Ph.D Thesis

Ph.D StudentRatner Vadim
SubjectElastic Manifolds in Image Processing
DepartmentDepartment of Electrical and Computer Engineering
Supervisor PROFESSOR EMERITUS Yehoshua Zeevi
Full Thesis textFull thesis text - English Version


The Telegrapher's (Telegraph-Diffusion, TeD) wave equation, uttt-div(k*grad(u))=0, where u is considered to be an evolving image, is applied in the context of image processing and computer vision. We show that this partial differential equation provides a more powerful and flexible operator than the diffusion equation, which is widely used in image processing and computer vision in applications such as denoising, enhancement, segmentation and optical flow. To this end we show that both equations can be derived in a similar manner by functional minimization; and that the linear forms of the equations can be described by similar convolution kernels. We highlight the fact that the standard discretization, commonly used in the application of the diffusion equation, does, in fact, approximate the TeD equation. We, therefore, conclude that many applications that employ the diffusion equation should benefit from the introduction and analysis of the TeD equation in this context.

The main difference between the continuous diffusion and TeD equations is the velocity of information propagation. Whereas in the case of diffusion the velocity is infinite, in case of TeD it is finite. We claim that this difference has several important consequences that are useful in the context of image processing. First of all, in the continuous regime, finite information propagation speed results in increased stability. Indeed, whereas most of the continuous diffusion-based schemes do not produce solutions with converging energy, TeD-based versions of the same schemes can be proven to converge. Second, control over propagation velocity allows better detail preservation by the TeD denoising scheme, resulting in better quality of the processed images.

Furthermore, we offer an approximated solution of the ill-posed inverse problem of image deblurring, based on the TeD. Utilizing the above results with reference to the stability of denoising-TeD, we show that the proposed approximated deblurring process remains bounded energy-wise for any finite time period and for any given enhancement strength.

In order to test the theoretical results we adopt the diffusion-based denoising enhancement schemes in both grayscale and color-manifold domains for application in the TeD-based framework. We consider both temporal and spatial discretization schemes and analyze them in terms of convergence and efficiency.

It is argued that the approach presented in this work can be applied to other nonlinear and ill-posed schemes, besides image denoising-enhancement. Furthermore, the stabilization-by-approximation technique derived here can be applied to other, non-diffusion-related, ill-posed problems in image processing and computer vision (or elsewhere).