Studies On Image Segmentation Based On Partial Differential Equations

Image segmentation, extracting the objects of interest from images, is a mostfundamental and important problem in image processing. It has always been a hot but aformidable task in an image project. Recently, segmentation methods based on partialdifferential equation (PDE) have been widely paid attention by many researchers, due totheir variable form, flexible structure and excellent performance. The basic idea is todeform a curve, surface or image according to a PDE with initial and boundaryconditions, and obtain the desired segmentation results as the solution of the equation.The evolution PDE can be designed directly or indirectly according to imagecharacteristic and user demand.This thesis concentrates on PDE-based techniques for image segmentation. Themain work can be summarized as follows:1. Combining with entropy, we present a scheme of improvement on the RSF(Region-Scalable Fitting) model in terms of robustness to initialization and noiseThe RSF model is recently proposed to segment images with intensityinhomogeneity, such as MRI brain image. This model draws upon intensity informationin spatially varying local regions depending on a scale parameter, so it is able to dealwith intensity inhomogeneity accurately and efficiently. However, the RSF model easilygets stuck in local minimums which makes it sensitive to the contours initialization.Besides, the RSF model is also sensitive to high noise.To these issues, we proposed an improvement on the RSF model. First, theGaussian kernel for the RSF energy is replaced with a ‘‘mollifying’’ kernel withcompact support. Second, the RSF energy is redefined as a weighted energy integral,where the weight is local entropy deriving from a grey level distribution of image. Thetotal energy functional is then incorporated into a variational level set formulation withtwo extra internal energy terms. The new RSF model not only handles better intensityinhomogeneity, but also allows for more flexible initialization and more robustness tonoise compared to the original RSF model.2. Studying on the initialization problem of level set function, we proposed anadaptive level set evolution equation starting with a constant functionFor the segmentation technique based on partial differential equations,segmentation can be regarded as a process of seeking the numerical solution of a partial differential equation with initial condition. Because the segmentation results typicallydepend on the selection of initial contours, most of the existing methods need userintervention to define the initial contours professionally. This means that they may befraught with the problems of how and where to define the initial contours. Up to now, itis still a great challenge to find an efficient way to tackle the contour initializationproblem.Combining the TV (Total Variation) regularization, we proposed an adaptive levelset evolution equation starting with a constant function. The formulation is composedby an adaptive driving force and a TV-based regularizing force. The adaptive drivingforce makes the level set function to have the opposite sign along the edges atconvergence and the regularizing force is used to smooth the level set function. Due tothe adaptive driving force, the level set function can be initialized to a constant function,which completely eliminates the need of initial contours. This implies that the newformulation is robust to initialization or even free of manual initialization. In addition,the evolution PDE can be solved numerically via a simple explicit finite differencescheme with a significantly larger time step. The proposed model is fast enough for nearreal-time segmentation applications while still retaining enough accuracy; in general,only a few iterations are needed to obtain segmentation results accurately.3. In the framework of level set method, we developed a nonlinear diffusionequation directly for image segmentationNonlinear diffusion equations have received a lot of attention in the area of imageanalysis and computer vision. However, diffusion-based segmentation methods areproposed directly for edge-preserving smoothing; segmentation is integrated intosmoothing process which generates a piecewise constant approximation to thegeometrical description of image. So, segmentation results rely closely on theperformance of smoothing. Besides, a smoothing algorithm with good performance inpreserving image features (edges) usually needs to design intricate diffusion term,which may introduce complex computation that may lead to the inefficiency of thewhole process.Based on the mechanism of nonlinear diffusion, we develop a nonlinear diffusionequation (with the initia and boundary conditions) directly for image segmentation. Thediffusion term in our equation is response for the smoothness of the level set functionduring the evolution. The source term is used for identifying object and its backgroundwith “source” and “sink”. The level set function can be initialized to any bounded function, e.g., a zero function, which completely eliminates the need of initial contours.This implies that our model is robust to initialization or even free of manualinitialization. The zero contour line of level set function starting with a zero functioncan be smoothly generated, and quickly come to a steady state which separates objectfrom its background. This work constitutes a framework for further investigations onnonlinear diffusion equations directly for segmentation.4. Propose a nonlinear diffusion equation and apply it successfully on thedocument image binarization produced by camerasDocument image binarization, as an important preprocessing step in the documentimage analysis, has been an active research area and has been extensively studied fordecades. Many binarization methods have been presented, but they almost allsubordinate to thresholding techniques, whose differences are the information theyexploit in determining the threshold.In this thesis, following the structure and the idea of the nonlinear diffusionequation, we present an evolution equation-based binarization method for documentimages produced by cameras. The idea behind our method is to evolve from an originalimage, a family of gradually binarized images derived from the solution of the proposedevolution partial differential equation. In our formulation, the evolution is controlled bya global force and a local force, both of which have opposite sign inside and outside theobject of interests in the original image. A simple finite difference scheme with asignificantly larger time step is used to solve the evolution equation numerically; thedesired binarization is obtained after only one or two iterations. Experiments verify thatthe proposed method is very appropriate for document images obtained by camerasunder normal and inadequate illumination. The main contribution of this study is thatthe present method provides an alternative binarization method for camera documentimages, while further extending the application of partial differential equation in imageprocessing.