# A Binary Characterization Method for Shape Convexity and Applications

arXiv: https://arxiv.org/abs/2203.11395 · HTML: https://mlsanity.com/paper/2203.11395/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 39 references that are in the map.
- First posted on arXiv 22 Mar 2022.

## Builds on (most cited first, 9 of 9)

- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [Encoder-Decoder with Atrous Separable Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1802.02611/) (2018)
- [Deep Vessel Segmentation By Learning Graphical Connectivity](https://mlsanity.com/paper/1806.02279/) (2018)
- [Deep Convolutional Neural Networks with Spatial Regularization, Volume and Star-shape Priori for Image Segmentation](https://mlsanity.com/paper/2002.03989/) (2020)
- [Convexity Shape Prior for Level Set based Image Segmentation Method](https://mlsanity.com/paper/1805.08676/) (2018)
- [Convexity Shape Constraints for Image Segmentation](https://mlsanity.com/paper/1509.02122/) (2015)
- [Efficient Projection Algorithms onto the Weighted l1 Ball](https://mlsanity.com/paper/2009.02980/) (2020)
- [Parametric Shape Optimization using the Support Function](https://mlsanity.com/paper/1809.00254/) (2018)
- [A level set representation method for N-dimensional convex shape and applications](https://mlsanity.com/paper/2003.09600/) (2020)

## Built on by (most cited first, 3 of 3)

- [Image Vectorization with Depth: convexified shape layers with depth ordering](https://mlsanity.com/paper/2409.06648/) (2024)
- [Topology preserving Image segmentation using the iterative convolution-thresholding method](https://mlsanity.com/paper/2503.17792/) (2025)
- [D-Convexity: A Unified Differentiable Convex Shape Prior via Quasi-Concavity for Data-driven Image Segmentation](https://mlsanity.com/paper/2605.19210/) (2026)

## Abstract (arXiv)

Convexity prior is one of the main cue for human vision and shape completion with important applications in image processing, computer vision. This paper focuses on characterization methods for convex objects and applications in image processing. We present a new method for convex objects representations using binary functions, that is, the convexity of a region is equivalent to a simple quadratic inequality constraint on its indicator function. Models are proposed firstly by incorporating this result for image segmentation with convexity prior and convex hull computation of a given set with and without noises. Then, these models are summarized to a general optimization problem on binary function(s) with the quadratic inequality. Numerical algorithm is proposed based on linearization technique, where the linearized problem is solved by a proximal alternating direction method of multipliers with guaranteed convergent. Numerical experiments demonstrate the efficiency and effectiveness of the proposed methods for image segmentation and convex hull computation in accuracy and computing time.