leq t\leq 1$, that is the shortest
path in terms of $W_{2}$ connecting them ($\rho_{0}=\mu$, $\rho_{1}=\nu$) and,
moreover, the length of $\rho_{t}$ is exactly $W_{2}(\mu,\nu)$. Furthermore,
as an infinite-dimensional manifold, $(\mathcal{P}(\Omega),W_{2})$ possesses a
Riemannian structure (its tangent spaces are equipped with an inner product).
A precise definition of these structures requires the dynamic approach
benamou2000computational of the Optimal Transport problem.

Kantorovich and Monge’s approaches provide static formulations of the
transport problem. Roughly speaking, transport plans and maps give a rule that
assigns to each initial mass a final point where it must be moved, but they do
not say how the system should evolve from the initial to final configurations.

In the dynamic formulation, we consider a curve of measures
$\rho\in\mathcal{P}([0,1]\times\Omega)$ that for each time $t\in[0,1]$ gives a
measure $\rho_{t}:=\rho(t,\cdot)\in\mathcal{P}(\Omega)$. To transport mass
from $\mu$ to $\nu$ we will require the curve to have initial and final
conditions $\rho_{0}=\mu$ $\rho_{1}=\nu$, to be sufficiently smooth, and to
satisfy a conservation of mass law: the curve $\rho_{t}$ together with a
velocity field $v_{t}:=v(t,\cdot)$ must satisfy the continuity equation with
boundary conditions888To impose $\rho$ to preserve the mass, we can state that
for every $B\subset\Omega$, we should have
$\frac{d}{dt}\int_{B}\rho_{t}dV=-\int_{\partial B}\rho_{t}v_{t}\cdot dS$,
where $dV$ and $dS$ denote volume and surface integrals, respectively. By
using the Divergence Theorem we obtain
$\frac{d}{dt}\int_{B}\rho_{t}dV=-\int_{B}\nabla\cdot\rho_{t}v_{t}dV$ from
where we can derive the continuity equation (5).

$\partial_{t}\rho+\nabla\cdot(\rho v)=0,\qquad\rho_{0}=\mu,\quad\rho_{1}=\nu.$
(5)

Then, the Optimal Transport problem can be stated as minimizing the following
kinetic energy

$OT(\mu,\nu)=\inf\left\\{\int_{0}^{1}\int_{\mathbb{R}^{d}}\|v_{t}(x)\|^{2}d\rho_{t}(x)dt:\,(\rho,v)\text{
satisfies \eqref{eq: continuity eq}}\right\\}$ (6)

Since the space $\mathcal{P}(\Omega)$ has the metric $W_{2}$, we can speak
about the length of curves. Formally, for a curve
$\rho\in\mathcal{P}([0,1]\times\Omega)$ we would define its length as the
limit

$\displaystyle\text{length}(\rho)$ $\displaystyle\approx\lim_{\Delta t\to
0}\sum\frac{W_{2}(\rho_{t+\Delta t},\rho_{t})}{\Delta t}\Delta
t\approx\int_{0}^{1}\lim_{\Delta t\to 0}\frac{W_{2}(\rho_{t+\Delta
t},\rho_{t})}{\Delta t}\ dt.$

In particular, for a curve $\rho$ solving (5) with velocity filed $v$, the
length of the curve can be computed as

$\text{length}(\rho)=\int_{0}^{1}\int_{\mathbb{R}^{d}}\|v_{t}(x)\|^{2}d\rho_{t}(x)dt.$

Under the assumption of the existence of an optimal Monge map $T_{\mu\to\nu}$,
an optimal solution $(\rho,v)$ for (6) can be given explicitly. If a particle
starts at position $x$ and finishes at position $T_{\mu\to\nu}(x)$, then for
$0<t<1$ it will be at the point $T_{t}(x)=(1-t)x+tT_{\mu\to\nu}(x).$ Varying
both the time $t\in[0,1]$ and $x\in\Omega$, the mapping $(t,x)\mapsto
T_{t}(x)$ can be interpreted as a flow whose time velocity is

$v_{t}(x)=T_{\mu\to\nu}(x_{0})-x_{0},\qquad\text{ for }x=T_{t}(x_{0}).$ (7)

Then, to obtain the curve of probability measures $\rho_{t}$, one can evolve
$\mu$ through the flow $T_{t}$ by using the formula

$\rho_{t}=(T_{t})_{\\#}\mu=((1-t)\mathrm{id}+tT_{\mu\to\nu})_{\\#}\mu,\qquad
0\leq t\leq 1.$ (8)

It holds that this pair $(\rho,v)$ satisfies the continuity equation (5) and
solves (6). The vector field $v_{t}$ can be viewed as the tangent vector to
the evolution curve at time $t$. With this interpretation, the field
$v_{0}(x)=T_{\mu\to\nu}(x)-x$ (which encodes the optimal displacement) is the
initial tangent vector to the optimal curve transporting $\mu$ to $\nu$. In
symbols, $v_{0}\in\mathrm{Tan}_{\mu}$. By determining $v_{0}$, from (7) we
know $v_{t}$ for every $t$. Also, it holds that
$\text{length}(\rho)=\int_{\mathbb{R}^{d}}\|v_{0}(x)\|^{2}d\rho_{0}(x)=\int_{\mathbb{R}^{d}}\|T_{\mu\to\nu}(x)-x\|^{2}d\mu(x)$.
If $\mu\ll\mathcal{L}$, by Brenier’s Theorem $T_{\mu\to\nu}=\nabla\varphi$
with $\varphi$ convex and $\int_{\Omega}\|\nabla\varphi\|^{2}d\mu<\infty$, and
so $v_{0}=\nabla u$ for $u(x)=\varphi(x)-\|x\|^{2}/2$, which is also convex
and with $\int_{\Omega}\|v_{0}\|^{2}d\mu<\infty$. Moreover, with the
Wasserstein metric $W_{2}$, the space $\mathcal{P}(\Omega)$ becomes a
Riemannian manifold ambrosio2005gradient ; ding2021geometry ; otto2001geometry
; otto2000generalization : the tangent space at a point
$\mu\in\mathcal{P}(\Omega)$ is

$\mathrm{Tan}_{\mu}=L^{2}(\mu):=\left\\{v:\Omega\to\mathbb{R}^{d}:\,\|v\|^{2}_{L^{2}(\mu)}:=\int_{\Omega}\|v(x)\|^{2}\,d\mu(x)<\infty\right\\}.$
(9)

Sliced Wasserstein Distance. Solving the optimal transport problem for
dimensions bigger than one is, in general, a computationally expensive
procedure. An alternative but equivalent metric to $W_{2}$ can be obtained by
reducing the problem into several one-dimensional transport minimizations. The
Sliced Wasserstein distance between probability measures $\mu$ and $\nu$ on
$\mathbb{R}^{d}$ consists of first obtaining a family of one-dimensional
measures through projections of $\mu$ and $\nu$ (slicing the measures), then
calculating one-dimensional Wasserstein distances, and finally averaging over
all the projections. Precisely, given $\theta\in\mathbb{S}^{d-1}$ (the unit
sphere in $\mathbb{R}^{d}$), consider the projection onto the
$\theta$-direction:

$\theta^{*}:\mathbb{R}^{d}\to\mathbb{R},\qquad\theta^{*}(x):=\langle
x,\theta\rangle,$

where $\langle\cdot,\cdot\rangle$ denotes the usual inner product in
$\mathbb{R}^{d}$. Then, the measure
$\theta^{*}_{\\#}\mu\in\mathcal{P}(\mathbb{R})$ is called the slice of $\mu$
with respect to $\theta$. This operation is related to the generalization of
the Radon Transform999The Radon transform maps a function $f\in
L^{1}(\mathbb{R}^{d})$ into a function $\mathcal{R}f\in
L^{1}(\mathbb{R}\times\mathbb{S}^{d-1})$. For
$(t,\theta)\in\mathbb{R}\times\mathbb{S}^{d-1}$, $\mathcal{R}f(t,\theta)$ is
the surface integral of $f$ over the hyperplane orthogonal to $\theta$ that
passes through $t\theta$:
$\mathcal{R}f(t,\theta)=\int_{\mathbb{R}^{d-1}}f(t\theta+U_{\theta}\xi)\,d\xi$
where $U_{\theta}\in\mathbb{R}^{d\times(d-1)}$ is any matrix such that its
columns form an orthonormal set of vectors in $\mathbb{R}^{d}$ perpendicular
to $\theta$.  $\mathcal{R}$ for measures:
$\mathcal{R}\mu\in\mathcal{P}(\mathbb{R}\times\mathbb{S}^{d-1})$ is the
measure that can be disintegrated according to slices
$[\mathcal{R}\mu](\cdot,\theta):=\theta^{*}_{\\#}\mu$ (see the Appendix). In
particular, if $\mu$ has density $f_{\mu}\in L^{1}(\mathbb{R}^{d})$, the
identity $\mathcal{R}f_{\mu}(\cdot,\theta)=\theta_{\\#}^{*}\mu$ holds in the
sense of distributions101010For every test function $\psi$,
$\int_{\mathbb{R}}\mathcal{R}f_{\mu}(t,\theta)\psi(t)\,dt=\int_{\mathbb{R}}\int_{\mathbb{R}^{d-1}}f_{\mu}(t\theta+U_{\theta}\xi)\,d\xi\psi(t)\,dt=\int_{\mathbb{R}^{d}}f_{\mu}(x)\psi(\langle
x,\theta\rangle)\,dx=\int_{\mathbb{R}^{d}}\psi(\langle
x,\theta\rangle)\,d\mu=\int_{\mathbb{R}^{d}}\psi(x)\,d\theta_{\\#}^{*}\mu,$
where in the second identity we changed variables $(t,\xi)\mapsto
x\in\mathbb{R}^{d}$ by noticing that $t=\langle
t\theta+U_{\theta}\xi,\theta\rangle=\langle x,\theta\rangle$.. Finally, the
Sliced Wasserstein metric is defined as,

$\displaystyle
SW_{2}(\mu,\nu)=\left(\int_{\mathbb{S}^{d-1}}W^{2}_{2}(\theta^{*}_{\\#}\mu,\theta^{*}_{\\#}\nu)\
d\theta\right)^{\frac{1}{2}},$

where $d\theta$ is the uniform measure on the sphere $\mathbb{S}^{d-1}$. For
$d=2$, we will parameterize $\mathbb{S}^{1}$ with angles $\theta\in[0,2\pi)$
and, in fact, it would be necessary to consider only angles in $[0,\pi]$.

######  Remark 2 (Equivalences between metrics)

In $\mathcal{P}_{2}(\mathbb{R}^{d})$, $SW_{2}\leq W_{2}$, and in
$\mathcal{P}(B(0,R))$, where $B(0,R)$ is the ball in $\mathbb{R}^{d}$ centered
at the origin $0$ with radius $R>0$, the Wasserstein distance and the Sliced
Wasserstein distances are topologically equivalent.

##  3 Embeddings or Transforms

Optimal transport has been used to establish bijective nonlinear
transformations and various transport-related metrics between measures,
encompassing signals, images, point clouds, and more aldroubi2021signed ;
bai2022sliced ; bai2023linear ; martin2023lcot ; kolouri2015radon ;
kolouri2016continuous ; kolouri2016sliced ; wang2010optimal ; wang2013linear ;
beier2021linear . Despite the nuances and technicalities of each transform,
the core formulation is based on the so-called Linear Optimal Transport (LOT)
transform. We will introduce the recipe to define LOT and then adapt it for
different scenarios.

###  3.1 Linear optimal transport (LOT) representation

In wang2010optimal ; wang2013linear , a linear optimal transport (LOT)
framework was proposed for efficiently analyzing large databases of images.
Their main contributions were the introduction of a new transport-related
distance (LOT-distance) and an embedding (LOT transform) of the space of
probability distributions into a linear space. The key point was to represent
convoluted manifolds of images as simpler spaces.

Let us consider a measure $\mu_{r}\in\mathcal{P}_{2}(\Omega_{r})$ absolutely
continuous with respect to the Lebesgue measure that we will call the
reference. Given a second measure $\nu\in\mathcal{P}_{2}(\Omega)$, there is a
unique optimal Monge map $T_{\mu_{r}\to\nu}$. By Theorem 2.1, it is
characterized as the unique map satisfying
$(T_{\mu_{r}\to\nu})_{\\#}\mu_{r}=\nu$ and $T_{\mu_{r}\to\nu}=\nabla\varphi$
for some convex function $\varphi$. Thus, the application

$\nu\to\widehat{\nu}:=T_{\mu_{r}\to\nu}$

is a one-to-one and onto map between $\mathcal{P}_{2}(\Omega)$ and
$L_{G}^{2}(\mu_{r})$ that will be called the LOT transform (or LOT embedding,
due to its injectivity property). Its inverse transform is the application
$T\to T_{\\#}\mu_{r}$, which by definition satisfies
$\nu=\widehat{\nu}_{\\#}\mu_{r}$. Recipe that defines the LOT transform:

  * •

Choose a reference measure $\mu_{r}\ll\mathcal{L}$.

  * •

Given $\nu\in\mathcal{P}_{2}(\Omega)$, Find $T_{\mu_{r}\to\nu}=\nabla\varphi$
(with $\varphi$ convex) such that $(T_{\mu_{r}\to\nu})_{\\#}\mu_{r}=\nu$, and
define the LOT transform of $\nu$ as $\widehat{\nu}:=T_{\mu_{r}\to\nu}$.

