# Mathematical morphology on directional data

Konstantin Hauch
Department of Mathematics,
RPTU Kaiserslautern-Landau,
Kaiserslautern, Germany
<EMAIL_ADDRESS>
&Claudia Redenbach
Department of Mathematics,
RPTU Kaiserslautern-Landau,
Kaiserslautern, Germany
<EMAIL_ADDRESS>

###### Abstract

We define morphological operators and filters for directional images whose
pixel values are unit vectors. This requires an ordering relation for unit
vectors which is obtained by using depth functions. They provide a centre-
outward ordering with respect to a specified centre vector. We apply our
operators on synthetic directional images and compare them with classical
morphological operators for grey-scale images. As application examples, we
enhance the fault region in a compressed glass foam and segment misaligned
fibre regions of glass fibre reinforced polymers.

_K_ eywords depth function $\cdot$ $\mu$CT imaging $\cdot$ image processing
$\cdot$ filtering $\cdot$ glass foam $\cdot$ glass fibre reinforced polymer

##  1 Introduction

Mathematical morphology combines nonlinear image processing and filtering
techniques based on ideas from (random) set theory, topology, and stochastic
geometry. It is widely applied for the analysis of spatial structures, e.g.
from geology, biology or materials science. The foundations of mathematical
morphology are laid in the books by Matheron [1], Serra [2, 3], and Soille
[4]. Sternberg [5] generalised mathematical morphology to numerical functions
via the umbra method, i.e., the application of set morphology to the graph of
the function. Ronse [6] and Goutsias et al. [7] derived a further
generalisation to complete lattices, that is, partially ordered sets with the
property that all subsets have a supremum and an infimum.

The progress of imaging methods from binary via grey-scale images to vector-
valued images, such as colour or ultra-spectral images, requires modified
morphological operators. In particular, diffusion tensor imaging (DTI) in
neuroimaging [8] results in vector-valued images showing the diffusion of
water molecules in the brain. These images can be used to better understand
neural diseases as the diffusion in diseased brains is disturbed or altered
compared to healthy brains. For instance, Zhang et al. [9] proposed a novel
Alzheimer’s disease multi-class classification framework with embedding
feature selection and fusion based on multi-modal neuroimaging.

In mathematical morphology, the step from univariate to multivariate pixel
values involves the challenging task of ordering vectors to define the notions
of minimum and maximum of a set of vectors. An ordering structure for vectors
can be derived from the concept of depth [10]. For a set of vectors, depth
functions assign to each vector a value that measures its "centrality" within
the set. The "centre" is the vector maximising the depth function. Thus, a
centre-outward ordering based on the depth values of each vector yields a
sound definition of minimum and maximum.

Velasco-Forero et al. [11] defined multivariate mathematical morphological
operators via random projection depth. They illustrated their approach on
colour and hyperspectral images. Concepts of depth for directional data, i.e.
unit vectors in $\mathbb{R}^{d}$, have been studied, for instance, by Liu et
al. [12] and Pandolfo et al. [13]. Ley et al. [14] introduced quantiles for
directional data and the angular Mahalanobis depth, and García-Portugués et
al. [15] developed optimal tests for rotational symmetry against new classes
of hyperspherical distributions.

Based on an ordering on the unit sphere ${\mathcal{S}^{d-1}}$, mathematical
morphology was extended to directional images by several authors. Roerdink
[16] introduced mathematical morphology on the sphere via generalised
Minkowski operations. His motivation is morphological processing of earth
pictures under consideration of the curvature. Morphological operators for
angle-valued images were introduced by Peters [17] and Hanbury et al. [18].
However, a generalisation of their results to the unit sphere
${\mathcal{S}^{d-1}}$ with $d>2$ is not straightforward. Frontera-Pons and
Angulo [19] defined morphological operators via a local partial ordering. They
used the Fréchet-Karcher barycenter as local origin $\mu$ on the sphere. The
maximum and minimum of a set of vectors are then found via projecting the
vectors into the tangent space at $\mu$. A drawback of their approach is that
the minimum and maximum derived this way are not necessarily elements of the
given vector set, which seems unnatural. Angulo [20] extended the concept of
multi-scale morphological operators, i.e. a scale-dependent morphology and
filters, from $\mathbb{R}$-valued to $\mathbb{R}^{d}$-valued images. He
defined morphological scale-space operators on metric Maslov-measurable spaces
for images supported on point clouds. An application of his operators to RGB-
valued point clouds, e.g. for object extraction, showed the usefulness of
applying the theory of multi-scale morphological operators to
$\mathbb{R}^{d}$-valued images.

In this work, we introduce morphological operators on directional images.
Section 2 contains basics about mathematical morphology for grey-scale images
and statistical depth functions on ${\mathcal{S}^{d-1}}$. In Section 3, we
introduce the concept of mathematical morphology on directional data using the
directional projection depth. These morphological operators are extended to
multi-scale operators in Section 4. We illustrate and interpret our findings
on synthetic and real-world $\mathcal{S}^{2}$-valued images in Section 5. A
short conclusion is given in Section 6.

##  2 Basics and Notation

###  2.1 Images

We denote by ${\mathcal{S}^{d-1}}=\\{x\in\mathbb{R}^{d}:\lVert
x\rVert_{2}=1\\}$ the $(d-1)$-dimensional unit sphere and $\lVert
x\rVert_{2}=\sqrt{x^{T}x}$ the Euclidean norm. An image is a mapping
$I:E\rightarrow S$ that is defined on a $q$-dimensional domain $E$ with
$E\subset\mathbb{R}^{q}$ or $E\subset\mathbb{Z}^{q}$. The set $S$ is the set
of pixel values. We call $I$ a binary image if $S=\\{0,1\\}$, a grey-scale
image if $S\subset\mathbb{R}$, a vector-valued image if
$S\subset\mathbb{R}^{d}$ and a directional (${\mathcal{S}^{d-1}}$-valued)
image if $S\subset{\mathcal{S}^{d-1}}$. Usually, $q,d\in\\{2,3\\}$. In this
paper, we consider examples of $q$-dimensional ${\mathcal{S}^{d-1}}$-valued
images with $q=2,3$ and $d=3$. We denote by

  * •

$\max I$ the maximal possible pixel value of an image $I$. For instance, $\max
I=1$ if $I$ is a binary image or $\max I=255$ if $I$ is an 8-bit grey-scale
image.

  * •

$\overline{\mathbb{R}}=\mathbb{R}\cup\\{-\infty,\infty\\}$ the extended real
numbers.

  * •

$SO(d)$ the rotation group on $\mathbb{R}^{d}$.

