# HandCT: hands-on computational dataset for X-Ray Computed Tomography and
Machine-Learning

Valat Emilien, Valat Loth

###### Abstract

Machine-learning methods rely on sufficiently large dataset to learn data
distributions. They are widely used in research in X-Ray Computed Tomography,
from low-dose scan denoising to optimisation of the reconstruction process.
The lack of datasets prevents the scalability of these methods to realistic 3D
problems. We develop a 3D procedural dataset in order to produce samples for
data-driven algorithms. It is made of a meshed model of a left hand and a
script to randomly change its anatomic properties and pose whilst conserving
realistic features. This open-source solution relies on the freeware Blender
and its Python core. Blender handles the modelling, the mesh and the
generation of the hand’s pose, whilst Python processes file format conversion
from obj file to matrix and functions to scale and center the volume for
further processing. Dataset availability and quality drives research in
machine-learning. We design a dataset that weighs few megabytes, provides
truthful samples and proposes continuous enhancements using version control.
We anticipate this work to be a starting point for anatomically accurate
procedural datasets. For instance, by adding more internal features and fine
tuning their X-Ray attenuation properties.

##  1 Introduction

## Background & Summary

Machine-learning (ML) methods are widely used in X-Ray Computed Tomography
(XCT), mainly for low-dose denoising, scarce-view CT and dual-energy CT. They
rely on data samples to be trained on, and their performance is conditioned by
the availability and quality of the datasets. At present, the limitations to
the existing datasets we identify are related to the purposes for which they
were designed as well as the difficulty to collect and use CT data.

Benchmarking reconstruction algorithms and training ML approaches require
different types of datasets. The Shepp-Logan phantom [14] is a computational
phantom used routinely when comparing reconstruction processes. It consists of
a set of ellipses with various X-Ray attenuation values and radius, and has
been designed to resemble to a human brain. As training a ML algorithm on one
data point is irrelevant, it cannot be used for that purpose. [6] consists in
a set of projections and an associated reconstruction of a single walnut,
chosen for its likeness to a human brain. As all these items relate to the
same data-point, it is unsuitable for ML purposes as well. The SophiaBeads
dataset [2] is used to benchmark reconstruction processes against a certain
level of angular scarcity in the projection set. Even though there are
different data-points in the angular domain, all the measures relate to one
actual sample. [16, 3, 15] have the same limitation, knowing that the number
of samples is to little to undertake ML procedures without over-fitting the
data. One might consider using those datasets for 2D CT by reconstructing the
3D volumes, slice and re-project them to produce as many samples as there are
slices. This can lead to data-leakage and should be avoided, as illustrated in
Fig. 1.

Figure 1: Example of data-leakage for 2D slices of a 3D volume. The color
indicate the level of correlation with the last few training samples. Green,
orange and red indicate strong, weak, poor correlation, respectively.

When training image improvement methods on the first 180 slices out of 200
that compose a volume from the Sophiabeads dataset, the samples are highly
correlated with their immediate neighbours. As such, when evaluating on the
last 10% of the dataset which corresponds to the last 20 slices, we observe
that the performance of an approach depends on the index of the slice
considered: the closer to the last slice of the training set a test sample is,
the better. Averaging the performance across the test dataset might mask this,
but is not very honest. Re-projecting data can be used in context where
several volumes are present, as in [5, 12].

CT datasets of real data are collected for and by CT practitioners. As
underlined by [4], they rely on proprietary software and data formats not
directly compatible with ML frameworks. For instance, [11] offers a collection
of hundreds of Computer Assisted Design drawings of skulls, obtained by expert
segmentation of CT scans of skulls. The use of this dataset requires
considerable amount of space and is not shipped with file conversion scripts.
The scientist coming from ML will often use unrealistic datasets made of
ellipsoids, as in [8, 1, 9]. The ease of use and development of these is out-
weighted by the coarse feature modelled. To the extent of our knowledge, the
only dataset that is designed for 3D Cone-Beam CT is [4]. It is made of a
considerable amount of scan data from 42 walnuts exactly fit for ML purposes.
Each sample consists on average of 6 Gb of projection data. As such, there are
few light-weight datasets of realistic computational phantoms for training
machine-learning algorithms on. To the best of our knowledge, only [13] offers
a procedure to generate foam-like phantoms for X-Ray CT and machine-learning.
To add to that literature, we develop HandCT, a procedural dataset that can
generate sufficiently many good-quality samples for data-driven approaches.

