In this paper, we propose a unified energy minimization model for segmentation of non-smooth image structures, e.g., textures, based on Mumford-Shah functional and linear patch model. We consider that image patches of a non-smooth image structure can be modeled by a patch subspace, and image patches of different non-smooth image structures belong to different patch subspaces, which leads to a computational framework for segmentation of non-smooth image structures. Motivated by the Mumford-Shah model, we show that this segmentation framework is equivalent to minimizing a piecewise linear patch reconstruction energy. We also prove that the error of segmentation is bounded by the error of the linear patch reconstruction, meaning that improving the linear patch reconstruction for each region leads to reduction of the segmentation error. In addition, we derive an algorithm for the linear patch reconstruction with proven global optimality and linear rate of convergence. The segmentation in our method is achieved by minimizing a single energy functional without requiring predefined features. Hence, compared with the previous methods that require predefined texture features, our method can be more suitable for handling general textures in unsupervised segmentation. As a by-product, our method also produces a dictionary of optimized orthonormal descriptors for each segmented region. We mainly evaluate our method on the Brodatz textures. The experiments validate our theoretical claims and show the clear superior performance of our methods over other related methods for segmentation of the textures.

This paper addresses the problem of rigid and nonrigid shape registration and recovery in the presence of shape deformation, missing parts and/or overlapping of multiple shapes. A novel shape evolution approach based on truncated warping transformation formulated in an Energy-Minimization-Curve-Evolution framework is proposed to solve this problem. We deterministically model the rigid and nonrigid shape deformation/registration as curve evolution by a warping function mapping. We also derive the curve evolution equation of warping to minimize functional energies. Hence, by selecting a prior shape as the initial curve for the curve evolution, the shape evolution for registration is within the shape space generated by the warping transformation of the prior shape. Based on the Fourier shape contour spectrum, local shape contour distortions that result in significant visual impact is considered to be largely contained in the changes of the high frequency components. Thus, the smoothing of the warping function by truncation is performed to recover the true shape. We adopted the Chan-Vese model and a truncated warping function to obtain our algorithm for shape registration and recovery. Experiments validated our model and algorithm quantitatively.

Partial Differential Equation(PDE) based curve evolution can stop far away from the stationary solution of the corresponding Euler-Lagrange(EL) equation, which is considered as a Pseudo-Stationary-Phenomenon(PSP)problem in this paper. The consequence of this is early termination of evolution of the deformable curve towards desired object boundary. In this paper, we attempted to provide a solution to this problem. Our contribution is twofold. First, from the theoretical point of view, the PSP problem in PDE based curve evolution is studied and a new geometric flow called the Equilibrium Flow(EF) is proposed to address the problem. We prove that by implementing the EF and the original curve evolution flow alternatively, the stationary solution to the corresponding EL equation can be attained in general. Second, we apply our idea to a general snakes model. We show that there is the PSP problem with a general snakes model. We then solve the PSP problem by applying the derived general solution to the snake model. Experimental results on both synthetic and real images are reported in this paper to demonstrate the PSP problem and illustrate our proposed method to overcome this problem. 1