  * •

$x\times y$ the cross product between two vectors $x,y\in\mathbb{R}^{d}$.

  * •

$\mathcal{GC}_{\mu,x}=\\{y\in\mathbb{R}^{d}:(x\times\mu)^{T}y=0\\}\cap{\mathcal{S}^{d-1}}$
the great circle containing the vectors $\mu$ and $x\in{\mathcal{S}^{d-1}}$.
For $d=3$, the great circle $\mathcal{GC}_{\mu,x}$ corresponds to a closed
curve on the surface of $\mathcal{S}^{2}$ created by the intersection of
$\mathcal{S}^{2}$ and a $2$-dimensional hyperplane $H$ passing through the
origin $0_{3}$ [21].

  * •

$\check{b}$ the reflection of a function
$b:\mathbb{R}^{q}\rightarrow\overline{\mathbb{R}}$, i.e., $\check{b}(i)=b(-i)$
for all $i\in\mathbb{R}^{q}$, and by $\check{B}$ the reflection of a set
$B\subset\mathbb{R}^{q}$, i.e., $\check{B}=-B$. A function $b$ is called
symmetric if $b=\check{b}$ and a set $B$ is symmetric if $B=\check{B}$.

###  2.2 Mathematical morphology

We first define morphological operators on grey-scale images $I$. See [1, 2,
4] for a detailed introduction. The two fundamental operations of mathematical
morphology are erosion $\tilde{\varepsilon}_{b}$ and dilation
$\tilde{\delta}_{b}$. They depend on a structuring function
$b:\mathbb{R}^{q}\rightarrow\overline{\mathbb{R}}$ and are defined as

$\displaystyle\tilde{\varepsilon}_{b}(I)(i)$ $\displaystyle=\inf_{j\in
E}\\{I(j)-b(j-i)\\},\quad i\in E,$ (1) $\displaystyle\tilde{\delta}_{b}(I)(i)$
$\displaystyle=\sup_{j\in E}\\{I(j)+b(j-i)\\},\quad i\in E.$ (2)

In the discrete case, supremum and infimum can be replaced by maximum and
minimum. We will not discuss edge effects here (see [22]). Throughout the
paper, we assume that $b$ is symmetric.

A well-known and widely used example for $b$ is the flat structuring function.
Given a structuring element $B\subset\mathbb{R}^{q}$, it is defined as

$\displaystyle b(i)=\begin{cases}0&i\in B\\\ -\infty&i\not\in B.\\\
\end{cases}$

We assume that $B$ is centred at the origin and symmetric. Morphological
operators with flat structuring functions are called flat operators. Symbols
for flat operators will be written with index $B$ rather than $b$, e.g., a
flat erosion with structuring element $B$ is denoted by
$\tilde{\varepsilon}_{B}$.

Non-flat or volumic structuring functions assign non-constant weights to the
pixel values [2]. For instance,

$\displaystyle b(i)$ $\displaystyle=-\left(\frac{\lVert
i\rVert_{2}}{2}\right)^{2}$

is a non-flat structuring function. Note that the grey-scale ranges of images
dilated or eroded by non-flat structuring functions are not bounded [4]. For
instance, an erosion with a non-flat structuring function of an image with
non-negative pixel values can result in negative pixel values. Flat
structuring functions do not suffer from that problem. The output of a flat
erosion or dilation is bounded by the grey-scale range of the input image.

The composition of erosion and dilation yields the morphological operators
opening $\tilde{\gamma}_{b}$ and closing $\tilde{\varphi}_{b}$

$\displaystyle\tilde{\gamma}_{b}(I)(i)$
$\displaystyle=\tilde{\delta}_{b}(\tilde{\varepsilon}_{b}(I))(i),$
$\displaystyle\tilde{\varphi}_{b}(I)(i)$
$\displaystyle=\tilde{\varepsilon}_{b}(\tilde{\delta}_{b}(I))(i).$

In applications, the opening is used to remove bright noise and the closing to
remove dark noise.

Some properties of the morphological operators are summarised in the
following, see also [1, 2, 22]. Let $i\in E$, $I,I^{\prime}$ grey-scale images
and $\\{I_{l}\\}_{l\in\mathbb{N}}$ a family of grey-scale images. We write
$I\leq I^{\prime}$ if $I(i)\leq I^{\prime}(i)$ for all $i\in E$. Furthermore,
$\bigvee$ denotes the point-wise maximum operator and $\bigwedge$ the point-
wise minimum operator.

  1. 1.

All the morphological operators are non-linear, i.e., in general

$\displaystyle\tilde{\Psi}(aI+bI^{\prime})\not=a\tilde{\Psi}(I)+b\tilde{\Psi}(I^{\prime}),$

where $a,b\in\mathbb{R}$ and
$\tilde{\Psi}=\tilde{\delta}_{b},\tilde{\varepsilon}_{b},\tilde{\gamma}_{b}$,
or $\tilde{\varphi}_{b}$.

  2. 2.

Dilation and erosion are dual w.r.t. complementation
$\tilde{\operatorname{C}}$ with $\tilde{\operatorname{C}}I(i)=\max I-I(i)$,
i.e.,

$\displaystyle\tilde{\delta}_{b}(I)$
$\displaystyle=\tilde{\operatorname{C}}\tilde{\varepsilon}_{b}(\tilde{\operatorname{C}}I)$

with $b(i)=b(-i)$ due to symmetry.

  3. 3.

Opening and closing are dual w.r.t. complementation
$\tilde{\operatorname{C}}$, i.e.

$\displaystyle\tilde{\gamma}_{b}(I)$
$\displaystyle=\tilde{\operatorname{C}}\tilde{\varphi}_{b}(\tilde{\operatorname{C}}I)$

  4. 4.

Opening and closing are idempotent, i.e.,

$\displaystyle\tilde{\gamma}_{b}(\tilde{\gamma}_{b}(I))$
$\displaystyle=\tilde{\gamma}_{b}(I),$
$\displaystyle\tilde{\varphi}_{b}(\tilde{\varphi}_{b}(I))$
$\displaystyle=\tilde{\varphi}_{b}(I).$

  5. 5.

The following distribution laws hold

$\displaystyle\tilde{\delta}_{b}\left(\bigvee_{l}I_{l}\right)$
$\displaystyle=\bigvee_{l}\tilde{\delta}_{b}(I_{l}),$
$\displaystyle\tilde{\varepsilon}_{b}\left(\bigwedge_{l}I_{l}\right)$
$\displaystyle=\bigwedge_{l}\tilde{\varepsilon}_{b}(I_{l}).$

  6. 6.

