# A Deep Learning-Based and Fully Automated Pipeline for Regurgitant Mitral
Valve Anatomy Analysis from 3D Echocardiography

Riccardo Munafò  Simone Saitta  Giacomo Ingallina  Paolo Denti  Francesco
Maisano  Eustachio Agricola  Alberto Redaelli  Emiliano Votta

###### Abstract

Echocardiography, specifically three-dimensional transesophageal
echocardiography (3DTEE), is the recommended method for diagnosing mitral
regurgitation (MR). 3DTEE provides a high-quality 3D image of the mitral valve
(MV), allowing for precise segmentation and measurement of the regurgitant
valve anatomy. However, manual TEE segmentations are time-consuming and prone
to intra-operator variability, affecting the reliability of the measurements.
To address this, we developed a fully automated pipeline using a 3D
convolutional neural network (CNN) to segment MV substructures (annulus,
anterior leaflet, and posterior leaflet) and quantify MV anatomy. The 3D CNN,
based on a multi-decoder residual U-Net architecture, was trained and tested
on a dataset comprising 100 3DTEE images with corresponding segmentations from
percutaneous MV interventions. Within the pipeline, a custom algorithm refines
the CNN-based segmentations and extracts MV models, from which anatomical
landmarks and features are quantified. The accuracy of the proposed method was
assessed using Dice score and mean surface distance (MSD) against ground truth
segmentations, and the extracted anatomical parameters were compared against
the semiautomated commercial software TomTec Image Arena. The trained 3D CNN
achieved an average Dice score of 0.79 and MSD of 0.47 mm for the combined
segmentation of the annulus, anterior and posterior leaflet. The proposed CNN
architecture outperformed a baseline residual U-Net architecture in MV
substructure segmentation, and the refinement of the predicted annulus
segmentation improved MSD by 8.36%. The annular and leaflet linear
measurements differed by less than 7.94 mm and 3.67 mm, respectively, compared
to the 3D measurements obtained with TomTec Image Arena. The proposed pipeline
was faster than the commercial software, with a modeling time of 12.54 s and a
quantification time of 54.42 s. This pipeline improves the MV segmentation and
anatomical quantification process, efficiently providing consistent and
repeatable results.

###### keywords:

Three dimensional transesophageal echocardiography , Mitral regurgitation ,
Automatic Segmentation , Convolutional Neural Network , Mitral valve anatomy
quantification , Mitral Valve

[Poli]organization=Department of Electronics, Information and Bioengineering,
Politecnico di Milano,city=Milan, country=Italy [OSR]organization=Unit of
Cardiovascular Imaging, IRCCS San Raffaele Hospital,city=Milan, country=Italy
[Chi-OSR]organization=Cardiac Surgery Department, IRCCS San Raffaele
Hospital,city=Milan, country=Italy [Uni-OSR]organization=Vita-Salute San
Raffaele University,city=Milan, country=Italy

##  1 Introduction

The mitral valve (MV) consists of two leaflets, named anterior and posterior
leaflet, respectively, inserted on a fibrous and muscular ring, called
annulus, that is in continuity with the wall of the left ventricle and of the
left atrium. The leaflets are connected by fibrous chordae tendineae to two
papillary muscles that stem from the mid region of the left ventricular wall.
In physiologic conditions, during ventricular systole the increase in
transvalvular pressure difference drives MV leaflets motion towards the
atrium; the synergistic function of all MV substructures allows for leaflet
coaptation and MV continence, thus preventing from blood backflow into the
atrium. When continence is lost due to the dysfunction of any MV substructure,
mitral regurgitation (MR) occurs; if untreated, MR typically worsens and leads
to heart failure [8]. The preferred treatment for MR is MV repair, owing to
its better short-term results and to significantly better long-term survival
rates in the elderly population as compared to MV replacement [7].
Echocardiography is the standard imaging modality in the clinical routine to
perform diagnosis of MV regurgitation, according to the guidelines of the
European Society of Cardiology (ESC) [23] and the American Heart Association
(AHA) [17]. In transesophageal echocardiography (TEE) the ultrasound probe is
inserted down the esophagus, resulting in higher-quality images of the heart
due to the proximity to the target structures. TEE allows to acquire two-
dimensional (2D) and three-dimensional (3D) images of the heart, 3DTEE being
recommended over 2DTEE as it yields more accurate and reproducible volumetric
measurements [23, 17]. Clear images of MV can be acquired with the TEE probe
at the midesophageal level, a standard view in 3DTEE examinations. Of note,
TEE is also used for intraoperative guidance in structural interventional
cardiology, i.e., in procedures devoted to treat intracardiac pathologies by
implanting ad hoc devices by means of a catheter that is inserted at a
peripheral access and driven through the bloodstream to the heart [26].
Precise MV segmentation and identification of its substructures (annulus and
leaflets) can facilitate accurate and quantitative measurements of the
regurgitant MV anatomy to support MR diagnosis, MV repair planning, and
intraprocedural maneuvers in structural interventional cardiology. However,
manual segmentation of TEE imaging is time-consuming and affected by intra-
operator variability that impacts the reliability of the resulting anatomical
measurements. In recent years, several works have introduced semi- or fully
automatic methods to detect and segment MV structures. Many early methods
proposed in the last decade are based on thin tissue detector and active
contour algorithms, which require human-in-the-loop-interactions to work
properly [3, 22]. In [20], Pouch et al. presented a semi-automatic framework
for 4D segmentation and geometric modeling of the MV based on multi-atlas
label fusion and template-based medial modeling. Ionasec et al. [11] proposed
a fully automatic system for valve modeling, adopting several machine learning
algorithms to fit mitral and aortic valve models on both CT [11] and 3DTEE
[15, 27] images. Methods based on deep learning, and in particular
Convolutional Neural Networks (CNNs), have become increasingly popular for MV
segmentation. In [2], Andreassen et al. proposed a mitral annulus segmentation
method that leverages 2D CNN-based predictions on individual image slices
extracted from a 3DTEE volume to generate a 3D annulus model through an
iterative post-processing algorithm. Similarly, Zhang et al. [28] applied a
deep reinforcement learning algorithm and a 2D CNN for annulus landmark
detection in combination with a spline fitting algorithm to infer the 3D
mitral annulus from 3DTEE images. In a recent work, Carnahan et al. [5]
proposed a fully automatic method adopting a 3D CNN fed by 3DTEE images for
segmentation of MV leaflets in diastole. Overall, previous approaches for MV
segmentation from 3DTEE are either time-consuming [3, 22], or semi-automatic
[20], or limited in their capability to capture anatomical variability [11].
On the other hand, the most recent CNN-based methods, which are fully
automatic and fast, focus only on the mitral annulus [2, 28] or segment the MV
without distinguish among the different substructures [5].
In this paper, we present a fully automated pipeline for MV substructures
segmentation and anatomical characterization from 3DTEE. Within our pipeline,
a trained 3D CNN is fed by 3DTEE images at end-systole and is able to
separately segment the mitral annulus, the anterior leaflet and the posterior
leaflet. Leveraging the CNN-based 3D MV model, our pipeline can
comprehensively quantify the MV anatomical features relevant to diagnosis,
pre-procedural planning or intra-procedural support.

##  2 Methods

###  2.1 Data Collection and Manual Segmentation

Intra-procedural 4DTEE acquisitions from 78 candidates to percutaneous MV
repair by means of MitraClip system (Abbott) at IRCCS San Raffaele Hospital
(Milan, Italy) were collected. Acquisitions were performed using Philips EPIQ
CVx scanner with an X8-2T transducer (Philips, Andover, MA, USA). Image data
were anonymized and exported in Cartesian format using QLab software (Philips,
Andover, MA, USA). The study was approved by the local ethics committee.
Midesophageal views zoomed on the MV were selected from the image data to
construct our dataset. Some 4DTEE acquisitions included more than one of these
views, which were all considered in our dataset. Acquisitions with serious
artifacts were excluded. Our final dataset counted 100 3DTEE volumetric and
time-dependent data with a mean voxel spacing of 0.37 $\times$ 0.55 $\times$
0.24 mm3 and a mean framerate of 22 per cardiac cycle. For each sequence, the
end-systolic frame, identified as the time frame just before aortic valve
closure, was selected.
3DTEE data were manually annotated to generate the training and test sets for
the 3D CNN. The manual annotations were performed using 3D Slicer [13] by
three independent and experienced operators. They segmented different images,
ensuring an almost balanced division of the dataset among them. First, each
user selected the end systolic frame and placed a series of contour points to
define the valve annulus. The contour points were placed at the connections
between the leaflets and the myocardium (Fig. 1a, left) on multiple rotating
long-axis views and then fitted with a smooth curve using the Slicer-Heart
software [16] (Fig. 1b, left) with a contour radius of 1 mm. The leaflets were
sketched manually in multiple parallel long-axis planes that spanned the valve
from commissure to commissure (Fig. 1a, right). The selected views in the
dataset had the centerline of the volume approximately coinciding with the
long axis of the left ventricle. The anterior and posterior leaflets were
identified as gray scale pixels with two different labels, between the
annulus, defined in the previous step, and the coaptation zone. This resulted
in a non-contiguous segmentation of the leaflets on several long-axis slices
(Fig. 1b, center), which were then interpolated (Fig. 1b, right) using a
morphological contour interpolator filter [1].

Figure 1: Intermediate and final result of the steps followed to manually
annotate 3DTEE images. (a) 3D rendering of the segmentation mask for each
step, and (b) corresponding cross sectional view overlapped on the 3DTEE
image.

