Computational Mathematics and Computer Modeling with Applications (CMCMA)

On two convex variational models and their iterative solutions for selective segmentation of images with intensity inhomogeneity

https://doi.org/10.52547/CMCMA.1.2.86
Volume 1, Issue 2 - Serial Number 2
December 2022
Pages 86-103
Computational Mathematics and Computer Modeling with Applications (CMCMA)

Document Type : Regular paper

Authors

Liam Burrows 1
Ke Chen 2
Francesco Torella 3

1 Centre for Mathematical Imaging Techniques and Department of Mathematical Sciences, University of Liverpool, Liverpool L19 7ZL, United Kingdom.

2 Department of Mathematical Sciences, University of Liverpool, Liverpool, UK

3 Liverpool Vascular & Endovascular Service, Royal Liverpool and Broadgreen University Hospitals NHS Trust, Liverpool, L7 8XP, United Kingdom

Abstract
Treating images as functions and using variational calculus,
mathematical imaging offers to design novel and continuous methods, outperforming traditional methods based on matrices, for modelling real life tasks in image processing.
Image segmentation is one of such fundamental tasks  as  many application areas demand a reliable segmentation method. Developing reliable selective segmentation algorithms is
particularly important in relation to training data preparation in modern machine learning as accurately isolating a specific object in an image with minimal user input is a valuable tool. When an image's intensity is consisted of mainly piecewise constants, convex models are available.
Different from previous works, this paper
proposes two convex models that are capable of segmenting local features defined by geometric constraints for images having intensity inhomogeneity.
Our new, local, selective and convex variants are extended from the non-convex Mumford-Shah model intended for global segmentation.
They have fundamentally improved on previous selective models that assume intensity  of piecewise constants. Comparisons with related models are conducted to illustrate the advantages of  our new models.

Keywords

Variational calculus
Inverse problems
Image segmentation
Mumford-Shah
Intensity inhomogeneity
Geometric constraints
Iterative methods
  • Receive Date 13 March 2023
  • Revise Date 05 May 2023
  • Accept Date 02 June 2023

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