Dilation is associative and erosion fulfils the chain rule, i.e.,

$\displaystyle\tilde{\delta}_{b}(\tilde{\delta}_{b^{\prime}}(I))$
$\displaystyle=\tilde{\delta}_{\tilde{\delta}_{b}(b^{\prime})}(I),$
$\displaystyle\tilde{\varepsilon}_{b}(\tilde{\varepsilon}_{b^{\prime}}(I))$
$\displaystyle=\tilde{\varepsilon}_{\tilde{\delta}_{b}(b^{\prime})}(I).$

  7. 7.

All morphological operators are increasing, i.e.,

$\displaystyle I\leq
I^{\prime}\Rightarrow\tilde{\Psi}(I)\leq\tilde{\Psi}(I^{\prime}),$

where
$\tilde{\Psi}=\tilde{\delta}_{b},\tilde{\varepsilon}_{b},\tilde{\gamma}_{b}$,
or $\tilde{\varphi}_{b}$.

  8. 8.

If $b$ is defined at the origin and $b(0)\geq 0$, dilation is extensive and
erosion is anti-extensive, i.e.,

$\tilde{\varepsilon}_{b}(I)\leq I\leq\tilde{\delta}_{b}(I).$

  9. 9.

Closing is extensive and opening is anti-extensive, i.e.,

$\tilde{\gamma}_{b}(I)\leq I\leq\tilde{\varphi}_{b}(I).$

The latter two properties can be combined to give the ordering relation

$\displaystyle\tilde{\varepsilon}_{b}(I)$
$\displaystyle\leq\tilde{\gamma}_{b}(I)\leq
I\leq\tilde{\varphi}_{b}(I)\leq\tilde{\delta}_{b}(I).$ (3)

###  2.3 Morphological scale-space

In the context of mathematical morphology for grey-scale images, erosion and
dilation can be written as scale-space operators [2, 23]

$\displaystyle\tilde{\varepsilon}_{b_{t}}(I)(i)$ $\displaystyle=\inf_{j\in
E}\\{I(j)-{b_{t}}(j-i)\\}$ (4) $\displaystyle\tilde{\delta}_{b_{t}}(I)(i)$
$\displaystyle=\sup_{j\in E}\\{I(j)+{b_{t}}(j-i)\\},$ (5)

with a scaled structuring function

$\displaystyle{b_{t}}:\mathbb{R}^{q}\rightarrow\overline{\mathbb{R}},\quad
t\geq 0$

with scale parameter $t$. Following [22, p.15], we assume ${b_{t}}$ to have
the following properties:

  1. 1.

$\\{{b_{t}}\\}_{t}$ is a one-parametric family of convex, continuous,
symmetric functions and fulfils the semi-group property, i.e.

$\displaystyle{b_{t}}\dot{+}{b_{s}}$ $\displaystyle={b_{t+s}}\qquad t,s\geq
0,$ (6)

where $\dot{+}$, $+$ are group operations [23]. For instance, when choosing
$b_{t}$ to be a flat structuring function with structuring element $tB$, the
group operations are the Minkowski addition and the standard addition in
$\mathbb{R}$:

$\displaystyle tB\oplus sB$ $\displaystyle=(t+s)B\qquad t,s\geq 0.$

  2. 2.

${b_{t}}$ is non-positive and monotonically decreasing with a global maximum
at the origin of value zero, i.e.,

$\displaystyle{b_{t}}(i)$ $\displaystyle\leq 0\qquad\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ t\geq 0,i\in\mathbb{R}^{q},$
(7) $\displaystyle{b_{t}}(i)$ $\displaystyle\geq{b_{t}}(j)\quad\lVert
i\rVert_{2}<\lVert j\rVert_{2},$ (8) $\displaystyle{b_{t}}(0)$
$\displaystyle=0.$ (9)

An example of a non-flat scaled structuring function is the Poweroid
structuring function [22, p.14, Def. 5]

$\displaystyle{b_{t}}(i)$ $\displaystyle=-t\left(\frac{\lVert
i\rVert_{2}}{t}\right)^{a}\qquad a\geq 0,t>0.$ (10)

###  2.4 Directional data

Directional data are given as a set of vectors on the unit sphere
${\mathcal{S}^{d-1}}$ for some $d\geq 2$. The vector
$x=(x_{1},\dots,x_{d})^{T}\in{\mathcal{S}^{d-1}}$ can be represented as a
point on the surface of ${\mathcal{S}^{d-1}}$. For $x$ in Cartesian
coordinates, a representation in angular coordinates for arbitrary $d$ is
given in [24].

For $d=3$, we use the usual notation for spherical coordinates

$\displaystyle x=x(\phi,\theta)$
$\displaystyle=\begin{pmatrix}\sin{(\theta)}\cos{(\phi)}\\\
\sin{(\theta)}\sin{(\phi)}\\\ \cos{(\theta)}\end{pmatrix}$ (11)

with co-latitude $\theta\in[0,\pi]$ and longitude $\phi\in[0,2\pi)$. A random
vector $X\in\mathcal{S}^{2}$ can be represented in spherical coordinates by
random variables $\Theta$ and $\Phi$ with realisations $\theta$ and $\phi$.

###  2.5 Statistical depth functions

Depth functions are applied in multidimensional non-parametric robust data
analysis and establish an ordering relation between vectors. See [10] for a
detailed introduction. The literature gives several depth functions, for
instance, halfspace depth [25], simplicial depth [26], projection depth [27],
spatial depth [28], or the Mahalanobis depth [10]. For defining a depth for
directional data, we initially restrict attention to the class
$\mathcal{F}_{\mu}$ of distributions on ${\mathcal{S}^{d-1}}$ with a bounded
density that admits a unique modal direction $\mu$. Examples of distributions
in $\mathcal{F}_{\mu}$ are the von Mises–Fisher distribution, the Kent
distribution or the Bingham distribution [21]. We further assume that $\mu$
coincides with the Fisher spherical median [29], that is

$\displaystyle\mu=\arg\min_{\gamma\in{\mathcal{S}^{d-1}}}E(\arccos(X^{T}\gamma)).$

Ley et al. [14] adapted the definition of a depth function given in [10,
Definition 2.1] for directional data as follows.

######  Definition 2.1.

A statistical depth function on ${\mathcal{S}^{d-1}}$ is a bounded, non-
negative mapping
$\mathcal{D}_{\cdot}(\cdot):\mathcal{F}_{\mu}\times{\mathcal{S}^{d-1}}\rightarrow\mathbb{R}$
satisfying

  1. 1.