###  2.2 Neural Network Architecture and Training

The trained CNN was based on a 3D U-Net [21] with encoding and decoding
branches of 5 resolution levels each, defined using residual units as shown in
[29]. In addition, PReLU activation layers [10] were adopted in each residual
unit and the number of decoding branches was extended to three, one for each
class detected (i.e., mitral annulus, anterior leaflet and posterior leaflet).
This resulted in a multi-decoder version of the residual U-Net.
For the neural network training, 90 volumes (90 %) with ground truth (GT)
annotations (training set) were randomly selected. The remaining 10 volumes
(10 %) were used as test set to evaluate the performance of the network. A
5-fold cross-validation was performed on the training set. A data augmentation
routine was adopted to increase the diversity of the data: transformations
taken from the MONAI framework [4] were used including, Gaussian noise,
cropping, flipping, rotation and elastic deformation; transformations were
applied randomly to the dataset. The model was implemented using the Pytorch
framework [18] and it was run on a NVIDIA A100 over 600 epochs with batch size
of 3. Novograd optimizer [9] was used with initial learning rate of 0.001,
halved every 200 epochs. Intensity of the input images was normalized with
values ranging between 0 and 1. A weighted-combination of Dice and Focal
losses [25] (with weight of 0.6 and 0.4, respectively) was computed at the
output block of each decoder branch. The sum of the loss functions of the
three decoders was used to train the neural network.

###  2.3 Pipeline Implementation

We implemented a fully automatic pipeline (Fig. 2), embedding the trained 3D
CNN, for MV segmentation and morphological characterisation. The 3D CNN is fed
by 3DTEE images at end-systole and segments the MV by separately recognising
the mitral annulus, the anterior leaflet and the posterior leaflet. Given the
multi-class segmentation mask provided by 3D CNN (Fig. 2, Segmentation), the
implemented pipeline allows to (i) smooth and correct the predicted
segmentation (Fig. 2, Refinement), (ii) calculate the position and orientation
of the valve, (iii) detect MV anatomical landmarks, (iv) build an approximate
model of the coaptation line (Fig. 2, extract MV Features) and (v) quantify MV
anatomy (Fig. 2, Quantification of MV Anatomy).

Figure 2: Schematic representation of the implemented automatic pipeline. SH =
saddle horn, PAM = posterior annulus mid-point, MC = medial commissure, LC =
lateral commissure, $D_{CC}$ = inter-commissural diameter, $D_{AP}$ = antero-
posterior diameter, $L_{A}$ = annulus length, $\Gamma_{A}$ = annulus surface,
Tips = leaflets tips $L_{PL}$ = posterior leaflet length, $L_{AL}$ = anterior
leaflet length, $\Gamma_{AL}$ = anterior leaflet surface, $\Gamma_{PL}$ =
posterior leaflet surface. Figure 3: Illustrative example of reconstructed MV
model with anatomical features and measurements extracted by the automatic
pipeline. (a) Reconstructed and refined mitral annulus ($\Omega_{A}$) together
with the annular anatomical landmarks (SH = saddle horn, PAM = posterior
annulus mid-point, MC = medial commissure, LC = lateral commissure), the best
fitting plane ($\Pi$) and the unitary vector (n) normal to $\Pi$. (b)
Reconstructed mitral leaflets ($\Omega_{AL}$=anterior leaflet surface, in red;
$\Omega_{PL}$=posterior leaflet surface, in blue), main leaflet landmarks
(Tips = leaflets tips), and the reconstructed model of the coaptation line
(green). (c) Leaflet 3D middle surfaces ($\Gamma_{AL}$, $\Gamma_{PL}$) and
leaflet 3D measurements ($L_{PL}$ = posterior leaflet length, $L_{AL}$ =
anterior leaflet length). (d) Color-coded 3D representation of the
reconstructed MV representing leaflet height. (e) 3D surface interpolating the
non planar annular profile ($\Gamma_{A}$) and defining the annular area, and
annulus 3D measurements ($D_{CC}$ = inter-commissural diameter, $D_{AP}$ =
antero-posterior diameter, $L_{A}$ = annulus length, $H_{A}$ = annulus
height).

#### Smoothing and Correction

A marching cubes algorithm [14] is applied to the multi-label segmentation
mask to extract three triangulated surface meshes of the mitral annulus
$\Omega\textsubscript{A}$, and of the anterior $\Omega\textsubscript{AL}$ and
posterior leaflet $\Omega\textsubscript{PL}$, respectively (Fig. 3a and b).
The surface meshes are then smoothed using a windowed sinc filter [19]. To
correct possible discontinuities in annulus segmentation the skeleton of
$\Omega\textsubscript{A}$ is reconstructed and processed to guarantee a closed
profile. Briefly, the valve orifice center xc is calculated by taking the
average of the coordinates xi = (xi, yi, zi) of the $n$ points on
$\Omega\textsubscript{A}$. Singular-value decomposition (SVD) on the set of
directions
$\textit{D}:\\{\textbf{d}\textsubscript{i}=\textbf{x}\textsubscript{i}-\textbf{x\textsubscript{c}},\quad\textbf{x}\textsubscript{i}\in\Omega\textsubscript{MA}\\}$
allows to define the principal directions of the annulus geometry. The radial
r and normal n directions are identified through SVD as the direction of
highest and lowest variance, respectively. xc and n uniquely identify the
plane $\Pi$, which is used to define the MV orifice plane in the subsequent
steps of the automated pipeline. Multiple-rotating planes are defined around
the normal direction n based on an angular offset
($\theta_{offset}=15^{\circ}$). The intersection between these planes and
$\Omega\textsubscript{A}$ identifies a sequence of 3D point sets whose centers
of mass outline the mitral annulus shape (Alg. 1). Finally, the skeleton is
obtained by interpolating the sequence of centers of mass with a cubic spline.
The skeleton of the annulus is the expanded radially by 1 mm to obtain the
corrected version of $\Omega\textsubscript{A}$, consistently with the contour
radius used for the GT label (Fig. 4).

Algorithm 1 Definition of skeleton of the mitral annulus from segmentation
mask

1:Given $\Omega_{A}$ and D

2:$\theta_{offset}=15^{\circ}$, $\theta=0^{\circ}$

3:n and r $\leftarrow$ SVD(D)

4:while $\theta<2\pi$ do

5: r$\theta$ $\leftarrow$ Rot$\theta_{offset}$(r)

6: $\Pi$ $\leftarrow$ $\textbf{n}\times\textbf{r}\textsubscript{$\theta$}$

7: Find set of intersecting points S between $\Pi$ and $\Omega_{A}$

8: Compute the center of mass of S

9: $\theta\leftarrow\theta_{offset}+\theta$

10:end while

Figure 4: Schematic representation of the three steps for annulus correction,
from top left to bottom right. Step one: incomplete annulus mask
$\Omega\textsubscript{A}$ (green) intersected by multiple planes passing
through the axis defined by the annulus center and the axis defining the
lowest variance of the annular point coordinates; the intersection between a
plane and $\Omega\textsubscript{A}$ is zoomed in, where the corresponding
point set defining the cross-sectional profile of $\Omega\textsubscript{A}$
(green) and their center of mass (red) are depicted. Step two: the centers of
mass obtained in step one are interpolated through a cubic spline yielding a
complete annulus skeleton (yellow). Step three: the complete skeleton is
expanded radially by 1 mm yielding the corrected annulus mask.

#### Detection of Mitral Valve Anatomical Landmarks

The pipeline automatically identifies the MV anatomical landmarks commonly
used to describe annulus and leaflet anatomy. This is done using the
reconstructed surface meshes $\Omega\textsubscript{A}$,
$\Omega\textsubscript{AL}$, and $\Omega\textsubscript{PL}$.
On the annulus, the saddle horn (SH) and the two commissures (LC, MC) are
identified respectively as the highest and the two lowest points of
$\Omega\textsubscript{A}$ skeleton with respect to $\Pi$, while the posterior
annular midpoint (PAM) is defined as the point opposite to SH in the posterior
portion of $\Gamma\textsubscript{A}$ skeleton (Fig. 3a).
On the leaflets, the leaflet tips are identified: a plane
($\Pi\textsubscript{SH-PAM}$) passing through SH and PAM and perpendicular to
$\Pi$ is defined; the intersection of this plane with
$\Omega\textsubscript{AL}$ and $\Omega\textsubscript{PL}$ identifies two sets
of points on the leaflet surface; among these, the leaflet tips (Fig. 3b) are
defined as the two points farthest from SH and PAM.

#### Coaptation Line Identification

The coaptation line is also automatically identified. First, two 3D middle
surfaces $\Gamma\textsubscript{AL}$ and $\Gamma\textsubscript{PL}$ are defined
using a 5th order spline radial basis function (RBF) interpolation [24] of
$\Omega\textsubscript{AL}$ and $\Omega\textsubscript{PL}$, respectively. Then,
the set of candidate points belonging to the coaptation zone are defined as:

$P:\\{(\textbf{x}\textsubscript{i},\textbf{x}\textsubscript{j})|\lVert\textbf{x}\textsubscript{i}-\textbf{x}\textsubscript{j}\rVert<\epsilon\\}$
(1)

with xi $\in\Gamma\textsubscript{AL}$, xj $\in\Gamma\textsubscript{PL}$ and
$\epsilon$ is a specific tolerance set as 0.001 mm. Finally, a least-square
fit of point set $P$ with 5th degree polynomial is used to obtain a smooth and
continuous line (Fig. 3b).