It consists of a left-hand surface mesh modelled in Blender and a random
morphological deformation routine in Python. Inspired by the Z-Anatomy project
[10], we chose to model an anatomically accurate object, that can be validated
by anatomists and enhanced by 3D artists. To this initial layer, we added a
Python script, implemented in Blender which offers a Python interface, that
creates a sample by applying a modification (pose of the hand, pose of
individual fingers, thickness of phalanxes) to the rig of the model, meshing
it and saving it to an obj. format. We added a script to convert the latter to
a Numpy matrix and to perform basic matrix operations (scaling and
translating). The main feature of this dataset is the realistic anatomic
feature modelling. The latter can be enhanced and refined by modifying the
open-source mesh, and reproducibility is ensured using version control on the
Zenodo and Github platforms. It is also important to note that the model
itself weighs few megabytes.

## Methods

HandCT relies on three steps. First, a base model is designed and rigged by a
3D artist. Then, a python script modifies the latter into a sample model using
random scaling of random parts of the rig. The sample’s mesh is saved as an
.obj file and the modifications reversed back to the initial state. Finally, a
python script converts the .obj file into a numpy array and deals with
absorption values, scaling of the volume in which the sample is defined and
changing the coordinate system. The model is now saved as an array of desired
size and orientation, ready to be used further. The interaction between base
model, sample model and array is described in Figure 2.

BlenderPythonBaseSampleArray Figure 2: Data processing pipeline. The default
model is first modified using random transformations (green arrow), then the
morphed model is saved as a .obj and the transformations reversed (blue
arrow). The parsing operation (red arrow) to an array is irreversible as edge
information is not stored.

### Hand model design and rigging

We build on a realistic 3D model initially designed for virtual and augmented
reality applications [7]. This model is composed of two rigged hands (rig and
mesh), but we kept only the left hand in this initial design. A rig is an
underlying skeleton, not rendered at display time, that allow animators to
give different poses to a model. Fig.3 displays the skeleton hierarchy of our
model. Compared to the original, we replaced the initial rigid bones by the
so-called “bendy bones”, allowing more finesse in the movements. There are six
controllers for the hand pose: one per finger and an extra one for the whole
hand at once. There are fourteen controllers for the phalanx thickness. The
scale attribute of each controller is bounded between 0.75 and 1.15 using
trial and error, to maintain a morphological coherence.

ptForearmHandHand
Flexibility$M_{f}$$M_{b}$$M_{1}$$M_{2}$$M_{3}$$I_{f}$$I_{b}$$I_{1}$$I_{2}$$I_{3}$$R_{f}$$R_{b}$$R_{1}$$R_{2}$$R_{3}$$P_{f}$$P_{b}$$P_{1}$$P_{2}$$P_{3}$$T_{f}$$T_{b}$$T_{1}$$T_{2}$
Figure 3: Hierarchical structure of the rig. Arrows indicate inheritance from
parent to child. gray, green and blue nodes indicate rigid bones, pose-control
and thickness-control nodes, respectively. Fingers are referred to with the
first letter of their name.

The mesh controls the visual aspect of the hand. In order to have a sufficient
resolution and visual smoothness, we used the “Subsurf” option to increase the
density if vertices on the mesh. Then, in order to be processed, the model has
to be remeshed to a high resolution point cloud. To perform this operation, we
use the “Block Remesh” tool and set the Octree Depth parameter to a value of
10 using trial and error. Our way to control this parameter was to visually
assess the quality of the model once loaded in Python, as shown in Fig. 4. As
a result, each model is composed of roughly 65k points.

Figure 4: Comparison of the base model (Left), with a low-poly version
(Middle) and the chosen resolution (Right).

### Python Processing Functions

We kept the Python-only part of the script at its minimum: the only functions
that it handles are:

  * •

Randomly selecting a bendy bone and apply a random scaling.

  * •

Parsing a .obj file to a point cloud.

