From the beginning of science, visual observations have been playing important roles. Advances in computer technology have made it possible to apply some of the most sophisticated developments in mathematics and the sciences to the design and implementation of fast algorithms running on a large number of processors to process image data. As a result, image processing and analysis techniques are now applied to virtually all natural sciences and technical disciplines ranging from computer sciences and electronic engineering to biology and medical sciences; and digital images have come into everyone's life. Mathematics has been playing an important role in image and signal processing from the very beginning. There are two major mathematical approaches for image restoration, namely, wavelet tight frame approaches and differential/variational approaches. The main research objective of this project is to investigate geometric aspects of the former approach by connecting it with the latter. It will give rise to new mathematical models and numerical algorithms that benefit researchers in academia, national research laboratories, as well as in industry. The understandings of the geometric aspects of the wavelet frames and the connections with differential operators will contribute to both the community of computational harmonic analysis and the community of variational techniques and numerical PDEs. The education plan will bring undergraduate and graduate students to the frontiers of research in computational mathematics, computer vision and medical imaging; and strengthen the collaborations among mathematicians, engineers, computer scientists and medical doctors.
Wavelet frames are systems of functions that provide linear representations of functions living in certain function spaces such as L2(Rn). In contrast to the classic (bi)orthogonal wavelet bases, such representations are generally redundant which is desirable in many applications. Although most theoretical aspects of wavelet frames have already been well understood in the literature, geometric meanings of the wavelet frame transform are still generally unknown. In fact, the lack of geometric interpretations is one of the major flaws of wavelet frames that prohibits the applications of wavelet frames in some important problems of data analysis that require geometric regularization of the objects-of-interest reside in the data. The main research objective of this proposal is to develop a generic geometric interpretation to the wavelet frame transform, by studying its relations with differential operators within various variational frameworks. Based on the geometric interpretation, we propose new models and algorithms for several important applications such image restoration (deblurring, inpainting, CT/MR imaging, etc.). Through both theoretical analysis and numerical experiments, we will explore the advantages of the proposed wavelet frame based models over the existing variational and differential models for different applications. The proposed research will focus on: (1) the approximation of the differential operators by the wavelet frame transform within general variational frameworks; (2) solving large-scaled ill-posed inverse problems (e.g., image restoration, blind deconvolution) through convex/nonconvex optimizations using wavelet frames; (3) designing and solving wavelet frame based models in real-world applications in imaging such as low-dose CT image reconstruction, removing blurs caused by camera shaking, etc. The study of the geometric meanings of the wavelet frame transform will interpret wavelet frames and their associated optimization models from a whole new angle. Such fundamental study enables us, for the very first time, to fully utilize the unique properties of wavelet frames in geometry-involved data analysis tasks and finding numerical solutions of PDEs. The practical advantages (such as the quality of restoration for inverse problems) of wavelet frame transform over standard finite difference approximations in various applications will become more evident after the proposed studies. Furthermore, this project will also bring new understandings to numerical methods solving variational models; and answers some fundamental and important questions of variational models that are unclear from the literature.