#### Quantification of Mitral Valve Anatomy

The final step of the implemented pipeline consists in the automatic
computation of MV morphological metrics based on the identified anatomical
landmarks. For the annulus, the anterior-posterior diameter ($D_{AP}$) and the
inter-commissural diameter ($D_{CC}$) are computed as the Euclidian distance
from SH to PAM and from MC to LC, respectively. Annular length (LA) is defined
as the length of the $\Omega\textsubscript{MA}$ skeleton. Finally, the height
of the annulus (HA) is defined as the height of the smallest hexahedral
bounding box encompassing $\Omega\textsubscript{MA}$. Also, a 3D surface
($\Gamma\textsubscript{A}$) representing the non-planar MV orifice is
reconstructed using a thin plate radial basis function interpolation [24] of
$\Omega\textsubscript{MA}$. Annular area is defined by as the surface area of
$\Gamma\textsubscript{A}$ (Fig. 3e).
For the leaflets, the length is defined as the length of the intersection
between each leaflet and $\Pi\textsubscript{SH-PAM}$, spanning from the SH to
the anterior tip and from PAM to the posterior tip, respectively. Leaflet area
is defined as the surface area of $\Gamma\textsubscript{AL}$ and
$\Gamma\textsubscript{PL}$, respectively (Fig. 3c). The local height of the
leaflets is computed as the signed distance of $\Gamma\textsubscript{AL}$ and
$\Gamma\textsubscript{PL}$, respectively, from the 3D orifice surface
$\Gamma\textsubscript{A}$ (Fig. 3d). Leaflet flail or prolapse would result in
a greater positive distance from $\Gamma\textsubscript{A}$.

###  2.4 Performance Evaluation

For each pair of the 5 training-validation splits, we computed Dice score and
Mean Surface Distance (MSD, in mm). Given the points $\mathbf{p}$ of a GT
surface $S$ and the points $\mathbf{p^{\prime}}$ of the corresponding inferred
segmentation $S^{\prime}$, MSD was defined as:

$MSD=\frac{1}{n_{S}+n_{S^{\prime}}}(\sum^{n_{S}}D(\mathbf{p},S^{\prime})+\sum^{n_{S^{\prime}}}D(\mathbf{p^{\prime}},S))$
(2)

with $D(\mathbf{p},S^{\prime})=\min_{\mathbf{p^{\prime}}\in
S^{\prime}}\lVert\mathbf{p}-\mathbf{p^{\prime}}\rVert$. The Dice score and the
MSD were computed individually for each label and for the complete
segmentation mask, i.e., for all the three labels considered together,
excluding the background. The network with the best performance over the five
folds was then evaluated on the test set using the same metrics. The distance
errors were recalculated after the smoothing and correction steps of our
pipeline.
The best-performing model was selected for use in our pipeline to extract the
relevant anatomical features of the mitral valve (MV) and to quantify its
anatomy on the test set. The results of the morphological analysis were
compared against measurements performed through TomTec Image Arena (TomTec
Imaging Systems GmbH, Unterschleissheim, Germany), a commercial software that
provides a semi-automated segmentation tool (4D MV Assessment, 4.6) considered
to be the gold standard for MV anatomy segmentation and assessment from 3DTEE
images.
The performance of our multi-decoder U-Net was compared to a baseline residual
U-Net with an equivalent encoder architecture and a single decoder. Both
networks were trained on the combined training and validation set (90 volumes)
using the same hyperparameters. The Dice score and MSD were calculated on the
test set and the segmentation masks provided by the two networks were visually
inspected. All performance metrics are expressed as average value and [min,
max] range.

##  3 Results

###  3.1 Cross Validation Results and Model Selection

We report in Table 1 the average Dice score and MSD computed with respect to
the GT segmentations for each pair of the 5 training-validation splits. Models
2 and 3 provided the highest Dice score for the complete segmentation mask
(0.79 [0.74, 0.84] and 0.79 [0.75, 0.85], respectively). With model 3, we
obtained higher average Dice scores for annulus (0.41 [0.30, 0.59]) and
anterior leaflet (0.80 [0.73, 0.87]). Accordingly, model 3 provided lower
average MSD errors for annulus (0.85 mm [0.60 mm, 1.23 mm]), anterior leaflet
(0.48 mm [0.27 mm, 0.63 mm]), and complete segmentation mask (0.47 mm [0.32
mm, 0.60 mm]). Therefore, model 3 was re-evaluated on the test set and
included in our pipeline.

| Annulus | Anterior | Posterior | Complete Mask
---|---|---|---|---
| Dice Score | MSD [mm] | Dice Score | MSD [mm] | Dice Score | MSD [mm] | Dice Score | MSD [mm]
1 | 0.43 | 0.92 | 0.77 | 0.52 | 0.67 | 0.77 | 0.77 | 0.48
2 | 0.35 | 1.03 | 0.78 | 0.55 | 0.72 | 0.67 | 0.79 | 0.50
3 | 0.42 | 0.85 | 0.80 | 0.48 | 0.67 | 0.76 | 0.79 | 0.47
4 | 0.27 | 1.28 | 0.78 | 0.52 | 0.68 | 0.84 | 0.77 | 0.56
5 | 0.43 | 0.86 | 0.75 | 0.69 | 0.65 | 0.98 | 0.76 | 0.50
Average | 0.38 $\pm$ 0.07 | 0.99 $\pm$ 0.18 | 0.78 $\pm$ 0.02 | 0.55 $\pm$ 0.08 | 0.68 $\pm$ 0.02 | 0.80 $\pm$ 0.11 | 0.78 $\pm$ 0.01 | 0.50 $\pm$ 0.04

Table 1: Average Dice scores and MSD in mm for each training on the 5
training-validation pairs. Performance metrics on the validation set are
reported for mitral annulus (Annulus), anterior leaflet (Anterior), posterior
leaflet (Posterior) and for complete segmentation mask (Complete Mask).
Average value across the 5 models and standard deviation are given in the last
row.

###  3.2 Test Set Results

Model 3 was used to run inference on a test set consisting of 10 3DTEE
volumes. The trained CNN was capable of accurately extracting the MV
substructures anatomy (as shown in Fig. 5). The average Dice scores for the
annulus, the anterior leaflet, and the posterior leaflet were 0.39 [0.21,
0.54], 0.78 [0.70, 0.84], and 0.67 [0.36, 0.79], respectively. For the
complete segmentation mask, the average Dice score was 0.79 [0.71, 0.84]. The
average MSD errors for the annulus, the anterior leaflet, and the posterior
leaflet were 1.01 mm [0.56 mm, 1.88 mm], 0.53 mm [0.32 mm, 0.80 mm], and 0.80
mm [0.51 mm, 1.48 mm], respectively. For the complete segmentation mask, the
average MSD error was 0.47 mm [0.35 mm, 0.73 mm]. Overall, in 8 test volumes
out of 10 the mitral valve was successfully segmented, with few cases in which
the mitral annulus presented discontinuities (P6 and P7 in Fig. 5). In one
case (P4), the CNN segmented the mitral annulus and the posterior leaflet only
partially, and only the inferred anterior leaflet looked plausible at visual
observation. Only one test volume (P5) showed sub-optimal segmentation of the
anterior and posterior leaflets with many mislabeled regions along the
coaptation zone, especially in the medial and lateral portion of the valve.
(Fig. 5)
The MSD error for the annulus label was recomputed after applying the
correction algorithm for all ten volumes, yielding an average value of 0.93 mm
[0.54 mm, 1.85 mm] ($\Delta$=-8.36%). In all ten test volumes, the refinement
of the inferred annulus segmentations yielded reasonable annular profiles,
also in those cases where the raw annulus segmentation was discontinuous or
incomplete, and reduced the surface error (Fig. 6).

Figure 5: Long-axis views of 3D TEE images and segmentation masks for each
volume in test set. In the first 3 rows the GT label is shown in light green,
and predicted label (CNN) is shown in dark green for annulus, anterior leaflet
and posterior leaflet, respectively. In the fourth row, a 3D representation of
predicted segmentation mask is shown, using a color code to distinguish
annulus (A), anterior leaflet (AL) and posterior leaflet (PL). White arrows on
long-axis views and black arrows on 3D representations highlight leaflet
defects (P2) or prolapse (P6) correctly identified in the predicted
segmentation masks. Figure 6: Heatmaps of the distance of the annulus from the
GT reference for the 10 patients of the testing set. Data are reported for the
annulus before (left column) and after (right column) the smoothing and
correction step of our pipeline.