  * •

Scaling and centering the point cloud in a $512^{3}$ Numpy array.

Fig. 5 shows three views of the base model, once parsed to a converted to a
Numpy matrix.

Figure 5: Comparison of the base model in front view (Left), side view
(Middle) and bottom view (Right).

## Data Records

The data is stored on Zenodo and GitHub. The files are the following:

  * •

base_model.blend. For the model and the morphological modifications script.

  * •

parse_obj _file.py. For the centering, scaling and numpy conversion.

  * •

demo.py. Can be run to display the base model once saved in Numpy format.

## Technical Validation

The purpose of this dataset is to give an easily modifiable yet accurate
modelling of a hand. We bounded the range of deformations, as well as the some
movements to preserve realism.

## Usage Notes

First, in Blender editor, edit the number of samples, the random seed used and
the save path. Then, run the script generate _samples.py from the Text Editor
area. Running the script will generate and save the different samples, that
can be further processed in Python using the command:

python obj_to_array.py ---path_to_file path_to_your_file

Alternatively, the function parse_obj_file can be used in a loop iterating on
the files path.

## Code availability

The dataset and the code are available on Zenodo and GitHub.

## References

  * [1] J. Adler and O. Öktem.  Learned Primal-dual Reconstruction.  IEEE Transactions on Medical Imaging, 37(6):1322–1332, 7 2018.
  * [2] S. Coban.  SophiaBeads Dataset Project [Data set] .  Technical report, University of Manchester, Manchester, 2015.
  * [3] F. De Carlo, D. Gürsoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijbers, and M. Rivers.  TomoBank: a tomographic data repository for computational x-ray science.  Measurement Science and Technology, 29(3):034004, 3 2018.
  * [4] H. Der Sarkissian, F. Lucka, M. Van Eijnatten, G. Colacicco, S. B. Coban, Kees, and J. Batenburg.  a cone-beam X-ray computed tomography data collection designed for machine learning.  Nature Scientific Data, 6(1):1–8, 2019.
  * [5] O. Grove, A. E. Berglund, M. B. Schabath, H. J. W. L. Aerts, A. Dekker, H. Wang, E. R. Velazquez, P. Lambin, Y. Gu, Y. Balagurunathan, E. Eikman, R. A. Gatenby, S. Eschrich, and R. J. Gillies.  Quantitative computed tomographic descriptors associate tumor shape complexity and intratumor heterogeneity with prognosis in lung adenocarcinoma.  PLoS ONE, 10(3), 7 2015.
  * [6] K. Hämäläinen, L. Harhanen, A. Kallonen, A. Kujanpää, E. Niemi, and S. Siltanen.  Tomographic X-ray data of a walnut.  Technical report, Department of Physics, Helsinki, 2 2015.
  * [7] D. Haupt.  Rigged and Animated Hand, 2012.
  * [8] K. H. Jin, M. T. McCann, E. Froustey, and M. Unser.  Deep Convolutional Neural Network for Inverse Problems in Imaging.  IEEE Transactions on Image Processing, 26(9):4509–4522, 11 2017\.
  * [9] B. Kelly, T. P. Matthews, and M. A. Anastasio.  Deep Learning-Guided Image Reconstruction from Incomplete Data.  Technical report, Washington University, Department of Computer Science, Saint Louis, 9 2017.
  * [10] G. Kervyn.  The First Open Source 3d Atlas of Human Anatomy.  Technical report, Department of Anatomy, Montpellier, 2022.
  * [11] J. Li, M. Krall, F. Trummer, A. R. Memon, A. Pepe, C. Gsaxner, Y. Jin, X. Chen, H. Deutschmann, U. Zefferer, U. Schäfer, G. v. Campe, and J. Egger.  MUG500+: Database of 500 high-resolution healthy human skulls and 29 craniotomy skulls and implants.  Data in Brief, 39:102171, 12 2021.
  * [12] C. McCollough.  TU-FG-207A-04: Overview of the Low Dose CT Grand Challenge.  Medical Physics, 43(6Part35):3759–3760, 6 2016.
  * [13] D. M. Pelt, A. A. Hendriksen, and K. J. Batenburg.  Foam-like phantoms for comparing tomography algorithms.  Journal of Synchrotron Radiation, 29:254–265, 1 2022.
  * [14] L. Shepp and B. Logan.  The Fourier Reconstruction of a Head Section.  IEEE Transactions on Nuclear Science, 1974.
  * [15] K. Singh, H. Menke, M. Andrew, C. Rau, B. Bijeljic, and M. J. Blunt.  Data descriptor: Time-resolved synchrotron X-ray microtomography datasets of drainage and imbibition in carbonate rocks.  Scientific Data, 5, 2018.
  * [16] J. A. Slotwinski, E. J. Garboczi, P. E. Stutzman, C. F. Ferraris, S. S. Watson, and M. A. Peltz.  Characterization of Metal Powders Used for Additive Manufacturing.  Journal of Research of the National Institute of Standards and Technology, 119:460–493, 2014.


