On Investigation Of Image Representation

The effective mathematical tools, closely related to image representation, play a veryimportant role in the numerous areas of science and technology. Image representationis one of the important study fields in image processing and computer vision. Thoughsparse representation, variational method and partial differential equation (PDE) nowoccupy the dominant position in image processing, there are still many problems. Thisthesis takes the image processing as a background, the mathematical imagerepresentation as the main objects. We will study some key problems in the relateddomain including sparse representation, Radon domain representation, the local andnonlocal description, quaternion domain description. Firstly, we explore sparserepresentation method based on dictionary learning, and formulate the multi-tasks,multi-dictionaries learning model. Secondly, we give a characterization of the inverseRadon transform associated with classical domain of type one. Thirdly, based on thelocal and nonlocal description of image, we present two new variational models.Finally, by introducing a particular non-commutative algebra: quaternion, we studyquaternion wavelet (including continuous wavelet and discrete wavelet), quaternionPDE and their applications in color image processing.The main research results are as follows:1. We propose multi-dictionaries learning (M-DL) models and a series of imagedecomposition methods. Specifically, space guided M-DL, non-coherence constrainedM-DL,“joint dictionary” based multi-task image processing method, andmulti-dictionaries and structure sparse based gene recognition method are presented.2. The unbounded realization of the classical domain of type one is Siegel domain oftype two. Its ilov boundary is a nilpotent Lie group of step two in general. Wecharacterize a subspace on which the Radon transform is a bijection. We also give aninversion formula of the Radon transform on the boundary by using continuouswavelet transform.3. Based on the local and nonlocal description of image, we propose two novelvariational models. Based on the local description, we present alternated split BregmanL1projection algorithm. The convergence of the iteration scheme is analyzed. Basedon the nonlocal description, we give a higher-oder nonlocal model. Numericalexperiments are presented to demonstrate the performance of the proposed models. 4. Quaternion is a special hyper complex number. From the view of the squareintegrable representation of groups, using real valued inner product on quaternion set, we decompose L~2(2,; dx) into the direct sum of the irreducible invariantsubspaces and give the characterization of the admissibility condition. On quaterniondomain, we establish the wavelet Parseval’s formula which holds on an additionalcondition compared with the corresponding formula on complex domain. By using theequivalence of quaternion and complex duplex matrix, we design severalquaternion-valued filters and fast algorithms. Finally, the experiments are given toverify our theory.5. We propose two quaternion diffusion equations. Firstly, we extend the usualcomplex diffusion equation to quaternion domain. Based on this, another equation byusing the algebraic operations of quaternion as color filters is presented. We applythese two non-linear diffusion equations in color image processing and obtainpreliminary results.