###  3.3 Comparison vs. Semi-Automated Measurements

The differences between the 3D measurements yielded by the proposed automated
pipeline and those obtained by an experienced operator using the semi-
automated tool (4D MV Assessment 4.6) embedded in TomTec are detailed in Table
2. For the anatomy of the leaflets, the comparison was carried out upon
excluding the two test cases (P4 and P5) the trained 3D CNN failed to segment
for at least one leaflet. Overall, a good agreement was found when considering
length measurements, although dimensions as estimated by TomTec were larger as
compared to those estimated by our approach. The bias was within 7.94 mm for
annular measurements and within 3.67 mm for leaflet measurements. The highest
bias (250.23 mm2) was found for annulus area estimation. The commercial
software required a significant user interaction; for an experienced user, it
took around 3 minutes on average to process a single 3DTEE volume.

| Automated pipeline | TOMTEC | Bias | 95% limits of agreement
---|---|---|---|---
Annulus |  |  |  |
Inter-commisural diameter, mm | 36.24 | 43.79 | 7.55 | [-2.96, 18.07]
Anteroposterior diameter, mm | 39.78 | 42.64 | 2.86 | [-5.68, 11.40]
Height, mm | 10.37 | 10.97 | 0.60 | [-3.68, 4.88]
Length, mm | 136.79 | 144.73 | 7.94 | [-8.53, 24.41]
Area, mm2 | 1333.87 | 1584.10 | 250.23 | [-190.66, 691.12]
Leaflet |  |  |  |
Anterior leaflet length, mm | 32.60 | 35.40 | 2.79 | [-6.12, 11.70]
Posterior leaflet length, mm | 14.33 | 18.00 | 3.67 | [-4.96, 12.30]
Anterior leaflet surface area, mm2 | 1057.76 | 1151.37 | 93.61 | [-283.35, 470.58]
Posterior leaflet surface area, mm2 | 768.18 | 832.87 | 64.96 | [-325.74, 455.13]

Table 2: 3D MV morphology analysis. Comparison between measurements on test
set of annulus and leaflet geometry obtained with our automated pipeline and
TOMTEC. Mean differences (Bias) and 95% limits of agreement are reported.

###  3.4 Comparison with Baseline Residual U-Net

Our proposed Multi-Decoder U-Net architecture outperformed a baseline residual
U-Net in terms of segmentation mask quality. This is clearly exemplified in
Fig. 7, where the reconstruction for one MV is depicted for both
architectures: the baseline residual U-Net identified an annulus with noisy
boundaries and with unreasonable tracts at the insertion of the central
posterior cusp. The annular region identified by our Multi-Decoder U-Net was
not affected by these issues, although it was incomplete in some short tracts;
this specific issue was however solved by the subsequent smoothing and
correction step of our automated pipeline. The improved quality of the masks
yielded by our Multi-Decoder U-Net reflected into the improved average Dice
score and MSD errors computed on the test set as compared to those obtained
with the baseline residual U-Net, especially when considering the three masks,
i.e., annulus, anterior leaflet, and posterior leaflet, separately (Table 3).

| Baseline U-Net | Multi-Decoder U-Net
---|---|---
| Dice Score | MSD [mm] | Dice Score | MSD [mm]
Annulus | 0.29 [0.16, 0.38] | 1.69 [0.72, 3.42] | 0.38 [0.21, 0.53] | 1.45 [1.07, 2.84]
Anterior | 0.75 [0.67, 0.82] | 0.67 [0.45, 1.09] | 0.78 [0.63, 0.83] | 0.58 [0.40, 1.30]
Posterior | 0.58 [0.33, 0.76] | 2.02 [0.51, 10.92] | 0.63 [0.16, 0.77] | 0.97 [0.58, 1.50]
Complete Mask | 0.77 [0.68, 0.81] | 0.57 [0.45, 0.77] | 0.79 [0.71, 0.85] | 0.50 [0.45, 0.77]

Table 3: Dice score and MSD errors in mm computed on test set using the
baseline Residual U-Net and the multi-decoder Residual U-Net. Data are
expressed as mean value and [min, max] range. Figure 7: Illustrative example
of CNN-based segmentation mask using a baseline U-Net and a Multi-Decoder
U-Net. A color-coded 3D rendering of the segmentation mask and a corresponding
cross-sectional view overlapped on the 3DTEE image are shown.

###  3.5 Inference Time

The inference speed required for 3DTEE image segmentation and MV anatomical
features extraction was evaluated on the test set. The trained 3D CNN takes on
average 0.05 s [0.026 s, 0.22 s] to provide a raw multi-label segmentation
mask running on a GPU Nvidia RTX A4000. The automated pipeline took on average
66.98 s [51.39 s, 94.22 s] to perform inference on the complete process using
exclusively CPU (Intel Xenon W-2235). Specifically, 12.54 s [7.98 s, 25.42 s]
are required on average for MV model refinement, 52.58 s [40.40 s, 68.07 s]
for the extraction of anatomical landmarks and features and 1.84 s [0.59 s,
3.24 s] for quantification of MV anatomy. Overall, the proposed pipeline took
one third of the time required by an experienced user with TomTec Image Arena
software to perform MV segmentation and quantitative morphological analysis.

##  4 Discussion

We presented a deep learning-based pipeline to automatically segment MV
substructures (annulus and leaflets) in end-systolic configuration from 3DTEE
images, and extract the relevant anatomical landmarks and features to quantify
MV anatomy. The pipeline is based on a CNN with a multi-decoder residual U-Net
architecture which provides multi-structure end-to-end segmentation that does
not require any manual input nor parameter fine-tuning, thus ruling out any
dependency on the operator. The trained CNN selected for use in the pipeline
achieved an average Dice score of 0.79 (range 0.71-0.85) for the complete
segmentation mask, which is comparable to the Dice score (0.81 $\pm$ 0.07)
reported in [5]. However, Carnahan et al. [5] trained a 3D CNN to
automatically segment the MV in its entirety without distinguishing its
substructures. Moreover, they reconstructed the anatomy of the MV leaflets in
its open configuration; that choice inherently prevents the identification of
the leaflet coaptation line. This anatomical feature is key in supporting the
quantitative identification of coaptation defects, and it is a crucial
landmark in percutaneous MV repair, where the pose of the implantable device
must be set by the operator accounting for the location of the defect and for
the local orientation of the coaptation line.
As shown in Fig. 5, the predicted segmentation masks were overall consistent
with the corresponding GT segmentations. The mitral annulus is the label that
showed the least overlap with the GT datum, as suggested by the low Dice score
obtained for this label. However, it should be pointed out that the annulus is
the thinnest structure; thus, small discrepancies in annular profile may
reflect in a notable decrease in Dice score. On the other hand, the anterior
and posterior leaflets showed a good degree of overlap with the corresponding
GT segmentations, especially along the coaptation region. This result suggests
that our automated pipeline reliably identifies leaflet contact regions,
coaptation defects (P2 in Fig. 5) or prolapses (P6 in Fig. 5), which are
distinctive features of regurgitant valves candidates to surgical or
percutaneous MV repair. Of note, inaccuracies in our automated segmentation
were minor. The average surface distance to the GT data for the full
segmentation mask was of the order of the spatial resolution of the TEE
volumes in our dataset (0.30-0.70 mm/voxel). Furthermore, the reported
distance was almost equal to the inter-user variability typical of manual
segmentation, previously reported as 0.6 $\pm$ 0.17 mm [12], and compared
favorably with previous semi-automatic (0.59 $\pm$ 0.49 mm [22] and 0.60 $\pm$
0.20 mm [20]) and fully automated methods (1.54 $\pm$ 1.17 mm [11] and 0.59
$\pm$ 0.23 mm [5]). Average surface errors were slightly lower then surface
distance computed for the complete segmentation mask when calculated for each
label separately. This worsening of the MSD is mainly related to the lack of
intensity-based boundaries for MV structures in the 3DTEE images. The CNN
provided poor quality segmentation of the leaflets in one test volume (P5 in
Fig. 5) and segmented incompletely the mitral annulus and the posterior
leaflet in another test volume (P4 in Fig. 5). For both cases, the low
performance can be related to the scarce quality of the TEE data, which were
affected by a high speckle noise. Nevertheless, as shown in Fig. 6 the
implemented correction algorithm enabled a robust and complete reconstruction
of the annulus resulting in a significant improvement of performance metrics.
Moreover, based on the average distance error for the predicted annulus (0.93
mm [0.54 mm, 1.85 mm]), the proposed pipeline outperformed the approaches
proposed in [28] (2.74 mm) and [2] (2 mm) for mitral annulus segmentation.
Furthermore, our proposed multi-decoder residual U-Net architecture
outperformed a baseline residual U-Net in segmenting MV from 3DTEE images, as
it provided better quality segmentation masks, especially in identifying the
boundaries of MV structures (Fig. 7). This result suggests that the multi-
decoder architecture can mitigate the lack of intensity-based boundaries for
cardiac structures in ultrasound images that makes it hard to distinguish
between the different components of these structures with automatic methods.
The multiple decoders allow the network to incorporate information from
multiple scales and levels of abstraction, improving its ability to
distinguish different structures even in the absence of clear intensity-based
boundaries. This makes the multi-decoder U-Net architecture a more effective
tool for the automated segmentation of ultrasound images as compared to
simpler architectures.
The pipeline also demonstrated consistent identification of MV anatomical
landmarks and reconstruction of anatomical features extracted from the CNN-
based MV model. Overall, good agreement was found between our proposed method
and TomTec Image Arena, with differences in annular and leaflet linear
measurements of less than 7.94 mm and 3.67 mm, respectively. The highest bias
(250.23 mm2) was found for annulus area estimation. As a matter of fact,
TomTec Image Arena Software overestimated the annular area since it simply
identifies the mitral annulus as a closed line representing the outer border
of the mitral leaflets identified as single surfaces. Our automated pipeline,
on the other hand, represents the mitral annulus and leaflets with thickness,
providing a more realistic MV representation. The comparison with commercial
software is further proof of the accuracy and robustness of the proposed
segmentation method. To the best of our knowledge, only the very recent work
by Chen el al. [6] tested a CNN-based method for MV segmentation against
commercial software. However, their comparison was limited to the annulus, and
their results were similar to our own findings. It is worth noting that Chen
et al. used a larger dataset, consisting of twenty-eight 3D volumes, to
evaluate the accuracy of their approach, while our study only used ten 3D
volumes.
Finally, the proposed pipeline is fully automated and requires no manual input
at any of its steps. The resulting segmentations and subsequent analyses will
therefore be more reproducible than those yielded by manual or semi-automated
approaches. The proposed pipeline provided a full and refined MV model in
12.54 s on average (range [7.98 s, 25.42 s]); hence, it proved substantially
faster than the gold standard method herein considered for comparison, which
also required extensive user interaction.
To the best of our knowledge, none of the previously proposed methods for MV
segmentation integrated annulus and leaflets quantitative characterisation
into a single workflow like the pipeline presented here. As compared to
previous CNN-based approaches for segmentation of TEE images that focused only
on mitral annulus [28, 2] or leaflets segmentation [5], our trained 3D CNN
proved capable of segmenting and identifying all substructures of the MV.
Moreover, our approach addressed the challenging task of segmenting
regurgitant MVs at end-systole. The characterization of annulus and leaflets
anatomy at this time instant allows to extract relevant information to support
MR diagnosis and MV repair planning.