$\mathcal{D}_{F_{AX}}(AX)=\mathcal{D}_{F_{X}}(X)$ holds for any random vector
$X\in{\mathcal{S}^{d-1}}$ and any rotation matrix $A\in\mathbb{R}^{d\times
d}$.

  2. 2.

$\mathcal{D}_{F}(\mu)=\sup_{y\in{\mathcal{S}^{d-1}}}\mathcal{D}_{F}(y)$ holds
for any $F\in\mathcal{F}_{\mu}$.

  3. 3.

For any $F\in\mathcal{F}_{\mu}$,
$\mathcal{D}_{F}(x)\leq\mathcal{D}_{F}(c(\alpha))$ for the unique geodesic
$c:[0,1]\rightarrow{\mathcal{S}^{d-1}}$ with $c(0)=\mu$, $c(1)=x$ and
$\alpha\in[0,1]$.

  4. 4.

$\mathcal{D}_{F}(-\mu)=0$ for each $F\in\mathcal{F}_{\mu}$ where $-\mu$ is the
antipodal point of the centre $\mu$.

Property 1 implies that the depth of a vector should be independent of the
underlying coordinate system. Property 2 means that for any $F$ with unique
centre $\mu$, the depth function is maximal at $\mu$. Property 3 indicates
that for any point $X$ moving away from the centre $\mu$ along any geodesic
starting at $\mu$, the depth of $X$ decreases monotonically. Property 4
implies that the depth value of $X$ is zero at $-\mu$ since $-\mu$ is at
maximal distance from $\mu$ on ${\mathcal{S}^{d-1}}$.

Depth functions for directional data are, for instance, directional distance-
based depths [13] or the angular Mahalanobis depth [14]. For defining
morphological operators, we cannot use the latter since its application
violates the ordering property (3) for morphological operators. We rather use
a re-scaled version of the cosine distance depth from [13] as defined in the
following.

###  2.6 The angular projection depth

Let $X,X_{1},\dots,X_{n}\in{\mathcal{S}^{d-1}}$ be i.i.d. random vectors with
$X\sim F\in\mathcal{F}_{\mu}$. We define a depth for directional data by

$\displaystyle D^{proj}_{F}(X)=\frac{1+X^{T}\mu}{2},\quad
X\in{\mathcal{S}^{d-1}}.$ (12)

$D^{proj}_{F}(X)$ provides a centre-outward ordering with
$D^{proj}_{F}(\mu)=1$, $D^{proj}_{F}(-\mu)=0$ and is decreasing on a geodesic
from $\mu$ to $-\mu$. Let $A\in\mathbb{R}^{d\times d}$ be any rotation matrix.
Then the distribution of the transformed vector $AX$ has centre $A\mu$. It
follows that

$\displaystyle D^{proj}_{F_{AX}}(AX)$
$\displaystyle=\frac{1+(AX)^{T}A\mu}{2}=\frac{1+X^{T}A^{T}A\mu}{2}$
$\displaystyle=\frac{1+X^{T}\mu}{2}=D^{proj}_{F_{X}}(X).$

Thus, the properties of a directional depth function given in Definition 2.1
are fulfilled.

The empirical angular projection depth reads

$\displaystyle D^{proj}(x)$ $\displaystyle=\frac{1+x^{T}\hat{\mu}}{2},$

where $\hat{\mu}$ is the empirical Fisher spherical median [29]

$\displaystyle\hat{\mu}=\arg\min_{\gamma\in\mathcal{S}^{2}}\sum_{i=1}^{N}\arccos(X_{i}^{T}\gamma).$
(13)

##  3 Mathematical morphology on directional images using directional
projection depth

$\mu$$o$$D^{proj}$01$x_{3}$$x_{(3)}^{T}\mu$$x_{(2)}$$x_{(2)}^{T}\mu$$x_{(1)}$$x_{(1)}^{T}\mu$
Figure 1: Vector ordering by directional projection depth.

The definition of the standard operators of mathematical morphology requires
an ordering relation between pixel values, see (1) and (2). Multivariate
extensions of ordering and mathematical morphology were subject to several
studies [30, 31, 32, 7, 33]. In particular, orderings can be derived from
depth functions.

For ${\mathcal{S}^{d-1}}$ valued images, we use the projection depth
$D^{proj}$ to define an ordering relation between unit vectors. This allows
for a definition of erosion, dilation, and the morphological operators derived
from them as described below. For simplicity, we concentrate on flat
morphological operators with structuring element $B$. In a given application,
the assumption $I(i)\sim F\in\mathcal{F}_{\mu}$ may not be fulfilled.
Nevertheless, a suitable central direction $\mu$ may be derived from the
experimental setup. Hence, $\mu$ will be interpreted as a parameter in the
following. We will write $D_{\mu}^{proj}$ for the projection depth using
$\mu\in{\mathcal{S}^{d-1}}$ as the centre. The parameter $\mu$ can be selected
globally for the whole image or in a locally adaptive manner for parts of the
image. The latter is not considered here.

The erosion $\varepsilon_{B}$ of a directional image $I$ at pixel position
$i\in E$ is (implicitly) defined by

$\displaystyle D_{\mu}^{proj}(\varepsilon_{B}(I))(i)$
$\displaystyle=\tilde{\varepsilon}_{B}(D_{\mu}^{proj}(I))(i),$

where $\tilde{\varepsilon}$ is the flat erosion of a grey-scale image.
Analogously, the dilation $\delta_{B}$ of a directional image $I$ at pixel
position $i\in E$ is (implicitly) defined by

$\displaystyle D_{\mu}^{proj}(\delta_{B}(I))(i)$
$\displaystyle=\tilde{\delta}_{B}(D_{\mu}^{proj}(I))(i),$

where $\tilde{\delta}$ is the flat dilation of a grey-scale image. Hence, a
dilation will select the most central direction covered by the structuring
element while an erosion selects the most outlying direction.

The explicit definition reads

$\displaystyle\varepsilon_{B}(I)(i)$
$\displaystyle={D_{\mu}^{proj}}^{-1}\left(\tilde{\varepsilon}_{B}(D_{\mu}^{proj}(I))\right)(i),$
$\displaystyle\delta_{B}(I)(i)$
$\displaystyle={D_{\mu}^{proj}}^{-1}\left(\tilde{\delta}_{B}(D_{\mu}^{proj}(I))\right)(i),$

where ${D_{\mu}^{proj}}^{-1}$ refers to the preimage under ${D_{\mu}^{proj}}$.