# Effects of a background scalar field induced by the Lorentz symmetry
violation on Non Relativistic Quantum Mechanics

J. D. García-Aguilar<EMAIL_ADDRESS>Centro de Estudios Científicos y
Tecnológicos No 16, Instituto Politécnico Nacional, Pachuca: Ciudad del
Conocimiento y la Cultura, Carretera Pachuca Actopan km 1+500, San Agustín
Tlaxiaca, Hidalgo, México.

###### Abstract

The extension of Standard Model of the Fundamental Particles which consider
the Lorentz Symmetry Violation governed by a background tensor field presented
by Colladay and Kostelecky consider the hypothesis of having a privileged
direction in the space-time. Here we investigated a extension of the
Schrödinger Equation inspired on that model taking into account a background
scalar field. We analyze the behavior of this modified Schrödinger equation on
a particle trapped in a well and quantum harmonic oscillator.

Lorentz Violation, CPT Violation, Standard Model Extension, Bound states.

###### pacs:

11.30.Cp, 11.30.Er, 12.60.-i, 03.65.Ge

##  I Introduction

The Standard Model Extension which consider the CPT symmetry violation (SME)
[1, 2] is a proposal to explain some conflicts between
experimental/observational data and the theoretical predictions of the
Standard Model (SM) [3, 4, 5, 6, 7, 8]. The SME assumes that all possible
Lorentz violating terms arise from nonzero expectation values for Lorentz
tensors which are considered as spacetime background fields in a fundamental
theory. These background fields give rise to preferred space time directions.

Lorentz symmetry violation leads to some modification on particle properties
such as energy-dependent dispersion relations and field equations (see, e.g.,
Refs. [9, 10, 11, 12]) its effects can make evident considering the framework
of SME in the ultrarelativistic and non relativistic limits. Currently, from
the model building view, a large effort has been done in different branches of
physics to find possibles signal on Lorentz symmetry violation, and therefore,
provide experimental bounds on SME parameters. Some of these studios utilizing
neutrinos [13, 14, 15, 16, 17], photons [18, 19, 20, 21], kaons [22], harmonic
oscillator under influence of a Coulomb-like potential [23], Bose-Einstein
condensates [24, 25], supersymmetry [26, 27] and extra dimensions [28, 29].
However, a complete treatment of the SME in the non-relativistic quantum
mechanics sector remains an open issue.

In the present work, we seek to address this gap in the literature by
extending the existing work of SME in non relativistic quantum mechanic sector
to study its implications over Schrödinger equation, thereby opening the path
for additional searching for Lorentz violation. To achieve this goal, we have
investigated the effects of a background constant scalar field induced by
Lorentz symmetry breaking. Then, we search a bound state solutions considering
a time independent modified Schrödinger equation and shown that the Lorentz
violation terms can be eliminated through a redefinition on wavefunctions.

This papper is organized as follows: in section II we show how considering the
CPT-odd sector of SME the momentum operator is modified and this leads a
Schrödinger equation which is not invariant under parity transformation
$x\rightarrow-x$, we also determine the solution for this equation considering
a free particle. In section III, we solve the modified Schrödinger equation
for a particle trapped in a well whereas in section IV harmonic oscillator
case is treated. Finally, in section V we present our conclusions.