  * •

The inverse LOT transform is given by $T\mapsto\nu:=T_{\\#}\mu_{r}$.

Since $L^{2}_{G}(\mu_{r})\subset L^{2}(\mu_{r})$ and the LOT transform is
1-to-1, we can pull-back the metric from $L^{2}(\mu_{r})$ to
$\mathcal{P}_{2}(\Omega)$ and the define the LOT-distance as

$\displaystyle d_{LOT}^{2}(\nu_{1},\nu_{2})$
$\displaystyle:=\|\widehat{\nu_{1}}-\widehat{\nu_{2}}\|_{L^{2}(\mu_{r})}.$

Geometric interpretation: Recall that the tangent space Tan${}_{\mu_{r}}$ of
the Wasserstein manifold $(\mathcal{P}(\Omega_{r}),W_{2})$ at $\mu_{r}$ is the
linear space $L^{2}(\mu_{r})$. Given any measure $\nu_{i}$, we can associate a
vector in Tan${}_{\mu_{r}}$ by taking the initial velocity field of the
geodesic joining $\mu_{r}$ and $\nu_{i}$, which is exactly
$v_{0}^{i}(x)=\widehat{\nu_{i}}(x)-\mathrm{id}(x)$. The LOT-distance between
$\nu_{1},\nu_{2}$ coincides with the distance between their associated vectors
$v_{0}^{1},v_{0}^{2}\in\mathrm{Tan}_{\mu_{r}}$:

$\displaystyle\|\widehat{\nu_{1}}-\widehat{\nu_{2}}\|_{L^{2}(\mu_{r})}$
$\displaystyle=\|(\widehat{\nu_{1}}-\mathrm{id})-(\widehat{\nu_{2}}-\mathrm{id})\|_{L^{2}(\mu_{r})}$
$\displaystyle=\|v_{0}^{1}-v_{0}^{2}\|_{L^{2}(\mu_{r})}=\|v_{0}^{1}-v_{0}^{2}\|_{\text{Tan}_{\mu_{r}}}.$
(10)

This motivates the name “Linear Optimal Transform”, as the LOT-distance can be
viewed as a linearized version of the OT-distance (i.e., a linearized version
of the Wasserstein distance $W_{2}$), since it corresponds to the metric in a
tangent space of the Wasserstein manifold. We refer the reader to Figure 5 for
an illustration.

Figure 5: Illustration of the LOT-transform, its inverse, and the LOT-
distance. Let $\Omega\subset\mathbb{R}^{d}$ be convex and compact, then the
LOT transform can be interpreted as the embedding of the Wasserstein manifold
$(\mathcal{P}(\Omega),W_{2})$ into the tangent space
$(\mathrm{Tan_{\mu_{r}}},\|\cdot\|_{L^{2}(\mu_{r})})$ at the reference measure
$\mu_{r}$. The LOT-distance is defined as the distance between vectors in such
tangent space as shown in (3.1)
($d_{LOT}(\nu_{1},\nu_{2})=\|\widehat{\nu_{1}}-\widehat{\nu_{2}}\|_{{L^{2}(\mu_{r})}}$).
It follows that $d_{LOT}(\cdot,\cdot)$ preserves the Wasserstein metric when
one of the measures is the reference, that is,
$d_{LOT}(\mu_{r},\nu_{i})=W_{2}(\mu_{r},\nu_{i})$, for $i=1,2$, but in
general, $d_{LOT}(\nu_{1},\nu_{2})\not=W_{2}(\nu_{1},\nu_{2})$. Similarly, the
geodesic between $\mu_{r}$ and $\nu_{i}$ ($i=1,2$) in
$(\mathcal{P}(\Omega),W_{2})$ goes to a line-segment (geodesic) when applying
the LOT-transform, but the $W_{2}$-geodesic between general measures $\nu_{1}$
and $\nu_{2}$ is not preserved, in general, after applying LOT-transform (does
not go, in general, to a line-segment in the transformed space). Reversely,
line-segments (geodesics) in the LOT-transformed space (equivalently, in the
tangent space $\mathrm{Tan}_{\mu_{r}}$) between $\widehat{\mu_{r}}$ an
$\widehat{\nu_{i}}$ (for $i=1,2$) go to geodesics in the Wasserstein manifold
after applying the LOT-inverse transform, but in general, line-segments (LOT-
geodesics) between arbitrary ‘tangent vectors’ $\widehat{\nu_{1}}$ and
$\widehat{\nu_{2}}$ do not correspond with geodesics in the Wasserstein
manifold after applying the LOT-inverse transform.

###### Remark 3

When the reference measure $\mu_{r}$ is not absolutely continuous with respect
to the Lebesgue measure, there is no guarantee of the existence of an optimal
transport map. The proper extension of the LOT distance in the general case is
as the shortest generalized geodesic connecting $\mu$ and $\nu$
ambrosio2005gradient . We refer towang2013linear for the technical details.

It can be shown that the range of the LOT-transform, i.e.,
$L^{2}_{G}(\mu_{r})$, is a convex set included in $L^{2}(\mu_{r})$ (as convex
combinations of gradients of convex functions is a gradient of a convex
function). Since the latter is a geodesic space with geodesics given by
‘linear interpolation’ (i.e., line-segments), we can copy this structure with
the inverse LOT-transform. That is, with the LOT-distance, the geodesic
joining $\nu_{1}$ and $\nu_{2}$ is
$\rho_{t}:=\left((1-t)\widehat{\nu_{1}}+t\widehat{\nu_{2}}\right)_{\\#}\mu_{r}$,
for $0\leq t\leq 1$ which is the inverse of the geodesic (line-segment)
$(1-t)\widehat{\nu_{1}}+t\widehat{\nu_{2}}$ (for $t\in[0,1]$) between
$\widehat{\nu_{1}}$ and $\widehat{\nu_{2}}$ in the range of the transform.

In $\mathcal{P}_{2}(\Omega)$ we previously introduced the Wasserstein
distance. Thus, it is natural to ask how $d_{LOT}$ compares to $W_{2}$. The
answer is that, in general111111If the reference $\mu_{r}$ is such that
$\mu\ll\mathcal{L}$, given $\nu_{1},\nu_{2}\in\mathcal{P}_{2}$, assume that
that also $\nu_{1}\ll\mathcal{L}$ and that $T_{\mu\to\nu_{1}}$ is invertible,
then by using the change of variables $y=T_{\mu\to\nu_{1}}(x)$, we have
$d_{LOT}^{2}(\nu_{1},\nu_{2})=\int\|T_{\mu\to{\nu_{1}}}(x)-T_{\mu\to{\nu_{2}}}(x)\|^{2}d\mu(x)=\int\|y-T_{\mu\to{\nu_{2}}}(T_{\mu\to{\nu_{1}}}^{-1}(y))\|^{2}d({T_{\mu\to{\nu_{1}}}}_{\\#}\mu)(y)=\int\|y-T_{\mu\to\nu_{2}}(T_{\mu\to\nu_{1}}^{-1}(y))\|^{2}d\nu_{1}(y)\geq
W_{2}^{2}(\nu_{1},\nu_{2})$, where the last inequality holds since
${(T_{\mu\to\nu_{2}}\circ T_{\mu\to\nu_{1}}^{-1})}_{\\#}\nu_{1}=\nu_{2}$. In
general, $d_{LOT}(\nu_{1},\nu_{2})\geq W_{2}(\nu_{1},\nu_{2})$ as the LOT
transform comes from considering another transportation plan (as pointed in
Remark 3), not necessarily the optimal between $\nu_{1}$ and $\nu_{2}$. ,
$d_{LOT}$ induces a finer topology since it satisfies
$W_{2}(\nu_{1},\nu_{2})\leq d_{LOT}(\nu_{1},\nu_{2})$. For more details, we
refer the reader to moosmuller2023linear .

Often, the utility of different transforms is associated with how they behave
under certain actions. The LOT transform is particularly suited for actions of
certain groups of homeomorphisms of $\mathbb{R}^{d}$ on the space of measures
that preserve mass, i.e., utilizing the pushforward operation. Specifically,
consider the group
$\mathcal{G}_{LOT}=\\{g:\mathbb{R}^{d}\to\mathbb{R}^{d},g(x)=ax+b:\,a\in\mathbb{R}_{>0},b\in\mathbb{R}^{d}\\}$
representing translations and isotropic scalings. The LOT transform translates
actions in the domain of measures or the dependent variable into actions in
the independent variable: The pushforward action $g_{\\#}\nu$ on measures is
then transferred to the dependent variable by the rule
$\widehat{g_{\\#}\nu}=g\circ\widehat{\nu}$. A clear example of this can be
observed when considering $g(x)=x-\tau$ to be a pure translation. In this
case, $g_{\\#}\nu$ represents a translation in the dependant variable (i.e., a
‘horizontal’ shift of the measure $\nu$), while its transform is
$\widehat{\nu}+\tau$, that is, a translation in the independent variable
(i.e., a ‘vertical’ shift).

A summary of the preceding discussions can be found in the following box. A
similar summary will be given for the other transforms presented in this
chapter.

PROPERTIES OF LOT TRANSFORM Domain $=\mathcal{P}(\Omega)$, where
$\Omega\subset\mathbb{R}^{d}$ convex and compact. Codomain $=L^{2}(\mu_{r})$.

Range $=L_{G}^{2}(\Omega_{r})$. Inverse: $T\mapsto T_{\\#}\mu_{r}$.

Symmetry property: Consider the group of translations and isotropic scalings

$\mathcal{G}_{LOT}:=\left\\{g:\mathbb{R}^{d}\to\mathbb{R}^{d},g(x)=ax+b:\,a\in\mathbb{R}_{>0},b\in\mathbb{R}^{d}\right\\}$

Then, for every $g\in\mathcal{G}_{LOT}$, $\nu\in\mathcal{P}(\Omega)$

  * •

(at measure level) $\widehat{g_{\\#}\nu}=g\circ\widehat{\nu},$

  * •

(at density level) $\widehat{\mathrm{det}(Dg)\cdot(f_{\nu}\circ
g)}=g^{-1}\circ\widehat{f_{\nu}}.$

Relation with other distances: $d_{LOT}(\nu_{1},\nu_{2})\geq
W_{2}(\nu_{1},\nu_{2})$ $\forall\nu_{1},\nu_{2}\in\mathcal{P}(\Omega)$
and $d_{LOT}(\mu_{r},\nu)=W_{2}(\mu_{r},\nu)$
$\forall\nu\in\mathcal{P}(\Omega)$.

Geodesics:
$\rho_{t}=\left((1-t)\widehat{\nu_{1}}+t\widehat{\nu_{2}}\right)_{\\#}\mu_{r}$,
$0\leq t\leq 1$, is an $d_{LOT}$-geodesic between
$\nu_{1},\nu_{2}\in\mathcal{P}(\Omega)$ (which is a $W_{2}$-geodesic if
$\nu_{i}=\mu_{r}$ for some $i=1,2$).

###  3.2 Variants of LOT

We will describe the variants of the LOT transform in Euclidean domains
(measures defined on $\mathbb{R}^{d}$) or which we have a closed formula due
to its relation with OT in one dimension: The Cumulative Distribution
Transform (CDT), the Signed CDT (SCDT), the Radon CDT (RCDT), and the Signed
RCDT (RSCDT).

  * •

When $d=1$, the LOT transform is called CDT.

  * •

The extension of the CDT to signed measures is called SCDT.

  * •

For $d>1$, the variant of CDT that first pre-processes the signals by applying
the Radon transform and then uses the CDT, is called RCDT.

  * •

The extension of the RCDT to signed measures is called RSCDT.

One-dimensional probability measures: Cumulative Distribution Transform (CDT).
Let $I=[a,b]$ be a real interval, and consider $\nu\in\mathcal{P}(I)$.
Following the LOT-recipe and using (2), we fix continuous reference
$\mu_{r}\in\mathcal{P}(\mathbb{R})$ (i.e., $\mu_{r}\ll\mathcal{L}$) and define
the LOT transform of $\nu$ respect to $\mu$ and the LOT distance as

$\widehat{\nu}:=T_{\mu_{r}\to\nu}=F^{\dagger}_{\nu}\circ F_{\mu_{r}},$ (11)
$d_{LOT}(\nu_{1},\nu_{2})=\|F^{\dagger}_{\nu_{2}}\circ
F_{\mu_{r}}-F^{\dagger}_{\nu_{1}}\circ F_{\mu_{r}}\|_{L^{2}(\mu_{r})}.$ (12)

The expression (11) is also called the Cumulative Distribution Transform (CDT)
of $\nu$ with respect to $\mu_{r}$ park2018cumulative . Nevertheless, by the
change of variable formula, it results that the expression (12) coincides with

$\|F^{\dagger}_{\nu_{2}}-F^{\dagger}_{\nu_{1}}\|_{L^{2}(\mathcal{L}_{[0,1]})}$
(13)

where $\mathcal{L}_{[0,1]}$ is the Lebesgue measure on $[0,1]$, that is, the
uniform measure on $[0,1]$. In such a case, we will use the notation
$\|\cdot\|_{L^{2}(\mathcal{L}_{[0,1]})}=\|\cdot\|_{L^{2}({[0,1]})}$ This tells
us that the LOT-distance in 1D is independent of the choice of the reference
$\mu_{r}$. Moreover, the LOT transform achieves a simpler form when we use
$\mathcal{L}_{[0,1]}$ as the reference. Therefore, we define the 1D-version of
the LOT transform and the corresponding distance directly as follows.