As $D_{\mu}^{proj}$ is not injective, the preimage may consist of more than
one element. Consider vectors $x_{1},x_{2}\in{\mathcal{S}^{d-1}}$ with
$x_{1}^{T}\mu=x_{2}^{T}\mu$. Then,
$D_{\mu}^{proj}(x_{1})=D_{\mu}^{proj}(x_{2})$ but $x_{1}=x_{2}$ is not
necessarily the case. To resolve this issue we use a lexicographic order for
vectors of the same depth value which yields a total ordering [11]. Rotate the
vectors such that $\mu=(0,0,1)^{T}$. Among the (pairwise different) vectors
$\\{x_{i}\\}_{i=1,\dots,n}$ with
$D_{\mu}^{proj}(x_{i})=D_{\mu}^{proj}(x_{j})$, we choose the vector $x^{*}$
with the smallest longitude angle, i.e.,
$x^{*}=\arg\min\limits_{x_{i}}\phi_{i}$ with $(\theta_{i},\phi_{i})$ spherical
coordinates of $x_{i}$, $i=1,\dots,n$. Then rotate $x^{*}$ back.

Opening and closing are defined by

$\displaystyle\gamma_{B}(I)(i)$
$\displaystyle=\delta_{B}(\varepsilon_{B}(I))(i),$
$\displaystyle\varphi_{B}(I)(i)$
$\displaystyle=\varepsilon_{B}(\delta_{B}(I))(i).$

The theory of $h$-orderings [7] immediately implies that the operators defined
above fulfil the properties of morphological operators stated in Subsection
2.2, see [30]. In this theory, $h$ is a map from ${\mathcal{S}^{d-1}}$ into a
space with a partial order. Here we choose $h=D^{proj}$.

Of course, further morphological filters have their depth analogy. For
instance, the scalar difference between the depth of dilation and erosion
defines the morphological gradient

$\displaystyle\operatorname{g}_{D_{\mu}^{proj},B}(I)(i)$
$\displaystyle=D_{\mu}^{proj}\left(\delta_{B}(I)(i)\right)-D_{\mu}^{proj}\left(\varepsilon_{B}(I)(i)\right).$

The morphological Laplacian is defined by

$\displaystyle\Delta_{D_{\mu}^{proj},B}(I)(i)$
$\displaystyle=\Delta_{\delta}(i)-\Delta_{\varepsilon}(i)$

with
$\Delta_{\delta}(i)=D_{\mu}^{proj}\left(\delta_{B}(I)(i)\right)-D_{\mu}^{proj}(I)(i)$
and
$\Delta_{\varepsilon}(i)=D_{\mu}^{proj}(I)(i)-D_{\mu}^{proj}\left(\varepsilon_{B}(I)(i)\right)$.
The shock filter is defined by

$\displaystyle\operatorname{sf}_{D_{\mu}^{proj},B}(I)(i)=\begin{cases}\varepsilon_{B}(I)(i)&\Delta_{D_{\mu}^{proj},B}(I)(i)<0,\\\
\delta_{B}(I)(i)&\Delta_{D_{\mu}^{proj},B}(I)(i)>0,\\\ I(i)&\text{
otherwise.}\\\ \end{cases}$

Shock filtering is used to enhance edges. For grey-scale images, the idea is
to dilate near local maxima and erode near local minima. This way, the
contrast between regions of pixel values with high and low depth is enhanced.

##  4 Pseudo morphological multi-scale operators for directional images

In the next step, we want to extend morphological operators to a multi-scale
setting, leading to a morphological scale space (see [20]). A very simple
approach to scaling is scaling of a flat structuring element $B$ by a factor
$t$. Here, we will consider an alternative approach which yields a structuring
function which has a meaningful interpretation, fulfils the semi-group
property (at least partially), and results in bounded non-flat operators. We
call them pseudo morphological operators since their behaviour reminds of
erosion and dilation as demonstrated in Section 5.

###  4.1 Structuring function for directional images

In general, structuring functions ${b_{t}}:E\rightarrow\overline{\mathbb{R}}$
are unbounded. Thus, multi-scale dilation and erosion inherit this
unboundedness if we use $D_{\mu}^{proj}(I(i))+{b_{t}}(i)$ and
$D_{\mu}^{proj}(I(i))-{b_{t}}(i)$, respectively. However, we cannot associate
a vector with a depth value outside $[0,1]$ since no unit vector could be
found as preimage. In our approach we therefore use structuring functions
${b_{t}}:E\rightarrow SO(d)$. We will define the operation $I(i)+{b_{t}}(i)$
and $I(i)-{b_{t}}(i)$ to be a vector rotation of $I(i)$ where the rotation
angle depends on the scale $t$. Therefore, the depths of $I(i)+{b_{t}}(i)$ and
$I(i)-{b_{t}}(i)$ are both bounded.

The main idea is to rotate a pixel value $x=I(i)$, $i\in E$, about
$x\times\mu$ towards $\mu$ or away from $\mu$ on a great circle
$\mathcal{GC}_{\mu,x}$. This increases or decreases the depth value of the
rotated $x$ which gives our approach also a meaningful interpretation. Note
that $I(i)+{b_{t}}(i)$ and $I(i)-{b_{t}}(i)$, $i\in E$, result again in
directional images which seems natural.

$\mu$$\mathcal{GC}_{\mu,x}$$\mathcal{S}^{2}$$\alpha_{+}$$\alpha_{-}$$\theta$$x$$x-{b_{t}}$$x+{b_{t}}$
Figure 2:  Illustration of the operations $x-{b_{t}}$ given in (18) and
$x+{b_{t}}$ given in (19) with $d=3$ and $x=x(\phi,\theta)$ in spherical
coordinates (11). $x+{b_{t}}$ increases the geodesic distance between $x$ and
$\mu$ by $\alpha_{+}$. $x-{b_{t}}$ decreases the geodesic distance between $x$
and $\mu$ by $\alpha_{-}$. $x$ moves on the great circle
$\mathcal{GC}_{\mu,x}$.

For a formal definition, let $\mu\in{\mathcal{S}^{d-1}}$, $i\in E$,
$x=I(i)\in{\mathcal{S}^{d-1}}$ with $x^{T}\mu=\cos(\theta)$ and
$R_{\mu,x}(\alpha)\in SO(d)$ be a rotation matrix which rotates $x$ about the
cross product $\mu\times x$ and $R_{\mu,x}(\theta)x=\mu$. We define a
structuring function ${b_{t}}$ for directional images by a mapping

$\displaystyle{b_{t}}:E$ $\displaystyle\rightarrow SO(d)$ $\displaystyle i$
$\displaystyle\mapsto{b_{t}}(i):=R_{\mu,x}({\alpha_{t}(i)}),$ (14)

with rotation angle ${\alpha_{t}(i)}$, $t\geq 0$.

