# Deep Convolutional Neural Networks Meet Variational Shape Compactness Priors for Image Segmentation

arXiv: https://arxiv.org/abs/2406.19400 · HTML: https://mlsanity.com/paper/2406.19400/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 44 references that are in the map.
- First posted on arXiv 23 May 2024.

## Builds on (most cited first, 15 of 15)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Encoder-Decoder with Atrous Separable Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1802.02611/) (2018)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [Rethinking Atrous Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1706.05587/) (2017)
- [SegNet: A Deep Convolutional Encoder-Decoder Architecture for Image Segmentation](https://mlsanity.com/paper/1511.00561/) (2015)
- [UNet++: Redesigning Skip Connections to Exploit Multiscale Features in Image Segmentation](https://mlsanity.com/paper/1912.05074/) (2019)
- [Semantic Image Segmentation with Deep Convolutional Nets and Fully Connected CRFs](https://mlsanity.com/paper/1412.7062/) (2014)
- [ENet: A Deep Neural Network Architecture for Real-Time Semantic Segmentation](https://mlsanity.com/paper/1606.02147/) (2016)
- [Deep Convolutional Neural Networks with Spatial Regularization, Volume and Star-shape Priori for Image Segmentation](https://mlsanity.com/paper/2002.03989/) (2020)
- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
- [A Regularized Convolutional Neural Network for Semantic Image Segmentation](https://mlsanity.com/paper/1907.05287/) (2019)
- [An efficient iterative thresholding method for image segmentation](https://mlsanity.com/paper/1608.01431/) (2016)
- [Convex Shape Prior for Deep Neural Convolution Network based Eye Fundus Images Segmentation](https://mlsanity.com/paper/2005.07476/) (2020)
- [Double-well Net for Image Segmentation](https://mlsanity.com/paper/2401.00456/) (2023)
- [Unbiased Shape Compactness for Segmentation](https://mlsanity.com/paper/1704.08908/) (2017)

## Built on by (most cited first, 4 of 4)

- [A Mathematical Explanation of UNet](https://mlsanity.com/paper/2410.04434/) (2024)
- [A Mathematical Explanation of Transformers](https://mlsanity.com/paper/2510.03989/) (2025)
- [Architectural Design and Performance Analysis of FPGA based AI Accelerators: A Comprehensive Review](https://mlsanity.com/paper/2603.08740/) (2026)
- [A Unified Variational Framework for Deep Weakly Supervised Image Segmentation](https://mlsanity.com/paper/2607.19669/) (2026)

## Abstract (arXiv)

Shape compactness is a key geometrical property to describe interesting regions in many image segmentation tasks. In this paper, we propose two novel algorithms to solve the introduced image segmentation problem that incorporates a shape-compactness prior. Existing algorithms for such a problem often suffer from computational inefficiency, difficulty in reaching a local minimum, and the need to fine-tune the hyperparameters. To address these issues, we propose a novel optimization model along with its equivalent primal-dual model and introduce a new optimization algorithm based on primal-dual threshold dynamics (PD-TD). Additionally, we relax the solution constraint and propose another novel primal-dual soft threshold-dynamics algorithm (PD-STD) to achieve superior performance. Based on the variational explanation of the sigmoid layer, the proposed PD-STD algorithm can be integrated into Deep Neural Networks (DNNs) to enforce compact regions as image segmentation results. Compared to existing deep learning methods, extensive experiments demonstrated that the proposed algorithms outperformed state-of-the-art algorithms in numerical efficiency and effectiveness, especially while applying to the popular networks of DeepLabV3 and IrisParseNet with higher IoU, dice, and compactness metrics on noisy Iris datasets. In particular, the proposed algorithms significantly improve IoU by 20% training on a highly noisy image dataset.