SOLVING POISSON DIFFERENTIAL EQUATION USING VECTOR SPACE PROJECTIONS

This work presents a new approach to solving the
Poisson partial differential equation in the setting
of vector spaces and vector sets. It formulates a
very important signal-processing problem of phase
unwrapping as a Poisson differential equation
problem, and solves it using the vector space
projection (VSP) methods. The 2D phase unwrapping
problem in radar imaging and magnetic resonance
imaging (MRI) is described, and examples in both
fields are used to show the effectiveness of the
proposed vector space projection method. In the
second part of the work, the author approaches the
classical problem of solving the Poisson equation in
electrodynamics, formulating both the equation
itself, and the boundary conditions in the vector
space setting. Both the homogeneous and the non-
homogenous solutions are presented.
The book covers only two of the numerous
applications of the Poisson differential equation,
but it presents the essentials of the vector space
projection methods in solving the partial
differential equations. Other applications of VSP
methods are hinted, and suggested as the
extensions of this work.

Autorentext

Boris Marendic holds a PhD in EE from Illinois Institute of Technology in Chicago. He worked for Motorola Inc for seven years as a firmware engineer, and is currently a postdoctoral researcher at the IIT medical imaging research center. His research interests are in image reconstruction algorithms and statistical signal processing.

Klappentext

This work presents a new approach to solving the Poisson partial differential equation in the setting of vector spaces and vector sets. It formulates a very important signal-processing problem of phase unwrapping as a Poisson differential equation problem, and solves it using the vector space projection (VSP) methods. The 2D phase unwrapping problem in radar imaging and magnetic resonance imaging (MRI) is described, and examples in both fields are used to show the effectiveness of the proposed vector space projection method. In the second part of the work, the author approaches the classical problem of solving the Poisson equation in electrodynamics, formulating both the equation itself, and the boundary conditions in the vector space setting. Both the homogeneous and the non-homogenous solutions are presented.The book covers only two of the numerous applications of the Poisson differential equation, but it presents the essentials of the vector space projection methods in solving the partial differential equations. Other applications of VSP methods are hinted, and suggested as the extensions of this work.