Since $R_{\mu,x}$ depends on $x$, the structuring function is locally
adaptive. We choose

$\displaystyle{\alpha_{t}(i)}$ $\displaystyle=\min\left(\frac{\lVert
i\rVert_{2}^{2}}{t},\pi\right),$ (15)

where the truncation at $\pi$ is discussed in more detail in Remark 1. Note
the similarity of $\alpha_{t}$ with quadratic scaling as discussed in [34].

Let $f\Box g$ be the infimal convolution of two functions
$f,g:\mathbb{R}^{q}\to\mathbb{R}$ [35], i.e.,

$\displaystyle(f\Box
g)(i)=\inf_{j\in\mathbb{R}^{q}}\left\\{f(i-j)+g(j)\right\\}.$ (16)

Define

$\displaystyle({b_{t}}\dot{+}{b_{s}})(i):=R_{\mu,x}((\alpha_{t}\Box\alpha_{s})(i)).$
(17)

The structuring element ${b_{t}}$ fulfils the semi-group property (6), i.e.
${b_{t}}(i)\dot{+}{b_{s}}(i)={b_{t+s}}(i)$, for $\lVert
i\rVert_{2}^{2}/t,\lVert i\rVert_{2}^{2}/s<\pi$ as shown in the Appendix.

We illustrate the construction for $d=3$ and $x=x(\phi,\theta)$ in spherical
coordinates as given in (11): The rotation matrix $R_{\mu,x}$ rotates $x$ on
the great circle $\mathcal{GC}_{\mu,x}$. We use $R_{\mu,x}$ to move a pixel
value $x\in\mathcal{S}^{2}$ towards $\mu$ or away from $\mu$ on
$\mathcal{GC}_{\mu,x}$. Thus, the rotation matrix $R_{\mu,x}$ causes a change
of the angle between $x$ and $\mu$, here the co-latitudinal angle $\theta$.
The longitude angle $\phi$ is not changed. Therefore, we define a rotation of
the vector $x$ depending on the structuring function ${b_{t}}$ to this change
of $\theta$ as follows (see Figure 2):

$\displaystyle x(\phi,\theta)-{b_{t}}(i)$
$\displaystyle:=x(\phi,\theta-\alpha_{-})$ (18)

with

$\displaystyle\alpha_{-}$
$\displaystyle=\begin{cases}{\alpha_{t}(i)},&\text{for
}{\alpha_{t}(i)}\in[0,\theta]\\\ \theta,&\text{for
}{\alpha_{t}(i)}>\theta.\end{cases}$ $\displaystyle x(\phi,\theta)+{b_{t}}(i)$
$\displaystyle:=x(\phi,\theta+\alpha_{+})$ (19)

with

$\displaystyle\alpha_{+}$
$\displaystyle=\begin{cases}{\alpha_{t}(i)},&\text{for
}{\alpha_{t}(i)}\in[0,\pi-\theta]\\\ \pi-\theta,&\text{for
}{\alpha_{t}(i)}>\pi-\theta.\end{cases}$

$x-{b_{t}}$ has a smaller geodesic distance to $\mu$ than $x$, and $x+{b_{t}}$
has a larger geodesic distance to $\mu$ than $x$.

######  Remark 1 (Truncation of rotation).

The restriction of $\alpha_{-}$ to $\theta$ and of $\alpha_{+}$ to
$\pi-\theta$ are necessary to achieve an analogous ordering relation as in
(3). Otherwise, $x$ can be rotated beyond $\mu$ which would decrease
$D_{\mu}^{proj}(x-{b_{t}}(i))$. Analogously, $x$ can be rotated beyond $-\mu$
which would increase $D_{\mu}^{proj}(x+{b_{t}}(i))$. To avoid this, we fix the
vector at $\mu$ or $-\mu$ respectively, such that no roation beyond is
possible.

With truncation, the semi-group property is no longer valid. However,
truncation ensures that

$\displaystyle D_{\mu}^{proj}(x+{b_{t}})$ $\displaystyle\leq
D_{\mu}^{proj}(x)\leq D_{\mu}^{proj}(x-{b_{t}}).$ (20)

We obtain (in some sense) analogue properties to (7)$-$(9):

  1. 1.

${b_{t}}$ is non-positive w.r.t. $D_{\mu}^{proj}$ in the sense that for
rotations of $x$ away from $\mu$ its depth decreases, i.e.

$\displaystyle D_{\mu}^{proj}(x+{b_{t}}(i))-D_{\mu}^{proj}(x)$
$\displaystyle\leq 0.$

  2. 2.

${b_{t}}$ is monotonically decreasing w.r.t. $D_{\mu}^{proj}$ in the sense
that for increasing distance between pixel positions the depth decreases, i.e.

$\displaystyle D_{\mu}^{proj}(x+{b_{t}}(i))$ $\displaystyle\geq
D_{\mu}^{proj}(x+{b_{t}}(j)),\quad\lVert i\rVert_{2}<\lVert j\rVert_{2}.$

  3. 3.

${b_{t}}$ has a global maximum at the the origin $o\in\mathbb{R}^{q}$ w.r.t.
$D_{\mu}^{proj}$ with

$\displaystyle D_{\mu}^{proj}(x+{b_{t}}(o))=D_{\mu}^{proj}(x)$

###  4.2 Pseudo morphological multi-scale operators for directional images

Let ${b_{t}}$ be as in Equation (14). The multi-scale erosion
$\varepsilon_{b_{t}}$ of a directional image $I$ at pixel position $i\in E$ is
(implicitly) defined by

$\displaystyle D_{\mu}^{proj}(\varepsilon_{b_{t}}(I))(i)$
$\displaystyle=\inf_{j\in E}\\{D_{\mu}^{proj}(I(j)-{b_{t}}(j-i))\\},$ (21)

where $I(j)-{b_{t}}(j-i)$ is defined in Equation (18). Analogously, the multi-
scale dilation $\delta_{b_{t}}$ of a directional image $I$ at pixel position
$i\in E$ is (implicitly) defined by

$\displaystyle D_{\mu}^{proj}(\delta_{b_{t}}(I))(i)$ $\displaystyle=\sup_{j\in
E}\\{D_{\mu}^{proj}(I(j)+{b_{t}}(j-i))\\},$ (22)

where $I(j)+{b_{t}}(j-i)$ is defined in Equation (19). The interpretation of
the multi-scale operators is as follows: We want to rotate the vector $I(j)$
depending on $||i-j||_{2}$ and the scale $t$ towards $\mu$ ($D_{\mu}^{proj}$
increases) or towards $-\mu$ ($D_{\mu}^{proj}$ decreases).