##  II The modified Schrödinger equation

In order to obtain the modified Schrödinger equation, we perform the study
considering the scalar CPT-odd sector of the Lorentz Violation Standard Model
Extension [1] whose Lagrangian is given by

$\mathcal{L}=\left(D_{\mu}\phi\right)^{\dagger}D^{\mu}\phi+\mu^{2}\phi^{\dagger}\phi-\frac{\lambda}{3!}\left(\phi^{\dagger}\phi\right)^{2}+ik^{\mu}\phi^{\dagger}D_{\mu}\phi+h.c.,$
(1)

where $\phi$ denote the Higgs doublet, $D_{\mu}$ is the covariant derivative
and $k^{\mu}$ is the coefficient which quantifies the Lorentz violation which
has dimensions of mass and obeys the relation $k_{\mu}k^{\mu}\ll 1$.

For our purposes we are going to consider a massless scalar field without
interactions. Therefore, the equation 1 is reduced to

$\mathcal{L}=\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi+ik_{\mu}\phi^{\dagger}\partial^{\mu}\phi-
ik^{\mu}\phi\partial_{\mu}\phi^{\dagger},$ (2)

which can be rewritten as

$\mathcal{L}=\partial^{\prime}_{\mu}\phi^{\dagger}\partial^{\prime\mu}\phi,$
(3)

considering

$\partial^{\prime}_{\mu}=\partial_{\mu}+ik_{\mu},$ (4)

so, the redefinition for the derivative operator allows to eliminate the CPT
violating term.

Hence, the redefinition 4 establish the possibility to investigate the effects
of the momentum operator redefinition given by

$\hat{p}=-i\hbar\left(\frac{d}{dx}+i\alpha\right)$ (5)

where $\alpha$ is a background constant scalar field whose value quantifies
the effects of Lorentz symmetry violation.

The equation 5 leads to rewrite the Schrödinger equation for the free particle
in the form $\left(\hbar=1\right)$

$-\frac{1}{2m}\left(\frac{d^{2}\psi}{dx^{2}}+2i\alpha\frac{d\psi}{dx}-\alpha^{2}\psi\right)=E\psi,$
(6)

where, it is straightforward to see how the equation 6 is not invariant under
$x\rightarrow-x$ parity transformation.

We can rewrite the equation 6 in the form

$\frac{d^{2}\psi}{dx^{2}}+2i\alpha\frac{d\psi}{dx}=-\left(k^{2}-\alpha^{2}\right)\psi,$
(7)

where we define $k=\sqrt{2mE}$.

By imposing that $k^{2}>\alpha^{2}$, we obtain the solution

$\psi\left(x\right)=Ae^{i\left(k-\alpha\right)x}+Be^{-i\left(k+\alpha\right)x},$
(8)

where, we can see how the effects of Lorentz symmetry violation is present on
wavefunctions whereas energy eigenvalues is independent of background scalar
field.

Observe that in the limit $\alpha\rightarrow 0$ the Lorentz symmetry is
restored, so the equations 5, 6 and 8 are analogous to the usual equations to
describe the quantum free particle [30, 31]. This is to be expected because if
we make a wavefunction redefinition

$\psi\left(x\right)\rightarrow e^{-i\alpha x}\psi\left(x\right),$ (9)

then the equation 6 becomes

$-\frac{1}{2m}\frac{d^{2}\psi}{dx^{2}}=E\psi,$ (10)

which is the canonical Schrödinger equation for the free particle.

##  III Particle on a box

In this section, we investigate the effects of the background constant
constant field over a particle trapped in a well with potential given by

$V\left(x\right)=\left\\{\begin{array}[]{lr}0&0\leq x\leq L\\\
\infty&\mbox{otherwise}\end{array}\right.$ (11)

and hence a particle is completely free, except at the two ends and therefore
the wavefunction is only different zero inside the well.

We know from section II that the solution for the free particle considering
the operator 5 (in order to consider the Lorentz symmetry violation) and
subject to the condition

$\psi\left(0\right)=\psi\left(L\right)=0,$ (12)

is given by

$\psi_{n}\left(x\right)=Ae^{-i\alpha x}\sin\left(k_{n}x\right),$ (13)

where

$k_{n}=\frac{n\pi}{L}$ (14)

with $n=1,2,3,\ldots$.

