PDE Techniques in Wavelet Based Image Processing

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The goal of this research is to develop mathematical principlesand new methodologies for problems in and applications of digitalimage 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 severalhigh level mathematical tools including geometrical PDE's,multi-resolution harmonic analysis and variational frameworkstogether with some state-of-the-art engineering methods in imagecoding, such as group testing wavelet (GTW) algorithms for imagecompression. The key is to exploit the power of multi-resolutionproperties of wavelets while at the same time use PDE techniques tosystematically and explicitly control the geometrical informationin the image so that salient features, such as edges and corners,can be recovered in the reconstructions. Efficient computation algorithmswill be designed and analyzed.Image compression and inpainting are two typical tasks of imageprocessing. Due to the large number of pixels of digital images,most of them have to be stored in compressed format. The currentinternational compression standard (JPEG2000), which is largelybased on wavelet representation, is one of the most popular schemes.Image compression has been used everywhere in our daily life, examplesinclude images transmitted on the Internet, wireless communications,medical images (such as MRI), satellite images. Efficient compressionmethods are highly desirable and the universal goal is to represent themost salient features (typically geometrical structures) using minimalpossible resources. This is one of the objectives of this project.Image inpainting refers to automatic procedures to fillin incomplete, missing or damaged image information. Such lossis often unavoidable in many applications such as wirelesstransmission, intelligence, homeland security (airports' screening),robotic path finding in unknown environment, three-dimensionalobject reconstruction from two-dimensional medical images. Becausemany digital images are stored in wavelet formats, and damages to suchformatted images correspond to loss of wavelet coefficients,wavelet inpainting demands special attention. Again, a key objectivein filling in the missing information is to restore missinggeometrical properties. We will investigate new models and methods forwavelet inpainting and related mathematical theories.