Variational Models For Image Segmentation And Numerical Approaches

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. Recently,segmentation methods based on variational models have been widely paid attention bymany researchers, due to their variable form, flexible structure and excellentperformance. In general, variational models composed of several energy functionals,each of which is reflected to some of the characteristics of image to be segmented (gray,edge, color and texture, etc.), can also be subject to the shape and characteristics of thetarget of a priori knowledge. Minimizing the energy function to make segmentationcurve (or surface) evolution, and improve the alignment between the current imagesegmentation and image data, thus obtain the desired segmentation result finally.This thesis concentrates on variational models for image segmentation andnumerical approaches. Our studies can be summarized as the following three topics:1. Studying on the problem of images with intensity inhomogeneity andPoisson noise, we proposed a globally convex segmentation model with local fittingIntensity inhomogeneity often occurs in real images, especially in medical images,such as X-ray radiography/tomography and magnetic resonance (MR) images, and maycause considerable difficulties in image segmentation. Poisson noise also appears in awide class of real real-world applications, e.g., positron emission tomography inmedical imaging, fluorescence microscopy and radiography images. In particular, due totechnical limitations or artifacts introduced by the object being imaged, the radiographimages, such as X-ray is often created with intensity inhomogeneity. Li et al. recentlyproposed a region-scalable fitting (RSF) active contour model (originally termed aslocal binary fitting (LBF) model) in a variational level set formulation, which canhandle intensity inhomogeneity efficiently. However, the RSF model is derived from theMumford-Shah model that implicitly assumes the given image to be biased by additiveGaussian noise; thus it is not suitable for images with signal-dependent noise (e.g.,Poisson noise). Besides, it easily gets stuck in local minima for most of the contourinitializations. This makes the RSF model sensitive to the selection of initial contours. In order to handle images with intensity inhomogeneity and Poisson noise, weproposed a globally convex segmentation model with local fitting. An energy functionalis first proposed, which uses a data-fidelity term deduced from Poisson distributioninstead of the usualL2-norm as a measure of fidelity. Due to the new data-fidelitymeasure, this energy functional can fit the image intensity more accurately while it candiminish the influence of Poisson noise on segmentation results. We then reformulatethe energy function as globally convex formulation, which allows for more flexibleinitialization. The final convex energy functional is minimized via the dual formulationinstead of the usually used gradient descent method.2. Integrating clustering with level set method to solve piecewise constantMumford-Shah modelThe Mumford-Shah model for image segmentation is a powerful and robustregion-based method; however, the numerical method for solving the model is difficultto implement when direct implementations are performed. Therefore, in practice, one ofthe major challenges is to develop efficient algorithms to compute high-qualityminimizes of this functional. Based on the level set method, a very successful method isfirst introduced by Chan and Vese to solve the piecewise constant piecewise constantMumford-Shah model. As a result, a number of generalizations have been developed toimprove both its applicability and efficiency. However, the Chan-Vese method and theimproved version of Chan-Vese have an intrinsic limitation; these methods solving theMumford-Shah functional involve alternating optimization of the reconstructionfunction and the contour. The alternating optimization method may make them sensitiveto the contours initialization, and inefficiently segment images when implementednumerically. In this paper, we integrate clustering with level set method to solve thetwo-phase piecewise constant case of the Mumford-Shah model for image segmentation,pursuing the mechanism of the alternating optimization. Numerical results demonstratedsome advantages of the proposed method, such as robustness to the locations of initialcontour and the high computation efficiency.3. Based on Fuzzy C-means clustering algorithm, we proposed a convex activecontour model for image segmentation with clustering-based constraintsActive contour models is a description of contours which evolve under anappropriate energy functional to move toward desired object boundaries, and have beenwell established and widely used in various image applications. However, the activecontour models have slightly some new limitations:○1the active contour models may get stuck in local minimums which makes it sensitive to the contours initialization.○2the active contour models is also sensitive to high noise.○3In some cases, only certainobject(s) is/are the desired object(s) we wish to segment; but the active contour modelsmay fail to extract the desired objects. In this paper, a convex active contour model thatembeds the clustering-based constraints is proposed for image segmentation. Theclustering algorithm is an unsupervised pixel classification method, which is extremelyfast and simple to detect multiple components concurrently. So the clustering algorithmcan be easy to use to constrain the characters of objects without learned the priorinformation of the objects. For the realization of the segmentation method, we use avariational formulation of the GAC energy while an addedL1-norm data fidelity termuses to incorporate clustering-based constraints. To minimize the energy functional ofour model efficiently, we choose the dual formulation, which avoids the instability andthe non-differentiability of traditional gradient descent method.