# COM3D: Leveraging Cross-View Correspondence
and Cross-Modal Mining for 3D Retrieval

###### Abstract

In this paper, we investigate an open research task of cross-modal retrieval
between 3D shapes and textual descriptions. Previous approaches mainly rely on
point cloud encoders for feature extraction, which may ignore key inherent
features of 3D shapes, including depth, spatial hierarchy, geometric
continuity, etc. To address this issue, we propose COM3D, making the first
attempt to exploit the cross-view correspondence and cross-modal mining to
enhance the retrieval performance. Notably, we augment the 3D features through
a scene representation transformer, to generate cross-view correspondence
features of 3D shapes, which enrich the inherent features and enhance their
compatibility with text matching. Furthermore, we propose to optimize the
cross-modal matching process based on the semi-hard negative example mining
method, in an attempt to improve the learning efficiency. Extensive
quantitative and qualitative experiments demonstrate the superiority of our
proposed COM3D, achieving state-of-the-art results on the Text2Shape dataset.

††footnotetext: This work was partially supported by Guangzhou-HKUST(GZ) Joint
Funding Program (Grant No.2023A03J0008), Education Bureau of Guangzhou
Municipality, Guangdong Science and Technology Department, and the Foshan
HKUST Projects (FSUST21-FYTRI01A).

Index Terms—  Scene representation transformer, 3D retrieval

##  1 Introduction

Interactive 3D scenarios across entertainment, architecture, and automotive
industries yield vast 3D, image, and text data. The integration of these
cross-modal data representations is pivotal for enhanced comprehension and
interaction with the three-dimensional aspects of the physical world. To
facilitate a more intuitive interaction process, it is crucial to connect the
realms of 3D data, image data, and linguistic data. Normally, 3D shapes and
linguistic data can serve as complementary representations of object
information. As the capabilities of Large Language Models (LLMs) advance
rapidly, harnessing their robust capabilities to bridge the gap between
textual descriptions and 3D features is increasingly critical. The task of
aligning 3D shapes with corresponding textual information has emerged as a key
challenge in the field.

Within this context, 3D retrieval emerges as one of the crucial tasks.
Existing 3D retrieval methods [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] focus on
learning a unified joint embedding space for linguistic data and other data
types such as 3D shapes or images, employing metric or contrastive learning
approaches for matching. However, in many practical scenarios, the 3D features
jointly characterized by 3D shapes and linguistic data are often incomplete
and ambiguous. This inadequacy can result in models suffering from poor local
feature matching and slow computational speeds. It emphasizes the necessity to
enhance the representation of 3D data in the joint embedding process.

Another key aspect involves the matching of cross-modal embeddings using
similarity metrics. While these techniques can be straightforward, the
inherent gap between modalities like images or 3D data and text poses
significant challenges. This disparity complicates the effective use of both
global and local 3D semantic information, often resulting in poor matching
performance and limited generalization. Meanwhile, the accuracy of
conventional similarity measures in capturing cross-modal data nuances remains
questionable. Consequently, devising an efficacious method to assess cross-
modal similarity poses a significant challenge.

In response to these improvement opportunities, we introduce a novel text-3D
retrieval framework that leverages the multi-view correspondence features
derived from point clouds. COM3D employs the Scene Representation transformer
(SRT) [8] to extract features across different views using multi-view images,
along with their associated camera poses and rays. The SRT facilitates the
learning of joint 3D features from both point clouds and their multi-view
images, thereby enriching the inherent 3D representation with aspects such as
depth, spatial hierarchy, and geometric continuity. Additionally, we design a
semi-hard negative mining method for our similarity metric, which combines the
Earth Mover’s Distance (EMD) with cosine similarity.
Our contributions can be summarized as:

  * •

We enhance our 3D shape representations by uncovering the semantic
correspondence between multi-view point cloud images with the SRT transformer.

  * •

We improve the text-3D retrieval learning efficacy and efficiency with the
proposed semi-hard negative mining technique.

  * •

We demonstrate our proposed COM3D framework outperforms several state-of-the-
art methods for text-3D retrieval.

##  2 Related Work

###  2.1 Joint Learning of Linguistic Data and 3D Shapes

Recently, text-3D datasets such as ShapeNet [1] and Parts2words [2] enhance
the joint learning of linguistic and 3D data, significantly advancing the
geometric and semantic analysis in the 3D world. Text2shape [3] introduces a
framework based on joint embedding concepts to generate colored 3D shapes from
linguistic data. Parts2words represents 3D shapes as point clouds and learns
the joint embedding of point clouds and text while optimizing the feature
space for enhanced matching, yet it overlooks the critical depth information
and varying three-dimensional perspectives essential in text-3D tasks.
Y2seq2seq [4] proposes a view-based model that bridges the semantic meanings
embedded in both text and 3D data, focusing on global matching but neglecting
the local features crucial to text-3D tasks. Tricolo [11] utilizes contrastive
learning methods in joint representation learning for 3D and text, aiming to
enhance the connection between these two modalities.

PointCLIP and Clip2Point [12, 13] transfer the ideas of CLIP [14] and
contrastive learning to 3D visual tasks, establishing joint embeddings of
text, images, and point clouds based on rendered images. Uni3DL [15]
introduces a unified model for 3D and language understanding, which extends
the range of 3D-supported tasks through direct manipulation of point clouds,
facilitating a variety of 3D visual and language-related tasks.

###  2.2 Image-based 3D Shape Retrieval

High-quality image-3D datasets like Pix3D [16] and 3D-FUTURE [17] provide
ample data support for the research of image-3D retrieval tasks. Typically,
image-3D tasks aim to utilize a joint embedding space of 2D and 3D data and a
similarity metric defined in this space to facilitate retrieval and other
visual tasks [5, 6]. In this vein, multi-view Stereo (MVS) methods, such as
MVSnet [7], introduce an end-to-end deep learning model for depth inference to
achieve image-3D matching, employing 3D convolutional networks. MVSTER [18]
utilizes the epipolar Transformer to efficiently learn 2D semantics and 3D
spatial associations, thereby reducing the network’s running time. However,
due to limitations in image clarity or tasks requiring high granularity, MVS
methods normally performs below expectations.

Subsequently, SRT [8] emerges as a potential solution in these areas. The
model is an end-to-end supervised system designed to minimize the
reconstruction error of new 3D views, effectively integrating multi-view
correspondence with 3D shapes to achieve exceptional speed and recognition
accuracy. Building upon SRT, works like RUST [19] have transcended the
limitations of prior pose information and have expanded to handle complex
real-world scenarios while maintaining quality performance.

Fig. 1: An overview of our proposed COM3D. It is composed of three main
embeddings: multi-view embedding, shape embedding, and text embedding. The
multi-view embedding is derived from multi-view images using a Scene
Representation Transformer (SRT) Encoder, considering the corresponding camera
poses and rays; Shape embedding is extracted by PointNet++. Semi-Hard Negative
Mining enhances the matching of text and 3D by focusing on moderately
challenging samples within the semi-hard range, marked by orange lines, with
anchor depicted by red borders.

##  3 Method

Overview As shown in Fig. 1, the COM3D represents an innovative framework
designed for the 3D retrieval task, comprising two essential modules: a joint
encoder module and a matching module. To construct a unified joint embedding
representation space for matching features of 3D shapes, multi-view images,
and text, we initially apply an encoder to embed the multi-view images from 3D
shapes. Concurrently, a joint embedding is accomplished by combining a 3D
shape encoder and a text encoder, thereby achieving joint embedding across
modalities.

To optimize the matching between the fusion
embedding$~{}\mathcal{F}_{\text{fused}}$ which comprises 3D shapes, multi-view
images, and text embeddings$~{}\mathcal{T}$, we introduce a matching module to
score the 3D shapes and linguistic data. After that, we utilize a matching
loss composed of Earth Mover’s Distance (EMD) and Cosine similarity, training
our model based on the semi-hard negative mining method.

###  3.1 Preliminary

####  3.1.1 The Point-cloud Encoder

Point-cloud encoder such as PointNet++ [20] aims to extract features from
point cloud data $\mathcal{P}$. The feature propagation in PointNet++ is
achieved through a hierarchical process, interpolating features across varying
resolutions of the point set. For a given set of points $\mathcal{P}$, with
each point $x_{i}$ having its associated feature vector $f_{i}$, the
interpolated feature $\hat{f_{i}}$ at each point is computed by a distance-
based weighted average of the features from its $k$ nearest neighbors.

The feature propagation is formalized as follows:

$\hat{f_{i}}(x)=\frac{\sum_{j=1}^{k}w_{ij}(x)f_{j}(x)}{\sum_{j=1}^{k}w_{ij}(x)},\quad\text{where}\quad
w_{ij}(x)=\frac{1}{d(x,x_{j})^{p}}$ (1)

and $d(x,x_{j})$ denotes the Euclidean distance between the points $x$ and
$x_{j}$, with $p$ controlling the influence of the distance on the weighting
function. This process iteratively updates the feature vectors for each point
in $\mathcal{P}$, yet yielding a set of enriched feature vectors
$\mathcal{F}=\\{f_{i}\\}$ that capture the complex structure of the point
cloud.