Multi-scale opening, closing, morphological gradient and shock filter can be
defined as analogues to their flat counterparts.

##  5 Examples

In this section, we investigate the effect of the morphological operators on
synthetic $\mathcal{S}^{2}$-valued images and compare them to their standard
grey-scale counterparts. As real application examples, we detect misaligned
regions in a glass fibre reinforced composite and enhance changes in the
displacement field of a compressed glass foam.

###  5.1 Flat morphological operators

To investigate the newly defined flat morphological operators, we generate a
2-dimensional $\mathcal{S}^{2}$-valued image ${I}$ mimicking direction vectors
obtained from two fibres on a homogeneous background. That is, we assume that
the image contains two objects consisting of vectors with a small angular
deviation from $\mu$, see Figure 3. In the following, we will interpret those
as foreground. The background is formed by vectors directed along a plane
perpendicular to $\mu$. Flat dilation and flat erosion (Figure 4) show similar
behaviour as their standard grey-scale counterparts. Dilation expands objects
directed along $\mu$. That is, the directions of background pixels that are
close to the edge of these objects are rotated towards $\mu$. An erosion
shrinks objects, i.e., object vectors at the edge are rotated so that they are
assigned to the image background.

In a similar manner, the flat opening removes foreground objects that are
smaller than the structuring element. Figure 4 reveals that object vectors in
an object of size smaller than the structuring element $B$ are rotated such
that we assign them to the image background. The flat closing removes small
holes in the foreground. Vectors with large angular deviation from $\mu$
within a background object of size smaller than $B$ are rotated to become part
of the image foreground.

The results of shock filtering of ${I}$ are shown in Figure 5. The edges
between the two objects and the background pixels are enhanced.

Figure 3:  Original $\mathcal{S}^{2}$-valued image ${I}$. The direction
vectors within the fibres are rotationally symmetric around $\mu$ (large red
vector). Their angular deviation from $\mu$ is uniformly drawn from
$(0,\pi/8)$. Background vectors are also rotationally symmetric around $\mu$.
Their angular deviation from $\mu$ is uniformly drawn from $[3/8\pi,5/8\pi]$.

(a) $\delta_{B}({I})$, $B$ is $3\times 3$

(b) $\delta_{B}({I})$, $B$ is $5\times 5$

(c) $\varepsilon_{B}({I})$, $B$ is $3\times 3$

(d) $\varepsilon_{B}({I})$, $B$ is $5\times 5$

(e) $\gamma_{B}({I})$, $B$ is $3\times 3$

(f) $\gamma_{B}({I})$, $B$ is $5\times 5$

(g) $\varphi_{B}({I})$, $B$ is $3\times 3$

(h) $\varphi_{B}({I})$, $B$ is $5\times 5$

Figure 4:  Flat dilation $\delta_{B}$ (a,b), flat erosion $\varepsilon_{B}$
(c,d), flat opening $\gamma_{B}$ (e,f) and flat closing $\varphi_{B}$ (g,h)
with $B$ a square as a structuring element. The large red vector corresponds
to $\mu$.

(a) $\operatorname{sf}_{B}({I})$, $B$ is $3\times 3$

(b) $\operatorname{sf}_{B}({I})$, $B$ is $5\times 5$

Figure 5:  Flat shock filter $\operatorname{sf}_{B}({I})$ with $B$ a square as
structuring element.

###  5.2 Pseudo morphological multi-scale operators

We illustrate the multi-scale dilation and erosion on a directional image
${I}$. $\delta_{b_{t}}({I})$ and $\varepsilon_{b_{t}}({I})$ behave like their
grey-scale counterparts: Figure 6 shows that $\delta_{b_{t}}({I})$ enlarges
objects in the foreground and Figure 7 shows that $\varepsilon_{b_{t}}({I})$
shrinks objects in the foreground. Of course, the enlargement and shrinkage
depend on the continuous scale parameter $t$.

(a) $t=0.1$ (equals input image)

(b) $t=0.5$

(c) $t=0.7$

(d) $t=0.9$

(e) $t=1.1$

(f) $t=2$

Figure 6: $\delta_{b_{t}}({I})$ at scales $t=0.1,0.5,0.7,0.9,1.1,2$. The large
red vector corresponds to $\mu$.

(a) $t=0.1$

(b) $t=0.5$

(c) $t=0.7$

(d) $t=0.9$

(e) $t=1.1$

(f) $t=2$

Figure 7: $\varepsilon_{b_{t}}({I})$ at scales $t=0.1,0.5,0.7,0.9,1.1,2$. The
large red vector corresponds to $\mu$.

###  5.3 Segmentation of aligned and misaligned fibre regions of glass fibre
reinforced polymers

Glass fibre reinforced polymers (GFRP) are frequently used in lightweight
construction of, e.g., cars, planes, and wind power plants. In civil
engineering, they have emerged as an alternative to steel reinforcement as
they do not corrode in aggressive conditions such as marine environments [36].
GFRP materials are anisotropic and characterised by high tensile strength in
the direction of the reinforcing fibres. When manufactured by injection
moulding, the fibres follow the injection direction. However, the fibre
orientation in a central layer differs depending on the production parameters.
Analysing fibre orientation and quantifying misalignment, for instance by
directional filtering [37], may help to optimise the production process and to
improve GFRP.

We examine a $\mu$CT image of GFRP produced and imaged by the Leibniz
Institute for Composite Materials (IVW) in Kaiserslautern (Germany) and
discussed in [38]. See Figure 8(a) for an illustration. Pixel-wise fibre
orientations can be determined by using partial second derivatives as
described in [39] resulting in an $\mathcal{S}^{2}$-valued image.

Our goal is to segment the region aligned along the injection direction, as
desired, and the misaligned fibre region. Figure 8(b) shows a grey value
encoding of the scalar product between the pixelwise fibre directions and
$\mu$. The correctly aligned regions are at the left and right borders while
fibres in the central part are misaligned. The higher the value (brighter the
image pixel), the more aligned the vectors are along $\mu$. Simply
thresholding this image results in a very noisy segmentation. Therefore, the
directional image is smoothed prior to the segmentation by a morphological
closing along $\mu=(0,1,0)$ with $B$ a $7\times 7\times 7$ cube.

We choose a threshold value of 1.5 to segment areas aligned along $\mu$, see
Figure 8(b).