###  4.1 Limitations and Future work

Our trained 3D CNN provided sub-optimal MV substructures segmentations in two
test images out of ten. As previously pointed out, this failure can be
attributed to the low image quality and speckle noise that affected these
images. Excluding images affected by this type of noise from the dataset would
certainly ensure better performance in the quality of MV structure
segmentation. However, such a dataset would not be consistent with real world
applications, where the above mentioned noise is inherent to ultrasound images
and cannot be avoided during acquisitions. Future efforts will have to be
devoted to include a wider variation in image quality by expanding our data
set. Extending the method to a temporal segmentation problem could also help
increasing its accuracy, as well as provide a comprehensive assessment of MV
anatomy over the entire cardiac cycle. In fact, some MV substructures (e.g.,
the leaflets) are better imaged at specific phases of the cardiac cycle. By
enforcing the anatomical continuity of segmented MV structures in each frame
during 3D CNN training, we could improve our results and overcome limitations
due to image quality. Finally, as all CNN-based segmentation methods, our
pipeline would be in principle well suited for intraoperative use in
percutaneous MV repair. Indeed, the automatic localization of the copatation
line and the quantification of valve orientation would visualization of the
target defects and hence facilitate optimal catheter positioning before the
actual device deployment. However, the time required by the proposed pipeline
to extract MV characteristics is still not compatible with real-time
applications. Further work is needed to overcome this limitations and may
include the exploitation of GPU support throughout the entire workflow.

##  5 Conclusion

We presented a fully automated, 3D CNN-based pipeline for MV segmentation from
3DTEE and identification of anatomical landmarks and relevant features in
regurgitant MVs. The designed pipeline improves the workflow for MV
segmentation and anatomical quantification of the mitral annulus and leaflets,
providing consistent and repeatable results in a comprehensive and fast way.
This pipeline will potentially lead to a more reproducible and time-efficient
quantification of MV anatomy to support the diagnosis of MR and planning of MV
repair, with a potential future extension to intraoperative support in
transcatheter MV repair surgeries.

## Acknowledgements

This work has received funding from the European Union’s Horizon 2020research
and innovation programme under grant agreement No. 101017140,the ARTERY
project.

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# Fractional Revivals in Elliptical Atomtronics

Sriganapathy Raghav1 Suranjana Ghosh2 Luca Salasnich3 Jayanta Bera4 Utpal Roy1
<EMAIL_ADDRESS>1Indian Institute of Technology Patna, Bihta, Patna-801106,
India
2Indian Institute of Science Education and Research Kolkata, West Bengal,
India
3 Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universitá di Padova,
Via Marzolo 8, 35131 Padova, Italy
4 C. V. Raman Global University, Bhubaneswar, Odisha-752054, India

###### Abstract

Fractional revivals are recently reported for circular atomtronics, but get
disturbed for a nonzero eccentricity of the waveguide geometry. Here, we
provide a mechanism for the elliptical atomtronics with arbitrary eccentricity
to restore fractional revivals. The uniform ground state of the circular
waveguide becomes nonuniform in the elliptical geometry. An appropriate
dispersion management can bring back the uniformity and we use the overlap
function numerically to identify the corresponding dispersion coefficients,
which match with our proposed analytical formula. The fact that, the cloud
spends mostly along the semimajor edges, is demonstrated by the survival
function. The said dispersion management recovers the desired fractional
revivals patterns, where the revival time-scale becomes independent of the
eccentricity. The present method paves the way also to observe other known
phenomena of circular atomtronics in elliptical atomtronics.

##  I Introduction

Guiding matter waves is a pivotal component in constructing matter wave
circuits, particularly within atom chip technology [1, 2, 3]. The precise
fabrication of these waveguides is critical in the field of atomtronics,
enabling coherent control and manipulation of matter waves [4, 5]. Techniques
like time-averaged adiabatic potential (TAAP) [6, 7, 8, 9, 10], intensity mask
[11], and digital holography [12] have been instrumental in creating
waveguides for ultracold atoms, contributing significantly to the advancement
of atomtronics. Among the various types of waveguides, the circular waveguide
stands out as the simplest spatially closed atomtronic circuit [13, 14],
widely applied in atom interferometry [15, 16, 17, 18, 19], quantum transport
[20, 21], quantum sensing [22, 23, 24], atom SQUID [25, 26], and other quantum
technological applications [2]. The curvature-induced potential (CIP) of a
waveguide is proportional to the square of its curvature, implying a constant
CIP for a circular waveguide [27, 28, 29], which bears a uniformly distributed
ground state along its circumference [30].

The major utility of a circular waveguide stems from its constant curvature,
which facilitates matter wave interference, an integral component of the above
applications. The matter wave interference in a circular waveguide also gives
rise to fractional revivals (FR), which is the spawning of multiple replicas
of the initial matter wave packet along the perimeter of the circular
waveguide [31, 32, 33]. The FR instances are succeeded by the revival of the
initial condensate in shape and location. The FR time instances and revival
depends on a characteristic parameter linked to the radius of the waveguide
[32]. Moreover, higher-order FR patterns can provide a platform for studying
multiple source interference, which is usually studied using optical lattices
[34, 35, 36, 37]. In contrast, due to the variable width along the
circumference and non-constant CIP, elliptical waveguides result in a non-
uniform ground state along the perimeter [30, 38]. Unlike a circular
waveguide, elliptical counterpart lacks support for Talbot oscillations [39].

Concurrently, dispersion engineering has emerged as a significant method for
controlling and manipulating the dynamics of Bose-Einstein condensates (BEC)
in external traps [40, 41, 42]. Dispersion management (DM) in atomic BECs is
achieved by introducing optical lattices [43, 44] or manipulating spin-orbit
coupling [45, 46]. One can achieve desired BEC dynamics in a matter-wave
circuit of waveguides by tuning the nonlinearity and the dispersion through
Feshbach resonance and DM, as in the case of optical solitons [47, 48].
Recently, the nonlinearity of the BEC was used to nullify the effects of
curvature in an elliptical waveguide [38]. Here, we delve into engineering the
dispersion of BEC inside an elliptical waveguide to regain the properties of
circular atomtronics. Notably, the utilization of matter wave dispersion
serves as a means to counteract the effects of non-constant width in an
elliptical waveguide, offering potential strategies for manipulating and
optimizing matter wave behaviour in atomtronics applications.

The paper is organized as follows: Section II introduces the theoretical model
for BEC in an elliptical waveguide. This section also discusses the role of
waveguide geometry in the dispersion of matter waves. In Sec. III, we derive
the dynamical equation for BEC with tunable dispersion in an elliptical
waveguide and also outline the numerical methods applied to obtain both the
ground state solution and FR-dynamics within the waveguide. The ground state
solution of BEC in an elliptical waveguide, characterized by different
eccentricities, are demonstrated. We also determine the suitable dispersion
coefficients for matter waves, aiming to neutralize the effects of variable
thickness in an elliptical waveguide. Section IV delves into exploring the
impact of eccentricity on the FR-dynamics of matter waves and the subsequent
restoration of FR signatures through dispersion management, which elucidates
on how manipulating dispersion can counteract the challenges posed by the non-
constant width in elliptical waveguides. Finally, we conclude in Sec. V,
summarizing the key findings and future outlook.

##  II Bose-Einstein Condensate inside an Elliptical Waveguide

We consider a BEC of $N$ number of 23Na atoms, loaded in an elliptical
waveguide of eccentricity $\varepsilon$. A three-dimensional Gross Pitaevskii
equation (3D-GPE) [49, 50, 51, 52, 53] describes the evolution of the
macroscopic wavefunction $\Psi\equiv\Psi(r,t)$ of a BEC in an external trap:

$\displaystyle i\hbar\frac{\partial\Psi}{\partial
t}=\Bigg{[}-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{4\pi
Na_{s}}{m}|\Psi|^{2}+V(r)\Bigg{]}\Psi,$ (1)

where $V(r)=V(z)+V(x,y)$ with $V(z)=\frac{1}{2}m\omega_{\perp}z^{2}$ and

$\displaystyle
V(x,y)=V_{0}\Bigg{\\{}1-\exp{\Bigg{[}-\frac{1}{\gamma^{2}}\Bigg{(}\sqrt{x^{2}+\frac{y^{2}}{1-\varepsilon^{2}}}-a\Bigg{)}^{2}\Bigg{]}}\Bigg{\\}}.$

Here, $\omega_{\perp}$ is the frequency of the transverse harmonic trap,
$V_{0}$ is the depth of the waveguide, $\gamma$ is the width of the waveguide,
and $\varepsilon=\sqrt{1-\frac{b^{2}}{a^{2}}}$ is the eccentricity. For a
circular waveguide with a radius $a$, we take $\varepsilon=0$, and for an
elliptical waveguide with a semi-major radius $a$, we consider a non-zero
eccentricity.