####  3.1.2 The Text Encoder

Given a text sequence or caption $\mathcal{S}$, consisting of $m$ words, we
define the embedding of the $t$-th word as $\mathbf{e}_{t}$. The text encoder
computes the hidden states for each word using a bidirectional GRU, resulting
in a sequence of word embeddings:

$\displaystyle h_{t}^{f}$ $\displaystyle=\text{GRU}^{f}(e_{t},h_{t-1}^{f}),$
$\displaystyle t$ $\displaystyle\in[1,m]$ (2) $\displaystyle h_{t}^{r}$
$\displaystyle=\text{GRU}^{r}(e_{t},h_{t+1}^{r}),$ $\displaystyle t$
$\displaystyle\in[m,1]$ $\displaystyle w_{t}$
$\displaystyle=\frac{h_{t}^{f}+h_{t}^{r}}{2},$ $\displaystyle t$
$\displaystyle\in[1,m]$

Where$~{}\mathbf{h}_{t}^{f}$ is the forward hidden state of the GRU for the
$t$-th word.$~{}\mathbf{h}_{t}^{r}$ is the reverse hidden state of the GRU for
the $t$-th word.$~{}\mathbf{w}_{t}$ is the final embedding for the $t$-th
word, obtained by averaging its corresponding forward and reverse hidden
states. These embeddings $~{}\\{\mathbf{w}_{t}\\}_{t=1}^{m}$ are then used to
represent the text $\mathcal{S}$ within our model.

###  3.2 The Cross-modal Joint Encoder

####  3.2.1 Scene Representation Transformer Encoder

The Scene Representation Transformer (SRT) [8] was developed to quickly infer
3D scene representations from posed or unposed images for rendering novel
views. Given the point cloud data $\mathcal{P}$, our model first extracts a
set of multi-view images $\mathcal{I}=\\{I_{i}\\}_{i=1}^{V}$, where $V$ is the
number of views, and each image $I_{i}$ is associated with a specific camera
pose $c_{i}$ and ray $r_{i}$ from the set $\mathcal{C}$ and $\mathcal{R}$,
respectively.

In the SRT, image data and associated ray information are initially
transformed into feature vectors through a RayEncoder that employs positional
encoding. This encoding process incorporates multiple octaves, beginning at a
designated starting octave, which is then utilized to generate an encoded
feature vector $x$ for each image. Image features are processed through
convolutional layers, doubling dimensions until a set size, and transformed
into patch embeddings. These are enhanced with positional data and camera-
specific characteristics, then unified by a transformer architecture into a
comprehensive 3D scene representation. The mathematical representation of the
SRT encoder can be formalized as:

$\mathcal{M}=Trans\left(\text{Concat}\left(\mathcal{I}_{\text{conv}}+\mathcal{E}_{\text{pos}},\mathcal{C}_{\text{emb}}\right)\right)$
(3)

where $\mathcal{I}_{\text{conv}}$ denotes the image features after
convolutional processing, $\mathcal{E}_{\text{pos}}$ is the positional
embedding for pixels, $\mathcal{C}_{\text{emb}}$ is the camera position
embedding, and $Trans$ represents the transformer module.

####  3.2.2 Cross-modal Joint Encoder

Leveraging the information from multiple distinct modalities can fully exploit
the characteristics of three dimensions, optimizing subsequent 3D-Text
matching. This necessitates the construction of an appropriate unified joint
embedding space to thoroughly fuse the features of 3D shapes, multi-view
images, and text. Consequently, we propose a Cross-modal Joint Encoder,
composed of encoders corresponding to three different types of information: a
3D shape encoder, a multi-view encoder, and a text encoder. These encoders
embed the data into the same representational space, setting the stage for 3D
Retrieval.

###  3.3 The Retrieval Module

####  3.3.1 Optimal Matching Loss Function

In our retrieval module, we adopt the EMD to assess the alignment between 3D
shape embeddings $\mathcal{F}$ and textual embeddings $\mathcal{T}$.
Additionally, we integrate a cosine similarity loss to enhance the retrieval
accuracy. The 3D shape embeddings are first merged with the multi-view
embeddings $\mathcal{M}$ through a transformation, which consists of a fully
connected layer and a subsequent ReLU activation function. This results in a
fused embedding $\mathcal{F}_{\text{fused}}$.

After obtaining $\mathcal{F}_{\text{fused}}$, the cosine similarity $sim$ with
the text embeddings $\mathcal{T}$ is calculated, followed by the application
of the EMD framework to optimize the matching flow between the shape and text
representations. The cosine similarity is explicitly incorporated into the EMD
optimization as an additional term in the loss function, guiding the model
towards embeddings that are not only transport-efficient but also
directionally aligned in the embedding space. The combined loss is represented
as:

$\mathcal{L}_{\text{r}}=\text{$EMD$}(\mathcal{F}_{\text{fused}},\mathcal{T})-\lambda\sum_{i=1}^{n}\sum_{j=1}^{m}\text{$sim$}(\mathcal{F}_{\text{fused}_{i}},\mathcal{T}_{j})\cdot\hat{x}_{ij}$
(4)

where $\lambda$ is a balancing coefficient, $\hat{x}_{ij}$ denotes the optimal
transport plan computed via the Sinkhorn algorithm, and
$\text{$sim$}(\mathcal{F}_{\text{fused}_{i}},\mathcal{T}_{j})$ is the cosine
similarity between the fused shape embedding $\mathcal{F}_{\text{fused}_{i}}$
and the text embedding $\mathcal{T}_{j}$.

####  3.3.2 Semi-hard Negative Mining in Retrieval

Semi-hard negative mining is a strategy used in machine learning, especially
in metric learning and deep learning tasks, aiming to improve the
discriminative power of models. It involves selecting negative samples that
are neither too easy nor too hard for the model to classify — these are
referred as semi-hard negatives. This technique is crucial in the context of
triplet loss, where the goal is to ensure that an anchor is closer to positive
samples than to negative samples in the feature space.

Utilizing semi-hard negative mining within the representation space based on
3D shapes, multi-view images, and text joint embedding not only enhances the
matching effectiveness of text but also facilitates the model’s ability to
learn more robust and generalizable features. By focusing on semi-hard
negatives—those that are neither the easiest nor the most challenging to
distinguish—the method demonstrates exceptional performance in learning
difficult-to-classify target points and ambiguous local boundary features.
This focus leads to improved performance in tasks that require fine-grained
discrimination between similar classes. In our work, semi-hard negative mining
is applied during the training stage, where it directly influences the loss
functions of EMD and Cosine similarity, integral to the model’s learning
process. Taking EMD based mining as an example, the steps typically involved
in applying semi-hard negative mining are as follows:

Select Semi-hard negatives: Select negative samples that are far enough from
the positive sample but not the furthest. This is done by comparing each
negative sample’s EMD score to the positive sample’s EMD score, selecting
those with higher but not the highest scores as semi-hard negatives.

Apply Margin: With semi-hard negatives selected, a predefined margin is used
for pattern calculation adjustment. Calculation formula:

$\mathcal{L}_{\text{semi-
hard}}=\max(0,\text{$margin$}+\text{$score$}_{\text{$neg$}}-\text{$score$}_{\text{$pos$}})$
(5)

where $\text{$score$}_{\text{$pos$}}$ is the positive sample’s EMD score and
$\text{$score$}_{\text{$neg$}}$ is the semi-hard negative’s EMD score. By
minimizing this loss, the model learns to distinguish semi-hard negative
samples from positive samples, thus improving its ability to differentiate
between different types of sample categories.

Method | S2T | T2S
---|---|---
| RR@1 | RR@5 | NDCG@5 | RR@1 | RR@5 | NDCG@5
Text2Shape [3] | 0.83 | 3.37 | 0.73 | 0.40 | 2.37 | 1.35
Y2Seq2Seq [4] | 6.77 | 19.30 | 5.30 | 2.93 | 9.23 | 6.05
TriCoLo [11] | 16.33 | 45.52 | 12.73 | 10.25 | 29.07 | 19.85
Parts2Words [2] | 19.38 | 47.17 | 15.30 | 12.72 | 32.98 | 23.13
COM3D | 20.03 | 48.32 | 15.62 | 13.12 | 33.48 | 23.89
Table 1: Comparison of different methods on S2T and T2S tasks. The results
uses additional semantic part segmentation data in Parts2Words [2].

###  3.4 Overall Objective

The overarching goal of COM3D is to effectively retrieve 3D shapes by matching
them with corresponding textual descriptions. To achieve this, we employ a
comprehensive approach that involves feature extraction from both the
geometric and visual domains, as well as from the textual domain. Our method
synthesizes these multimodal inputs into a cohesive representation that is
conducive to accurate retrieval.

The joint embeddings from the 3D shapes and multi-view images are processed
through a sequence of transformational layers to produce fused embeddings
$\mathcal{F}_{\text{fused}}$. Concurrently, text embedding $\mathcal{T}$ are
extracted and prepared for comparison. The retrieval process is framed as an
optimization problem that leverages the Earth Mover’s Distance (EMD) while
incorporating a cosine similarity metric to refine the matching between 3D
shapes and texts.