(a) 2D slice of the original reconstructed 3D $\mu$CT image. The image size is
1100$\times$500 pixels with a pixel spacing of 4 $\mu$m.

(b) Left: 2D slice of a 3D grey scale image encoding the angular deviation of
the fibre direction in each pixel from $\mu=(0,1,0)$. The left and right side
are aligned with $\mu$ (correctly aligned). In the inner region, the fibres
are misaligned. Right: Closing of the directional image (with $B$ a $7\times
7\times 7$ cube) and binarisation of the resulting angular deviation image
(threshold 1.5) highlight the correctly aligned areas. The image size is 1100
$\times$ 1500 pixels.

Figure 8: Sectional $\mu$CT image of GFRP produced and imaged by the Leibniz
Institute for Composite Materials (IVW) in Kaiserslautern (Germany), the
corresponding original and closed directional image.

###  5.4 Enhancement of fault zones in compressed glass foam

To characterise the structural behaviour of complex materials during loading,
mechanical tests and simultaneous $\mu$CT imaging (in situ CT) can be used to
estimate local displacements of the material on the micro-scale. We analyse a
glass foam which consists of very thin struts.

The $\mu$CT images were recorded while the glass foam was compressed in the
z-direction. The foam is expected to fail very suddenly due to its thin
struts. Nogatz et al. [40] computed the displacement field for the transition
from strain level 1% to 3.8% which is shown in Figure 9(a). During
compression, a fault zone forms which is now post-processed by directional
morphology. To this end, our operators are applied to the directional
component of the displacement vectors. That is, the vectors were normalised
for the calculation of the morphological operators and were subsequently
scaled back to their original length. As we mostly expect compression along
the loading direction, we choose $\mu=(0,0,1)$.

We enhance the fault zone with the morphological gradient, see Figure 9(b). If
motion estimation algorithms fail to reconstruct these sharp edges, we can
enhance them by erosion, as shown in Figure 9(c). Some other materials however
are known to show creep before failure. Here, a smoother transition seems more
reasonable, which is obtained by a dilation, see Figure 9(d).

Closing (Figure 9(e)) and opening (Figure 9(f)) remove misalignment in the
transition between the two regions.

(a) Displacement field

(b) Morphological gradient

(c) Erosion

(d) Dilation

(e) Opening

(f) Closing

Figure 9: XZ-slices of displacement field based on [40]. Yellow colour
indicates movement along the z-direction, and blue refers to the opposite
direction. The computed displacement field reflects the influence on the
microstructure during compression (a). The morphological gradient (with $B$ a
$3\times 3$ square) enhances the fault zone (b). The results of other
morphological operators are given in (c-f). We chose $B$ as a $15\times 15$
square.

##  6 Conclusion

We have defined morphological operators for ${\mathcal{S}^{d-1}}$-valued
images. The required ordering for unit vectors is derived from the angular
projection depth which enables a sound definition of
${\mathcal{S}^{d-1}}$-valued morphological operators and filters. Relations of
these morphological operators to their grey-scale counterparts are emphasised.
Additionally, we segmented misaligned fiber regions of glass fiber reinforced
polymers and enhanced the fault region in a compressed glass foam.

Future work could address the removal of disturbances or the highlighting of
directional changes in DTI images by morphological operations [8, 9].

## Acknowledgment

We thank Tessa Nogatz for providing the displacement field of the glass foam
and Tin Barišin for providing the glass fibre reinforced polymers images.

## Appendix

Given (17) the structuring element ${b_{t}}$ fulfils the semi-group property
(6) for

$\displaystyle\frac{\lVert i\rVert_{2}^{2}}{t},\frac{\lVert
i\rVert_{2}^{2}}{s}<\pi$ (23)

as follows: Let $\overline{oi}$ be the line segment from the origin
$o\in\mathbb{R}^{q}$ to $i\in\mathbb{R}^{q}$. For $t,s>0$, let $j^{*}$ be a
point on $\overline{oi}\subset\mathbb{R}^{q}$ such that

$\displaystyle\lVert j^{*}\rVert_{2}$ $\displaystyle=\frac{s}{t+s}\lVert
i\rVert_{2}$ (24) $\displaystyle\lVert i-j^{*}\rVert_{2}$
$\displaystyle=\frac{t}{t+s}\lVert i\rVert_{2}.$ (25)

Then,

$\displaystyle(\alpha_{t}\Box\alpha_{s})(i)$
$\displaystyle\overset{\eqref{eq:infimal
convolution}}{=}\inf_{j\in\mathbb{R}^{q}}\left\\{\alpha_{t}(i-j)+\alpha_{s}(j)\right\\}$
$\displaystyle\overset{\eqref{eq:alpha}}{=}\inf_{j\in\mathbb{R}^{q}}\left\\{\min\left(\frac{\lVert
i-j\rVert_{2}^{2}}{t},\pi\right)+\min\left(\frac{\lVert
j\rVert_{2}^{2}}{s},\pi\right)\right\\}$
$\displaystyle\overset{\eqref{eq:norm_t_s}}{=}\inf_{j\in\mathbb{R}^{q}}\left\\{\frac{||i-j||_{2}^{2}}{t}+\frac{\lVert
j\rVert_{2}^{2}}{s}\right\\}$
$\displaystyle\overset{j^{*}\in\overline{oi}}{=}\frac{\lVert
i-j^{*}\rVert_{2}^{2}}{t}+\frac{\lVert j^{*}\rVert_{2}^{2}}{s}$ (26)
$\displaystyle\overset{\eqref{eq:||j^*||_2},\eqref{eq:||i-j^*||_2}}{=}\frac{t\lVert
i\rVert_{2}^{2}}{(t+s)^{2}}+\frac{s\lVert i\rVert_{2}^{2}}{(t+s)^{2}}$
$\displaystyle=\frac{\lVert i\rVert_{2}^{2}}{(t+s)}$
$\displaystyle\overset{\eqref{eq:norm_t_s}}{=}\min\left(\frac{\lVert
i\rVert_{2}^{2}}{(t+s)},\pi\right)$
$\displaystyle\overset{\eqref{eq:alpha}}{=}\alpha_{t+s}(i).$ (27)

Thus,

$\displaystyle{b_{t}}(i)\dot{+}{b_{s}}(i)$
$\displaystyle\overset{\eqref{eq:btsd(i) dot+
bssd(i)}}{=}R_{\mu,x}((\alpha_{t}\Box\alpha_{s})(i))\overset{\eqref{eq:angle
infimal convolution}}{=}R_{\mu,x}(\alpha_{t+s}(i))\overset{\eqref{eq:bt-
def}}{=}{b_{t+s}}(i).$

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