######  Definition 1 (CDT and CDT-distance)

When $\nu\in\mathcal{P}(I)$, with $I=[a,b]\subset\mathbb{R}$, the LOT
transform is called Cumulative Distribution Transform (CDT) and has the closed
form

$\widehat{\nu}:=F_{\nu}^{\dagger}.$

It is exactly the optimal Monge map from $\mathcal{L}_{[0,1]}$ to $\nu$. By
abuse of notation, when $\nu$ has a density $f_{\nu}$, the CDT transform can
be defined at the density level as
$\widehat{f_{\nu}}=\widehat{\nu}=F_{\nu}^{\dagger}$. The CDT-distance
$d_{CDT}:\mathcal{P}(I)\times\mathcal{P}(I)\to\mathbb{R}_{\geq 0}$ takes the
form given in (13), that is,

$d_{CDT}(\nu_{1},\nu_{2})=\|F^{\dagger}_{\nu_{2}}-F^{\dagger}_{\nu_{1}}\|_{L^{2}({[0,1]})}.$

PROPERTIES OF CDT

Domain $=\mathcal{P}(I)$, where $I=[a,b]\in\mathbb{R}$. Codomain
$=L^{2}([0,1])$.

Range $=\left\\{T:[0,1]\to I\text{ non-decreasing and right-
continuous}\right\\}$.

Inverse: $T\mapsto\nu:=T_{\\#}\mathcal{L}_{[0,1]}$, with
$f_{\nu}=(T^{\dagger})^{\prime}$ if the density exists.121212The derivative is
in the sense of distributions. If $f_{\nu}$ is a pdf, then
$\left(\widehat{f_{\nu}}^{\dagger}\right)^{\prime}=f_{\nu}.$

Symmetry property: Consider the group of homeomorphisms

$\mathcal{G}_{CDT}:=\left\\{g:I\to I:\,g\text{ is a strictly increasing
bijection}\right\\}$

Then, for every $g\in\mathcal{G}_{CDT}$, $\nu\in\mathcal{P}(I)$

  * •

(at measure level) $\widehat{g_{\\#}\nu}=g\circ\widehat{\nu},$

  * •

(at density level) $\widehat{g^{\prime}\cdot(f_{\nu}\circ
g)}=g^{-1}\circ\widehat{f_{\nu}}.$

Relation with other distances:
$d_{CDT}(\nu_{1},\nu_{2})=W_{2}(\nu_{1},\nu_{2})$
$\forall\nu_{1},\nu_{2}\in\mathcal{P}(I)$.

Geodesics:
$\rho_{t}=((1-t)\widehat{\nu_{1}}+t\widehat{\nu_{2}})_{\\#}\mathcal{L}_{[0,1]}$,
$0\leq t\leq 1$, is a $d_{CDT}$-geodesic between
$\nu_{1},\nu_{2}\in\mathcal{P}(I)$, which coincides with the Wasserstein
geodesic.

######  Theorem 3.1 (Characterizing Property.)

If a transform $\,\widehat{\cdot}\,$ defined on $\mathcal{P}(I)$ is such that
for some $\nu\ll\mathcal{L}$, we have $\widehat{\nu}$ increasing and
satisfying the symmetry property

$g\circ\widehat{\nu}=\widehat{g_{\\#}\nu}\qquad\forall g\text{ non-
decreasing},$ (14)

then $\,\widehat{\cdot}\,$ is the transform defined in (11), that is, the CDT
with respect to some reference $\mu_{r}\ll\mathcal{L}$. [We refer the reader
to the Appendix for the proof.]

One dimensional signed measures. The Signed Cumulative Distribution Transform
(SCDT). Let $I=[a,b]$ be a real interval, and consider $\nu\in\mathcal{M}(I)$
the space of signed measures over $I$. The Signed Cumulative Distribution
Transform (SCDT) aldroubi2021signed extends the CDT from $\mathcal{P}(I)$ to
$\mathcal{M}(I)$. It consists of taking the Hahn-Jordan decomposition of a
given measure $\nu=\nu^{+}-\nu^{-}$, normalizing its components, and applying
the CDT to each term separately.

###### Definition 2

When $\nu\in\mathcal{M}(I)$ with $I=[a,b]\subset\mathbb{R}$, the SCDT is
defined as

$\widehat{\nu}=\left(\widehat{\nu^{+N}},|\nu^{+}|,\widehat{\nu^{-N}},|\nu^{-}|\right),$

where $\nu^{\pm N}:=\nu^{\pm}/|\nu^{\pm}|$ are the normalized components of
$\nu$. If $\nu^{+}\equiv 0$, we define
$\widehat{\nu}=(0,0,\widehat{\nu^{-N}},|\nu^{-}|)$, respectively for the case
$\nu^{-}\equiv 0$, we define
$\widehat{\nu}=(\widehat{\nu^{+N}},|\nu^{+}|,0,0)$, and if $\nu\equiv 0$, then
$\widehat{\nu}=(0,0,0,0)$. By abuse of notation, when $\nu$ has a ‘density’
(i.e., Radon-Nicodym derivative) $f_{\nu}\in L^{1}(I)$, the transform can be
defined at the density level as

$\widehat{f_{\nu}}=\widehat{\nu}=\left(\widehat{f_{\nu}^{+N}},\|f_{\nu}^{+}\|_{1},\widehat{f_{\nu}^{-N}},\|f_{\nu}^{-}\|_{1}\right),$

where $f_{\nu}^{\pm N}=f_{\nu}^{\pm}/\|f_{\nu}^{\pm}\|_{1}$ and
$f_{\nu}^{+}(x)=\max\\{f(x),0\\}$, $f_{\nu}^{-}(x)=\max\\{-f(x),0\\}$ and
$\|f_{\nu}\|_{1}=\int|f_{\nu}(x)|dx$. As before, if $f_{\nu}^{\pm}\equiv 0$,
we define $f_{\nu}^{\pm N}\equiv 0$ and $\widehat{f_{\nu}^{\pm N}}\equiv 0$.

The SCDT-distance is defined as

$\displaystyle d^{2}_{SCDT}$
$\displaystyle(\nu_{1},\nu_{2})=\|\widehat{\nu_{1}}-\widehat{\nu_{2}}\|_{(L^{2}([0,1])\times\mathbb{R})\times(L^{2}([0,1])\times\mathbb{R})}^{2}$
$\displaystyle=d_{CDT}^{2}(\nu_{1}^{+N},\nu_{2}^{+N})+d_{CDT}^{2}(\nu_{1}^{-N},\nu_{2}^{-N})+\left|\nu_{1}^{+}-\nu_{2}^{+}\right|^{2}+\left|\nu_{1}^{-}-\nu_{2}^{-}\right|^{2}$
$\displaystyle=d_{CDT}^{2}(\nu_{1}^{+N},\nu_{2}^{+N})+d_{CDT}^{2}(\nu_{1}^{-N},\nu_{2}^{-N})+\left(|\nu_{1}^{+}|-|\nu_{2}^{+}|\right)^{2}+\left(|\nu_{1}^{-}|-|\nu_{2}^{-}|\right)^{2}.$

PROPERTIES OF SCDT

Domain $=\mathcal{M}(I)$, where $I=[a,b]\in\mathbb{R}$. Codomain
$=\left(L^{2}([0,1]\times\mathbb{R})\right)^{2}$.

Range
$=\left\\{(T,r,U,s)\in\mathcal{T}\times\mathcal{T}:\,T_{\\#}\mathcal{L}_{|_{[0,1]}}\perp
U_{\\#}\mathcal{L}_{|_{[0,1]}}\right\\}$,
where
$\mathcal{T}:=\mathrm{Range}({CDT})\times\mathbb{R}_{>0}\cup\left\\{(0,0)\right\\}$.131313The
first coordinate of the tuple $(0,0)$ denotes the function identically zero
and the second coordinate denotes the real number zero.

Inverse:
$(T,r,U,s)\mapsto\nu:=r\left(T_{\\#}\mathcal{L}_{[0,1]}\right)-s\left(U_{\\#}\mathcal{L}_{[0,1]}\right)$
with $f_{\nu}=r(T^{\dagger})^{\prime}-s(U^{\dagger})^{\prime}$ if the density
exists.

Symmetry property: If $g\in\mathcal{G}_{CDT}$, then

  * •

(at measure level)
$\widehat{g_{\\#}\nu}=\left(g\circ\widehat{\nu^{+N}},|\nu^{+}|,g\circ\widehat{\nu^{-N}},|\nu^{-}|\right),$

  * •

(at density level) $\widehat{g^{\prime}\cdot(f_{\nu}\circ
g)}=\left(g^{-1}\circ\widehat{f_{\nu}^{+N}},\|f_{\nu}^{+}\|_{1},g^{-1}\circ\widehat{f_{\nu}^{-N}},\|f_{\nu}^{-}\|_{1}\right)$.

Relation with other distances: $d^{2}_{SCDT}(\nu_{1},\nu_{2})=$

$=W_{2}^{2}(\nu_{1}^{+N},\nu_{2}^{+N})+W_{2}^{2}(\nu_{1}^{-N},\nu_{2}^{-N})+\left(|\nu_{1}^{+}|-|\nu_{2}^{+}|\right)^{2}+\left(|\nu_{1}^{-}|-|\nu_{2}^{-}|\right)^{2}$.

Geodesics: $(\mathcal{M}(I),d_{SCDT})$ is not a geodesic space (see
(li2022geodesic, , Thm 2.5): as a counterexample consider
$f_{\nu_{1}}=\mathbbm{1}_{[-1,0]}-\mathbbm{1}_{[0,1]}$, and
$f_{\nu_{2}}=-f_{\nu_{1}}$). However, given non-null finite positive measures
$\nu_{1},\nu_{2}\in\mathcal{M}_{+}(I)$, then
$\rho_{t}:=\left((1-t)|\nu_{1}|+t|\nu_{2}|\right)\left[\left((1-t)\widehat{\nu_{1}^{N}}+t\widehat{\nu_{2}^{N}}\right)_{\\#}\mathcal{L}_{[0,1]}\right]$,
where $\nu_{i}^{N}=\nu_{i}/|\nu_{i}|$ ($i=1,2$), is a geodesic in
$\mathcal{M}_{+}(I)$ with respect to $d_{SCDT}$ (see the Appendix).

Two-dimensional densities. The Radon Cumulative Distribution (RCDT). When
$\nu\in\mathcal{P}_{2}(\Omega)$ with $\Omega\subset\mathbb{R}^{2}$, to take
advantage of the closed formula of the CDT, one can use the Radon transform to
project to one-dimensional measures and then applying the CDT on each
projection. This gives rise to the Radon Cumulative Distribution Transform
(RCDT) kolouri2015radon .

###### Definition 3

When $\nu\in\mathcal{P}(\Omega)$ with $\Omega\subset\mathbb{R}^{2}$ compact,
the RCDT is defined as

$\displaystyle\widehat{\nu}^{RCDT}(\xi,\theta)$
$\displaystyle=\widehat{[\mathcal{R}\nu](\cdot,\theta)}(\xi)=\widehat{\theta_{\\#}^{*}\nu}(\xi)=F_{\theta_{\\#}^{*}\nu}^{\dagger}(\xi)\quad\theta\in[0,\pi],\xi\in[0,1]$
(15)

where in the RHS of (15) we are considering the CDT of
$[\mathcal{R}\nu](\cdot,\theta)=\theta_{\\#}^{*}\nu\in\mathcal{P}(\mathbb{R})$.
By abuse of notation, when $\nu$ has a density $f_{\nu}$ the transform can be
defined at the density level as
$\widehat{f_{\nu}}^{RCDT}(\xi,\theta)=\widehat{[\mathcal{R}f_{\nu}](\cdot,\theta)}(\xi)$,
where $\mathcal{R}f_{\nu}$ is the classical Randon transform of
$L^{1}$-functions. The RCDT-distance is defined as

$\displaystyle d_{RCDT}^{2}(\nu_{1},\nu_{2})$
$\displaystyle=\|\widehat{\nu_{1}}^{RCDT}-\widehat{\nu_{2}}^{RCDT}\|_{L^{2}([0,1]\times[0,\pi])}^{2}$
(16)
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{1}|\widehat{\nu_{1}}^{RCDT}(\xi,\theta)-\widehat{\nu_{2}}^{RCDT}(\xi,\theta)|^{2}d\xi
d\theta$

For simplicity, we are considering $\Omega\subset\mathbb{R}^{2}$ compact.
Thus, there exists $R>0$ such that $\Omega\subset B(0,R)$ and therefore, for
each $\theta\in\mathbb{S}^{1}$, we have
$\theta_{\\#}^{*}\nu\in\mathcal{P}([-R,R])$. Therefore, without loss of
generality, we will assume $\Omega=B(0,R)$ by extending any function or
measure by zero outside its original domain.