The final loss function encapsulates both the transport efficiency and the
alignment in the feature space, formulated as a weighted sum of the EMD and
the cosine similarity losses. It is given by:

$\mathcal{L}_{\text{r}}=\alpha\mathcal{L}_{\text{EMD}}+(1-\alpha)\mathcal{L}_{\text{cos}}$
(6)

where $\alpha$ is a hyperparameter that trades off the contribution of the
retrieval loss $\mathcal{L}_{\text{EMD}}$ and the cosine similarity loss
$\mathcal{L}_{\text{cos}}$. This combined loss function is minimized during
training to ensure that the resulting embeddings for the 3D shapes and the
textual descriptions are closely aligned, facilitating accurate retrieval in
response to textual queries.

Fig. 2: text-to-3D retrieval results by our COM3D. For each query sentence, we
show the top-5 ranked shape.

##  4 Experiments

###  4.1 Experimental settings

Data Preparation In the 3D retrieval task, we employ a dataset [2] that
provides both 3D representations and corresponding textual descriptions to
enable a comprehensive evaluation of our framework. We utilize the well-
constructed 3D-Text cross-modal dataset, which is specifically designed to
bridge the modalities of 3D shapes and natural language descriptions.

This dataset is split into a training set consisting of 11,498 3D shapes, and
a test set with 1,434 shapes. Each shape is accompanied by an average of five
textual descriptions, enabling the model to learn and predict from multiple
descriptive angles. It stands out due to its utilization of detailed 3D models
from ShapeNet [1] and PartNet [21]. ShapeNet provides the fundamental 3D
shapes, while PartNet offers a more intricate layer of fine-grained, instance-
level segmentation annotations. Such a combination allows for a rich
representation of objects that our model can leverage.

Evaluation metrics For the evaluation of our 3D retrieval model, we adopt the
standard metrics of Recall Rate at $k$ (RR@$k$), specifically for $k=1$ and
$k=5$, and the Normalized Discounted Cumulative Gain (NDCG). RR@$k$ quantifies
the likelihood of the correct match appearing within the top $k$ retrieved
results. NDCG assesses the ranking quality of the retrieval process, taking
into account the position of relevant items within the retrieval list.

Parameter Setting we configure the shape encoder to process point clouds fixed
at 2500 points each, segmented according to a granular scheme of 17 classes.
We employed PointNet++ [20] architectures within our shape encoder module to
evaluate their performance in feature extraction. The aggregation of features
was facilitated through average pooling in our group pooling module,
deliberately ignoring segments constituting less than 1% of the point cloud to
prioritize salient features. For the 3D retrieval task, embeddings were
standardized to a 1024-dimensional space. Textual encoding was performed using
a single-layer Bi-directional GRU, deliberately trained from scratch, owing to
negligible vocabulary benefits observed from pre-training techniques. The
optimization of our model employed a semi-hard negative mining strategy within
its triplet ranking loss, with an established margin of 0.2. An Adam optimizer
guided the learning process, initiated with a learning rate of 0.001.

###  4.2 Comparison results

We evaluate our framework in the dataset from part2words [2], which includes
additional semantic part segmentation data for the 3D point cloud.
Accordingly, our comparison results include scenarios both with and without
part segmentation. We compare with many previous text-3D retrievals methods
such as Text2Shape [3], Y2seq2seq [4], and TriCoLo [11]. our framework showed
consistent superiority and achieved state-of-the-art (SOTA) performance in
various settings.

Part2words showcases specific optimizations for part segmentation features. In
contrast, our framework enhances 3D semantic features through the inherent
characteristics of multi-view images in point clouds facilitated by the Scene
Representation Transformer (SRT). This strategy led to a marked improvement in
performance, enabling our method to surpass part2words and achieve state-of-
the-art (SOTA) results, as detailed in Tab. 1.

Our method outperforms previous approaches in experiments without part
segmentation, as shown in Tab. 2. This outcome underscores the remarkable
generalization capability of our model, evidencing the efficacy of the SRT
optimization across various settings. Our model consistently delivers SOTA
performance in different configurations, showcasing its robustness and wide
applicability. Besides, the visualization results are shown in Fig. 2.

###  4.3 Ablation study

EMD and cosine loss We evaluate the impact of the Earth Mover’s Distance (EMD)
and cosine loss on our retrieval module, as detailed in Tab. 2. In our
ablation analysis with COM3D, we observe a noticeable performance decline when
either of the losses is removed. Notably, the absence of EMD results in a more
pronounced drop in performance. This can be attributed to EMD’s focus on
optimizing local details, while cosine similarity concentrates on the overall
alignment in the vector space. These findings underscore the synergistic
importance of both losses in enhancing the retrieval process.

SRT The SRT is a cornerstone of our approach, extracting multi-view features
from point clouds. Compared with our baseline, the addition of SRT
significantly boosts performance. This improvement is attributed to the
ability of SRT to utilize information overlooked by PointNet++ and to enhance
the matching process with text from the multi-image perspective.

Semi-hard negative mining Semi-hard negative mining is an optimization
technique that we analyze in contrast to the commonly used hardest negative
mining. Replacing semi-hard with the hardest negative mining leads to a sharp
decline in performance. This may be due to the unsuitability of the hardest
negative mining for retrieval tasks and the specific variable characteristics
of our model. The effectiveness of semi-hard negative mining is thus validated
by these comparative results.

Method | T2S
---|---
| RR@1 | RR@5 | NDCG@5
Text2Shape | 0.40 | 2.37 | 1.35
Y2Seq2Seq | 2.93 | 9.23 | 6.05
Part2Words | 5.06 | 17.21 | 11.25
COM3D | 5.64 | 18.50 | 12.09
\- cosine loss | 5.12 | 17.20 | 11.54
\- EMD loss | 4.83 | 16.32 | 10.42
\- SRT | 5.07 | 17.92 | 11.52
\- semi-hard | 0.93 | 1.92 | 1.42

Table 2: Comparison of different methods on T2S tasks and the ablation study
on our COM3D. The results use the Text2Shape[3] ShapeNet subset without
semantic part segmentation.

##  5 Conclusion

In this paper, we introduce an efficient end-to-end cross-modal 3D retrieval
model. This model utilizes cross-modal data to achieve bidirectional matching
between different types, implementing text-3D retrieval. We initially propose
a cross-modal joint embedding model to fully integrate the features of text,
multi-view images, and 3D shapes. Moreover, we measure the features across
different dimensions using cosine similarity in the joint embedding space and
employ a hard sample mining approach to achieve improved matching results. Our
findings have achieved state-of-the-art (SOTA) results on the past2word
dataset. One limitation of our model is that it only considers point cloud
features due to the lack of sufficient datasets. However, the concept of joint
learning and matching based on different types of data is highly efficient,
endowing our model with the potential to extend to various 3D representations
such as mesh.

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##  Appendix A More experiment result

More visualization results of our COM3D are shown in Fig. 3 and Fig. 4.

Fig. 3: text-to-3D retrieval results by our COM3D. For each query sentence, we
show the top-5 ranked shape.  Fig. 4: 3D-to-text retrieval results by our
COM3D. For each query sentence, we show the top-5 ranked shape.

##  Appendix B More details of the proposed method

###  B.1 The Cross-modal Joint Encoder

####  B.1.1 Scene Representation Transformer Based Encoder

The Scene Representation Transformer (SRT) [8] has been innovatively adapted
to process 3D scenes from various perspectives, enabling novel view rendering.
In our framework, this transformer plays a pivotal role in encoding multi-view
images derived from 3D point clouds. For a set of point cloud data
$\mathcal{P}$, we extract a collection of multi-view images
$\mathcal{I}=\\{I_{i}\\}_{i=1}^{V}$, where $V$ represents the total number of
views. Each image $I_{i}$ is linked with a distinct camera pose $c_{i}$ and
ray $r_{i}$.

The SRT encoder operates in the following stages:

  1. 1.

Ray Encoding: Each image and its associated ray information are encoded using
a RayEncoder, applying positional encoding to create a feature vector $x$ for
each image.

  2. 2.

Convolutional Processing: The encoded image features undergo a series of
convolutional transformations, with the hidden dimension successively doubling
until reaching the target size.

  3. 3.

Patch Embedding: Post-convolutional features are reshaped into patch
embeddings and combined with positional embeddings and camera-specific
embeddings.

  4. 4.

Transformation: A transformer architecture integrates all patch embeddings,
resulting in a comprehensive representation of the 3D scene.

Mathematically, the SRT encoder is expressed as:

$\begin{split}\mathcal{M}=\text{Transformer}\Big{(}&\text{Concat}\big{(}\text{Conv}(\mathcal{I})\\\
&+\text{PosEmbed}(\mathcal{I}),\text{CamEmbed}(\mathcal{C})\big{)}\Big{)}\end{split}$
(7)

where $\text{Conv}(\mathcal{I})$ denotes convolutional features of the images,
$\text{PosEmbed}(\mathcal{I})$ represents positional embeddings, and
$\text{CamEmbed}(\mathcal{C})$ corresponds to the camera position embeddings.
The transformer module is denoted as Transformer.

####  B.1.2 Cross-modal Joint Encoder

To effectively harness the characteristics of 3D data, we propose a Cross-
modal Joint Encoder, integrating encoders for 3D shape, multi-view images, and
text into a unified embedding space. Our approach consists of:

  1. 1.

3D Shape Encoder: Utilizing PointNet++ [20] for extracting features from point
clouds.

  2. 2.

Multi-view Encoder: Based on the SRT, it synthesizes diverse views of 3D
shapes into a single scene representation.

  3. 3.

Text Encoder: Employs a Bi-directional GRU for bidirectional text feature
extraction.

