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Harmonic Beltrami Signature Network: a Shape Prior Module in Deep Learning Framework

Chenran Lin, Lok Ming Lui

TL;DR

It is demonstrated how HBSN can be directly incorporated into existing deep learning segmentation models, improving their performance through the use of shape priors, and the utility of HBSN as a general-purpose module for embedding geometric shape information into computer vision pipelines is confirmed.

Abstract

This paper presents the Harmonic Beltrami Signature Network (HBSN), a novel deep learning architecture for computing the Harmonic Beltrami Signature (HBS) from binary-like images. HBS is a shape representation that provides a one-to-one correspondence with 2D simply connected shapes, with invariance to translation, scaling, and rotation. By exploiting the function approximation capacity of neural networks, HBSN enables efficient extraction and utilization of shape prior information. The proposed network architecture incorporates a pre-Spatial Transformer Network (pre-STN) for shape normalization, a UNet-based backbone for HBS prediction, and a post-STN for angle regularization. Experiments show that HBSN accurately computes HBS representations, even for complex shapes. Furthermore, we demonstrate how HBSN can be directly incorporated into existing deep learning segmentation models, improving their performance through the use of shape priors. The results confirm the utility of HBSN as a general-purpose module for embedding geometric shape information into computer vision pipelines.

Harmonic Beltrami Signature Network: a Shape Prior Module in Deep Learning Framework

TL;DR

It is demonstrated how HBSN can be directly incorporated into existing deep learning segmentation models, improving their performance through the use of shape priors, and the utility of HBSN as a general-purpose module for embedding geometric shape information into computer vision pipelines is confirmed.

Abstract

This paper presents the Harmonic Beltrami Signature Network (HBSN), a novel deep learning architecture for computing the Harmonic Beltrami Signature (HBS) from binary-like images. HBS is a shape representation that provides a one-to-one correspondence with 2D simply connected shapes, with invariance to translation, scaling, and rotation. By exploiting the function approximation capacity of neural networks, HBSN enables efficient extraction and utilization of shape prior information. The proposed network architecture incorporates a pre-Spatial Transformer Network (pre-STN) for shape normalization, a UNet-based backbone for HBS prediction, and a post-STN for angle regularization. Experiments show that HBSN accurately computes HBS representations, even for complex shapes. Furthermore, we demonstrate how HBSN can be directly incorporated into existing deep learning segmentation models, improving their performance through the use of shape priors. The results confirm the utility of HBSN as a general-purpose module for embedding geometric shape information into computer vision pipelines.
Paper Structure (26 sections, 2 theorems, 18 equations, 16 figures, 3 tables)

This paper contains 26 sections, 2 theorems, 18 equations, 16 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Traditional segmentation algorithms
  4. Shape prior segmentation algorithms
  5. Deep learning segmentation networks
  6. Shape prior deep learning segmentation networks
  7. Quasi-conformal theory
  8. Theoretical basis
  9. Quasi-conformal mapping and Beltrami equation
  10. HBS
  11. Proposed network
  12. Problem setting
  13. Overall architecture
  14. Backbone
  15. Pre- and Post-STN
  16. ...and 11 more sections

Key Result

Theorem 1

There is a one-to-one correspondence between $\mathcal{B}$ and $\mathcal{S}$. In particular, given $[B]\in \mathcal{B}$, its associated shape $\Omega$ can be determined up to a translation, rotation, and scaling.

Figures (16)

  • Figure 1: Quasi-conformal maps infinitesimal circles to ellipses. The Beltrami coefficient measures the distortion or dilation of the ellipse under the QC map.
  • Figure 2: The illustration of HBS. (a) shows the shape $\Omega$ and conformal maps $\Phi_1$ and $\Phi_2$; (b) is the conformal welding $f = \Phi_1^{-1} \circ \Phi_2$; (c) is the Harmonic extension $H$ of conformal welding $f$; (d) is the GHBS $B$ corresponding to $H$.
  • Figure 3: The illustration of HBSN $N_\theta$. It receives 2D rectangle binary images $I_1, \cdots, I_N$ with simply connected shapes and then calculates their HBS $B_1, \cdots, B_N$.
  • Figure 4: Overall architecture of the proposed HBSN.
  • Figure 5: The architecture of the backbone.
  • ...and 11 more figures

Theorems & Definitions (3)

  • Definition 1
  • Theorem 1
  • Theorem 2