From the expression 13 we obtain the eigenvalues for the Energy in the form

$E_{n}=\frac{n^{2}\pi^{2}}{2mL^{2}}$ (15)

and the normalization constant sets

$A=\sqrt{\frac{2}{L}}.$ (16)

Assuming that $\alpha=1/10$ and $L=10$ for purposes of demonstration we obtain
the plot of the first four graphs of the real part to the equation 13 shown in
figure 1; here we see the dependence of wavefunctions on $\alpha$ value. As we
expected in the limit $\alpha\rightarrow 0$, the solution have the same
features when is taking into account the usual definition for the momentum
operator on the system [30, 31].

Figure 1: First four plots of the real parts of $\psi_{n}\left(x\right)$ for
$\alpha=1/10$ and $L=10$.

We have know an important result: the effects of Lorentz symmetry violation
appears only over wave functions for this system, the observables are
independent of background scalar field. Therefore, the Lorentz Violation does
not lead to alterations on the Heisenberg uncertainty principle [32].

##  IV The Harmonic Oscillator case

In this section, our aim is to solve the Schrödinger equation taking into
account the operator 5 and the potential given by

$V=\frac{1}{2}m\omega^{2}x^{2}.$ (17)

Thus, the time independent Schrödinger equation to solve is given in the form

$-\frac{1}{2m}\left(\frac{d^{2}\psi}{dx^{2}}+2i\alpha\frac{d\psi}{dx}-\alpha^{2}\psi\right)+\frac{1}{2}m\omega^{2}x^{2}\psi=E\psi.$
(18)

which we can solve considering the following ladder operators

$a_{\pm}=\frac{1}{\sqrt{2m\omega}}\left(\mp\left(\frac{d}{dx}+i\alpha\right)+m\omega
x\right),$ (19)

where direct calculation reveals that

$\left[a_{-},a_{+}\right]=1.$ (20)

With this, equation 18 becomes

$\omega\left(a_{+}a_{-}+\frac{1}{2}\right)\psi=E\psi,$ (21)

and through the annihilation operator we can determine the ground state
($\psi_{0}$) of the system

$a_{-}\psi_{0}=\frac{1}{\sqrt{2m\omega}}\left(\left(\frac{d}{dx}+i\alpha\right)+m\omega
x\right)\psi_{0}=0.$ (22)

The solution for this differential equation is straightforward to obtain and
is given by

$\psi_{0}\left(x\right)=Ae^{-\frac{m\omega}{2}x^{2}}e^{-i\alpha x}$ (23)

where the normalization constant take the value

$A=\left(\frac{m\omega}{\pi}\right)^{1/4},$ (24)

and to determine the energy we substitute the equation 23 into right side of
the equation 18, then

$E_{0}=\frac{\omega}{2},$ (25)

we can see how the last two results does not depend on the $\alpha$ parameter.

To generate the excited states we can use the creation operator $a_{+}$ over
ground state repeatedly

$\psi_{n}\left(x\right)=A_{n}a_{+}\psi_{0}\left(x\right),$ (26)

where $A_{n}$ is the normalization constant and the energy is incremented by
$\omega$ with each step

$E_{n}=\left(n+\frac{1}{2}\right)\omega.$ (27)

In figure 2 we plot the first three graphs of real part of the wavefunctions
given by the equation 26 using the equivalence showed in appendix A
considering $m=\omega=1$ and $\alpha=10$. In figure 3 we can see explicitly
the effects of Lorentz violation over wave functions because the imaginary
part of them it is different to zero .

Figure 2: First three plots of the real parts of $\psi_{n}\left(x\right)$
considering $\alpha=1/10$ and $m=\omega=1$. Figure 3: First three plots of the
imaginary parts of $\psi_{n}\left(x\right)$ considering $\alpha=1/10$ and
$m=\omega=1$.

In this section we have seen that similar to the previous section, the effects
of Lorentz symmetry violation is presented only over the wave functions, the
energy levels are independent on $\alpha$ parameter.

##  V Conclusions

This papper deals with time independent Schrödinger equation under effect of
the CPT-odd Lorentz violation. We have shown that momemtum operator is
modified and, consequently provides a modified Schrödinger equation to be
investigated. In this sense, we present the wavefunctions for two different
systems.