These encoders collectively map their respective inputs into a harmonized
feature space, facilitating efficient 3D-text retrieval.

###  B.2 The Retrieval Module

####  B.2.1 Optimal Matching Loss Function

Our retrieval module combines EMD and cosine similarity to optimize the
matching process.


aainstitutetext: University of Colorado Boulder,
2000 Colorado Ave, Boulder, CO 80309 United States of America

# Interacting CFTs for all couplings: Thermal versus Entanglement Entropy at
Large $N$

S. Grable<EMAIL_ADDRESS>
###### Abstract

In this paper, I calculate the large $N$ limit of marginal $O(N)$ models with
non-polynomial potentials in arbitrary odd dimensions $d$. This results in a
new class of interacting pure conformal field theories (CFTs) in $d=3+4n$ for
any $n\in\mathbb{Z}_{+}$. Similarly, in $d=3+4n$ I calculate the thermal
entropy for all couplings on $R^{2+4n}\times S^{1}$ for $n=0,1,2,3$. In 2+1
dimensions I find the strong-to-weak coupling ratio of the thermal entropy to
be 4/5, matching recent results, and further extend this analysis to higher
odd dimensions. Next, I calculated the vacuum entanglement entropy
$s^{d}_{\text{EE}}$ on $S^{d-2}$ for all couplings in arbitrary odd $d$ in the
large N limit. I find the vacuum entanglement entropy on $S^{d-2}$ to be not
only solvable but also constant for all couplings $\lambda$. Thus, in the
large $N$ limit, the vacuum entanglement entropy on $S^{d-2}$ for odd $d$ is
constant for all $\lambda$, in contrast to the thermal entropy which is shown
to also be monotonically decreasing with $\lambda$ in $d=3+4n$.

##  1 Introduction

Pure CFTs are a sub-class of conformal field theories (CFTs). CFTs main
characteristic is a vanishing beta-function and can be found at critical
parameter values within a larger parent theory. However, pure CFTs maintain
conformal invariance for all values of a parameter, such as the coupling
$\lambda$. Further, some CFTs in the large N-limit give exact analytic results
for all $\lambda$. A well know example of pure CFT is a $\mathcal{N}=4$ Super-
Yang-Mills theory in 3+1 dimensions whose free energy at finite temperature
and infinite coupling, via its conjectured gravity dual maldacena , is 3/4 of
the Stefan-Boltzmann free theory gubser dewolfe . However, the entropy density
$s$ of the $\mathcal{N}$ = 4 SYM theory in four dimensions is not solvable for
all $\lambda$ romatschke blaizot . Nonetheless, in the large N limit, I find a
large class of field theories, in $d=3+4n$ for any $n\in\mathbb{Z}_{+}$, that
are exactly solvable for all couplings. It is no secret that purely solvable
models for all couplings are rare, and as with quantum mechanics more broadly,
analytically solvable models often bring new insight.

In the large N limit Ref romatschke  shows a $\phi^{4}$ finite-temperature
$O(N)$ model in $2+1$ dimensions on flat space is solvable for all $\lambda$,
and has a weak-to-strong entropy density ratio of $4/5$. This result was then
extended to large $N$ supersymmetric Wess-Zumino model in $2+1$ dimension
where the weak-to-strong entropy density ratio was found to be $\frac{31}{35}$
dewolfe .
In this spirit I continue to investigate large $N$ pure CFT’s in higher odd
dimensions. I start by presenting a generalized marginal vector model in
arbitrary odd dimensions. All theories under this generalization lack
logarithmic divergences, and theories in $d=3+4n$ yield gap equations with
non-trivial finite solutions. This results in a new class of interacting pure
CFTs. Further, applying this generalized theory to $2+1d$ produces a strong-
to-weak thermal entropy density ratio of $\frac{4}{5}$, matching the results
of quartic and sextic theories in Refs. romatschke romatschkeshear  and pinto
.

In Refs. casini  and Myers  it was shown that the vacuum entanglement entropy
on a spherical region in $R^{d-1}\times S^{1}$ maps to the thermal entropy in
$S^{d}$. Further, it was shown that the non-universal piece of the vacuum
entanglement entropy on the boundary of a spherical region is positively
divergent. Removing the non-universal piece via regularization leaves a finite
universal piece which can be either positive or negative depending on
dimensionality. With this, I show in the large $N$ limit that the entanglement
entropy on $S^{d-2}$ defined as $s^{d}_{\text{EE}}$, is constant for all $d$,
further implying the free propagator is vanishing in all such theories.
Finally, $s^{d}_{\text{EE}}$ in the large N limit is shown to match $N$ copies
of the thermal entropy of a free theory on $S^{d}$ for any odd $d$, given by
Ref. klebanov .

The contrast between results of various geometries prompts further discussion
on large $N$ theories on curved spacetime, particularly on how the d-sphere
acts as a local minimum of entanglement entropy among all shapes mezei , and
the possibility of conformally deforming the minimized entanglement entropy
$S^{d}_{\text{EE}}$ to other topologies in the large $N$ limit. Further,
investigation can be extended to Fermionic large $N$ theories, and to fully
solvable higher-spin gravity duals of large $N$ theories.

##  2 The Model

###  2.1 A Marginal Theory on Flat Space in Odd Dimensions

Consider a marginal $O(N)$ bosonic scalar theory in odd dimensions where
$\vec{\phi}=(\phi_{1},\phi_{2}...\phi_{N}$) in the limit $N\rightarrow\infty$,
with the Euclidean action

$A^{(d)}=\frac{1}{2}\int_{x}\bigg{[}\partial_{\mu}\vec{\phi}\partial_{\mu}\vec{\phi}+m_{0}^{2}\vec{\phi}^{2}+\bigg{(}\frac{\lambda}{N}\bigg{)}^{\frac{2}{d-2}}\big{(}\vec{\phi}\cdot\vec{\phi}\big{)}^{\frac{d}{d-2}}\bigg{]}.$
(1)

As standard practice, let the Euclidean dimension of time be compactified to a
thermal circle of radius $\beta$ laine . Next the partition function
$Z^{(d)}(T)=\int\mathcal{D}\vec{\phi}e^{-A^{(d)}}$ is multiplied by the
identity romatschke

$1=\int\mathcal{D}\xi\mathcal{D}\sigma
e^{i\int_{x}\frac{1}{2}\xi(\sigma-\frac{\vec{\phi}\cdot\vec{\phi}}{N})},$ (2)

and for simplicity bare mass term $m_{0}$ is tuned to zero. Further let
$\xi\rightarrow N\xi$ and $\sigma\rightarrow\frac{\sigma}{\lambda}$ to evenly
distribute the coupling in $Z(T)$. Next saddle point integrals are taken
around the mean-field values of $\xi$ and $\sigma$, $\bar{\xi}$ and
$\bar{\sigma}$, which are exact in the large N limit. Applying the saddle
point condition for the $\sigma$ field I keep only the positive solution in
terms of $\bar{\xi}$, which is equivalent to keeping only positive effective
mass terms. Now $Z(T)$ is

$\begin{split}Z^{(d)}(T)=\int\mathcal{D}\phi
d\bar{\xi}\exp\bigg{[}-N\int_{x}\Bigg{(}\frac{1}{2}\phi\big{(}\partial_{\mu}\partial_{\mu}+i\bar{\xi}\big{)}\phi-\frac{(i\bar{\xi})^{d/2}}{\lambda}\bigg{(}\frac{2}{d}\bigg{)}\bigg{(}\frac{d-2}{d}\bigg{)}^{\frac{d-2}{2}}\Bigg{)}\bigg{]},\end{split}$
(3)

giving a marginally coupled theory for any $d$, however, this paper is
specifically investigating theories in odd dimensions. Letting
$\bar{i\xi}\rightarrow m^{2}$ in noting $\bar{i\xi}$ acts as the effective
mass, up to an overall constant the partition function is

$Z^{(d)}(T)=\exp\bigg{[}N\beta
V\Bigg{(}P^{(d)}_{\text{free}}\big{(}T,m\big{)}+\frac{m^{d}}{\lambda}\bigg{(}\frac{2}{d}\bigg{)}\bigg{(}\frac{d-2}{d}\bigg{)}^{\frac{d-2}{2}}\Bigg{)}\bigg{]},$
(4)

where $P^{(d)}_{\text{free}}\big{(}T,m\big{)}$, the pressure of a free scalar
field theory at temperature $T$ in $d$ dimensions, is given by

$P^{(d)}_{\text{free}}(T,m)=-\int\frac{d^{d-1}k}{(2\pi)^{d-1}}\bigg{[}\frac{E_{k}}{2}+T\ln(1-e^{-\beta
E_{k}})\bigg{]}.$ (5)

Using dimensional regularization $P^{(d)}_{\text{free}}$ evaluates to an exact
analytic result of the form:

$P^{(d)}_{\text{free}}=-\frac{1}{2}\frac{m^{d}\Gamma(-\frac{d}{2})}{(4\pi)^{\frac{d-1}{2}}\Gamma(-\frac{1}{2})}+\frac{T\Omega_{d-1}}{(2\pi)^{d-1}}\sum_{n=1}^{\infty}\int_{m}^{\infty}\frac{1}{n}(k^{2}-m^{2})^{\frac{d-3}{2}}ke^{-n\beta
k}dk,$ (6)

where $\Omega_{d}=\frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}$ is the solid angle,
and $d$ still holds as the dimensionality of theory. The integral in equation
(6) can be evaluated for any $d$ and $m>0$ to give,

