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Abstract
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Image restoration is one of the most important areas in imaging science. Mathematical tools have been widely used in image restoration, where the wavelet frame based approach is one of the successful examples due to its capability of capturing singularities from noisy and degraded images. In addition, their multiresolutional analysis (MRA) structure enables us to provide a rigorous connection between the wavelet frame based image restoration model in discrete and the variational model in continuum, which have long been seemingly unrelated. It also gave birth to many innovative and more e ective image restoration models and algorithms. Following the recent developments in wavelet frame, this talk consists of two parts; one is the edge driven wavelet frame based image restoration model for piecewise smooth functions. With an implicit representation of image singularities sets, the proposed model inicts di erent strength of regularization on smooth and singular image regions and edges. The proposed edge driven model is robust to both image approximation and singularity estimation. The implicit formulation also enables an asymptotic analysis of the proposed models and a rigorous connection between the discrete model and a general continuous variational model. The other part consists of the approximation property of wavelet frame based image restoration model. Given incomplete (degraded) measurements, we present the error between the underlying original discrete image and the approximate solution which has the minimal `1 norm of the canonical wavelet frame coecients among all possible solutions. Then we further connect the error estimate for the discrete model to the approximation to the underlying function from which the underlying image comes.
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