Image Segmentation Using Bayesian Inference for Convex Variant Mumford–Shah Variational Model

Xu XIAO, Youwei WEN*, Raymond CHAN, Tieyong ZENG

*Corresponding author for this work

Research output: Journal PublicationsJournal Article (refereed)peer-review


The Mumford–Shah model is a classical segmentation model, but its objective function is nonconvex. The smoothing and thresholding (SaT) approach is a convex variant of the Mumford–Shah model, which seeks a smoothed approximation solution to the Mumford–Shah model. The SaT approach separates the segmentation into two stages: first, a convex energy function is minimized to obtain a smoothed image; then, a thresholding technique is applied to segment the smoothed image. The energy function consists of three weighted terms and the weights are called the regularization parameters. Selecting appropriate regularization parameters is crucial to achieving effective segmentation results. Traditionally, the regularization parameters are chosen by trial-and-error, which is a very time-consuming procedure and is not practical in real applications. In this paper, we apply a Bayesian inference approach to infer the regularization parameters and estimate the smoothed image. We analyze the convex variant Mumford–Shah variational model from a statistical perspective and then construct a hierarchical Bayesian model. A mean field variational family is used to approximate the posterior distribution. The variational density of the smoothed image is assumed to have a Gaussian density, and the hyperparameters are assumed to have Gamma variational densities. All the parameters in the Gaussian density and Gamma densities are iteratively updated. Experimental results show that the proposed approach is capable of generating high-quality segmentation results. Although the proposed approach contains an inference step to estimate the regularization parameters, it requires less CPU running time to obtain the smoothed image than previous methods.
Original languageEnglish
Pages (from-to)248-272
Number of pages25
JournalSIAM Journal on Imaging Sciences
Issue number1
Publication statusPublished - 31 Mar 2024
Externally publishedYes

Bibliographical note

Funding: This work is supported in part by NSFC grant 12361089; the National Key R&D Program of China under grants 2021YFE0203700, NSFC/RGC N_CUHK 415/19, ITF MHP/038/20, CRF 8730063, RGC 14300219, 14302920, 14301121, and CUHK Direct Grant for Research, HKRGC grants CityU11301120, CityU11309922, C1013-21GF, CityU grant 9380101; HKRGC-NSFC grant N_CityU214/19.


  • image segmentation
  • Mumford–Shah model
  • Bayesian inference
  • mean field variational approximation
  • regularization parameters


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