In our first analysis, we considered a particle trapped in a well, where we
have shown that the effects of the Lorentz violation appears only on
wavefunctions whereas the energy levels and normalization constant are similar
to the case where Lorentz symmetry is invariant. In the second analysis the
harmonic oscilator is solved through a redefinition on the ladder operators
and we find that similarly to the first analysis, the Lorentz violation is
present only on the wavefunctions.

As we seen the presence of Lorentz symmetry violation in the Schrödinger
equation for a particle in a well and harmonic oscillator can not induce
modifications over energy levels of the system and therefore, it is not
possible to expect signal of SME in this energy scale. It is worth mentioning
that our analysis can be considered beyond one dimensión and it is in our
interest as future perspective to analize the two and three dimensional case.

###### Acknowledgements.

We would like to thank Pablo Paniagua López for their comments and discussión
on the manuscript. The author appreciate the facilities given by IPN through
the SIP project number 20201313. This work was partially supported by a PAPIIT
grant IN109321.

##  Appendix A Equivalence formula

It is to interesting to consider what happens when some redefinitions over the
wavefunctions is used before applies the ladder operators of the equation 19 .
Since the ground state give by the equation 23 can be written as

$\psi_{0}\left(x\right)=e^{-i\alpha x}\psi^{\prime}_{0}\left(x\right)$ (28)

where

$\psi^{\prime}_{0}\left(x\right)=\left(\frac{m\omega}{\pi}\right)^{1/4}e^{-\frac{m\omega}{2}x^{2}},$
(29)

is the usual ground state for quantum oscillator where Lorentz symmetry is
present.

Through algebraic manipulations

$\displaystyle\psi_{1}\left(x\right)$ $\displaystyle=$ $\displaystyle
A_{1}a_{+}\psi_{0}\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{A_{1}}{\sqrt{2m\omega}}\left(-\left(\frac{d}{dx}+i\alpha\right)+m\omega
x\right)e^{-i\alpha x}\psi^{\prime}_{0}\left(x\right)$ $\displaystyle=$
$\displaystyle e^{-i\alpha
x}\left(\frac{A_{1}}{\sqrt{2m\omega}}\left(-\frac{d}{dx}+m\omega
x\right)\psi^{\prime}_{0}\left(x\right)\right)$ $\displaystyle=$
$\displaystyle e^{-i\alpha x}\psi^{\prime}_{1}\left(x\right),$

where $\psi^{\prime}_{1}\left(x\right)$ is the first excited state for usual
quantum oscillator [30, 31].

Then, we can determine by similar way the general equation for the excited
states

$\psi_{n}\left(x\right)=e^{-i\alpha x}\psi^{\prime}_{n}\left(x\right).$ (31)

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# Musical Form Generation

Lilac Atassi
Department of Music
University of California, San Diego
<EMAIL_ADDRESS>

###### Abstract

While recent generative models can produce engaging music, their utility is
limited. The variation in the music is often left to chance, resulting in
compositions that lack structure. Pieces extending beyond a minute can become
incoherent or repetitive. This paper introduces an approach for generating
structured, arbitrarily long musical pieces. Central to this approach is the
creation of musical segments using a conditional generative model, with
transitions between these segments. The generation of prompts that determine
the high-level composition is distinct from the creation of finer, lower-level
details. A large language model is then used to suggest the musical form.

##  1 Introduction

Music generative models have recently demonstrated their ability to produce
remarkable musical compositions, potentially attributed to their proficiency
as sequence decision-makers [1] However, a significant hindrance to their
practical utility lies in their deficiency of composition-level structure.
This becomes evident after approximately one minute of generated music.
Neither a musician nor a novice user could use them to create a long,
appealing piece of music."

End-to-end music generative models often focus more on small-scale structures,
such as chords and phrases, over large-scale structures like musical form.
This emphasis can hinder a machine learning model’s ability to innovate at the
composition level. Segmenting music into dual structural levels can improve
control over its form. The concept of hierarchical music models dates back at
least to the work of Lerdahl and Jackendoff [2]. Conditional generative models
pave the way to distinguish between controlling composition-level music and
managing small-scale structures. By combining music segments generated from
distinct prompts, one can craft an extended piece. The prompt sequence
dictates the form, and each individual prompt guides the small-scale
structures within its associated segment. This distinction allows the
generative model to more abstractly conceptualize musical form.