The 3D GPE in Eq.1 is reduced to effective 2D GPE by considering a strong trap
in the transverse direction by writing the wavefunction as
$\Psi(r,t)=\psi(x,y,t)\Phi(z)$. Here, the function $\psi(x,y,t)$ describes the
dynamics of BEC in the elliptical waveguide and $\Phi(z)$ denotes the ground
state of the strong axial trap, which is a Gaussian wavefunction with width
$a_{\perp}$ [54]. The effective 2D GPE in the dimensionless form is obtained
after integrating out the z component and by scaling position, time and energy
by $a_{\perp}$, $\frac{1}{\omega_{\perp}}$ and $\hbar\omega_{\perp}$,
respectively. Here $a_{\perp}=\sqrt{\frac{\hbar}{m\omega_{\perp}}}$ and
$\omega_{\perp}$ are the harmonic oscillator length and frequency in the
transverse direction, respectively.

$i\frac{\partial\psi}{\partial
t}=\Bigg{[}-\frac{1}{2}\nabla_{x,y}^{2}+g|\psi|^{2}+V(x,y)\Bigg{]}\psi$ (2)

Here, $g=\frac{2\sqrt{\pi}Na_{s}}{a_{\perp}}$ and, $V(x,y)$ is the potential
of the elliptical waveguide scaled by $\hbar\omega_{\perp}$. This elliptical
waveguide has varying thickness for non-zero eccentricity and manifests
nontrivial dynamics.

Figure 1: Elliptical ring trap with varying width along its circumference. The
semimajor radius is taken as $a=10a_{\perp}$, and the semiminor radius is
$b=4.36a_{\perp}$. The corresponding eccentricity is $\varepsilon=0.9$. $x$
and $y$ are in the units of $a_{\perp}=2.32\;\mu$m and, $t$ is in the units of
$1/\omega_{\perp}=1.95\;$ms.

###  II.1 Waveguide Geometry and Matter Wave Dispersion

A circular waveguide with $\varepsilon=0$, formed by a ring Gaussian
potential, has constant width [55] unlike an elliptical waveguide, as shown in
Fig.1. This varying width of the ring affects the dispersion across different
waveguide segments. To elucidate the influence of nonzero eccentricity on
dispersion, we study the dynamics of a Gaussian wavepacket, placed at two
different positions inside the elliptical waveguide, namely $(a,0)$ and
$(0,b)$. The wavepacket is expressed as

$\chi(s,t)=e^{-i\frac{E_{k}t}{\hbar}}\chi(s,0).$ (3)

By decomposing the initial wavefunction into its constituent Fourier modes, we
write

$\displaystyle\chi(s,t)=\Bigg{(}\frac{d}{\sqrt{\pi}}\Bigg{)}^{\frac{1}{2}}\int_{-\infty}^{\infty}\frac{dk}{\sqrt{2\pi}}e^{-\frac{d^{2}k^{2}}{2}}e^{-i\frac{E_{k}t}{\hbar}}e^{iks},$
(4)

where, $d$ is related to the width of the wavepacket, and $k$ relates the
energy: $E_{k}=\hbar^{2}k^{2}/(2m)$, which allows us to simplify the
wavefunction in Eq.4 as,

$\displaystyle\chi(s,t)=\Bigg{(}\frac{D}{\sqrt{\pi}d}\Bigg{)}^{\frac{1}{2}}e^{-\frac{Ds^{2}}{2d^{2}}}$
(5)

with $D=\frac{1}{(1+i\frac{\hbar t}{md^{2}})}$. Consequently, the width at
time $t$ becomes

$\displaystyle
w_{t}^{2}=\int_{-\infty}^{\infty}\chi^{*}(s,t)s^{2}\chi(s,t)ds=w_{0}^{2}+\frac{\hbar^{2}t^{2}}{4m^{2}w_{0}^{2}}.$
(6)

Here, the initial width is given by $w_{0}=\frac{d}{\sqrt{2}}$ and the above
expression reduces to a dimensionless form as

$w_{t}^{2}=w_{0}^{2}+\frac{t^{2}}{4w_{0}^{2}}.$ (7)

For analysing the dispersions corresponding to the initial positions, $(a,0)$
and $(0,b)$, along the elliptical waveguide, we write the respective widths of
the waveguide as $d_{a}$ and $d_{b}$, having ratio
$\frac{d_{b}}{d_{a}}=\sqrt{1-\varepsilon^{2}}$. An intuitive understanding and
numerical trials suggest us to take the transverse widths of the wavepacket as
$d_{a}$ and $d_{b}$, respectively, to prevent inhomogeneous dispersion.

Accordingly, the normalization of the wavepacket at these places dictates that
their longitudinal widths, $w_{a}$ and $w_{b}$, should be inversely related to
$d_{a}$ and $d_{b}$:

$\frac{w_{a}}{w_{b}}=\frac{d_{b}}{d_{a}}=\sqrt{1-\varepsilon^{2}},$ (8)

which facilitates us to write their relation at time $t$, following Eq.7 as

$\displaystyle
w_{b,t}^{2}=\frac{1}{1-\varepsilon^{2}}\Bigg{[}w_{a}^{2}+\frac{t^{2}(1-\varepsilon^{2})^{2}}{4w_{a}^{2}}\Bigg{]}.$
(9)

Figure 2: Ground State solution of BEC in (a) a circular waveguide,
$\varepsilon=0$ and elliptical waveguides of eccentricity (b)
$\varepsilon=0.25$, (c) $\varepsilon=0.5$, (d) $\varepsilon=0.75$ and (e)
$\varepsilon=0.9$. The circular ring radius and the semimajor radius are taken
as $a=10a_{\perp}$. $x$ and $y$ are in the units of $a_{\perp}=2.32\;\mu$m
and, $t$ is in the units of $1/\omega_{\perp}=1.95\;$ms.

It becomes apparent that, the widths of the wavepacket at the semi-major and
semi-minor edges deviate from their initial ratio (Eq.8). The factor
$(1-\varepsilon^{2})^{2}$ introduces the inhomogeneity of the widths over the
time. This demands a modification of $E_{k}$ by the same factor, such that
$E_{k}=\frac{1}{(1-\varepsilon^{2})}\frac{\hbar^{2}k^{2}}{2m}$, to compensate
the deviation, caused by the nonzero eccentricity. Hence, the time dependent
widths maintain the initial ratio:

$\displaystyle w_{b,t}^{2}=$
$\displaystyle\frac{1}{1-\varepsilon^{2}}\Bigg{[}w_{a}^{2}+\frac{t^{2}}{4w_{a}^{2}}\Bigg{]},$
(10) $\displaystyle w_{a,t}^{2}=$
$\displaystyle(1-\varepsilon^{2})w_{b,t}^{2}.$ (11)

To mitigate the effects of nonzero eccentricity, it is necessary to manage
dispersion such that the dispersion along the $y$-direction is slower than
that along the $x$-direction by a factor of $(1-\varepsilon^{2})$. This can be
experimentally achieved by introducing a weak 2D optical lattice (OL) along
with the elliptical waveguide [40, 56] with their lattice vectors satisfying
$\frac{k_{x}}{k_{y}}=\sqrt{1-\varepsilon^{2}}$ [57, 58]. Moreover, the
influence of ellipticity can be counteracted by adjusting the depth of the
elliptical waveguide. The impact of varying width along the circumference can
be nullified by modulating the waveguide’s depth.

##  III Dispersion Management of Ground States

After incorporating the appropriate dispersion management, the effective
Gross-Pitaevskii equation becomes [40, 56, 59, 60, 61, 62, 63, 64]:

$\displaystyle i\frac{\partial\psi}{\partial
t}=\Bigg{[}-\frac{\alpha}{2}\frac{\partial^{2}}{\partial
x^{2}}-\frac{\beta}{2}\frac{\partial^{2}}{\partial
y^{2}}+g|\psi|^{2}+V(x,y)\Bigg{]}\psi.$ (12)

Here, $\alpha$ and $\beta$ are the dispersion coefficients in $x$\- and $y$\-
directions, respectively, whereas $V(x,y)$ is the potential of the elliptical
waveguide.

Figure 3: (a) Variation of overlap of the actual and desired wavefunction with
the dispersion coefficient for various values of eccentricities. Solid lines
with circles, squares, diamonds and triangles represent the eccentricities
$\varepsilon=0.9$, $\varepsilon=0.75$, $\varepsilon=0.5$ and
$\varepsilon=0.25$, respectively. (b) Numerical (dot) and theoretical (solid
line) values of dispersion coefficient for various eccentricities. The
circular ring radius and the semimajor radius are taken as $a=10a_{\perp}$.