$P^{(d)}_{\text{free}}=\frac{m^{d}\Gamma(\frac{-d}{2})}{2^{d+1}\pi^{\frac{d}{2}}}+\frac{2^{1-\frac{d}{2}}m^{d}}{(\pi
m\beta)^{\frac{d}{2}}}\sum_{n=1}^{\infty}\big{(}\frac{1}{n}\big{)}^{\frac{d}{2}}K_{\frac{d}{2}}\Big{(}mn\beta\Big{)},$
(7)

One can see that the dimensionally regulated piece of (7) has no logarithmic
divergence in odd dimensions. To apply (7) consider $d=2+1$ resulting in a
$\phi^{6}$ theory with marginal coupling, the free pressure evaluates to

$P^{(3)}_{\text{free}}=\frac{T^{3}}{2\pi}\big{[}\text{Li}_{3}(e^{-\beta
m})+\frac{m}{T}\text{Li}_{2}(e^{-\beta m})+\frac{m^{3}}{6T^{3}}\big{]}.$ (8)

The saddle condition of the $\xi$-field is given in terms of the effective
mass as

$\frac{2m}{\sqrt{3}\lambda}=\frac{m}{4\pi}+\frac{1}{2\pi}T\ln(1-e^{-\beta
m}).$ (9)

Applying the saddle point condition (9) to (4) gives

$P^{(3)}(T,\lambda)=N\bigg{[}\frac{T^{3}}{2\pi}\big{[}\text{Li}_{3}(e^{-\beta
m})+\frac{m}{T}\text{Li}_{2}(e^{-\beta m})-\frac{m^{2}}{3T^{2}}\ln(1-e^{-\beta
m})\big{]}\bigg{]}.$ (10)

The thermal entropy density is found via $s_{\text{therm}}=\frac{\partial
P}{\partial T}$, resulting in

$s^{(3)}_{\text{therm}}(T,\lambda)=N\bigg{[}\frac{3T^{2}}{2\pi}\big{[}\text{Li}_{3}(e^{-m\beta})+\frac{m}{T}\text{Li}_{2}(e^{-m\beta})-\frac{m^{2}}{3T^{2}}\ln(1-e^{-m\beta})\big{]}\bigg{]}.$
(11)

$s^{(3)}_{\text{therm}}(T,\lambda)$ matches the bosonic contribution of the
entropy density of a Nambu–Jona-Lasinio–Yukawa model given in Ref. pinto .
Considering equation (9), the gap equation at infinite coupling is

$-\beta m=2\ln(1-e^{-\beta m}).$ (12)

This gap equation is solved in terms of the golden ratio
$\phi=\frac{1+\sqrt{5}}{2}$ giving

$\beta m=2\ln(\phi),$ (13)

which matches the infinite coupling gap equation of Ref. romatschke . In the
zero coupling limit, we have $\beta m=0$. Then evaluating (11) with (13) and
$\beta m=0$, $s^{(3)}_{\text{therm}}(T,0)$ and
$s^{(3)}_{\text{therm}}(T,\infty)$ are given as

$\begin{split}&s^{(3)}_{\text{therm}}(T,0)=\frac{3NT^{2}\zeta(3)}{2\pi}=s_{\text{therm}}^{\text{free}^{(3)}},\\\
&s^{(3)}_{\text{therm}}(T,\infty)=\frac{6NT^{2}\zeta(3)}{5\pi}=s_{\text{therm}}^{\infty^{(3)}}.\end{split}$
(14)

These coupling limits match Ref. romatschke pinto , and romatschkeshear , and
the strong coupling limit was recognized earlier in Refs. sachdev drummond .
Together the equations of (14) give a strong-to-weak coupling ration of,

$\frac{s_{\text{therm}}^{\infty^{(3)}}}{s_{\text{therm}}^{\text{free}^{(3)}}}=\frac{4}{5}.$
(15)

This strong-to-weak entropy density ratio matches the results of a $\phi^{4}$
theory in $R^{2}\times S^{1}$ and has further been shown to be universal in a
large class of pure CFTs with only bosonic degrees of freedom in $R^{2}\times
S^{1}$ romatschke .

###  2.2 Higher Dimensions on Flat Space

To find the gap equations for higher dimensions I will start at $d=3$ and move
up in increments of 4 dimensions. We can call this “the rule of fours.” The
reasoning for this strategy goes as follows: in the $\lambda\rightarrow\infty$
limit the finite temperature piece of equation (7) gives a series of
polylogarithms of the form $\text{Li}_{n}(e^{-m\beta})$. Now, for all finite
$m$, $n$, and $d$:

$\begin{split}&\frac{\partial}{\partial
m^{2}}\text{Li}_{n}(e^{-m\beta})=-\text{Li}_{n-1}(e^{-m\beta})<0,\end{split}$
(16)

and,

$\lim_{m\to\infty}-\text{Li}_{n-1}(e^{-m\beta})=0.$ (17)

Further, the mass derivative of the zero temperature piece of the equation (7)
is proportional to $\Gamma(-\frac{d}{2})$

$\frac{\partial}{\partial
m^{2}}\frac{m^{d}\Gamma(-\frac{d}{2})}{2^{d+1}\pi^{\frac{d}{2}}}\sim\Gamma\bigg{(}-\frac{d}{2}\bigg{)}.$
(18)

Finally, $\Gamma\big{(}-\frac{d}{2}\big{)}>0$ for $d=3,7,11,15...$, for all
$d=3+4n$ with $n\in\mathbb{N}$. Thus, the gap equations in dimensions $d=3+4n$
are satisfied by some non-trivial finite mass value for any $n\in\mathbb{N}$
at any finite temperature $T$. This analysis can be extended to finite
$\lambda$ to include the coupling term in equation (4) which is overall
positive for $d>2$. For example, the gap equation in the limit
$\lambda\rightarrow\infty$ for $d=7$ is given as

$\beta^{5}m^{5}=30\beta^{2}m^{2}\text{Li}_{3}\left(e^{-m\beta}\right)+90\beta
m\text{Li}_{4}\left(e^{-m\beta}\right)+90\text{Li}_{5}\left(e^{-m\beta}\right),$
(19)

which is numerically solved around $\beta m\approx 2.178$. Likewise, the gap
equation in the zero temperature limit is

$\left(\frac{25\sqrt{\frac{5}{7}}}{49\lambda}+\frac{1}{480\pi^{3}}\right)m^{5}=0$
(20)

which satisfied at $m=0$ for all $\lambda>0$. Given that the gap equations for
dimensions $d=3+4n$ can be solved at finite temperature for all couplings
$\lambda$, and the corresponding theories contain no logarithmic divergences;
in the large $N$ limit, marginal theories in $d=3+4n$ generate a large class
of interacting pure CFTs. Figure (1) shows $P^{(d)}(T,m)$ for $d=3,7,11,15$.
Likewise the chart below gives $s^{\text{free}}_{\text{therm}}$ and
$s_{\text{therm}}^{\infty}/{s_{\text{therm}}^{\text{free}}}$ values for
$d=3,7,11,15$.

Thermal Entropy Relations
---
Dimensions |  $s^{\text{free}}_{\text{therm}}$ |  $s_{\text{therm}}^{\infty}/{s_{\text{therm}}^{\text{free}}}$
$d=3$ |  $\frac{3NT^{2}\zeta(3)}{2\pi}$ |  = 0.800
$d=7$ |  $\frac{15NT^{7}\zeta(7)}{8\pi^{3}}$ |  $\approx 0.649$
$d=11$ |  $\frac{945T^{11}\zeta(11)}{32\pi^{5}}$ |  $\approx 0.578$
$d=15$ |  $\frac{135135T^{15}\zeta(15)}{128\pi^{7}}$ |  $\approx 0.466$

Figure 1: The thermal entropy density of
$s_{\text{therm}}/s^{\text{free}}_{\text{therm}}$ are shown above for all
temperatures/couplings ranging in order from $d=3$ at the bottom to $d=15$.

However, $\Gamma\big{(}-\frac{d}{2}\big{)}<0$ for all $d=1+4n$ with
$n\in\mathbb{N}$. Thus, gap equations for dimensions $d=1+4n$ in the
$\lambda\rightarrow\infty$ limit are overall negative, giving only trivial
solutions at finite temperature. Consider the $\lambda\rightarrow\infty$ limit
for $d=5$, the gap equation is given as

$\frac{m^{3}\beta^{3}}{6}+\beta
m\text{Li}_{2}\left(e^{-m\beta}\right)+\text{Li}_{3}\left(e^{-m\beta}\right)=0.$
(21)

By physical demands $m>0$ and $\beta>0$ and because
$\text{Li}_{2}\left(e^{-m\beta}\right)>0$ this gap equation is only trivially
solved by $m=0$ as $\beta\rightarrow\infty$. We can also note that for
$d=1+4n$ in the zero temperature limit there is a critical coupling
$\lambda_{\text{crit}}$ which satisfies the saddle condition for all in medium
mass giving a constant zero pressure for all $m$. For $d=5$ in the
$T\rightarrow 0$ limit, the saddle condition is

$m^{3}\left(\frac{3\sqrt{\frac{3}{5}}}{5\lambda}-\frac{1}{48\pi^{2}}\right)=0,$
(22)

which is satisfied for all $m$ at
$\lambda_{\text{crit}}=\frac{144}{5}\sqrt{\frac{3}{5}}\pi^{2}$ giving
$P(0,m,\lambda_{\text{crit}})=0\;\forall\;m$. The physical implications of
this critical phenomenon can be explored in future work.

