Image Segmentation Models Based On Partial Differential Equations

Image segmentation is one of fundamental and important tasks in image analysisand computer vision. Given an image, the segmentation goal is to separate the imagedomain into dissimilar regions, each of which has a consistent trait (intensity, color ortexture, etc) throughout that is different from other regions in the image. Once adecision is made on the desired trait, various methods are available to reach thesegmentation goal. This paper will focus on variational level set methods and partialdifferential equation (PDE) methods for image segmentation.The basic idea behind these methods is explained as follows. The segmentationproblem is formulated in terms of minimizing (or at least finding critical points of) anenergy functional that takes a level set function. The level set functions evolveaccording to an evolution partial differential equation (PDE), which is derived from theminimization of the energy functional by calculating the L~2(ordinary) gradient ofenergy functional and using continuous gradient descent method.The research of the thesis including four parts is described as follows:1) A signed level set method to solve the Mumford-Shah modelThe Mumford-Shah model for image segmentation is a powerful and robustregion-based technique; however, the numerical method for solving the Mumford-Shahmodel is difficult to implement. Chan and Vese solved a particular case of theMumford-Shah model using the curve evolution and level set method for imagesegmentation, where the binary case of two regions was considered. As a result, anumber of generalizations have been developed to improve both its applicability andefficiency. However, the Chan-Vese method based on the traditional level set methodhas slightly some intrinsic limitations:1) The Dirac function has to be involved in theassociated gradient descent equation when minimizing with respect to level set function;2) The reinitializing procedure is quite complicated and expensive, and is fraught withits own problems, such as when and how to reinitialize;3) The numerical approximationof the evolution equation has to utilize a complex semi-implicit scheme. In this paper,we present a signed level set method to solve the two-phase piecewise constant case ofthe Mumford-Shah model for image segmentation, pursuing the mechanism of thetraditional level set method. The proposed method avoids some intrinsic limitations ofsolving methods in the traditional level set framework, and allows for more robustness to the locations and sizes of initial contour and more computational efficiency.Numerical results demonstrated that the proposed method is fast enough for nearreal-time bimodal segmentation applications while still retaining enough accuracy.2) Variational level set methods for image segmentation based on both L~2andSobolev gradientsVariational level set methods for image segmentation involve minimizing energyfunctional over a space of level set functions using continuous gradient descent method.The functional includes the internal energy (curve length, usually) for regularization andthe external energy that aligns the curves with object boundaries. Current practice is ingeneral to minimize the energy functional by calculating the L~2gradient of the totalenergy. However, Sobolev gradient is particularly effective for minimizing the curvelength functional by gradient descent method in that it produces the solution in a singleiteration. In this paper, we thus propose to use the Sobolev gradient for the internalenergy, while still using L~2gradient for the external energy. The test results show thatthe “L~2plus Sobolev” gradient scheme has much more computational efficiency thanthe methods only based on L~2gradient.3) Implicit active contour model with local and global intensity fitting energyIntensity inhomogeneities often occur in real-world images and may causeconsiderable difficulties in image segmentation. To handle intensity inhomogeneityefficiently, some localised region-based models have been proposed recently. Forexample, Li et al. recently proposed a region-scalable fitting (RSF) active contourmodel. Very recently, Zhang et al. proposed a novel active contour model driven bylocal image fitting energy, which also can handle intensity inhomogeneity efficiently.However, these models easily get stuck in local minimums for most of contourinitializations. This makes it need user intervention to define the initial contoursprofessionally. In this study, we propose a new active contour model, which integrates alocal intensity fitting (LIF) energy with an auxiliary global intensity fitting (GIF) energy.The LIF energy is responsible for attracting the contour toward object boundaries and isdominant near object boundaries, while the GIF energy incorporates global imageinformation to improve the robustness to initialization of the contours. The proposedmodel can efficiently handle intensity inhomogeneity, while allowing for more flexibleinitialisation and maintaining the sub-pixel accuracy.4) Implicit active contours based on mean curvature motion The implicit active contours have proved to be an efficient framework for imagesegmentation. This implicit model is derived from motion by mean curvature and usesthe image gradient to stop the evolution process. We propose a new formulation ofimplicit active contours based on mean curvature motion. The proposed model has fourmain advantages: First, it doesn’t use image gradient to stop the evolution process.Second, it allows robustness to initialization or even is free of manual initializationsince the level set function can be initialized to a binary function that contains bothpositive and negative values. Third, the zero-level line of level set function starting withsuch binary function finally comes to a unique steady state, thus it allows setting atermination criterion on the algorithm by determining the binary length of zero-levelline at each of iterations. Fourth, the evolution PDE is easily resolved numerically bythe use of the semi-implicit additive operator splitting (AOS) scheme introduced byWeichert et al. to nonlinear diffusion filtering, which remains numerically stable for alarge time step and so less iteration numbers are needed to converge to the steady statesolution. The proposed algorithm has been successfully applied to both synthetic andreal images with homogeneous intensity regions.