The investigators will study new computational techniques with applications to inverse problems and image analysis. They will seek new methods that combine variational arguments with ideas from computational harmonic analysis and partial differential equations in order to overcome limitations of existing methods. The research will have three objectives: propose new models for cartoon and texture separation in images by working with spaces of distributions; propose completely new techniques for multiscale hierarchical decomposition of images; propose efficient algorithms for solving inverse problems.
From the proposed research and educational program, computational mathematics, image processing, as well as more general areas of science and engineering will benefit. Applications include image analysis, medical imaging, satellite imaging, material science, and terrain data analysis, surveillance and inverse problems.
The principal investigators, their students and collaborators proposed several variational models and their numerical implementations for image processing applications. In particular, functional spaces for modeling texture have been investigated. Image restoration models for denoising and deblurring have been introduced that take into account one dimensional Sobolev norm profiles of noise and texture. The level set method and image segmentation techniques have been extended to the detection of open edges in images and to the motion of junctions. For example, blood vessels and long-like structures can be detected by the proposed method. Applications to medical imaging have been considered, such as denoising and deblurring of magneto-resonance images degraded by Rician noise and blur, and to the denoising of high angular resonance diffusion imaging degraded by Rician noise. Students and postdoctoral fellows have been trained by working on interdisciplinary projects supported by the award and by participating to research seminars and meetings on current literature in applied mathematics. The investigators and their students presented their findings to national and international conferences. Numerous research publications resulted from this award. Seminars and visits by international collaborators have been supported by the award. The principal investigator organized several special sessions and workshops at SIAM conferences and the Fields Institute.