When working with the RCDT we often push measures to measures in a slice-to-
slice manner. To do so, we start with a family of maps indexed by the angle
$\theta$ that we express as a function of two variables $T(t,\theta)$. Yet, if
we want to push slices of a two dimensional measure $\mu$ into slices of
another two dimensional measure, we cannot do $T_{\\#}\mu$ since the output
would be a one dimensional measure. Thus, we need to introduce a new notation:
Given $T:\mathbb{R}\times[0,\pi]\to\mathbb{R}$ such that
$\forall\theta,T(\cdot,\theta)$ is non-decreasing, we define

$L^{T}(t,\theta):=\left(T(t,\theta),\theta\right)$ (17)

which can be visualized in Figure 6. While the function $T$ above encodes how
to move each slice, $L^{T}$ adds a second component encoding that slices
remain at the same angle level. That is, if by abuse of notation
$\mu(\cdot,\theta)$ denotes the slice at angle $\theta$ (see the Appendix for
the precise definition), then $L^{T}$ pushes slices $\mu(\cdot,\theta)$ to
slices of $L^{T}_{\\#}\mu$ by the rule
$(L^{T}_{\\#}\mu)(\cdot,\theta)=T(\cdot,\theta)_{\\#}\mu(\cdot,\theta)$.

Figure 6: Visualization of the function $L^{T}$ defined by (17) acting as a
pushforward of the measure $\mu$: For each $\theta\in[0,\pi]$, it pushes
forward the $\theta$-slice of $\mu$ by $T(\cdot,\theta)$ resulting in the
measure $\nu$.

PROPERTIES OF RCDT

Domain $=\mathcal{P}(\Omega)$, where $\Omega=B(0,R)\subset\mathbb{R}^{2}$.
Codomain $=L^{2}([0,1]\times[0,\pi])$.

Range $=$ Measurable functions $T:[0,1]\times[0,\pi]\to[-R,R]$ such that
$\forall\theta,T(\cdot,\theta)$ is non-decreasing and
$L^{T}_{\\#}\mathcal{L}_{|_{[0,1]\times[0,\pi]}}\in\mathrm{Range}(\mathcal{R})$.

Inverse:
$T\mapsto\mathcal{R}^{-1}\left(L^{T}_{\\#}\mathcal{L}_{|_{[0,1]\times[0,\pi]}}\right)$
or $T\mapsto\mathcal{R}^{-1}h$,
$\,h(\xi,\theta):=(T(\cdot,\theta)^{\dagger})^{\prime}(\xi)$.

Symmetry property: Consider the group of slice homeomorphisms

$\displaystyle\mathcal{G}_{RCDT}:=\\{$ $\displaystyle
g:[-R,R]\times[0,\pi]\to[-R,R]:\,\forall\theta\in[0,\pi],$ $\displaystyle
g_{\theta}(\xi)=g(\xi,\theta)\text{ is a strictly increasing bijection}\\}$

For every $g\in\mathcal{G}_{RCDT}$, $\nu\in\mathcal{P}(\Omega)$, and pdf
$f_{\nu}$,

  * •

(at measure level) let
$\nu^{g}=\mathcal{R}^{-1}(L^{g}_{\\#}(\mathcal{R}(\nu)))$ then,

$\widehat{\nu^{g}}^{RCDT}(\xi,\theta)=g_{\theta}(\widehat{\nu}^{RCDT}(\xi,\theta))=g\left(\widehat{\nu}^{RCDT}(\xi,\theta),\theta\right).$

  * •

(at density level) let
$f_{\nu}^{g}=\mathcal{R}^{-1}\left(\mathrm{det}(DL^{g})\cdot\mathcal{R}(f_{\nu})\circ
L^{g}\right)$, then

$\widehat{f_{\nu}^{g}}^{RCDT}(\xi,\theta)=g_{\theta}^{-1}\left(\widehat{f_{\nu}}^{RCDT}(\xi,\theta)\right).$

Relation with other distances:

$d^{2}_{RCDT}(\nu_{1},\nu_{2})=\frac{1}{\pi}\int_{0}^{\pi}d_{CDT}^{2}\left({\mathcal{R}\nu_{1}(\cdot,\theta)},{\mathcal{R}\nu_{2}(\cdot,\theta)}\right)d\theta=SW_{2}^{2}(\nu_{1},\nu_{2})$.

######  Remark 4 (Translations and isotropic dilations)

In particular, when working with translations and anisotropic dilations, the
symmetry property reads as follows:

  * •

Let $b\in\mathbb{R}^{2}$ be a shift parameter, and consider $\tau_{b}(x):=x+b$
for all $x\in\mathbb{R}^{2}$, then

$\widehat{(\tau_{b})_{\\#}\nu}^{RCDT}(\xi,\theta)=\widehat{\nu}^{RCDT}(\xi,\theta)+\theta^{*}(b).$

  * •

Let $a\in\mathbb{R}_{>0}$ be a scaling parameter, and consider $D_{a}(x):=ax$
for all $x\in\mathbb{R}^{2}$ and $d_{a}(\xi):=a\xi$ for all
$\xi\in\mathbb{R}$, then

$\widehat{(D_{a})_{\\#}\nu}^{RCDT}=d_{a}\circ\widehat{\nu}^{RCDT}.$

###### Remark 5

In the literature (see, for e.g., kolouri2015radon ; gong2023radon ), the
definition of the RCDT and the RDCT-distance is often written in a more
general way, by taking a reference measure $\mu_{r}\in\mathcal{P}(B(0,R))$.
Such a new distance coincides with the given RCDT-distance in (16). See the
Appendix for a detailed discussion.

Two-dimensional signed measures. The Radon Signed Cumulative Distribution
Tranform (RSCDT). For simplicity, consider $\Omega=B(0,R)$ the ball centered
at zero and with radius $R$ in $\mathbb{R}^{2}$. The Radon Signed Cumulative
Distribution Transform (RSDCT) gong2023radon extends the RCDT from
$\mathcal{P}(\Omega)$ to $\mathcal{M}(\Omega)$. It consists of taking the
Radon transform and then SCDT for each projection angle.

###### Definition 4

When $\nu\in\mathcal{M}(\Omega)$ with $\Omega\subset\mathbb{R}^{2}$ compact,
the RSCDT is defined as

$\displaystyle\widehat{\nu}^{RSCDT}$
$\displaystyle(\xi,\theta)=\widehat{[\mathcal{R}\nu](\cdot,\theta)}(\xi)=\widehat{\theta_{\\#}^{*}\nu}(\xi)$
$\displaystyle=\left(\widehat{[\mathcal{R}\nu](\cdot,\theta)^{+N}}(\xi),|[\mathcal{R}\nu](\cdot,\theta)^{+}|,\widehat{[\mathcal{R}\nu](\cdot,\theta)^{-N}}(\xi),|[\mathcal{R}\nu](\cdot,\theta)^{-}|\right)$
$\displaystyle=\left(\widehat{[\theta_{\\#}^{*}\nu]^{+N}}(\xi),|[\theta_{\\#}^{*}\nu]^{+}|,\widehat{[\theta_{\\#}^{*}\nu]^{-N}}(\xi),|[\theta_{\\#}^{*}\nu]^{-}|\right)\quad\forall\theta\in[0,\pi],\xi\in[0,1]$

where the $\,\widehat{\cdot}\,$ used in the first equality is the SCDT, while
in the last equality, it is the CDT141414We note that, even though
$\theta_{\\#}^{*}\nu=\theta_{\\#}^{*}(\nu^{+})-\theta_{\\#}^{*}(\nu^{-})$, it
doesn’t hold that $\theta_{\\#}^{*}(\nu^{\pm})=(\theta_{\\#}^{*}\nu)^{\pm}$
since the projections $\theta_{\\#}^{*}(\nu^{+})$ and
$\theta_{\\#}^{*}(\nu^{-})$ are not necessarily mutually singular.. As for the
SCDT, if $[\theta_{\\#}^{*}\nu]^{\pm}=0$, then we define
$[\theta_{\\#}^{*}\nu]^{\pm N}=0$ and $\widehat{[\theta_{\\#}^{*}\nu]^{\pm
N}}=0$. By abuse of notation, when $\nu$ has a ‘density’ (i.e., a Radon-
Nicodym derivative) $f_{\nu}\in L^{1}(\Omega)$ the transform can be defined at
the density level as

$\displaystyle\widehat{f_{\nu}}^{RSCDT}(\xi,\theta)=\widehat{[\mathcal{R}f_{\nu}](\cdot,\theta)}(\xi)$
$\displaystyle=\left(\widehat{[\mathcal{R}f_{\nu}](\cdot,\theta)^{+N}},\|[\mathcal{R}f_{\nu}](\cdot,\theta)^{+N}\|_{L^{1}(\mathbb{R})},\widehat{[\mathcal{R}f_{\nu}](\cdot,\theta)^{-N}},\|[\mathcal{R}f_{\nu}](\cdot,\theta)^{-N}\|_{L^{1}(\mathbb{R})}\right),$

where $\mathcal{R}f_{\nu}$ is the usual Radon transform defined for
$L^{1}$-functions.

The RSCDT-distance is defined as

$\displaystyle
d_{RSCDT}^{2}(\nu_{1},\nu_{2})=\|\widehat{\nu_{1}}^{RSCDT}-\widehat{\nu_{2}}^{RSCDT}\|_{(L^{2}([0,1]\times[0,\pi]))^{4}}^{2}$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{1}|\widehat{\nu_{1}}^{RSCDT}(\xi,\theta)-\widehat{\nu_{2}}^{RSCDT}(\xi,\theta)|^{2}d\xi
d\theta$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{1}|\widehat{[\theta_{\\#}^{*}\nu_{1}]^{+N}}(\xi)-\widehat{[\theta_{\\#}^{*}\nu_{2}]^{+N}}(\xi)|^{2}+|\widehat{[\theta_{\\#}^{*}\nu_{1}]^{-N}}(\xi)-\widehat{[\theta_{\\#}^{*}\nu_{2}]^{-N}}(\xi)|^{2}d\xi
d\theta$
$\displaystyle\quad+\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{1}|[\theta_{\\#}^{*}\nu_{2}]^{+}-[\theta_{\\#}^{*}\nu_{2}]^{+}|^{2}+|[\theta_{\\#}^{*}\nu_{2}]^{-}-[\theta_{\\#}^{*}\nu_{2}]^{-}|^{2}d\theta
d\xi.$

Given a non-negative function $r:[0,\pi]\to\mathbb{R}_{\geq 0}$, let us denote
by $\mathcal{L}_{|_{[0,1]\times[0,\pi]}}^{r}$ the measure supported on
$[0,1]\times[0,\pi]$ with density $(t,\theta)\mapsto r(\theta)$. This and the
notation used for RCDT allow us to summarize the properties of the RSCDT as
follows.

PROPERTIES OF RSCDT

Domain $=\mathcal{M}(\Omega)$, $\Omega=B(0,R)\subset\mathbb{R}^{2}$. Codomain
$=\left(L^{2}([0,1]\times[0,\pi])\right)^{4}$.

Range $=$ 4-tuple $(T,r,U,s)$ where $(T,r)$ is a pair of measurable functions
$T:[0,1]\times[0,\pi]\to[-R,R]$, $r:[0,\pi]\to\mathbb{R}_{\geq 0}$ such that
for each $\theta$ $\left[T(\cdot,\theta)\text{ is non-decreasing and
}r(\theta)>0\right]\text{or }\left[T(\cdot,\theta)=0\text{ and
}r(\theta)=0\right]$ (analogously for $(U,s)$), with the conditions:
$\forall\theta$,
$(T(\cdot,\theta))_{\\#}\mathcal{L}_{|_{[0,1]}}\perp(U(\cdot,\theta))_{\\#}\mathcal{L}_{|_{[0,1]}}$
and also
$L^{T}_{\\#}\mathcal{L}_{|_{[0,1]\times[0,\pi]}}^{r}-L^{U}_{\\#}\mathcal{L}_{|_{[0,1]\times[0,\pi]}}^{s}\in\mathrm{Range}(\mathcal{R})$.

Inverse:
$(T,r,U,s)\mapsto\mathcal{R}^{-1}\left(L^{T}_{\\#}\mathcal{L}_{|_{[0,1]\times[0,\pi]}}^{r}-L^{U}_{\\#}\mathcal{L}_{|_{[0,1]\times[0,\pi]}}^{s}\right)$
or

$(T,r,U,s)\mapsto\mathcal{R}^{-1}(h)$,
$h(\xi,\theta):=r(\theta)(T(\cdot,\theta)^{\dagger})^{\prime}(\xi)-s(\theta)(U(\cdot,\theta)^{\dagger})^{\prime}(\xi)$.

Symmetry property: If $g\in\mathcal{G}_{RCDT}$, then

  * •

(at measure level) given $\nu\in\mathcal{M}(\Omega)$, let
$\nu^{g}=\mathcal{R}^{-1}(L^{g}_{\\#}(\mathcal{R}(\nu)))$ and assume
$\widehat{\nu}^{RSCDT}(\xi,\theta)=(T(\xi,\theta),r(\theta),U(\xi,\theta),s(\theta))$,
then

$\widehat{\nu^{g}}^{RSCDT}(\xi,\theta)=(g(T(\xi,\theta),\theta),r(\theta),g(U(\xi,\theta),\theta),s(\theta)).$

  * •

(at density level) given a $f_{\nu}\in L^{1}(\Omega)$,
$f_{\nu}^{g}=\mathcal{R}^{-1}\left(\mathrm{det}(DL^{g})\cdot\mathcal{R}(f_{\nu})\circ
L^{g}\right)$ and assume
$\widehat{f_{\nu}}^{RSCDT}(\xi,\theta)=(T(\xi,\theta),r(\theta),U(\xi,\theta),s(\theta)),$
then

$\widehat{f_{\nu}^{g}}^{RSCDT}(\xi,\theta)=(g_{\theta}^{-1}(T(\xi,\theta)),r(\theta),g_{\theta}^{-1}(U(\xi,\theta)),s(\theta)).$

Relation with other distances:

$d^{2}_{RSCDT}(\nu_{1},\nu_{2})=\frac{1}{\pi}\int_{0}^{\pi}d^{2}_{SCDT}(\mathcal{R}\nu_{1}(\cdot,\theta),\mathcal{R}\nu_{1}(\cdot,\theta))\
d\theta$.

