טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentOri Honigman
SubjectImage Processing Using Diffusion with Schrodinger's
Potential
DepartmentDepartment of Electrical Engineering
Supervisor Professor Emeritus Zeevi Yehoshua
Full Thesis textFull thesis text - English Version


Abstract

The complex diffusion process, recently introduced in image processing and computer vision by combining the linear diffusion equation and the 'free-particle' Schrödinger equation, is further generalized by incorporating the Schrödinger potential. It is well known from physics that the Schrödinger equation with potential generates an oscillatory response of a particle to the conditions imposed by a specific structure of the potential. This behavior of the Schrödinger equation highlights the idea that incorporating the potential into the complex diffusion equation, may introduce some kind of 'dynamic boundary conditions' that may serve as a filtering or even enhancing mechanism for textures. We show that this generalized complex diffusion equation is inherently endowed with processing properties suitable for dealing with textures in a naturally coupled manner. The Schrödinger potential is self-adopting to the specific properties of an image at hand, in that it implements an image-specific wavelet shrinkage algorithm. Results indicate that the generalized complex diffusion processing scheme not only preserves textures better than demonstrated by previously-reported results, but can even enhance textures by adjusting the coefficient that determines the magnitude of the potential, relative to the potentialless complex diffusion.

    Real-valued linear diffusion is controlled by the heat diffusion equation. In physics the generalized heat diffusion equation also handles the presence of a heat source, this is achieved by adding a heat source term to the partial differential equation (PDE). We incorporate a heat source term to the Perona and Malik (PM) scheme for nonlinear diffusion equation. This heat source term is derived using the wavelet shrinkage algorithm. Results obtained by the application of this type of generalized PM operator show that this processing benefits from both the smoothness model, assumed in the nonlinear diffusion, and the preservation of textures inherent in the wavelet shrinkage. In our framework we actually fuse the unique information produced by the nonlinear diffusion process and the wavelet shrinkage.

    Finally, we show that by adding a heat source term to the Beltrami flow we can process color images and other multidimensional inputs while benefitting from both worlds - the joint channel information and the textural information derived by the nonlinear wavelet process, which is used to construct the heat source term.

    Convergence of the different numerical schemes, presented in our work, to the PDEs they originate from is proved by analysis of the stability and consistency. Furthermore determining the different parameters that control the processes is discussed.