##  3 Entanglement Entropy on a d-Sphere

The action for a vector model marginally and conformally coupled on $S^{d}$ is
given byklebanov romatschke  as:

$A^{(d)}=\frac{1}{2}\int_{s^{d}}dx\sqrt{-g}\bigg{[}\nabla_{\mu}\vec{\phi}\nabla^{\mu}\vec{\phi}+\frac{R(d-2)}{4(d-1)}\vec{\phi}^{2}+\Big{(}\frac{\lambda}{N}\Big{)}^{\frac{2}{d-2}}\big{(}\vec{\phi}\cdot\vec{\phi}\big{)}^{\frac{d}{d-2}}\bigg{]},$
(23)

where $g\equiv\det g_{\nu\mu}$ is the determinant of the metric tensor of
$S^{d}$, $R$ is the Ricci scalar term defined as $R=\frac{d(d-1)}{r^{2}}$ with
$r$ being the radius of the d-sphere. Again, the mass has been tuned to zero,
resulting in a critical O(N) theory with conformal invariance on an odd
d-dimensional Euclidean manifold under all couplings $\lambda$. Further the
entanglement entropy of a spherical region in $S^{d-1}$ embedded in
$R^{d-1}\times S^{1}$ lives on the boundary in $S^{d-2}$, as shown in figure
(2), and maps directly to the thermal entropy of a theory on $S^{d}$ klebanov
Myers .

I will refer to the vacuum entanglement entropy of a region $S^{d-1}$ which
lives on $S^{d-2}$, as $s_{\text{EE}}^{(d)}$, and

$s_{\text{EE}}^{(d)}=\log|Z^{(d)}|=-\frac{1}{2}\log\big{(}\det{\mathcal{O}^{(d)}}\big{)},$
(24)

klebanov  casini , with,
$\mathcal{O}^{(d)}\equiv-\nabla^{2}+\frac{R(d-2)}{4(d-1)}$, and $Z^{(d)}$
being the partition function on $S^{(d)}$. Eigenvalues of minus the Laplacian
on a d-sphere for $d\geq 2$ are given by Shimakura klebanov  as

$\lambda_{n}=n(n+d-1)r^{-2},$ (25)

with multiplicities,

$m_{n}=\frac{(2n+d-1)(n+d-2)!}{(d-1)!n!},\quad n\geq 0.$ (26)

In odd dimension $Z^{(d)}$ has no dependence on $r$ giving a vanishing
$\beta$-function. Thus, for simplicity, the dependence of $r$ in $\mathcal{O}$
is manipulated by letting $r=1$. However, with hindsight of the effective mass
$m$ (associated with the auxiliary field $\xi$) being in the argument of
logarithmic functions, I will keep the dimensionful quantity of r associated
with $m$.

Figure 2: The entanglement entropy of a spherical region in $S^{d-1}$ embedded
in $R^{d-1}\times S^{1}$ is found on the boundary in $S^{d-2}$.$R^{d-1}\times
S^{1}$$S^{d-1}$$S^{d-2}$

###  3.1 Entanglement entropy for d=3

Considering $d=3$ equation (3) can be modified so the integrals are now over
$S^{3}$ give three-dimensional hypersphere volumes. Then,

$\begin{split}s_{\text{EE}}^{(3)}=N\bigg{[}\frac{1}{2}\log\big{(}\det[\mathcal{O}+(rm)^{2}]\big{)}-4\pi
r^{3}\bigg{(}\frac{2m^{3}}{9\sqrt{3}\lambda}\bigg{)}\bigg{]}.\end{split}$ (27)

Now the entanglement entropy of the free theory with the added auxiliary mass
field is

$s_{\text{EE}_{\text{free}}}^{(3)}=\frac{1}{2}\sum_{n=0}^{\infty}(n+1)^{2}\log\big{[}(n+1)^{2}-\frac{1}{4}+(rm)^{2}\big{]}.$
(28)

Taking the derivative of $S_{\text{EE}_{\text{free}}}^{(3)}$ w.r.t. $m^{2}$
and applying zeta-function regularization hawking  enables the saddle point of
$m$ to be evaluated:

$\begin{split}&\frac{\partial}{\partial
m^{2}}s_{\text{EE}_{\text{free}}}^{(3)}\\\
&=\frac{1}{2}\sum_{n=0}^{\infty}\frac{(n+1)^{2}}{(n+1)^{2}-\frac{1}{4}+(rm)^{2}},\\\
&=\frac{1}{2}\bigg{[}\zeta(0)+\frac{1}{4}(2-\sqrt{1-(2rm)^{2}}\pi\cot\big{(}\frac{1}{2}\sqrt{1-(2rm)^{2}}\pi\big{)}\bigg{]}.\end{split}$
(29)

Finally, utilizing equation (27) the saddle point condition for $m$ is

$\frac{1}{8}\sqrt{1-(2rm)^{2}}\pi\cot\big{(}\frac{1}{2}\sqrt{1-(2rm)^{2}}\pi\big{)}-\frac{4\pi
r^{3}m}{3\sqrt{3}\lambda}=0.$ (30)

Equation (30) is satisfied for all couplings $\lambda$ when $m=0$. In
evaluating (30), the saddle condition of $m$, call it
$C\big{(}m(\lambda)\big{)}$, is a functional of $m$ which is a function of
$\lambda$, and $C(m=0)=0$ for all $\lambda$. Therefore, $m(\lambda)$ cannot be
inverted into $\lambda(m)$. Thus, the in-medium mass is vanishing,
$s_{\text{EE}}^{(3)}$ is invariant to coupling variations in the large $N$
limit, and $s_{\text{EE}_{\text{free}}}^{(3)}/s_{\text{EE}_{\infty}}^{(3)}=1$.
Further, note satisfying (30) at $m=0$ is effectively the same as setting the
free propagator of the theory to zero. Recovering a functional form of
equation (27) is achieved by; integrating equation (29), evaluating for $m=0$,
and finally keeping the real parts of the result as demanded by the physical
nature of the entanglement entropy. Likewise, removing the pure imaginary of
integral result of (29) can be seen as removing an unspecified integration
constant, then,

$s_{\text{EE}}^{(3)}=-\frac{N}{16}\bigg{(}2\log(2)-\frac{3\zeta(3)}{\pi^{2}}\bigg{)}.$
(31)

The entanglement entropy is negative, however, this is since in regularization
we dropped the positively divergent non-universal contribution. This result
matches N copies of the thermal entropy of a free massless theory on $S^{3}$
given by klebanov . It has further been shown by Ref. mezei allais  that the
d-sphere entanglement entropy represents a minimum among all shapes. Thus, the
entanglement entropy on $S^{3}$ in the large $N$ limit is constant w.r.t. all
couplings $\lambda$ and minimized for all conformal geometries. Thus, while
the thermal entropy in $2+1d$ decreases monotonically with $\lambda$, the
entanglement entropy on a circular boundary in $2+1d$ is strictly constant as
shown in figure (3).

Figure 3: For $d=3$ the thermal entropy density of
$s_{\text{therm}}/s^{\text{free}}_{\text{therm}}$ is shown along with the
entanglement entropy for a circular region that lives in $R^{2}\times S^{1}$.
Results are shown in the compactified domain of
$\frac{1}{1+\sqrt{\lambda}}\in[0,1]$ starting at from the left at
$\lambda=\infty$.

###  3.2 Generalizations to arbitrary d

In order to generalize our theory to $S^{d}$ first we can generalize the
eigenvalues on $S^{d}$, let them be $\lambda^{(d)}_{n}$, and their associated
multiplicities $m^{(d)}_{n}$. The eigenvalues take the form

$\lambda^{(d)}_{n}=\Big{(}n+\frac{d-1}{2}\Big{)}^{2}-\frac{1}{4}+(rm)^{2},$
(32)

and the multiplicities generalize to,

$m^{(d)}_{n}=\frac{2}{(d-1)!}\prod_{j=0}^{\frac{d-3}{2}}\Big{[}\Big{(}n+\frac{d-1}{2}\Big{)}^{2}-j^{2}\Big{]}.$
(33)

Together equations (24), (32), and (33) give a general form for the saddle
condition in (29) as

$\frac{\partial}{\partial
m^{2}}s^{(d)}_{\text{EE}}=\frac{-1}{(d-1)!}\sum_{n=0}^{\infty}\prod_{j=0}^{\frac{d-3}{2}}\frac{\big{[}(n+\frac{d-1}{2})^{2}-j^{2}\big{]}}{(n+\frac{d-1}{2})^{2}-\frac{1}{4}+(rm)^{2}}=0.$
(34)

In anticipation of the gap equation being solved by $m=0$, the auxiliary field
term can be ignored. This is effectively the same as working in the infinite
coupling regime. In the case of $d=5$, evaluating (34) at $m=0$ gives