Generative models introduce musical variation through sampling. Typically, a
temperature parameter is fine-tuned, and the best results are often handpicked
to minimize repetition and ensure coherence. However, instead of leaving the
overarching structure to random chance, our method reduces the temperature
parameter during sampling. We introduce composition-level variations via text
prompts. Some samples demonstrating transitions between segments using our
approach can be accessed at lilac-code.github.io."

##  2 Method

Figure 1: In our proposed method, the transformer model produces new tokens by
sampling from a probability distribution. This distribution is a weighted
blend of two distinct distributions, each conditioned on a text prompt. Over
time, these weights adjust to facilitate a transition from one prompt to the
next.

Certain model architectures align more effectively with our proposed approach.
AudioLDM 2 [3] employs a language model grounded on text prompts to produce
tokens within the latent space of AudioMAE [4]. Following this, a latent
diffusion model conditioned on these tokens generates a latent vector, which
the VAE decoder then translates into a spectrogram. However, during our
experiments, we observed that the latent diffusion model in this architecture
struggled to produce transitions. This limitation primarily stems from the
diffusion model’s dependency on a robust classifier guidance scale to yield
high-quality audio. Moreover, the conditioning occurs over a global timeframe.
Consequently, when tasked with bridging two fixed endpoints to form a
transition, the diffusion model either results in a noisy sound with a
diminished guidance scale or a singular state when the guidance scale is
heightened.

The streamlined architecture of MusicGen [5] offers enhanced control over the
classifier-free guidance scale, which influences the generation of tokens. As
a result, the MusicGen model can be readily adapted to our method. In
MusicGen, a language model is trained to produce quantized latent vectors from
a given prompt. These vectors are subsequently transformed into audio by the
decoder of EnCodec [6].

We run the transformer inference using two conditioning vectors, derived from
two consecutive prompts. The next token is generated by sampling from the
weighted combination of the two probability distributions. Since the
transformer takes into account both past tokens and the two prompts when
generating each token, the mentioned jarring jump issue is eliminated with
this approach.

After obtaining the two probability distributions over all dictionary codes
from the two prompts, we compute their weighted average. Fig. 1 illustrates
this sampling process. We perform a linear interpolation between the two
distributions over an interval of 5 seconds or less. In instances where the
interval exceeds 5 seconds, our experiments showed a notable chance of audio
artifacts appearing. During this interpolation interval, we also boost the
temperature parameter by 50%. When using beam search as the sampling method,
we also double the top K parameter during the transition phase. In our tests,
depending on the distinctiveness of the prompts and their musical differences,
fine-tuning these two parameters proved essential for achieving seamless
transitions.

Our proposed method provides users with a dual level of control over musical
form. Users have the option to input a sequence of text prompts, specifying
the desired length for each musical segment. Alternatively, they can employ a
different system, such as a large language model with in-context learning, to
generate the sequence of text prompts for the music generative model. During
our experiments, we observed that when transitioning between two prompts that
represent a significant musical contrast, for example, moving from electronic
dance music to classical music, the model faced challenges. It either
hesitated to switch its musical context or produced a decrease in the quality
of the generated audio.

##  3 Conclusions

Current text-to-music models are limited by their reliance on a single text
prompt, though music generated under a minute can be engaging. This paper
proposes a method that extends the use of such models, offering control over
the musical form of longer pieces. Having high-level control over the musical
form facilitates research into models that operate at the abstract composition
level. This addresses a significant gap in the existing methods and tools for
music generation.

## Acknowledgments and Disclosure of Funding

I would like to express my gratitude for Shahrokh Yadegari, Robert Wannamaker,
and Mosalam Ebrahimi whose feedback helped shape the ideas that led to the
work presented in this paper. Additionally, I would like to thank NVIDIA
Corporation for donating the graphics cards used in this research.

## Ethical Considerations

The work showcased has ethical considerations in line with the foundational
generative models. As the public learn more about such models some of the
risks are mitigated.

## References

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