The goal of this research is to develop mathematical principles and new methodologies for problems in and applications of digital image processing, and study their associated mathematical foundations. The emphasis is on combining partial differential equation (PDE) techniques with multi-resolution wavelet representations. In particular, we focus on two different image processing applications: wavelet inpainting (filling in missing or damaged wavelet coefficients), and PDE image compression. To achieve the tasks, we will integrate several high level mathematical tools including geometrical PDE's, multi-resolution harmonic analysis and variational frameworks together with some state-of-the-art engineering methods in image coding, such as group testing wavelet (GTW) algorithms for image compression. The key is to exploit the power of multi-resolution properties of wavelets while at the same time use PDE techniques to systematically and explicitly control the geometrical information in the image so that salient features, such as edges and corners, can be recovered in the reconstructions. Efficient computation algorithms will be designed and analyzed.
Image compression and inpainting are two typical tasks of image processing. Due to the large number of pixels of digital images, most of them have to be stored in compressed format. The current international compression standard (JPEG2000), which is largely based on wavelet representation, is one of the most popular schemes. Image compression has been used everywhere in our daily life, examples include images transmitted on the Internet, wireless communications, medical images (such as MRI), satellite images. Efficient compression methods are highly desirable and the universal goal is to represent the most salient features (typically geometrical structures) using minimal possible resources. This is one of the objectives of this project. Image inpainting refers to automatic procedures to fill in incomplete, missing or damaged image information. Such loss is often unavoidable in many applications such as wireless transmission, intelligence, homeland security (airports' screening), robotic path finding in unknown environment, three-dimensional object reconstruction from two-dimensional medical images. Because many digital images are stored in wavelet formats, and damages to such formatted images correspond to loss of wavelet coefficients, wavelet inpainting demands special attention. Again, a key objective in filling in the missing information is to restore missing geometrical properties. We will investigate new models and methods for wavelet inpainting and related mathematical theories.
