Automatic prior shape selection for image edge detection with modified Mumford-Shah model
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 (\(\mathrm{Id}\) is the identity matrix), $D=[D^\prime_1,D^\prime_2]^\prime$ is the discrete gradient operator with the symmetric boundary conditions, $u$ is as in the Mumford-Shah model, $A\in\mathbb{R}^{mn\times l}$ is a set of $l$ prior shapes, $\psi$ belongs to a set of ``robust'' edge functions which are considered to approximate the possible edges for objects to be detected in $I$, \[ As+w-\psi=\sum_{i=1}^{l}A_{\cdot i}s_{i}+w-\psi, A=[A_{\cdot 1},A_{\cdot 2},\dots,A_{\cdot l}], \] we denotes the difference between $\psi$ and the sparse linear combination of prior dictionary forms, $A$, defined by $s$, and $\mu$, $\nu$, $\tau$, $\beta$, $\alpha$, and $\gamma$ are tuning parameters. Detailed literature on the subject is presented in Section 1. A detailed description of an algorithm for solving the minimization problem (1) is given in Section 3 of the paper. Subsequently, the convergence of the algorithm is rigorously analyzed. Finally, the power of the proposed method, which can be extended to object recognition and related problems in real-world images, is shown through six examples with simulated and real images.
scientific article; zbMATH DE number 7256681
scientific article