Automatic prior shape selection for image edge detection with modified Mumford-Shah model
type: Item
instance of: scholarly article
title: Automatic prior shape selection for image edge detection with modified Mumford-Shah model
zbMATH Open document ID: 1460.94013
author: Zhimei Huo
author: Jing Qin
author: Yilin Li
author: Yuying Shi
publication date: 2020-10-07T00:00:00Z
review text: This paper deals with one of the most relevant topics in image processing: the detection of edges with a given structure. Edge detection is not easy to do in order to accurately determine edges of objects, roads, railways, etc. Even more so when the image to be processed is noisy. Consider a brief mathematical description of the proposed technique. In the Mumford-Shah method, the images are considered as functions defined on an open and bounded set, $\Omega\subset\mathbb{R}^2$, it consists in finding $u_0$ and $\Gamma_0$ such that \[ E(u_0,\Gamma_0)\leq E(u,\Gamma):=\mu\int_{\Omega\backslash\Gamma}|\nabla u|^2.dxdy+(ν/2)\int_{\Omega}(u-I)^2.dxdy+|\Gamma|. \] where: $I$ is the given image, \(u\) belongs to a set of piecewise differentiable functions considered to be ``good'' approximations to $I$, $\Gamma$ is the union of the edges of objects to be determined, $|\Gamma|$ is the length of $\Gamma$, $\mu$ and $ν$ are positive tuning parameters. Taking into account subsequent modifications to this model proposed by various researchers, the authors of this paper propose the following variational model in which a dictionary of a priori shapes of objects to be detected is considered. Here the images are considered to be represented by real matrices when $M$ is an $m\times n$ matrix representing an image, \[ M=a_{11}\cdots a_{1n}\vdots\vdots\vdots a_{m1} \cdots a_{mn} \] is vectorized as follows \[ M=>a_{11}\vdots a_{m1}\vdots a_{1n}\vdots a_{mn} \] and Diag$(M)$ is the transformation of the vector $M$ into a diagonal matrix $mn\times mn$. Let's see the proposed model. Consider \(F\) as the function defined on a set of real vectors, $\Theta$, suitable for each situation given \[ F(u,w,s,\psi) =||D_{\psi}D_u||_2^2+||D_{\psi}D_2u||_2^2+(ν/2)||u-||_2^2+\tau||\psi||_1 +||D\psi||_1+\tag{1}\] \[+\beta||Dw||_1+\alpha||s||_1+(\gamma/2)||As+w-\psi||_2^2.\min\{F(u,w,s,\psi)/(u,w,s,\psi)\in\Theta\}. \] Here, $D_{\psi}=\mathrm{Id}-\mathrm{Diag}(\psi)$ is a $mn\times mn$ matrix (\(\math…
Mathematics Subject Classification ID: 94A08
Mathematics Subject Classification ID: 94A13
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 68U10
zbMATH DE Number: 7256681
zbMATH Keywords: edge detection
zbMATH Keywords: prior shape
zbMATH Keywords: ADMM
zbMATH Keywords: fixed-point iterative algorithm
zbMATH Keywords: automatic shape selection
zbMATH Keywords: modified Mumford-Shah model
reviewed by: Oscar Bustos
describes a project that uses: RecPF
MaRDI profile type: Publication
full work available at URL: j.camwa.2019.09.021
OpenAlex ID: W2978463336
cites work: Extracting grain boundaries and macroscopic deformations from images on atomic scale
cites work: Q5723170
cites work: Image Selective Smoothing and Edge Detection by Nonlinear Diffusion. II
cites work: Edge detection by adaptive splitting
cites work: Singularity detection and processing with wavelets
cites work: Image Selective Smoothing and Edge Detection by Nonlinear Diffusion
cites work: Optimal approximations by piecewise smooth functions and associated variational problems
cites work: Approximation of functional depending on jumps by elliptic functional via t-convergence
cites work: Q3992271
cites work: Numerical minimization of a second-order functional for image segmentation
cites work: Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models
cites work: A fast edge detection algorithm using binary labels
cites work: Nonparametric shape priors for active contour-based image segmentation
cites work: Automatic Prior Shape Selection for Image Segmentation
cites work: A New Alternating Minimization Algorithm for Total Variation Image Reconstruction
cites work: Iterative Methods for Total Variation Denoising
cites work: Acceleration methods for image restoration problem with different boundary conditions
cites work: On Two Multigrid Algorithms for Modeling Variational Multiphase Image Segmentation
cites work: Q4144736
cites work: Fast dual minimization of the vectorial total variation norm and applications to color image processing
cites work: Q4310146
cites work: Q4825126
cites work: Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers
cites work: Split Bregman Methods and Frame Based Image Restoration
cites work: Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models
cites work: Geometry of total variation regularized \(L^p\)-model
cites work: Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model
cites work: A dual algorithm for the solution of nonlinear variational problems via finite element approximation
cites work: An algorithm for total variation minimization and applications
cites work: Convergence and sparsity of Lasso and group Lasso in high-dimensional generalized linear models
cites work: An Iterative Regularization Method for Total Variation-Based Image Restoration
DOI: 10.1016/J.CAMWA.2019.09.021
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 7256681