Numerical Method: The ground state solution of Eq.[12] is numerically obtained
by implementing the imaginary time propagation (ITP) method, where the initial
wavefunction is allowed to evolve in imaginary time, $t=i\tau$. In this case,
any initial wavefunction under the action of time evolution operator,
$\exp{(-\tau\hat{H})}$, asymptotically converges to the ground state solution
as $t\rightarrow\infty$ [65]. The time dynamics of a localized matter wave
packet in the elliptical waveguide are obtained by the real time propagation
(RTP) method. In both ITP and RTP methods, the linear and non-linear parts of
the dynamical equation are treated separately, where the linear part is
evolved in the momentum space, and the non-linear part is evolved in the
coordinate space [66]. The $x$ and $y$ coordinates are equally divided into
512 grids with a step size of 0.1. The step size for time is 0.08, totalling
16384 grids. In our work, we have considered 23Na BEC of $N=1000$ atoms, with
parameters $m=3.816\times 10^{-26}\;$kg, $\omega_{\perp}=512\;$Hz,
$a_{\perp}=2.318\mu$m, and $a_{s}=2.75\times 10^{-9}$m [67, 55, 68]. The
initial condensate in the form of binary peaks is placed diametrically
opposite along the $x$-axis with the coordinates $(a,0)$ and $(-a,0)$,
respectively.

###  III.1 Ground State of BEC in an Elliptical Waveguide

For numerically finding the ground states, we consider a circular waveguide of
radius $a=10a_{\perp}$ and unity dispersion coefficients ($\alpha=\beta=1$),
for which the ground state density is uniform along the ring’s circumference.
Figure 2(a) shows the ground state of the circular waveguide. The solution can
be expressed as follows:

$\psi_{c}(x,y)=(\frac{\sqrt{V_{0}}}{\pi\gamma})^{\frac{1}{4}}e^{-\frac{\sqrt{V_{0}}(\sqrt{x^{2}+y^{2}}-a)^{2}}{2\gamma}},$
(13)

The ground state of the potential is a Gaussian ring since the cross-section
of the potential in the vicinity of the minima is harmonic in nature. Such
Gaussian ring condensate inside a circular waveguide has been discussed in
various experimental and theoretical works [69, 30]. More interesting things
happen when we increase the eccentricity of the ring waveguide from null. The
stationary states for waveguides with various eccentricities are shown in
Fig.2(b-e). The density is no longer uniformly distributed across the
circumference of the waveguide, whereas one could see the density accumulation
at the semi-major edges. The greater the eccentricity, the greater the density
accumulation at the edges, thereby making the waveguide behave like a double-
well potential. If the curvature effects are counterbalanced, one could obtain
a uniform stationary state in an elliptical waveguide, expressed quite similar
to Eq.13:

$\psi_{e}(x,y)=Ae^{-\frac{\sqrt{V_{0}}(\sqrt{x^{2}+\frac{y^{2}}{1-\varepsilon^{2}}}-a)^{2}}{2\gamma}}.$
(14)

Figure 4: 1D Cross-Sectional Densities in x and y directions for various
$\beta$ values (a)$\;0.7250$, (b)$\;0.7375$, (c)$\;0.75=\beta_{c}$,
(d)$\;0.7625$, (e)$\;0.7750$, (f)$\;0.7875$. The circular ring radius and the
semimajor radius are taken as $a=10a_{\perp}$. $x$ and $y$ are in the units of
$a_{\perp}=2.32\;\mu$m and, $t$ is in the units of
$1/\omega_{\perp}=1.95\;$ms.

###  III.2 Coefficients of Dispersion to Obtain Uniform Stationary State

As we increase the eccentricity of the waveguide, it gradually transforms to
an effective double-well potential, resulting into a non-uniform stationary
state along the circumference. As discussed earlier, to eliminate the effects
of nonzero eccentricities, we employ the method of dispersion management. The
dispersion coefficients, $\alpha$ and $\beta$, are tuned and here, we keep
$\alpha=1$ and vary $\beta$ to obtain a uniform stationary state in the
elliptical waveguide. For the purpose of determining $\beta$, we find the
overlap of the condensate density with an expected uniform density given in
Eq.14. The overlap between these two wavefunctions is defined by,

$\Lambda=\frac{[\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|\psi_{e}(x,y)|^{2}|\psi_{a}(x,y)|^{2}dxdy]^{2}}{\int_{-\infty}^{\infty}|\psi_{e}(x,y)|^{4}dxdy\int_{-\infty}^{\infty}|\psi_{a}(x,y)|^{4}dxdy}.$
(15)

Here, $\psi_{e}(x,y)$ is the expected wavefunction given in Eq.14 and
$\psi_{a}(x,y)$ is the actual wavefunction obtained numerically. Hence, unity
overlap function ($\Lambda=1$) will imply uniform distribution of condensate
density along the perimeter of the waveguide, whereas lower values of
$\Lambda$ indicate deformations. Figure 3(a) shows the variation of the
overlap function with $\beta$ for different eccentricities. The maxima in the
$\Lambda$ vs $\beta$ curves will give us the necessary $\beta_{c}$ values to
get the uniform ground state inside an elliptical waveguide. It is worth
observing that, $\beta_{c}$ becomes lower for higher eccentricities.

Figure 5: Survival function for two different initial orientations of the
initial clouds $(\pm a,0)$ and $(0,\pm a)$, with interatomic interaction $g=2$
and, for eccentricities (a) $e=0$, (b) $e=0.25$, (c) $e=0.75$ and (d) $e=0.9$.
The circular ring radius and the semimajor radius are taken as
$a=10a_{\perp}$. $t$ is in the units of $1/\omega_{\perp}=1.95\;$ms.

###  III.3 Dispersion Coefficients and Eccentricities

We have noticed that, for higher eccentricity one needs to take lower
$\beta_{c}$ to maintain the uniformity of the ground state. However, the exact
relationship between $\beta_{c}$ and eccentricity is not obtained. To obtain
this relation, we write Eq. 12 with the transformed variable,
$Y=\frac{y}{\sigma}$, where $\sigma=\sqrt{1-\varepsilon^{2}}$:

$\displaystyle\Bigg{[}i\frac{\partial}{\partial
t}+\frac{\alpha}{2}\frac{\partial^{2}}{\partial
x^{2}}+\frac{\beta}{2\sigma^{2}}\frac{\partial^{2}}{\partial
Y^{2}}-g|\psi|^{2}-V(x,Y)\Bigg{]}\psi=0,$

where $\psi\equiv\psi(x,Y)$. The potential of the elliptical waveguide
transforms to

$\displaystyle
V(x,Y)=V_{0}\Bigg{[}1-e^{-\frac{1}{\gamma^{2}}(\sqrt{x^{2}+Y^{2}}-a)^{2}}\Bigg{]}.$
(16)

Therefore, it becomes transparent to infer that, the dynamical equations for
circular and elliptic cases take identical form provided

$\displaystyle\frac{\beta_{c}}{\alpha_{c}}=1-\varepsilon^{2}.$ (17)

This confirms the earlier prediction from the width dynamics of a Gaussian
wavepacket in the vicinities of semi-major and semi-minor edges. The obtained
$\beta_{c}$ is plotted along with it numerically obtained values in Fig.3(b),
where the dots indicate the numerical values and the solid line indicates the
values obtained from Eq.17. It is clear that $\beta_{c}$ falls as we increase
the eccentricity, such that $\beta_{c}=1$ for a circular waveguide and
$\beta_{c}\rightarrow 0$ for higher eccentricities. The desired dispersion
coefficient $\beta_{c}$, being a maximum in the overlap $\Lambda$, indicates
that the ground state density below and above $\beta_{c}$ must be non-
uniformly distributed and different from each other. This is visualized in
Fig. 4 to find the cross-sectional densities in the elliptical ring along the
semi-major and semi-minor axes. The 1D cross-sectional (CS) densities along
the semi-major and semi-minor axes are denoted by $|\psi(x,0)|^{2}$ and
$|\psi(0,y)|^{2}$, respectively. Figure 4 is depicted for a variety of
eccentricities, where it is clear that at $\beta=\beta_{c}=0.75$, as shown in
Fig.4(c), the 1D cross-sectional densities along the semi-major and semi-minor
axis are almost equal, $|\psi(x,0)|^{2}\approx|\psi(0,y)|^{2}$. On the other
hand, for $\beta<\beta_{c}$, $|\psi(x,0)|^{2}<|\psi(0,y)|^{2}$ (see
Fig.4(a-b)) and for $\beta>\beta_{c}$, $|\psi(x,0)|^{2}>|\psi(0,y)|^{2}$ (see
Fig.4(d-f)). Without any dispersion management, the dispersion coefficient is
unity, $\beta=1$, corresponding to the complete density accumulation at the
semi-major edges and $|\psi(0,y)|^{2}\approx 0$.

Figure 6: Condensate density in circular waveguide $\varepsilon=0$ at time
instances $20.14\;$ms, $26.85\;$ms, $40.28\;$ms and $80.57\;$ms shown in a1,
a2, a3 and a4, respectively. Condensate density in an elliptical waveguide of
eccentricity $\varepsilon=0.75$ at time instances $14.50\;$ms, $19.30\;$ms,
$28.95\;$ms and $57.90\;$ms shown in b1, b2, b3 and b4, respectively. The
circular ring radius and the semimajor radius are taken as $a=10a_{\perp}$.
$x$ and $y$ are in the units of $a_{\perp}=2.32\;\mu$m and, $t$ is in the
units of $1/\omega_{\perp}=1.95\;$ms.

