Anisotropic Mesh Adaptation for Image Representation

Triangular meshes have gained much interest in image representation and have been widely used in image processing. This paper introduces a framework of anisotropic mesh adaptation (AMA) methods to image representation and proposes a GPRAMA method that is based on AMA and greedy-point removal (GPR) scheme. Different than many other methods that triangulate sample points to form the mesh, the AMA methods start directly with a triangular mesh and then adapt the mesh based on a user-defined metric tensor to represent the image. The AMA methods have clear mathematical framework and provides flexibility for both image representation and image reconstruction. A mesh patching technique is developed for the implementation of the GPRAMA method, which leads to an improved version of the popular GPRFS-ED method. The GPRAMA method can achieve better quality than the GPRFS-ED method but with lower computational cost.Comment: 25 pages, 15 figure

Anisotropic Mesh Adaptation for the Finite Element Solution of Anisotropic Diffusion Problems

Anisotropic diffusion problems arise in many fields of science and engineering and are modeled by partial differential equations (PDEs) or represented in variational formulations. Standard numerical schemes can produce spurious oscillations when they are used to solve those problems. A common approach is to design a proper numerical scheme or a proper mesh such that the numerical solution satisfies discrete maximum principle (DMP). For problems in variational formulations, numerous research has been done on isotropic mesh adaptation but little work has been done for anisotropic mesh adaptation. In this dissertation, anisotropic mesh adaptation for the finite element solution of anisotropic diffusion problems is investigated. A brief introduction for the related topics is provided. The anisotropic mesh adaptation based on DMP satisfaction is then discussed. An anisotropic non-obtuse angle condition is developed which guarantees that the linear finite element approximation of the steady state problem satisfies DMP. A metric tensor is derived for use in mesh generation based on the anisotropic non-obtuse angle condition. Then DMP satisfaction and error based mesh adaptation are combined together for the first time. For problems in variational formulations, two metric tensors for anisotropic mesh adaptation and one for isotropic mesh adaptation are developed. For anisotropic mesh adaptation, one metric tensor (based on Hessian recovery) is semi-a posterior and the other (based on hierarchical basis error estimator) is completely a posterior. The metric tensor for isotropic mesh adaptation is completely a posterior. All the metric tensors incorporate structural information of the underlying problem into their design and generate meshes that adapt to changes in the structure. The application of anisotropic diffusion filter in image processing is briefly discussed. Numerical examples demonstrate that anisotropic mesh adaptation can significantly improve computational efficiency while still providing good quality result. More research is needed to investigate DMP satisfaction for parabolic problems

Effect of Shear on Gelation of Polyacrylamide-Chromium Acetate Gelant in a Circular Tube

This research studies the gelation and flow properties of a partially hydrolyzed polyacrylamide (HPAM)-chromium acetate gel system under shear conditions. Gelation was observed in steady shear on a rheometer and during injection through a 1,031-ft long stainless steel tubing. Fluid experienced continuously non-uniform shear in the tubing (zero shear at the center and maximum shear at the tube wall) and the shear environment may not be reproduced by rheometers. Both in-line-mixed and preformed gels were injected through the tubing, and gel samples were collected to determine syneresis. Interesting results were observed due to the non-uniform shear rate distribution in the tube. During injection of in-line-mixed gelant solution, flow resistance developed with length down the tube to a steady resistance value. Shear retarded the gelation process as compared to a quiescent bulk gel. Spaghetti-like gel emerged from the tube, and the low apparent viscosity of the gel in the tube indicated the existence of lubricant solvent between the gel and the tube wall. During injection of preformed gels, flow resistance at upstream of the tube was higher than that of in-line-mixed gelant. However, it decreased down the tube to steady values that were lower than the steady values measured for in-line-mixed gelant. The effluent gel more easily broke into pieces than the effluent during the injection of in-line-mixed gelant solution. Effluent samples from both types of flow experiments underwent significant syneresis due to the shear experience in the tube; while bulk-mixed gel samples were stable. Samples placed under constant, steady shear in a rheometer for 10 hours did not form strong gels until the shear was stopped. This study shows that shear experienced by a gelant during displacement in the reservoir significantly affects the properties of the gel. Results of the study are applicable to gel placement in fractured reservoirs