###  3.3 Numerical methods and software

Numerical methods for computing the Monge and Kantorovich versions, based on
PDEs and linear programming have existed for decades (an excellent repository
can be found in the Python Optimal Transport (POT) library flamary2021pot ). A
detailed description of these is omitted for brevity and is beyond the scope
of this document. We do note that continuous formulations based on the
solution to the Monge-Ampere equations, as well as constrained minimization
methods, exist. For the discrete problem, traditional methods based on linear
programming have recently been enhanced through the use of entropy-based
regularization methods, facilitating a quick solution. A more complete
discussion can be found in cuturi2013sinkhorn ; kolouri2017optimal ;
peyre2019computational . As far as the transport transforms discussed above
derived from the LOT concept, software for computing the discrete
wang2013linear and continuous kolouri2016continuous versions is available in
the PyTransKit software package pytranskit . Software for computing the CDT,
SCDT, RCDT, RSCDT using numerical integration is also available in the
PyTranskit package pytranskit .

##  4 Applications

The transport-based signal transformation framework has been used to generate
state of the art results in numerous applications in data classification
park2018multiplexing ; neary2021transport ; shifat2021radon ;
kundu2018discovery ; basu2014detecting ; rubaiyat2021nearest ; liao2021multi ;
guan2019vehicle ; lee2021identifying ; kolouri2020wasserstein ;
kundu2020enabling ; ozolek2014accurate ; tosun2015detection ;
roy2020recognizing ; shifat2020cell ; zhuang2022local ;
thorpe2017transportation ; rabbi2022invariance ; kolouri2016radon ;
kolouri2016continuous ; kolouri2016sliced ; kolouri2018sliced ;
kolouri2019generalized ; kolouri2017optimal , as well as signal estimation
rubaiyat2020parametric ; nichols2019time , communications park2018multiplexing
; neary2021transport , image formation in turbulence nichols2018transport ,
optics nichols2019transport ; nichols2021vector , particle and high energy
physics cai2020linearized , system identification rubaiyat2024data ,
structural health monitoring wang2020fault ; zhang2024combining ;
rubaiyat2020parametric , reduced order modeling solutions of PDEs ren2021model
, and others basu2014detecting ; rahimi2017automatic ; cloninger2023linearized
. Here we focus our description on three broad areas: signal estimation,
classification, and transport-based morphometry.

###  4.1 Linearizing estimation problems

Numerous applications (e.g. radar, communications) require us to estimate a
certain signal characteristic or parameter that provides information about the
physical environment. Typically, signal parameters of interest include
amplitude, frequency, time delay (or spatial shift in the case of images).
Consider for example the radar problem where a known signal $f(t)$ (an
electromagnetic waveform) is emitted aimed at a moving object. The object
reflects the wave and the received signal is of the form

$h(t)=\alpha f(\omega t-\tau)+\eta(t)$

where $\eta(t)$ is a noise process, $\alpha$ refers to an arbitrary (i.e.,
calibration) amplitude parameter, $\omega$ a frequency shift, and $\tau$ a
certain time delay. Recovering the unknown parameters $\omega,\tau$ would give
us information about the object’s velocity and position (the same principle is
used to localize sources, such as a cell phone), and it is what is known as an
estimation problem.

We can note that the parameter $\alpha$ is unrelated to the quantities of
interest (position and speed of the object), and can be arbitrary (i.e.,
affected by the material properties of the reflecting object, calibration of
the system, and so on). One way to deal with it so as not to affect the
analysis is to simply work with the normalized energy density of the signal
$h^{2}/\|h\|_{2}^{2}$ (and normalizing the known template $f$ the same way) as
discussed in nichols2019time ; rubaiyat2020parametric . Assuming both $f$ and
$h$ have been converted into energy densities, we can now rewrite the
estimation problem as searching for the transformation $g(t)=\omega t-\tau$
such that $f^{g}(t)=g^{\prime}(t)f(g(t))=\omega f(\omega t-\tau)\sim h(t)$. To
do so, it is common to establish a metric $d(\cdot,\cdot)$ between the
received signal and parametric model and define the solution to the estimation
problem as

$\min_{g}d^{2}(f^{g},h)$

When $d^{2}(f^{g},h)=\|f^{g}-h\|_{2}^{2}$ this problem is typically nonlinear
and non-convex. Numerous approaches for finding the solution to such
estimation problems have been proposed including cross-correlation, ESPRIT,
MUSIC, and others (see, for example, jakobsson1998subspace ). Another commonly
used approach is to locate the peak of Wide-band Ambiguity Function (WBAF)
$A_{f^{g},h}(\omega,\tau)$ between the measured signal $h$ and the known
signal $f$ jin1995estimation ; niu1999wavelet ; tao2008two , which is
nonlinear, non-convex, and thus non-trivial to optimize.

As an alternative, rubaiyat2020parametric ; nichols2019time proposed to
utilize the Wasserstein metric with the aid of the Cumulative Distribution
Transform park2018cumulative . The idea is to note that, using the symmetry
property of the CDT,

$d^{2}_{CDT}(f^{g},h)=\|\widehat{f^{g}}-\widehat{h}\|_{2}^{2}=\|g^{-1}\circ\widehat{f}-\widehat{h}\|_{2}^{2}=\left\|\frac{1}{\omega}\widehat{f}+\frac{\tau}{\omega}-\widehat{h}\right\|_{2}^{2}.$

Renaming $a=1/\omega$, $b=\tau/\omega$, we can optimize for

$\min_{a,b}\|a\widehat{f}+b-\widehat{h}\|_{2}^{2}$

which we can note is a quadratic function on the unknown parameters $(a,b)$.
We can thus minimize it with the well-known linear least squares technique,
greatly simplifying the procedure. References nichols2019time ;
rubaiyat2020parametric have extensive analysis with respect to noise and
comparison between numerous techniques. Figure 7 gives an overview of the
simplification.

Figure 7: Depiction of a typical parametric signal estimation problem. A
frequency and time delay problem can be converted into a linear least squares
minimization problem by using transport transforms (CDT).

###  4.2 Linearizing classification problems

The symmetry property of transport transforms helps identify which types of
classification problems each transform can solve. The analysis comes from
understanding how classes are generated and represented in transform space.
Generally, classes consist of deformations of a template distribution $\nu$,
forming sets $\mathbb{S}_{\nu,G}=\\{\nu^{g}:\,g\in G\\}$, where $\nu^{g}$ and
$G$ depend on each specific transport transform framework, as summarized in
Table 4.2. These models for classes are often called in the literature as
algebraic generative models. When $G$ is a convex group, the classes will be
convex in transform space as they are essentially conformed by
$g\circ\widehat{\nu}$ such that $g\in G$. Roughly speaking, changes in the
independent variable in the original signal space transform into deformations
in the dependent variable. In this case, when working with finite samples, any
two distinct classes can be linearly separated in the transform space. We
refer the reader mainly to aldroubi2021partitioning ; kolouri2016radon .

Table 1:

$\nu$ |  $\nu^{g}$ |  density $f_{\nu}^{g}$ |  $\widehat{\cdot}$ |  $G$
---|---|---|---|---
$\nu\in\mathcal{P}(\mathbb{R})$ |  $\nu^{g}=g_{\\#}\nu$ |  $f_{\nu}^{g}=g^{\prime}\cdot f_{\nu}\circ g$ |  CDT |  $G\subseteq\mathcal{G}_{CDT}$
$\nu\in\mathcal{P}(\mathbb{R}^{2})$ |  $\nu^{g}=\mathcal{R}^{-1}(L^{g}_{\\#}(\mathcal{R}(\nu)))$ |  $f_{\nu}^{g}=\mathcal{R}^{-1}\left(\mathrm{det}(DL^{g})\cdot\mathcal{R}(f_{\nu})\circ L^{g}\right)$ |  RCDT |  $G\subseteq\mathcal{G}_{RCDT}$
$\nu\in\mathcal{P}(\mathbb{R}^{d})$ |  $\nu^{g}=g_{\\#}\nu$ |  $f_{\nu}^{g}=\mathrm{det}({Dg})\cdot f_{\nu}\circ g$ |  LOT |  $G\subseteq\mathcal{G}_{LOT}$

###### Theorem 4.1

By using the notation in each row of Table 4.2, we have the following: Given a
template $\nu$ with compact support, consider the class
$C_{\nu}=\\{\nu^{g}:\,g\in G\\}$, where $G$ is a convex subgroup of
$\mathcal{G}$, then $\widehat{\mathbb{S}_{\nu,G}}=\\{\widehat{\nu^{g}}:\,g\in
G\\}$ is a convex set in the transport transform space.

When working with real-world classification problems, verifying that classes
satisfy the algebraic generative model given above is often not possible. The
heuristic is that if elements in the same class can be considered as some
mass-preserving deformation of a template, then transport transforms could be
used. As an example, consider classifying images of segmented faces into faces
that are either neutral (class 1) or smiling (class 2). Figure 8 shows a few
example faces from each class, along with the projection of previously
separated test data onto the 2D penalized linear discriminant (PLDA) analysis
wang2011penalized subspace computed from training data (see kolouri2016radon
for more details). In the right panels, each dot corresponds to the projection
of one image. The top plot corresponds to PLDA projections computed using the
native signal intensities, while the bottom plot shows the projections
computed in RCDT kolouri2016radon space. The RCDT space offers a simplified
data geometry where machine learning can be more successful.

Figure 8: Classifying data in native image domain as opposed to in transport
transform domain (RCDT). Smiling and neutral faces are non-linearly
distinguishable from one another in the image domain. The data classes become
more convex in the transform domain, where they can be separated with a linear
classifier. Figure adapted from kolouri2016radon .

As another example, we demonstrate the ability of the transport signal
transformation framework to create a classifier capable of dealing with
channel nonlinearities due to turbulence in optical communications. A simple
nearest subspace classification method in RCDT domain shifat2021radon can be
used to effectively classify symbols that represent sequences of 5 bits (see
Figure 9 Top-Right). In this case, the classes consist of noisy deformations
of template symbols that occur during transmission over an optical channel as
represented in Figure 9 Top-Left. The transport-based classification method
can show substantial improvements with respect to state-of-the-art Deep
Learning in classification accuracy (bit error rate) as a function of the
amount of training data used (Fig. 9 bottom plot 1), out-of-distribution
performance (Fig. 9 bottom plot 1 plot 2) and computational performance (Fig.
9 bottom plot 3), where several orders of magnitude improvements are
obtained).

Figure 9: OAM patterns (top-right) are emitted by the transmitter, travel
through a turbulent channel, and must be decoded by the receiver (32 class
classification problem). The transport systems representation (red)
substantially outperforms deep learning techniques in terms of classification
accuracy as a function of the number of training data, out-of-distribution
performance, and number of floating points operations (FLOPS). Figure adapted
from shifat2021radon .

###  4.3 Transport-based Morphometry (TBM)

Beyond facilitating signal estimation and classification problems by rendering
data classes convex, the transport transformation framework can also be viewed
as a more general mathematical modeling technique. When applied to modeling
forms, structures, or morphology, the technique has been named Transport-based
Morphometry (TBM) basu2014detecting ; kundu2018discovery ; kundu2020enabling ;
kundu2024discovering ; ironside2024fully .

In medicine and biology, TBM has been applied to model subcellular mass
distributions basu2014detecting ; wang2010optimal ; ozolek2014accurate ;
tosun2015detection ; hanna2017predictive ; rabbi2023transport , gray and white
matter brain morphology kundu2018discovery ; kundu2024discovering ;
kundu2019assessing ; kundu2021investigating , brain hemorrhage morphology in
stroke ironside2024fully , and knee cartilage in osteoarthritis
kundu2020enabling , to name a few. The idea is simple and can be interpreted
in a 3-step pipeline as shown in Figure 10. Given a set of labeled data
(signals or images)
$\\{f_{1},y_{1}\\},\\{f_{2},y_{2}\\},\cdots,\\{f_{N},y_{N}\\}$, where $f_{k}$
corresponds to the signal and $y_{k}$ to a label such as patient age,
molecular measurement, class (diseased vs. non-diseased), we first choose a
suitable reference (such as the intrinsic mean of the dataset as suggested in
basu2014detecting ; kolouri2016continuous ) and use it to compute the
transport transform (LOT, CDT, RCDT, etc.) to obtain a representation in
transport domain
$\\{\widehat{f}_{1},y_{1}\\},\\{\widehat{f}_{2},y_{2}\\},\cdots,\\{\widehat{f}_{N},y_{N}\\}$.
Once we have a representation in transport transform space, different
statistical methods (PCA, LDA, PLDA, CCA, etc) can be employed to understand
the salient features in the dataset, the most important attributes associated
with each label or class, or other characteristics of interest. Interestingly
enough, the characterizations obtained in transform space can often be traced
back to physical quantities such as form or shape providing a natural view of
the dataset.

