# Deep Convolutional Neural Networks with Spatial Regularization, Volume and Star-shape Priori for Image Segmentation

arXiv: https://arxiv.org/abs/2002.03989 · HTML: https://mlsanity.com/paper/2002.03989/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2020.

## Built on by (most cited first, 30 of 31)

- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
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- [On Image Segmentation With Noisy Labels: Characterization and Volume Properties of the Optimal Solutions to Accuracy and Dice](https://mlsanity.com/paper/2206.06484/) (2022)
- [Deep Convolutional Neural Networks Meet Variational Shape Compactness Priors for Image Segmentation](https://mlsanity.com/paper/2406.19400/) (2024)
- [A Mathematical Explanation of UNet](https://mlsanity.com/paper/2410.04434/) (2024)
- [A Binary Characterization Method for Shape Convexity and Applications](https://mlsanity.com/paper/2203.11395/) (2022)
- [Connections between Operator-splitting Methods and Deep Neural Networks with Applications in Image Segmentation](https://mlsanity.com/paper/2307.09052/) (2023)
- [Detection-Guided Deep Learning-Based Model with Spatial Regularization for Lung Nodule Segmentation](https://mlsanity.com/paper/2410.20154/) (2024)
- [Contour Flow Constraint: Preserving Global Shape Similarity for Deep Learning based Image Segmentation](https://mlsanity.com/paper/2504.09384/) (2025)
- [A Mathematical Explanation of Transformers](https://mlsanity.com/paper/2510.03989/) (2025)
- [Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints](https://mlsanity.com/paper/2601.11409/) (2026)
- [Mappings for Marginal Probabilities with Applications to Models in Statistical Physics](https://mlsanity.com/paper/2208.05333/) (2022)
- [Noisy Image Segmentation With Soft-Dice](https://mlsanity.com/paper/2304.00801/) (2023)
- [Interpretable Small Training Set Image Segmentation Network Originated from Multi-Grid Variational Model](https://mlsanity.com/paper/2306.14097/) (2023)
- [Slender Object Scene Segmentation in Remote Sensing Image Based on Learnable Morphological Skeleton with Segment Anything Model](https://mlsanity.com/paper/2411.08592/) (2024)
- [A Registration-Based Star-Shape Segmentation Model and Fast Algorithms](https://mlsanity.com/paper/2508.07721/) (2025)
- [Mathematical Modeling and Convergence Analysis of Deep Neural Networks with Dense Layer Connectivities in Deep Learning](https://mlsanity.com/paper/2510.02049/) (2025)
- [JOINEDTrans: Prior Guided Multi-task Transformer for Joint Optic Disc/Cup Segmentation and Fovea Detection](https://mlsanity.com/paper/2305.11504/) (2023)
- [Inverse Evolution Layers: Physics-informed Regularizers for Deep Neural Networks](https://mlsanity.com/paper/2307.07344/) (2023)
- [Graph Similarity Regularized Softmax for Semi-Supervised Node Classification](https://mlsanity.com/paper/2409.13544/) (2024)
- [LMS-Net: A Learned Mumford-Shah Network For Few-Shot Medical Image Segmentation](https://mlsanity.com/paper/2502.05473/) (2025)
- [Topology preserving Image segmentation using the iterative convolution-thresholding method](https://mlsanity.com/paper/2503.17792/) (2025)
- [Contour Field based Elliptical Shape Prior for the Segment Anything Model](https://mlsanity.com/paper/2504.12556/) (2025)
- [Dual-Scale Volume Priors with Wasserstein-Based Consistency for Semi-Supervised Medical Image Segmentation](https://mlsanity.com/paper/2509.04273/) (2025)
- [Continuous-tone Simple Points: An ℓ0-Norm of Cyclic Gradient for Topology-Preserving Data-Driven Image Segmentation](https://mlsanity.com/paper/2604.28159/) (2026)
- [D-Convexity: A Unified Differentiable Convex Shape Prior via Quasi-Concavity for Data-driven Image Segmentation](https://mlsanity.com/paper/2605.19210/) (2026)
- [Geodesics with Unified Tangent-constrained Priors and Curvature Regularization](https://mlsanity.com/paper/2606.00139/) (2026)
- [A Unified Variational Framework for Deep Weakly Supervised Image Segmentation](https://mlsanity.com/paper/2607.19669/) (2026)
- [From level set evolution to threshold optimization: A grayscale level set framework for image segmentation](https://mlsanity.com/paper/2607.22255/) (2026)

## Abstract (arXiv)

We use Deep Convolutional Neural Networks (DCNNs) for image segmentation problems. DCNNs can well extract the features from natural images. However, the classification functions in the existing network architecture of CNNs are simple and lack capabilities to handle important spatial information in a way that have been done for many well-known traditional variational models. Prior such as spatial regularity, volume prior and object shapes cannot be well handled by existing DCNNs. We propose a novel Soft Threshold Dynamics (STD) framework which can easily integrate many spatial priors of the classical variational models into the DCNNs for image segmentation. The novelty of our method is to interpret the softmax activation function as a dual variable in a variational problem, and thus many spatial priors can be imposed in the dual space. From this viewpoint, we can build a STD based framework which can enable the outputs of DCNNs to have many special priors such as spatial regularity, volume constraints and star-shape priori. The proposed method is a general mathematical framework and it can be applied to any semantic segmentation DCNNs. To show the efficiency and accuracy of our method, we applied it to the popular DeepLabV3+ image segmentation network, and the experiments results show that our method can work efficiently on data-driven image segmentation DCNNs.