##  IV Fractional Revivals in an Elliptical Waveguide and Dispersion
Management

When a localised cloud of BEC is placed in a circular waveguide, it disperses
and interferes with itself, forming interference fringes. The time at which
the interference brings out the revival of the dispersed cloud in shape and
position is termed the revival time $T_{r}$. At fractional multiples of the
revival time, we can have multiple replicas of the initial condensate. This
phenomenon is called fractional revivals. In this work, the initial condensate
in the form of binary peaks is placed at $(\pm a,0)$. From the physics of
dispersion, the revival time and FR time scales of two clouds in a circular
waveguide of circumference, $C=2\pi r_{0}$, are given by [32]

$\displaystyle T_{r}=\frac{C^{2}}{4\pi},\;\;\;t=\frac{p}{q}T_{r},$ (18)

where $p$ and $q$ are mutually prime integers. However, things are different
when one places the binary peaks of BEC inside an elliptical waveguide of high
eccentricity.

###  IV.1 Revival Time Scale for an Elliptical Waveguide

The revival dynamics are conventionally studied by the time-dependent
characteristic functions such as the autocorrelation function $A(t)$ or the
survival function $S(t)$ [70, 71, 72]. The survival function is the
probability of finding the condensate in its initial state. In other words, it
is the absolute square of the Autocorrelation function $A(t)$, which is
defined as follows:

$\displaystyle S(t)=|A(t)|^{2},$ (19) $\displaystyle
A(t)={\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\psi^{*}(x,y,0)\psi(x,y,t)dxdy}.$
(20)

It is a time series that quantifies the overlap of the wavefunction at a later
time with that of the initial wavefunction, where its value closer to $1$
indicates full revival, and the smaller peaks indicate FR instances. In a
circular waveguide, at half revival $T_{r}/2$ and its odd integral multiples,
$|A(t)|^{2}$ becomes zero since the clouds are at position $(0,\pm a)$, which
is spatially orthogonal to the initial location $(\pm a,0)$. Figure 5 shows
the survival function $|A(t)|^{2}$ for BEC in an elliptical waveguide of
different eccentricities $(a)\;\varepsilon=0$, $(b)\;\varepsilon=0.25$,
$(c)\;\varepsilon=0.75$, and $(d)\;\varepsilon=0.9$. Here, one could note that
for $\varepsilon=0$, there is a clear signature of FR, as pointed out in Fig.
5 (a). However, the signature of FR disappears at higher eccentricities. At
eccentricity as high as $\varepsilon=0.9$, one could see many peaks with
similar heights and the minimum of the survival function no longer touches
zero, indicating that the cloud spends most of its time at the semi-major
edges $(\pm a,0)$. This is delineated by showing the survival function for two
different initial orientations of the clouds, namely $(\pm a,0)$ and $(0,\pm
a)$, in Fig.5. At $\varepsilon=0$, the survival functions coincide for these
two different orientations, whereas at higher eccentricities, the survival
functions are no longer identical. While the survival function for $(0,\pm a)$
orientation touches zero more often, the survival function for $(\pm a,0)$
orientation hardly touches zero. Therefore, irrespective of where the initial
clouds are placed inside the elliptical waveguide, the cloud tends to spend
most of its time in the semi-major edges. This clearly indicates the
disruption of FR instances in an elliptical waveguide. However, one would
still get FR instances at very low eccentricities, and the corresponding time
scales are worth finding out.

###  IV.2 Restoration of Fractional Revivals through Dispersion Management

When the eccentricity of the waveguide is non-zero, the revival time takes the
form,

$T_{r}=\frac{a^{2}[\int_{0}^{2\pi}\sqrt{1-\varepsilon^{2}\sin^{2}\phi}d\phi]^{2}}{4\pi},$
(21)

since the circumference of an ellipse is given by,

$\displaystyle C=a\int_{0}^{2\pi}\sqrt{1-\varepsilon^{2}\sin^{2}\phi}d\phi,$
(22)

where $\phi\in[0,2\pi]$ is the azimuthal coordinate, and $a$, $b$ are semi-
major and semi-minor radii, respectively.

Figure 7: Snapshots of DM condensate density in elliptical waveguide of eccentricity $\varepsilon=0.75$ at time instances $20.14\;$ms, $26.85\;$ms, $40.28\;$ms and $80.57\;$ms shown in b1, b2, b3 and b4, respectively. The semimajor radius is $a=10a_{\perp}$. $x$ and $y$ are in the units of $a_{\perp}=2.32\;\mu$m and, $t$ is in the units of $1/\omega_{\perp}=1.95\;$ms. Eccentricity |  $\varepsilon=0$ |  $\varepsilon=0.25$ |  $\varepsilon=0.75$ |  $\varepsilon=0.9$
---|---|---|---|---
$T_{r}$ before DM (ms) | $161.10$ | $156.07$ | $115.80$ | $95.86$
$T_{r}$ after DM (ms) | $161.10$ | $161.10$ | $161.10$ | $161.10$
Table 1: Revival time for different eccentricities before and after dispersion
management. The circular ring radius and the semimajor radius are taken as
$a=10a_{\perp}$.

At low eccentricities, the daughter condensates of the FR are spatially
resolved, and the revival time can be defined by Eq.21. However, at higher
eccentricities, the multiple splits of the FR are no longer spatially
resolved, and the FR patterns are disrupted. This is evident from Fig. 6,
where the condensate densities at times $\frac{T_{r}}{8}$, $\frac{T_{r}}{6}$,
$\frac{T_{r}}{4}$, and $\frac{T_{r}}{2}$ are denoted by numbers
$1,\;2,\;3,\;4$, respectively. Figures for the two distinct cases
($\varepsilon=0$ and $0.75$) are consequently leveled by $(a)$ and $(b)$. One
can observe no-FR for $\varepsilon=0.75$ and the cloud tends to spend more
time at the semi-major edges, irrespective of the initial placement of the
cloud. In the previous section, we showed that choosing the appropriate
dispersion coefficients could nullify the effects of the non-constant
curvature and produce uniformly distributed ground states inside an elliptical
waveguide. We apply this technique to restore fractional revivals of matter
waves in an elliptical waveguide. We choose the dispersion coefficients
$\alpha=1$ and $\beta=1-\varepsilon^{2}$, as obtained in Eq. 17. The spatial
resolution of the daughter condensates at FR instances, for
$\varepsilon=0.75$, is far lower than that in $\varepsilon=0.5$. In such a
case, the significance of dispersion management is greatly evident. Figure 7
shows the snapshots of dispersion managed condensate density for eccentricity
$\varepsilon=0.75$ at FR instances
$\frac{Tr}{8},\;\frac{Tr}{6},\;\frac{Tr}{4}$ and, $\frac{Tr}{2}$ denoted by
$b1,\;b2,\;b3$ and, $b4$, respectively. The snapshots confirm the restoration
of FR patterns of dispersion-managed BEC in an elliptical waveguide. The
daughter condensates at the FR instances are spatially resolved in the case of
dispersion-managed BEC. Interestingly, for a dispersion-managed matter wave,
the revival time and FR times no longer depend on the eccentricity, unlike the
non-dispersion-managed case. Table 1 shows the revival times for different
eccentricities without and with dispersion management (DM). One could note
that after DM, the BEC revives at times independent of the eccentricity of the
waveguide. The eccentricity-dependent dispersion coefficient (Eq.17) balances
the eccentricity-dependent time scale (Eq.21) of the matter wave. In other
words, the ellipticity-induced effects are nullified through dispersion
management, and the matter-wave in the elliptical waveguide of semi-major
radius $a$ behaves like that in a circular waveguide of radius $a$.

##  V Conclusion

We investigated the influence of ellipticity on the ground state of a BEC
within an elliptical waveguide and its impact on the FR instances in a
localized matter wave. The elliptical waveguide exhibits behaviour reminiscent
of a double-well potential. Notably, an increase in eccentricity correlates
with a heightened concentration of condensate density at these semi-major
edges. We effectively manage dispersion to counteract the effects of variable
thickness within the elliptical waveguide. We achieve a uniform ground state
by identifying optimal dispersion coefficients from the overlap function.
Interestingly, these coefficients are found to be contingent upon the
waveguide’s eccentricity.

In the subsequent phase of our investigation, we explore the disruption of FR
patterns in a localized matter wave confined within an elliptical waveguide.
We employ the time-dependent characteristic function known as the survival
function to analyze the perturbed dynamics of the FR instances. Further, using
the determined dispersion coefficients, we restore the FR instances of the
Bose-Einstein condensate within the elliptical waveguide. This comprehensive
study sheds light on the intricate interplay between ellipticity and
dispersion of matter waves and offers exchanging the physical merits between
the atomtronics applications with zero and non-zero eccentricities.

##  VI Acknowledgement

LS is partially supported by the European Union-NextGenerationEU within the
National Center for HPC, Big Data and Quantum Computing [Project No.
CN00000013, CN1 Spoke 10: Quantum Computing], by the BIRD Project Ultracold
atoms in curved geometries of the University of Padova, by Iniziativa
Specifica Quantum of Istituto Nazionale di Fisica Nucleare, by the European
Quantum Flagship Project PASQuanS 2, and by the PRIN 2022 Project Quantum
Atomic Mixtures: Droplets, Topological Structures, and Vortices, and by the
Project Frontiere Quantistiche within the 2023 funding programme ’Dipartimenti
di Eccellenza’ of the Italian Ministry for Universities and Research. UR
acknowledges the support through the project on ’Realistic Exact Model for
Quantum Logic Gates and Quantum Computation using Bose-Einstein Condensate’
(project No. CRG/2022/007467) by the Science and Engineering Research Board,
India.

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Realizability in Semantics-Guided Synthesis Done Eagerly

Roland Meyer

TU Braunschweig

Jakob Tepe

TU Braunschweig

Sebastian Wolff

New York University

get arXiv to do 4 passes: Label(s) may have changed. Rerun