Figure 10: Transport-based morphometry (TBM) can be understood as a three-step
process. In the first step, a reference measure is chosen, and the transport
transforms of the signals/images are computed. In the second step, the data
represented in transform space is modeled according to different goals. The
third step consists of visualizing the model obtained by the framework by
taking the inverse transforms of the representations in the previous step.

For example, given a large set of images $f_{1},f_{2},\cdots,f_{N}$ scientists
are often interested in organizing the data into its main principal components
- i.e., a decomposition of different modes describing the variation in forms
in the dataset. Principal component analysis (PCA) anderson1958introduction is
a common way to do this, and has been extensively employed in medical image
analysis to visualize ‘registration’ modes (see frangi2002automatic for
example).

Though the procedure can be described in the continuous sense, here we
describe its discrete implementation, and thus we take $\widehat{f}_{k}$ to be
(sampled) discrete vectors. The idea in PCA is to compute a set of orthonormal
basis functions $\\{\widehat{\phi_{k}}\\}_{k=1}^{M}$ which can reconstruct the
given dataset in the least squares sense. Since we apply PCA in the transport
transform space, the procedure involves computing the covariance matrix of the
transformed dataset
$S=1/N\sum_{k=1}^{N}(\widehat{f}_{k}-\bar{m})(\widehat{f}_{k}-\bar{m})^{T}$,
where $\bar{m}=1/N\sum_{k=1}^{N}\widehat{f}_{k}$ is its mean. It can be shown
that the top $M<N$ eigenvectors $\widehat{\phi_{k}}$ of $S$ correspond to the
solution of the least squares approximation problem, subject to the constraint
that the set $\\{\widehat{\phi_{k}}\\}_{k=1}^{M}$ be orthonormal. Each
$\widehat{\phi_{k}}$ can be thought to encode a transport ‘mode of variation’
explaining the dataset. To visualize the deformation mode $\widehat{\phi_{k}}$
we can invert the line
$\widehat{f_{\alpha,k}}:=\bar{m}+\alpha\widehat{\phi_{k}}$,
$\alpha\in\mathbb{R}$, in transform space to obtain the respective functions
$f_{\alpha,k}$ in native space. This procedure is demonstrated in a nuclear
structure dataset in Figure 11 (see basu2014detecting ; rabbi2023transport for
more details). In this database, each image corresponds to a nucleus segmented
from a microscopy image. The intensity at each pixel is proportional to the
relative amount of chromatin present at that location. The left panel shows
the raw images and the right panel shows the deformation modes (obtained by
the TBM method) explaining shape and texture variations present in the
dataset. The first mode corresponds to size differences, the second to
elongation, the third chromatin displacements towards the northeast/southwest
direction, the fourth contains variations in chromatin from the center of the
nucleus to the periphery in a radial fashion, and so on. These help us
visualize and understand the variations present in the dataset in terms of
language that can be associated with specific physical characteristics.

Figure 11: Transport-based morphometry (TBM) used to visualize variations in
texture and shape of nuclear chromatin in cancer.

In addition to visualizing the main modes of variation within a dataset, we
can also use TBM to visualize the main modes of _discrimination_ between
different classes. Let the dataset be as defined above
$\\{f_{1},y_{1}\\},\\{f_{2},y_{2}\\},\cdots,\\{f_{N},y_{N}\\}$, where the
labels $y_{k}$ contain class information (e.g. malignant vs. benign cancer).
We can use the TBM processing pipeline to discover modes of variation that
characterize differences between these classes. Indeed, we can use
regression/classification methods on
$\\{\widehat{f}_{1},y_{1}\\},\\{\widehat{f}_{2},y_{2}\\},\cdots,\\{\widehat{f}_{N},y_{N}\\}$
to discover meaningful differences between the classes. One such procedure
could be the so-called Penalized Linear Discriminant Analysis (PLDA) method
wang2011penalized , which is an improvement upon the classic Linear
Discriminant Analysis (LDA). The new directions $\widehat{\phi_{k}}$ obtained
by PLDA are characterized by the solution of the generalized eigenvalue
problem $S\widehat{\phi}=\lambda(S_{W}+\gamma I)\widehat{\phi}$, with $S$ as
above in the PCA procedure, and where $S_{W}$ is the so-called ‘within’
covariance matrix which is simply the sum of the covariance matrices
corresponding to each class. The procedure can be understood as a ‘mixture’
between PCA and LDA, with $\gamma$ a regularization term. When
$\gamma\rightarrow 0$, we have the traditional LDA procedure fisher1936use ,
and when $\gamma\rightarrow\infty$, we recover PCA. As opposed to PCA which
optimizes the least squares reconstruction, we now have an orthogonal
decomposition into multiple components $\widehat{\phi_{k}}$ that optimizes for
the most discriminating directions between the classes. Figure 12 shows an
example of TBM using PLDA applied to discover discriminating information
between benign and malignant cancer nuclei. The linear classifier $\phi_{1}$
(the first direction retrieved by the PLDA procedure) can be reconstructed
back in image space as before by inverting the line
$\widehat{f_{\alpha,1}}=\bar{m}(x)+\alpha\widehat{\phi_{1}}$. The
visualization is provided in the right portion of the figure, demonstrating
that malignant nuclei tend to have their chromatin more evenly spread
throughout the nuclei than their benign counterparts. Whereas the benign
nuclei tend to have their chromatin more concentrated near the periphery of
the nucleus.

The TBM procedure is not limited to PCA, LDA, or PLDA. In fact, it can be used
more generally. Instead of using binary regression (classification) as in the
example above, we can also perform regression onto any continuous labels
$y_{1},y_{2},\dots$ that may be available in the original dataset. Canonical
component analysis (CCA), for example, can be used to find canonical
components $\widehat{w_{k}},v_{k}$ in transport transform space and label
space, respectively. The canonical components $\widehat{w_{k}},v_{k}$ maximize
the correlation coefficients of the projections
$\text{Corr}(\widehat{w_{k}}^{T}\widehat{f},v_{k}^{T}y)$. As such they can be
used to recover meaningful relationships between transport modes and
functional data. Once the relationships are obtained, they can also be
reconstructed and visualized back in signal/image space. Examples are omitted
here for brevity, but they can be found in basu2014detecting ;
kundu2018discovery ; kundu2019assessing ; kundu2021investigating where
subcellular mass concentrations have been correlated with drug dosage, brain
white and gray matter measurements correlated with ageing, cardiorespiratory
fitness, reaction times and others.

Figure 12: TBM is applied to model discriminating information between nuclei
from benign and malignant tissues. Nuclear images are first transformed into
transport space, where penalized linear discriminant analysis is applied. The
resulting classifier (orange line) is inverted and visualized in image space,
along with the projection of the dataset. In this case, TBM suggests the
hypothesis that in malignant nuclei, nuclear chromatin is often more dispersed
throughout the nucleus, in contrast to benign nuclei, which have more
chromatin positioned at the periphery of the nucleus. Figure adapted from
basu2014detecting .

##  5 Summary

This chapter reunites the most important properties of the Linear Optimal
Transport (LOT) transform together with its variants based on Euclidean one-
dimensional OT. We decided to write it using measure theory since it greatly
simplifies the notation and is better suited to understanding the spaces where
these transforms are defined. We took the viewpoint of starting with a
bijective map into a Euclidean space (based on OT) and then copying the
structure from the Euclidean space. Although in some cases this bijection has
a very clear interpretation from OT, in some others it is a combination of
techniques that aim to balance the most useful aspects of Optimal Transport
and Hilbert Spaces methods. The simplicity of this approach is sometimes
overlooked in the literature in favor of linearizing in the strict sense of
finding a linear geometric structure of different variants of OT. The
exposition is not exhaustive since other transport-based transforms (or
frameworks) exist. For example, Linearized Optimal Transport on manifolds
sarrazin2023linearized which includes the Linear Circular Optimal Transport
(LCOT) embedding martin2023lcot (i.e., LOT for one-dimensional circular
domains), Linearized Hellinger-Kantorovich (LHK) approach cai2022linearized ,
Linear Optimal Partial Transport (LOPT) embedding bai2023linear , linear
Gromov-Wasserstein distance beier2021linear .

## Acknowledgments

This work was funded in part by ONR award N000142212505 and NIH award
GM130825. Authors wish to offer Professor Akram Aldroubi their sincere thanks
for his teaching and support throughout many years.

## Appendix

LOT-geodesics. Consider the curve of measures
$\rho_{t}:=\left((1-t)\widehat{\nu_{1}}+t\widehat{\nu_{2}}\right)_{\\#}\mu_{r}$
for $t\in[0,1]$. First, when $t=0$,
$\rho_{0}=(\widehat{\nu_{1}})_{\\#}\mu_{r}=\nu_{1}$, and when $t=1$,
$\rho_{1}=(\widehat{\nu_{2}})_{\\#}\mu_{r}=\nu_{2}$. Secondly, for each fixed
$t$, by Brenier’s Theorem the map $(1-t)\widehat{\nu_{1}}+t\widehat{\nu_{2}}$
is the optimal transport map between $\mu_{r}$ and $\rho_{t}$, since it pushes
$\mu_{r}$ to $\rho_{t}$ by definition of $\rho_{t}$, and it is convex because
is a non-negative linear combination of convex functions. Finally, $\rho_{t}$
is a constant speed geodesic:

$\displaystyle d_{LOT}(\rho_{t},\rho_{s})$
$\displaystyle=\|\widehat{\rho_{t}}-\widehat{\rho_{s}}\|_{L^{2}(\mu_{r})}$
$\displaystyle=\|(1-t)\widehat{\nu_{1}}+t\widehat{\nu_{2}}-(1-s)\widehat{\nu_{1}}-s\widehat{\nu_{2}}\|_{L^{2}(\mu_{r})}$
$\displaystyle=\|(s-t)(\widehat{\nu_{1}}-\widehat{\nu_{2}})\|_{L^{2}(\mu_{r})}=|s-t|d_{LOT}(\nu_{1},\nu_{2}).$

Characterization property of CDT.

######  Proof (Proof of Theorem 3.1)

Since $\widehat{\nu}$ is non-decreasing, by Theorem 2.1 and (2), it is the
transport map from the reference $\mu_{r}:=(\widehat{\nu}^{\dagger})_{\\#}\nu$
to $\nu$ (since $(\widehat{\nu}\circ\widehat{\nu}^{\dagger})_{\\#}\nu=\nu$).

Now, given any $\sigma\in\mathcal{P}(I)$ there exists a unique non-decreasing
$g$ such that $g_{\\#}\nu=\sigma$ (such function $g$ is the unique optimal
Monge map). Thus, by property (14),
$\widehat{\sigma}=\widehat{g_{\\#}\nu}=g\circ\widehat{\nu}$.

Since $g\circ\widehat{\nu}$ is non-decreasing and satisfies
$(g\circ\widehat{\nu})_{\\#}\mu_{r}=\sigma$, using the uniqueness in Theorem
2.1, we have that $\widehat{\sigma}=g\circ\widehat{\mu}$ is the optimal
transport map from $\mu_{r}$ to $\sigma$. This proves that the transform of
any $\sigma$ is exactly the CDT with respect to the same reference $\mu_{r}$.

Geodesic property of SCDT. The curve $\rho_{t}$ is a curve of Radon measures,
that is, $\rho_{t}\in\mathcal{M}_{+}(I)$ with —ρ_t—=(1-t)—ν_1—+t—ν_2— since
$\left((1-t)\widehat{\nu_{1}^{N}}+t\widehat{\nu_{2}^{N}}\right)_{\\#}\mathcal{L}_{[0,1]}$
is a probability measure. Thus, given $s,t\in[0,1]$

$d_{SCDT}^{2}(\rho_{t},\rho_{s})=d_{CDT}^{2}(\rho_{t}^{N},\rho_{s}^{N})+\left(|\rho_{t}|-|\rho_{s}|\right)^{2}.$

On the one hand,

$\left||\rho_{t}|-|\rho_{s}|\right|=|t-s|\left||\nu_{1}|-|\nu_{2}|\right|.$

On the other hand, notice that ρ_t^N=((1-t)^ν_1^N+t^ν_2^N)_#L_[0,1] is the
geodesic curve between $\nu_{1}^{N}$ and $\nu_{2}^{N}$ with respect to
$d_{CDT}$ and $W_{2}$. Thus,

$d_{CDT}(\rho_{t}^{N},\rho_{s}^{N})=|t-s|d_{CDT}(\nu_{1}^{N},\nu_{2}^{N}).$

Combining all together we obtain:

$\displaystyle d_{SCDT}^{2}(\rho_{t},\rho_{s})$
$\displaystyle=(t-s)^{2}d_{CDT}^{2}(\nu_{1}^{N},\nu_{2}^{N})+(t-s)^{2}\left(|\nu_{1}|-|\nu_{2}|\right)^{2}$
$\displaystyle=(t-s)^{2}d_{SCDT}^{2}(\nu_{1},\nu_{2}).$

Inverse formula and symmetry property of RCDT.

