Duan, Jinming
(2018)
Variational and PDE-based methods for image processing.
PhD thesis, University of Nottingham.
Abstract
In this thesis, we study modern variational and partial differential equation (PDE)-based methods for three image analysis applications, namely, image denoising, image segmentation, and surface reconstruction from point clouds. A common feature these applications have is the use of novel variational formulations.
For image denoising, we focus on higher order variational functionals in which the regulariser incorporates second order derivatives or is a sophisticated combination of first and second order derivatives. We study seven representative first and/or second order functionals, implement them using the efficient split Bregman algorithm, and compare their performances. With the knowledge of the main properties of each of the denoising approaches, we can then select and adapt them for image segmentation.
For image segmentation, we are in particular interested in images of three types: red blood cell (RBC) images, histology images of the microglial cells, and optical coherence tomography (OCT) images of the retina. For RBC images we develop an automated and accurate image analysis framework for an image-based cytometer that uses variational total generalised variation, adaptive thresholding and support vector machine. The framework can 1) detect and numerically count malaria parasite infected RBCs acquired from Giemsa-stained smears; 2) classify all parasitic subpopulations by quantifying the area occupied by the parasites within the infected cells; 3) predict if the RBC image has been infected by malaria parasites. We show the effectiveness of the framework by quantifying and classifying both RBC and infected RBC images.
For histology images of the microglial cells, we introduce an automated image segmentation method that is capable of efficiently extracting microglial cells from the images. The method uses variational Mumford-Shah total variation and split Bregman for image denoising and segmentation and is fast, accurate and robust against noise and inhomogeneity in the image. We evaluate the method on the image data from wild type mice and transgenic mouse models of Alzheimer's disease. The method is scalable to large datasets, allowing microglia analysis in regions of interest and across the whole brain.
For OCT images of the retina, we propose a novel and accurate geodesic distance method to segment healthy and pathological OCT images, in both two and three dimensions. The method uses a weighted geodesic distance by an exponential function, taking into account horizontal and vertical intensity variations. The fast sweeping method is used to derive the geodesic distance from an Eikonal equation, a special case of Hamilton-Jacobi equations that belongs to the family of nonlinear PDEs. Segmentation is then achieved by solving an ordinary differential equation using the resulting geodesic distance. The proposed method is also extensively compared with the parametric active contour model and graph theoretic methods.
Finally, we study surface reconstruction from point clouds. We treat this reconstruction problem as an image segmentation problem and hence develop a novel variational level set method. Th method is capable of reconstructing implicit surfaces from unorganised point clouds while preserving fine details of the surfaces. A distance function, derived from the point cloud using the fast sweeping algorithm, is used as an edge indicator function and to find an initial image enclosed by the point cloud. A novel variational segmentation functional is then proposed that effectively integrates the initial image and edge indicator. Gradient descent optimisation finally minimises the functional and ensures an accurate and smooth reconstruction.
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