The objective of this project is to develop a set of tools capable of performing truly localized Fourier analysis and synthesis of objects of interest in a given image that have smooth boundaries but of general shape. To do so, we will bring in tools in the traditionally different fields such as boundary detection and delineation algorithms (computer vision), image compression and denoising schemes (image processing), elliptic boundary value problem solvers and potential function computations (scientific computing), and Fourier analysis and fast algorithms (computational harmonic analysis). Our primary concern is analysis (e.g., extraction and characterization of spatial frequency features) and synthesis (e.g., reconstruction from the compressed representation of objects after their boundaries are detected and they are segmented either manually by a human interpreter using a pointing device or automatically by the algorithms proposed by other researchers. On the one hand, the boundary of an object provides important information: geometry and shape of the object. On the other hand, it becomes a nuisance for other tasks such as the Fourier analysis of the internal information (e.g., textures) of the object because it creates spurious interference patterns due to the Gibbs phenomenon that masks the important internal information of the object. We will decouple the geometry and internal information of the object by solving the elliptic boundary value problem on the domain where the object is supported. More precisely, for the analysis of the object, we embed the detected and segmented object in an otherwise empty rectangular domain, and smoothly extend the object to the outside of the object boundary by solving the Poisson equation with the homogeneous Dirichlet boundary condition at the edges of the covering rectangle. Since the values on the edges of the covering rectangle vanish, this smoothly extended component can be expanded into the Fourier sine series with quickly decaying coefficients, which enable us to effectively characterize and compress the internal information of the object. Finally, we subtract this component from the original object on the supported domain to obtain the component responsible for geometric information of the object, which turns out to be a solution of the Laplace equation on that domain. For the synthesis or reconstruction of the object, we need to store the Fourier sine coefficients of the extended component and the boundary coordinates and the original values of the object at those points in the analysis stage. Then, the original object is recovered by adding the geometric component (which is recovered by evaluating the single and double layer potentials on the domain) and the smoothly extended component (which is easily recovered from the Fourier sine coefficients). In order to solve the Laplace/Poisson equations, we will fully utilize the advanced Laplace/Poisson solvers based on Fast Multipole Methods. We will also extend our analysis and synthesis paradigm to an object with holes (i.e., multiply-connected domains), investigate the effect of noise on the boundary shapes and values, and investigate the effect of compression of the boundary information to the quality of the reconstructed images. Furthermore, we will develop a gradient and directional derivative estimation algorithm equipped with regularization (high-frequency attenuation) capabilities, which will provide good boundary conditions and consequently improve the performance of the boundary detection and delineation algorithms as well as the accuracy of the solutions of the Laplace/Poisson equations.
Potential applications of our methodology include biometrics and image-based diagnostics in medicine and other fields such as geology and material sciences. Biometrics has recently become a tremendously important subject for homeland security reasons. Since our paradigm provides both geometric/shape information and internal texture information of an object of interest in an separate manner for images obtained by various sensors and imaging modalities, it may allow data examiners to characterize the features of the objects of interest much more reliably compared to the methods which solely use either shape or texture information. For example, characterizing and diagnosing cancerous cells in various image modalities including Pap smear test images in gynecology may benefit from using the tools we will develop in our project (e.g., object-based storage, cataloging, compression, and analysis). Similarly, extracting quantitative information from optical images of sections of rock core samples (e.g., the size and internal texture/spatial frequency information of some fossils), which is important in earth science including oil and gas exploration industry, may also benefit from our research. We envision that scientists in completely different disciplines such as medicine, biology, and geology, will start noticing the importance of computational harmonic analysis and certain partial differential equations (PDEs) if they use our tools to be developed in this project and feel that these are useful for their own tasks. This is a great thing we, as applied and computational mathematicians, can hope for. In terms of the educational impact, this project will create a common meeting ground among students in the different fields: applied mathematics, computer science, electrical engineering, statistics, and neuroscience. We expect lively interchanges of ideas among such students who will participate in this research project or attend the associated courses and seminars we are developing. Students participating in our project will also learn computational harmonic analysis, the basic theory and fast computational algorithms of certain PDEs, and image analysis, which will become indispensable for the future applied mathematicians and scientists working in the area of imaging science, and which will be surely helpful for their future career, either in academia or in industry.