###### Remark 6

We notice that if $\mu$ is a probability measure on $\mathbb{R}^{2}$, then
$\nu:=\mathcal{R}(\mu)$ is a probability measure defined on
$\mathbb{R}\times[0,\pi]$. By the disintegration theorem in classic measure
theory, there exists a $\nu-$a.s. unique set of measures
$(\nu_{\theta})_{\theta\in[0,\pi]}\subset\mathcal{M}(\mathbb{R})$ such that
for any $\psi\in C_{0}(\mathbb{R}\times[0,\pi])$, we have

$\displaystyle\int_{\mathbb{R}\times[0,\pi]}\psi(t,\theta)\,d\nu(t,\theta)=\int_{[0,\pi]}\int_{\mathbb{R}}\psi(t,\theta)\,d\nu_{\theta}(t)\,d\theta.$
(18)

Given a fixed $\theta\in[0,\pi]$, $\mathcal{R}(\mu)_{\theta}$ describes the
projected measure of $\mu$ into the 1D space indexed by $\theta$
(bonneel2015sliced, , Proposition 6), that is: R(μ)_θ=ν_θ=θ_#^* μ∈P(R). By
abuse of notation, we will use ν_θ=ν(⋅,θ)  and so  R(μ)_θ=R(μ)(⋅,θ)=θ_#^* μ.
i.e., $(\mathcal{R}(\mu)(\cdot,\theta))_{\theta\in[0,\pi]}$ also denotes the
‘disintegration’ of $\mathcal{R}(\mu)$.

###### Remark 7

By using disintegration of measures, the inverse formula of RCDT can be
written as

$\displaystyle
T\mapsto\mathcal{R}^{-1}\left[\left(T(\cdot,\theta)_{\\#}\mathcal{L}_{|_{[0,1]}}\right)_{\theta\in[0,\pi]}\right]=\mathcal{R}^{-1}\left[\left(CDT^{-1}(T(\cdot,\theta))\right)_{\theta\in[0,\pi]}\right].$

Now, we can decide the formula of the inverse RCDT: The inverse formula of the
RCDT follows from the identity:

$T(\cdot,\theta)_{\\#}[\sigma(\cdot,\theta)]=[L^{T}_{\\#}\sigma](\cdot,\theta)$
(19)

that holds for every measure $\sigma\in\mathcal{P}(\Omega)$,
$\Omega\subset\mathbb{R}^{2}$. Notice that, in particular, for
$\sigma=\mathcal{L}_{|_{[0,1]\times[0,\pi]}}$, we have
$\mathcal{L}_{|_{[0,1]\times[0,\pi]}}(\cdot,\theta)=\mathcal{L}_{|_{[0,1]}}$.

Then, the symmetry property of the RCDT can be deduced as follows: By using
Remark 6, we obtain

$\displaystyle[\mathcal{R}\nu^{g}](\cdot,\theta)$
$\displaystyle=[\mathcal{R}\mathcal{R}^{-1}(L^{g}_{\\#}\mathcal{R}\nu)](\cdot,\theta)=[L^{g}_{\\#}\mathcal{R}\nu](\cdot,\theta)$
$\displaystyle={(g_{\theta})}_{\\#}\left([\mathcal{R}v](\cdot,\theta)\right)={(g_{\theta})}_{\\#}\left([\mathcal{R}v]_{\theta}\right).$

By taking the CDT of such dimensional measure, and using the symmetry property
of the CDT, we get

$\widehat{{(g_{\theta})}_{\\#}\left([\mathcal{R}v]_{\theta}\right)}=g_{\theta}\circ\widehat{[\mathcal{R}\nu]_{\theta}}.$

Thus,

$\displaystyle\widehat{\nu^{g}}^{SCDT}(\theta,\xi)$
$\displaystyle=\widehat{[\mathcal{R}\nu^{g}](\cdot,\theta)}(\xi)=g_{\theta}(\widehat{[\mathcal{R}v]_{\theta}}(\xi))$
$\displaystyle=g(\widehat{[\mathcal{R}v](\cdot,\theta)}(\xi),\theta)=g(\widehat{\nu}^{SCDT}(\xi,\theta),\theta).$

RCDT-distance is independent from the choice of a reference. Here we will
discuss Remark 5. First, notice that the RCDT is characterized by satisfying
the property

$\int_{-\infty}^{\widehat{f_{\nu}}^{RCDT}(\xi,\theta)}\mathcal{R}f_{\nu}(t,\theta)dt=\xi=\int_{0}^{\xi}1\,dt\qquad\forall\theta\in\mathbb{[}0,\pi],\,\xi\in[0,1]$
(20)

In the literature (see, for e.g., kolouri2015radon ; gong2023radon ), the
left-most RHS is taken in a more general way: Given a reference $r$ (a
function $r:B(0,R^{\prime})\to\mathbb{R}_{\geq 0}$ or a measure $r$ supported
on a ball $B(0,R^{\prime})$), we consider the Radon Cumulative Distribution
Transform with respect to $r$, denoted by $RCDT_{r}$, as the transformation
that satisfies the property

$\int_{-\infty}^{\widehat{f_{\nu}}^{RCDT_{r}}(\xi,\theta)}\mathcal{R}f_{\nu}(t,\theta)dt=\int_{0}^{\xi}\mathcal{R}r(t,\theta)dt=F_{\mathcal{R}r(\cdot,\theta)}(\xi)$

where $F_{\mathcal{R}r(\cdot,\theta)}$ is the CDF o the probability density
$\mathcal{R}r(\cdot,\theta)$. So,

$\displaystyle\widehat{f_{\nu}}^{RCDT_{r}}(\xi,\theta)$
$\displaystyle=\widehat{[\mathcal{R}f_{\nu}](\cdot,\theta)}^{CDT_{[\mathcal{R}r](\cdot,\theta)}}(\xi),\qquad\xi\in[-R^{\prime},R^{\prime}],\theta\in[0,\pi]$

The inverse formula in this case is

$\displaystyle T\mapsto\mathcal{R}^{-1}(L^{T}_{\\#}(\mathcal{R}r))\qquad\text{
where }L^{T}(t,\theta)=(T(t,\theta),\theta).$

The new RCDT-distance with respect to the reference $r$ is given by

$\displaystyle d_{RCDT_{r}}^{2}$
$\displaystyle(\nu_{1},\nu_{2})=\|\widehat{\nu_{1}}^{RCDT_{r}}-\widehat{\nu_{2}}^{RCDT_{r}}\|^{2}_{L^{2}(d\mathcal{R}r)}$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\int_{-R^{\prime}}^{R^{\prime}}|\widehat{\nu_{1}}^{RCDT_{r}}(\xi,\theta)-\widehat{\nu_{2}}^{RCDT_{r}}(\xi,\theta)|^{2}[\mathcal{R}r](\xi,\theta)\,d\xi
d\theta$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}d^{2}_{CDT_{\mathcal{R}r(\cdot,\theta)}}(\mathcal{R}\nu_{1}(\cdot,\theta),\mathcal{R}\nu_{1}(\cdot,\theta))\
d\theta$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}d^{2}_{CDT}(\mathcal{R}\nu_{1}(\cdot,\theta),\mathcal{R}\nu_{1}(\cdot,\theta))\
d\theta=SW_{2}^{2}(\nu_{1},\nu_{2})=d_{RCDT}^{2}(\nu_{1},\nu_{2})$

thus, the RCDT-distance does not depend on the choice of the reference.

Inverse formula of RSCDT. By using disintegration of measures, the inverse
formula of the RCDT can be written as follows:

$\displaystyle((T,a),(U,b))\mapsto$
$\displaystyle\mathcal{R}^{-1}\left[\left(SCDT^{-1}\left(T(\cdot,\theta),a(\theta),U(\cdot,\theta),b(\theta)\right)\right)_{\theta\in[0,\pi]}\right]$
$\displaystyle=\mathcal{R}^{-1}\left[\left(a(\theta)\left(T(\cdot,\theta)_{\\#}\mathcal{L}_{|_{[0,1]}}\right)-b(\theta)\left(U(\cdot,\theta)_{\\#}\mathcal{L}_{|_{[0,1]}}\right)\right)_{\theta\in[0,\pi]}\right].$

