A Novel Euler’s Elastica-Based Segmentation Approach for Noisy Images Using the Progressive Hedging Algorithm

Abstract

Eulers elastica based unsupervised segmentation models have strong capability of completing the missing boundaries for existing objects in a clean image, but they are not working well for noisy images. This paper aims to establish a Eulers elastica based approach that properly deals with the random noises to improve the segmentation performance for noisy images. We solve the corresponding optimization problem via using the progressive hedging algorithm (PHA) with a step length suggested by the alternating direction method of multipliers (ADMM). Technically, all the simplified convex versions of the sub problems derived from the major framework of PHA can be obtained by using the curvature weighted approach and the convex relaxation method. Then an alternating optimization strategy is applied with the merits of using some powerful accelerating techniques including the fast Fourier transform (FFT) and generalized soft threshold formulas. Extensive experiments have been conducted on both synthetic and real images, which validated some significant gains of the proposed segmentation models and demonstrated the advantages of the developed algorithm.

Nonlinear diffusion based image segmentation using two fast algorithms

“…As described in [25,7], the projection φ (•) in equation ( 11) is a simple truncation of φ k+1 to the interval [0, 1]. For k = 0, 1, ... the minimizers of variables Algorithm 1: ADMM for the proposed model 1.…”

“…where F − 1(•) denotes the discrete inverse Fourier transform, (•) represents the real part of a complex number. After obtaining the value of φ k+1 obtained by (27), we use a simple projection [25,7] to make sure φ k+1 ∈ [0, 1] as in Equation ( 11), that is:…”

“…The third family based on variational label function method is an extended version of the second one. The typical representative is the global convex segmentation model, which was transformed from the Chan-Vese model with constraints and the relaxation-based method by Bresson et al [25,7].…”

“…The proposed model adopts diffusion terms as the regularization term and therefore, can obtain a higher quality of segmentation results. Instead of solving high order nonlinear PDEs, alternating direction method of multipliers (ADMM) [28,16,27] (i. e. augmented Lagrangian method (ALM) [2,6,35]) is applied to transform the energy minimization problem of proposed model into three subproblems, which are then solved by fast Fourier transform (FFT) [14], projection formula [27], analytical soft thresholding equation [26,35] and threshold method [25,35]. Moreover, we creatively propose a new fast algorithm (NVPM) based on normal vector projection and alternating optimization method to solve our model.…”