$\begin{split}&\frac{\partial}{\partial
m^{2}}S^{(5)}_{\text{EE}}\bigg{|}_{m=0}\\\
&=\frac{-1}{(4)!}\sum_{n=0}^{\infty}\frac{(n+2)^{2}\big{[}(n+2)^{2}-1\big{]}}{(n+2)^{2}-\frac{1}{4}+(rm)^{2}}\bigg{|}_{m=0}\\\
&=\frac{-1}{(4)!}\big{[}\zeta(-2)-1-\zeta(0)\frac{3}{4}+\frac{3}{4}-\frac{1}{8}]\\\
&=0.\end{split}$ (35)

The integral of (35) evaluated at $m=0$ further produces $s^{(5)}_{\text{EE}}$
as

$s^{(5)}_{\text{EE}}=\frac{N}{2^{8}}\bigg{(}2\log(2)+\frac{2\zeta(3)}{\pi^{2}}-\frac{15\zeta(5)}{\pi^{4}}\bigg{)},$
(36)

matching N copies of the thermal entropy of a free theory in $S^{5}$ given by
Ref. klebanov . This recipe can be repeated for $d=7$ giving

$s^{(7)}_{\text{EE}}=\frac{-N}{2^{12}}\bigg{(}4\log(2)+\frac{82\zeta(3)}{15\pi^{2}}-\frac{10\zeta(5)}{\pi^{4}}-\frac{63\zeta(7)}{\pi^{6}}\bigg{)}$
(37)

Note when evaluating $s^{(d)}_{\text{EE}}$ for $d=3$, $d=5$, and $d=7$ the
process of zeta-function regularization produced a rational number which
canceled when we set $m=0$. For example, in equation (29) we have

$\begin{split}&\frac{1}{2}\bigg{[}\zeta(0)+\frac{1}{4}(2-\sqrt{1-(2rm)^{2}}\pi\cot\big{(}\frac{1}{2}\sqrt{1-(2rm)^{2}}\pi\big{)}\bigg{]}\Bigg{|}_{m=0}\\\
&=\frac{1}{2}\bigg{[}\zeta(0)+\frac{1}{4}(2)\bigg{]}=0.\end{split}$ (38)

This was likewise the case for $d=5$ and $d=7$. It is reasonable to ask if the
effective mass is vanishing in the saddle condition for all odd dimensions
$d$, which is equivalent to asking if all rational terms produced in zeta
function regularization will always perfectly cancel when setting $m=0$. To
explore this, modifying equation (34), the derivative of the
$s^{d}_{\text{EE}}$ for arbitrary $d$ can be expressed as

$\frac{\partial}{\partial
m^{2}}\big{(}S^{(d)}_{\text{EE}}\big{)}=-\frac{1}{2}\sum_{n=0}^{\infty}\frac{\Gamma(n+d-1)}{\Gamma(d)\Gamma(1+n)}\frac{2n+d-1}{n(n+d-1)+d/2(d/2-1)+(rm)^{2}},$
(39)

which evaluates analytically to

$\begin{split}&\frac{\partial}{\partial
m^{2}}\big{(}S^{(d)}_{\text{EE}})=-\frac{8\cos(d\pi/2)\cos(1/2\sqrt{1-4(rm)^{2}}\pi)}{\pi(d(d-2)+4(rm)^{2}))}\Gamma(1-d)\\\
&\times\Gamma[1/2(1+d-\sqrt{1-4(rm)^{2}})]\Gamma[1/2(1+d+\sqrt{1-4(rm)^{2}})].\end{split}$
(40)

Prior to regularization, it can be shown that for all odd positive integer
values of $d$ the saddle condition of (33) is satisfied in the limit
$m\rightarrow 0$, and likewise in the limits where $d\rightarrow\infty$ and
$m\rightarrow 0$;

$\begin{split}&\lim_{m\rightarrow 0}\frac{\partial}{\partial
m^{2}}\big{(}S^{(d)}_{\text{EE}}\big{)}=0\quad\forall
d\in\\{2n-1|n\in\mathcal{N}\\}\\\ &\And\\\ &\lim_{m\rightarrow
0}\frac{\partial}{\partial m^{2}}\big{(}S^{\infty}_{\text{EE}})=0.\end{split}$
(41)

Further, note that the integrating equation (30) and keeping the real parts
gives a function that is monotonically decreasing for all $m>0$ and that all
$s^{d}_{\text{EE}}$ will likewise take the form of (30) before applying saddle
conditions. Thus, there is only one global minimum for any
$s^{d}_{\text{EE}}$, and together with (41) this implies $m=0$ is the only
solution to the saddle point condition for odd $d$ on $S^{d}$. This also
implies that all rational terms generated via zeta-function regularization
will perfectly cancel for all odd $d$, such as in the cases of $d=3$, $d=5$,
and $d=7$. Thus, via iteratively applying zeta-function regularization the
entanglement entropy can be expressed as the real part of the integral form:

$\begin{split}&s^{(d)}_{\text{EE}}\\\ &=\lim_{m\rightarrow
0}\frac{N}{(d-1)!}(-1)^{(\frac{d+1}{2})}\Bigg{[}\int\prod_{j=0}^{\frac{d-3}{2}}\big{[}(rm)^{2}-\frac{1}{4}+j^{2}\big{]}\sum_{n=1}^{\infty}\frac{1}{n^{2}-\frac{1}{4}+(rm)^{2}}dm^{2}-C\Bigg{]},\end{split}$
(42)

where $C$ is simply the rational term that is generated from the integral
which will cancel with all rational terms generated via zeta-function
regularization. Altogether, taking the limit and the integral in equation
(42), and then removing the rational and imaginary pieces will generate
$s^{d}_{\text{EE}}$ values that match the list of F-coefficients for free
theories on $S^{d}$ in Ref.klebanov  up a factor of $-N$. Further, in this
form, it can be shown that the entanglement entropy goes to zero in the limit
$d\rightarrow\infty$ with letting $C=0$.

##  4 Conclusions

In section 2) I derived previously known results for marginal coupling in the
large N limit on $R^{2}\times S^{1}$, via a generalized method which was then
applied to higher dimensions. This resulted in a large class of CFTs, with
non-polynomial potentials, which are solvable and monotonically decreasing for
all couplings $\lambda$ in $d=3+4n$. In section 3 I further extended large $N$
methods to $S^{3}$ and showed the vacuum entanglement entropy
$s^{(3)}_{\text{EE}}$ is both solvable and constant for all couplings. This
was then shown to hold for all $s^{(d)}_{\text{EE}}$ given odd $d$. The
results of $s^{(3)}_{\text{EE}}$ could be experimentally verified given a
Bose-gas with a vanishing effective mass in 2+1 dimensions. Further, O(N)
models are conjectured to have a high spin gravity dual by ref. klebanov2002 .
Thus this work introduces the possibility of arbitrary even dimensional high
spin gravity models which are solvable for all couplings.

Generally, this work prompts further questions on how coupling behaves on
other novel manifolds in the Large N limit. Likewise, what are the physical
motivations of this coupling invariant behavior, and is it unique to the
hypersphere or is it found to exist on other manifolds more generally such as
on a hyperbolic geometry? In Refs. allais  and mezei  perturbative and
computational methods were used to generate entanglement entropy via small
deformations of a sphere, however, no computational or analytic method has
been provided to calculate the entanglement entropy of generally deformed
spheres. Such as one can imagine deforming a sphere into flat space by letting
the major axis of the deformed sphere go to infinity. Such methods could
potentially conformally map the flat to the spherical theories presented in
this paper, and further, bolster a general theory between field theory
geometry and entanglement entropy. Further, this investigation of scalar
theories could be extended to consider supersymmetric behavior in a large N
Wess-Zumino model in $S^{d}$ via methods similar to those used in dewolfe , or
consider numerical calculations of a $\phi^{4}$ theory on $S^{3}$ to better
understand the conformal coupling invariance phenomena. In the extension to
Fermionic theories, if it likewise holds that the effective mass is vanishing
for all odd dimensional theories on $S^{d}$, graphene could be used to
experimentally test these theories. Likewise, one could extend this
generalization of large N models to theories with even dimensions where the
$\beta$-function is non-vanishing, and perhaps, likewise extend this large
class of fully solvable models with non-polynomial potentials to even
dimensional theories.

###### Acknowledgements.

Many thanks to Paul Romatschke for the wonderful experience gained from
working on this project together, and to Alex Flournoy for encouraging and
supporting my independent study of QFT. This work was partially supported by
the DOE, award number DE-SC0017905.