## References

  * (1) Aldroubi, A., Li, S., Rohde, G.K.: Partitioning signal classes using transport transforms for data analysis and machine learning.  Sampling Theory, Signal Processing, and Data Analysis 19(1), 1–25 (2021)
  * (2) Aldroubi, A., Martin, R.D., Medri, I., Rohde, G.K., Thareja, S.: The signed cumulative distribution transform for 1-d signal analysis and classification.  Foundations of Data Science 4, 137–163 (2022)
  * (3) Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows: in metric spaces and in the space of probability measures.  Springer Science & Business Media (2005)
  * (4) Anderson, T.W., Anderson, T.W., Anderson, T.W., Anderson, T.W., Mathématicien, E.U.: An introduction to multivariate statistical analysis, vol. 2.  Wiley New York (1958)
  * (5) Bai, Y., Medri, I.V., Diaz Martin, R., Shahroz, R., Kolouri, S.: Linear optimal partial transport embedding.  In: International Conference on Machine Learning, pp. 1492–1520. PMLR (2023)
  * (6) Bai, Y., Schmitzer, B., Thorpe, M., Kolouri, S.: Sliced optimal partial transport.  arXiv preprint arXiv:2212.08049 (2022)
  * (7) Barrett, H.H., Myers, K.J.: Foundations of image science.  John Wiley & Sons (2013)
  * (8) Basu, S., Kolouri, S., Rohde, G.K.: Detecting and visualizing cell phenotype differences from microscopy images using transport-based morphometry.  Proceedings of the National Academy of Sciences 111(9), 3448–3453 (2014)
  * (9) Beier, F., Beinert, R., Steidl, G.: On a linear gromov–wasserstein distance.  IEEE Transactions on Image Processing 31, 7292–7305 (2022)
  * (10) Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the monge-kantorovich mass transfer problem.  Numerische Mathematik 84(3), 375–393 (2000)
  * (11) Bonneel, N., Rabin, J., Peyré, G., Pfister, H.: Sliced and Radon Wasserstein barycenters of measures.  Journal of Mathematical Imaging and Vision 51, 22–45 (2015)
  * (12) Cai, T., Cheng, J., Craig, N., Craig, K.: Linearized optimal transport for collider events.  Physical Review D 102(11), 116019 (2020)
  * (13) Cai, T., Cheng, J., Schmitzer, B., Thorpe, M.: The linearized hellinger–kantorovich distance.  SIAM Journal on Imaging Sciences 15(1), 45–83 (2022)
  * (14) Cloninger, A., Hamm, K., Khurana, V., Moosmüller, C.: Linearized Wasserstein dimensionality reduction with approximation guarantees.  arXiv preprint arXiv:2302.07373 (2023)
  * (15) Cuturi, M.: Sinkhorn distances: Lightspeed computation of optimal transport.  Advances in neural information processing systems 26 (2013)
  * (16) Diaz Martin, R., Medri, I., Bai, Y., Liu, X., Yan, K., Rohde, G.K., Kolouri, S.: Lcot: Linear circular optimal transport.  In: International Conference on Learning Representations (2024)
  * (17) Ding, H., Fang, S.: Geometry on the wasserstein space over a compact riemannian manifold.  Acta Mathematica Scientia 41(6), 1959–1984 (2021)
  * (18) Fisher, R.A.: The use of multiple measurements in taxonomic problems.  Annals of eugenics 7(2), 179–188 (1936)
  * (19) Flamary, R., Courty, N., Gramfort, A., Alaya, M.Z., Boisbunon, A., Chambon, S., Chapel, L., Corenflos, A., Fatras, K., Fournier, N., Gautheron, L., Gayraud, N.T., Janati, H., Rakotomamonjy, A., Redko, I., Rolet, A., Schutz, A., Seguy, V., Sutherland, D.J., Tavenard, R., Tong, A., Vayer, T.: Pot: Python optimal transport.  Journal of Machine Learning Research 22(78), 1–8 (2021).  URL http://jmlr.org/papers/v22/20-451.html
  * (20) Frangi, A.F., Rueckert, D., Schnabel, J.A., Niessen, W.J.: Automatic construction of multiple-object three-dimensional statistical shape models: Application to cardiac modeling.  IEEE transactions on medical imaging 21(9), 1151–1166 (2002)
  * (21) Gong, L., Li, S., Pathan, N.S., Rohde, G.K., Rubaiyat, A.H.M., Thareja, S., et al.: The radon signed cumulative distribution transform and its applications in classification of signed images.  arXiv preprint arXiv:2307.15339 (2023)
  * (22) Guan, S., Liao, B., Du, Y., Yin, X.: Vehicle type recognition based on radon-cdt hybrid transfer learning.  In: 2019 IEEE 10th International Conference on Software Engineering and Service Science (ICSESS), pp. 1–4. IEEE (2019)
  * (23) Hanna, M.G., Liu, C., Rohde, G.K., Singh, R.: Predictive nuclear chromatin characteristics of melanoma and dysplastic nevi.  Journal of Pathology Informatics 8 (2017)
  * (24) Helgason, S., et al.: Integral geometry and Radon transforms.  Springer (2011)
  * (25) Ironside, N., VISTA-ICH, El Naamani, K., Rizvi, T., El-Rabbi, M.S., Kundu, S., Becceril-Gaitan, A., Chen, C.J., Mayer, S.A., Connolly, E.S., et al.: Fully automated hematoma expansion prediction from non-contrast computed tomography in spontaneous ich patients.  medRxiv pp. 2024–05 (2024)
  * (26) Jakobsson, A., Swindlehurst, A.L., Stoica, P.: Subspace-based estimation of time delays and Doppler shifts.  IEEE Transactions on Signal Processing 46(9), 2472–2483 (1998)
  * (27) Jin, Q., Wong, K.M., Luo, Z.Q.: The estimation of time delay and doppler stretch of wideband signals.  IEEE Transactions on Signal Processing 43(4), 904–916 (1995)
  * (28) Kolouri, S., Naderializadeh, N., Rohde, G.K., Hoffmann, H.: Wasserstein embedding for graph learning.  arXiv preprint arXiv:2006.09430 (2020)
  * (29) Kolouri, S., Nadjahi, K., Simsekli, U., Badeau, R., Rohde, G.: Generalized sliced wasserstein distances.  Advances in neural information processing systems 32 (2019)
  * (30) Kolouri, S., Park, S., Thorpe, M., Slepčev, D., Rohde, G.K.: Transport-based analysis, modeling, and learning from signal and data distributions.  arXiv preprint arXiv:1609.04767 (2016)
  * (31) Kolouri, S., Park, S.R., Rohde, G.K.: The radon cumulative distribution transform and its application to image classification.  IEEE transactions on image processing 25(2), 920–934 (2015)
  * (32) Kolouri, S., Park, S.R., Rohde, G.K.: The radon cumulative distribution transform and its application to image classification.  IEEE transactions on image processing 25(2), 920–934 (2016)
  * (33) Kolouri, S., Park, S.R., Thorpe, M., Slepcev, D., Rohde, G.K.: Optimal mass transport: Signal processing and machine-learning applications.  IEEE signal processing magazine 34(4), 43–59 (2017)
  * (34) Kolouri, S., Rohde, G.K., Hoffmann, H.: Sliced wasserstein distance for learning gaussian mixture models.  In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3427–3436 (2018)
  * (35) Kolouri, S., Tosun, A.B., Ozolek, J.A., Rohde, G.K.: A continuous linear optimal transport approach for pattern analysis in image datasets.  Pattern recognition 51, 453–462 (2016)
  * (36) Kolouri, S., Zou, Y., Rohde, G.K.: Sliced wasserstein kernels for probability distributions.  In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5258–5267 (2016)
  * (37) Kundu, S., Ashinsky, B.G., Bouhrara, M., Dam, E.B., Demehri, S., Shifat-E-Rabbi, M., Spencer, R.G., Urish, K.L., Rohde, G.K.: Enabling early detection of osteoarthritis from presymptomatic cartilage texture maps via transport-based learning.  Proceedings of the National Academy of Sciences 117(40), 24709–24719 (2020)
  * (38) Kundu, S., Ghodadra, A., Fakhran, S., Alhilali, L., Rohde, G.: Assessing postconcussive reaction time using transport-based morphometry of diffusion tensor images.  American Journal of Neuroradiology 40(7), 1117–1123 (2019)
  * (39) Kundu, S., Huang, H., Erickson, K.I., McAuley, E., Kramer, A.F., Rohde, G.K.: Investigating impact of cardiorespiratory fitness in reducing brain tissue loss caused by ageing.  Brain Communications 3(4), fcab228 (2021)
  * (40) Kundu, S., Kolouri, S., Erickson, K.I., Kramer, A.F., McAuley, E., Rohde, G.K.: Discovery and visualization of structural biomarkers from mri using transport-based morphometry.  NeuroImage 167, 256–275 (2018)
  * (41) Kundu, S., Sair, H., Sherr, E., Mukherjee, P., Rohde, G.: Discovering the gene-brain-behavior link in autism via generative machine learning.  Science Advances p. in press (2024)
  * (42) Lee, J., Nishikawa, R.M.: Identifying women with mammographically-occult breast cancer leveraging gan-simulated mammograms.  IEEE Transactions on Medical Imaging (2021)
  * (43) Li, S., Hasnat, A., Rubaiyat, M., Rohde, G.K.: Geodesic properties of a generalized wasserstein embedding for time series analysis.  In: Topological, Algebraic and Geometric Learning Workshops 2022, pp. 216–225. PMLR (2022)
  * (44) Liao, B., He, H., Du, Y., Guan, S.: Multi-component vehicle type recognition using adapted cnn by optimal transport.  Signal, Image and Video Processing pp. 1–8 (2021)
  * (45) Mallat, S.: A wavelet tour of signal processing.  Elsevier (1999)
  * (46) Mallat, S.: Group invariant scattering.  Communications on Pure and Applied Mathematics 65(10), 1331–1398 (2012)
  * (47) Moosmüller, C., Cloninger, A.: Linear optimal transport embedding: provable wasserstein classification for certain rigid transformations and perturbations.  Information and Inference: A Journal of the IMA 12(1), 363–389 (2023)
  * (48) Neary, P.L., Nichols, J.M., Watnik, A.T., Judd, K.P., Rohde, G.K., Lindle, J.R., Flann, N.S.: Transport-based pattern recognition versus deep neural networks in underwater oam communications.  JOSA A 38(7), 954–962 (2021)
  * (49) Nichols, J., Emerson, T., Cattell, L., Park, S., Kanaev, A., Bucholtz, F., Watnik, A., Doster, T., Rohde, G.: Transport-based model for turbulence-corrupted imagery.  Applied optics 57(16), 4524–4536 (2018)
  * (50) Nichols, J., Emerson, T., Rohde, G.: A transport model for broadening of a linearly polarized, coherent beam due to inhomogeneities in a turbulent atmosphere.  Journal of Modern Optics 66(8), 835–849 (2019)
  * (51) Nichols, J., Nickel, D., Bucholtz, F.: Vector beam bending.  arXiv preprint arXiv:2110.14582 (2021)
  * (52) Nichols, J.M., Hutchinson, M.N., Menkart, N., Cranch, G.A., Rohde, G.K.: Time delay estimation via wasserstein distance minimization.  IEEE Signal Processing Letters 26(6), 908–912 (2019)
  * (53) Niu, X.X., Ching, P.C., Chan, Y.T.: Wavelet based approach for joint time delay and doppler stretch measurements.  IEEE Transactions on Aerospace and Electronic Systems 35(3), 1111–1119 (1999)
  * (54) Otto, F.: The geometry of dissipative evolution equations: the porous medium equation.  Communications in Partial Differential Equations (2001)
  * (55) Otto, F., Villani, C.: Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality.  Journal of Functional Analysis 173(2), 361–400 (2000)
  * (56) Ozolek, J.A., Tosun, A.B., Wang, W., Chen, C., Kolouri, S., Basu, S., Huang, H., Rohde, G.K.: Accurate diagnosis of thyroid follicular lesions from nuclear morphology using supervised learning.  Medical image analysis 18(5), 772–780 (2014)
  * (57) Park, S.R., Cattell, L., Nichols, J.M., Watnik, A., Doster, T., Rohde, G.K.: De-multiplexing vortex modes in optical communications using transport-based pattern recognition.  Optics express 26(4), 4004–4022 (2018)
  * (58) Park, S.R., Kolouri, S., Kundu, S., Rohde, G.K.: The cumulative distribution transform and linear pattern classification.  Applied and Computational Harmonic Analysis 45(3), 616–641 (2018)
  * (59) Peyré, G., Cuturi, M., et al.: Computational optimal transport: With applications to data science.  Foundations and Trends® in Machine Learning 11(5-6), 355–607 (2019)
  * (60) Rabbi, M.S.E., Ironside, N., Ozolek, J.A., Singh, R., Pantanowitz, L., Rohde, G.K.: Transport-based morphometry of nuclear structures of digital pathology images in cancers.  ArXiv (2023)
  * (61) Rabbi, M.S.E., Zhuang, Y., Li, S., Rubaiyat, A.H.M., Yin, X., Rohde, G.K.: Invariance encoding in sliced-wasserstein space for image classification with limited training data.  Pattern Recognition, in press (2023)
  * (62) Rahimi, A.M., Kolouri, S., Bhattacharyya, R.: Automatic tactical adjustment in real-time: Modeling adversary formations with radon-cumulative distribution transform and canonical correlation analysis.  In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp. 83–90 (2017)
  * (63) Ren, J., Wolf, W.R., Mao, X.: Model reduction of traveling-wave problems via radon cumulative distribution transform.  Physical Review Fluids 6(8), L082501 (2021)
  * (64) Rohde, G.: Pytranskit.  https://github.com/rohdelab/PyTransKit (2021)
  * (65) Roy, A.R., Arif, A.S.M.: Recognizing bangla handwritten numerals: A hybrid model.  In: 2020 3rd International Conference on Intelligent Sustainable Systems (ICISS), pp. 636–643. IEEE (2020)
  * (66) Rubaiyat, A.H.M., Hallam, K.M., Nichols, J.M., Hutchinson, M.N., Li, S., Rohde, G.K.: Parametric signal estimation using the cumulative distribution transform.  IEEE Transactions on Signal Processing 68, 3312–3324 (2020)
  * (67) Rubaiyat, A.H.M., Li, S., Yin, X., Shifat-E-Rabbi, M., Zhuang, Y., Rohde, G.K.: End-to-end signal classification in signed cumulative distribution transform space.  IEEE Transactions on Pattern Analysis and Machine Intelligence (2024)
  * (68) Rubaiyat, A.H.M., Thai, D.H., Nichols, J.M., Hutchinson, M.N., Wallen, S.P., Naify, C.J., Geib, N., Haberman, M.R., Rohde, G.K.: Data-driven identification of parametric governing equations of dynamical systems using the signed cumulative distribution transform.  Computer Methods in Applied Mechanics and Engineering 422, 116822 (2024)
  * (69) Rubaiyat, A.H.M., Zhuang, Y., Li, S., Rohde, G.K., et al.: Nearest subspace search in the signed cumulative distribution transform space for 1d signal classification.  arXiv preprint arXiv:2110.05606 (2021)
  * (70) Santambrogio, F.: Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs and Modeling.  Birkhäuser (2015)
  * (71) Sarrazin, C., Schmitzer, B.: Linearized optimal transport on manifolds.  arXiv preprint arXiv:2303.13901 (2023)
  * (72) Shifat-E-Rabbi, M., Yin, X., Fitzgerald, C.E., Rohde, G.K.: Cell image classification: a comparative overview.  Cytometry Part A 97(4), 347–362 (2020)
  * (73) Shifat-E-Rabbi, M., Yin, X., Rubaiyat, A.H.M., Li, S., Kolouri, S., Aldroubi, A., Nichols, J.M., Rohde, G.K.: Radon cumulative distribution transform subspace modeling for image classification.  Journal of mathematical imaging and vision 63, 1185–1203 (2021)
  * (74) Siebert, W.M.: Circuits, signals, and systems.  MIT press (1986)
  * (75) Stein, E.M., Shakarchi, R.: Fourier analysis: an introduction, vol. 1.  Princeton University Press (2011)
  * (76) Tao, R., Zhang, W.Q., Chen, E.Q.: Two-stage method for joint time delay and doppler shift estimation.  IET Radar, Sonar & Navigation 2(1), 71–77 (2008)
  * (77) Thorpe, M., Park, S., Kolouri, S., Rohde, G.K., Slepčev, D.: A transportation $\ell_{p}$ distance for signal analysis.  Journal of mathematical imaging and vision 59(2), 187–210 (2017)
  * (78) Tosun, A.B., Yergiyev, O., Kolouri, S., Silverman, J.F., Rohde, G.K.: Detection of malignant mesothelioma using nuclear structure of mesothelial cells in effusion cytology specimens.  Cytometry Part A 87(4), 326–333 (2015)
  * (79) Villani, C.: Topics in optimal transportation.  58\. American Mathematical Soc. (2003)
  * (80) Villani, C.: Optimal transport: old and new.  Springer (2009)
  * (81) Wang, B., Baraldi, P., Lu, X., Zio, E., et al.: Fault detection based on optimal transport theory.  In: 30th European Safety and Reliability Conference, ESREL 2020 and 15th Probabilistic Safety Assessment and Management Conference, PSAM 2020, pp. 1764–1771. Research Publishing Services (2020)
  * (82) Wang, W., Mo, Y., Ozolek, J.A., Rohde, G.K.: Penalized fisher discriminant analysis and its application to image-based morphometry.  Pattern recognition letters 32(15), 2128–2135 (2011)
  * (83) Wang, W., Ozolek, J.A., Slepčev, D., Lee, A.B., Chen, C., Rohde, G.K.: An optimal transportation approach for nuclear structure-based pathology.  IEEE transactions on medical imaging 30(3), 621–631 (2010)
  * (84) Wang, W., Slepčev, D., Basu, S., Ozolek, J.A., Rohde, G.K.: A linear optimal transportation framework for quantifying and visualizing variations in sets of images.  International journal of computer vision 101(2), 254–269 (2013)
  * (85) Zhang, M., Liao, B., Wang, H.: Combining deep learning with r-cdt for solar defect recognition.  In: Eighth International Conference on Energy System, Electricity, and Power (ESEP 2023), vol. 13159, pp. 674–679. SPIE (2024)
  * (86) Zhuang, Y., Li, S., Yin, X., Rubaiyat, A.H.M., Rohde, G.K., et al.: Local sliced-wasserstein feature sets for illumination-invariant face recognition.  arXiv preprint arXiv:2202.10642 (2022)