#### Note added.

## References

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  * (13) Blaizot, Jean-Paul and Iancu, Edmond and Kraemer, Ulrike and Rebhan, Anton _Hard thermal loops and the entropy of supersymmetric Yang-Mills theories_ , _Journal of High Energy Physics_ , IOP 2007 (2007) pg. 035
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# STA_RX Paper

Thenmozhi Elango (Department of Electrical Engineering,
Anna University, Chennai, Tamilnadu,India)
Samiyuktha Kalalii (Department of Electrical Engineering,
Indian Institute of Technology Hyderabad)
P. Rajalakshmi (Department of Electrical Engineering,
Indian Institute of Technology Hyderabad)

(Aug 5, 2014)

# A Novel Approach to Ultrasound Beamforming using Synthetic Transmit Aperture
with Low Complexity and High SNR for Medical Imaging

Thenmozhi Elango (Department of Electrical Engineering,
Anna University, Chennai, Tamilnadu,India)
Samiyuktha Kalalii (Department of Electrical Engineering,
Indian Institute of Technology Hyderabad)
P. Rajalakshmi (Department of Electrical Engineering,
Indian Institute of Technology Hyderabad)

(Aug 5, 2014)

##  1 Abstract

This paper presents an architecture for Ultrasound Beamforming using Synthetic
Transmit Aperture with Low Complexity and High SNR for medical imaging.
Synthetic Transmit Aperture is a novel approach in ultrasound imaging system
by which frame rate and image quality is increased significantly on less data-
transfer and computational requirements. The real-time beamforming performance
of Phased Array(PA) method is limited by high computation and cost. Thus STA
method(data-transfer rate-8MB/frame) advances over the Phased Array
Method(data-transfer rate-95MB/frame) with comparitively much higher frame
rate and Signal to Noise Ratio(SNR. In this paper, we have implemented receive
beamforming using Synthetic Transmit Aperture (STA) method for eight channels
and have obtained the sample data for reconstruction of image. The
experimental results are compared with the conventional phased array and
linear array beamforming, where it can be observed that the reduction in
memory requirement and high SNR.

##  2 INTRODUCTION

Ultrasound imaging is a technique that has become much more prevalent than
other medical imaging techniques since this technique is more accessible, less
expensive, safe, simpler to use and produces images in real-time. However,
images produced by an ultrasound imaging system, must be of sufficient quality
to provide accurate clinical interpretation. The most commonly used image
quality measures are spatial solution, image contrast and frame rate. In
conventional ultrasound imaging system, when one transducer is used, the
quality of images directly depends on the transducer acoustic field.

Also in conventional ultrasound imaging the image is acquired sequentially one
image line at a time that puts a strict limit on the frame rate that is
important in real-time imaging system. Thus the moving structures are not
easily imaged and diagnosis may be impaired. This limitation will seriously
halt the image reconstruction when going for portable devices. An alternative
way to obtain an appropriate spatial resolution, without decreasing of the
frame rate, is to use the Synthetic Transmit Aperture Technique [10].

The basic idea of the STA method is to combine information from missions close
to each other. It provides the full dynamic focusing, both in transmit and
receive modes, yielding the highest imaging quality[8]. In the STA method at
each time one array element transmits a pulse and all elements receive the
echo signals. The data acquired simultaneously from all directions over a
number of emissions, and the full image can be reconstructed from these data.
The advantage of this approach is that a full dynamic focusing can be applied
to the transmission and the receiving, giving the highest quality of image
[1].

Figure 1: Architecture of STA ultrasound beamforming

This is a contrast to the conventional beamforming, where only imaging along
one line in receiving is used [6]. In the STA method every scan line is imaged
as many times as the number of elements used. This will create an equal amount
of low resolution images which are summed up to create one high resolution
image.The performance of a system is directly proportional to the size of the
active aperture, which translates into a large number of (primarily receive)
channels. The cost of the system is also directly proportional to the number
of channels. Ideally we would like to create a large imaging aperture without
an increase in the number of processing channels for memory concerns.

Current systems increase the number of active elements in the beam former and
better engineering makes it possible to increase the transducer center
frequency for the same penetration depth, which lowers the frame rate, if the
image quality has to be maintained.When comes for flow estimation the problem
is still increased, since several pulse-echo lines have to be used per
estimate and this correspondingly lowers the frame rate. Also, conventional
beam forming methods suffer from impedance mismatch between the transducer and
receiver circuit which results in low SNR. This can be overcome by exciting
multi element sub aperture method.

A further problem in conventional imaging is the single transmit focus, so
that the imaging is only optimally focused at one depth. The transmit focusing
is thus synthesized by combining the low resolution images, and the focusing
calculation makes the transmit focus dynamic for all points in the image. the
focus is therefore, both dynamic in transmit and receive and the highest
possible resolution for delay-sum beam forming is obtained everywhere in the
image.

For single-element aperture, the transmit aperture is split into N
subapertures. Each subaperture transmits a prefocused beam or non-focused beam
and dynamically receives the echo signals. Therefore, Low Resolution
Images(LRIs) is formed at each transmission, and the High Resolution
Image(HRI) is formed by combining all LRIs when all the elements have
transmitted. For N transmissions, there are N number of LRIs to form a HRI.
Since the HRI of STA is formed by LRIs, the STA image is susceptible to
motions [9].

The experiments conducted for an eight channel phased array transducer shows
the time required to transmit a beam and receive the echoes through the
focused organ is comparatively less. The number of sample data for image
reconstruction is also greater when compared to the conventional beamforming.

##  3 PROPOSED ARCHITECTURE FOR STA AND MSTA

The proposed architecture for Ultrasound Beamforming using Synthetic Transmit
Aperture with Low Complexity and High SNR for medical imaging is shown in Fig.
1. From the architecture we infer that, with the selection of number of
channels at transmit side, receive beamformingwith STA correspondingly
generates the scan line sample data which is further required for image
reconstruction.

We designed a reconfigurable low complex and high frame rate Synthetic
transmit Aperture method for eight channels, which can be extended to any
number of channels. The main idea of our invention is to reduce the memory
requirement for producing the sample data when going for more transducer
elements. Also, for portable ultrasound imaging systems it is of high priority
to form a clear and excellent image quality for clinical interpretation.

The proposed architecture is designed at a frequency of 20 MHz clock, and an
inbuilt delay circuit for the receive beamforming which includes the coarse
and fine delays, selection lines to select the transducer element to activate
the channel. Once the transmit beamforming circuit excites the pulses to form
a ultrasound signal through a single element, the receiver circuit will be
active to receive the echoes from all the channels of the transducer. Thus it
provides a full length data construction for imaging.

Figure 2: STA method for single transducer elements

The proposed architecture can be applicable for all the modes of ultrasound
imaging which includes Amplitude mode (A - mode), Brightness mode (B - mode),
Colour mode (C - mode) and motion mode (M- mode). Single Element STA ¯ : In
the STA method focusing is performed by finding the geometric distance from
the transmitting element to the imaging point and back to the receiving
element. The structure of the synthetic aperture and geometric relation
between the transmit and receive element combination is shown in Fig. 2. When
a short pulse is transmitted by element m and the echo signal is received by
element n, as shown in Fig. 2, a round-trip delay is T(m,n) = T(m) + T(n),
where (m,n) is a transmit and receive combination [7]. Multi element STA : The
major drawback of STA imaging is the low SNR. Since a single element is used
at each transmission the SNR is very low compared to linear array imaging,
which significantly limits its clinical application. To overcome this, a
subaperture consisting of multiple grouped elements can be used to emulate a
spherical wave, and hereby increase the SNR. This concept is called
multielement synthetic transmit aperture imaging. Both groups show that by
properly delaying the individual elements in the subaperture, good
approximation to a spherical wave can be obtained along with a significant
improvement in SNR [3]. Applications which involve large penetration depth,
lateral resolution and high SNR (Signal to Noise Ratio) requires more than one
element to generate the ultrasound signals[2]. This method of invention is
called Multi element Synthetic Transmit Aperture (MSTA). In this method small
number of elements are used to transmit a pulse but all array elements receive
the echo signals. In practice, it is not very expensive to build a large
transmit aperture, but it is very complex to form a large receive aperture.
For a transmit pulse (from all transmit subaperture elements), the RF echoes
for all receive elements are stored in memory [5]. When all RF echo signals
have been acquired, the total RF sum is formed by coherently adding them. The
multi-element STA method is proposed to increase the system frame rate and the
speed of the image acquisition is determined by the number of transmissions.

Figure 3: Multi element STA

The geometrical locations of the transmit elements in a multi-element array
system impacts the radiation pattern of that system, which in its turn impacts
lateral resolution of the image, whereas the number of active transmit
elements directly influences the transmitted energy and the SNR. These
parameters define the ultrasound image quality. Therefore, the optimization of
a multi-element STA imaging system can be formulated as an optimization
problem of the location and the number of the transmit elements. The main
optimization criterion in the multi-element STA method is the minimal width of
the main-lobe combined with the minimum side-lobe level[11]. This optimization
leads to increasing image penetration depth and high lateral resolution [4].

Figure 4: Simulation results for STA Ultrasound for eight channels

Depending upon the application and penetration depth to form a focal point,
our design provides an accessibility for performing STA and MSTA through the
selection elements. This correspondingly reduces the time taken to form frame
rate for imaging, utilizing less memory resources. Near field and Far field
applications such as cardio vascular applications requires high lateral
resolution and penetration depths provides less accurate frame rates for image
reconstruction. This can be overcome with the MSTA method of invention.

Figure 5: Synthesized architecture for STA receive beamforming

##  4 EXPERIMENTAL RESULTS

The proposed dessign for Ultrasound beamforming with STA is developed using
verilog HDL on Xilinx ISE 14.4 tool. Fig. 4 shows the simulation results for
the proposed architecture designed for eight channels at a clock frequency of
20 MHz. Fig. 5 shows the synthesized RTL implementation generated through
Xilinx ISE 14.4 tool. The obtained results show that the design can be easily
configured into hardware for practical applications.

##  5 CONCLUSION

In this paper we proposed the design of ultrasound beamforming with Synthetic
transmit aperture method. The proposed architecture utilizes less memory with
increased frame rate for medical imaging purposes.

## References

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