## Image de-noising on the manifold of patches: a spectral approach

#### Francois MeyerUniversity of Colorado, Boulder

One of the most fundamental problems in imaging remains the construction of sparse representation for images. Although multiscale decompositions have been somewhat successful the improvements over blocked Fourier transform are only minor. In parallel to the development of multiscale transforms, several studies have analyzed the statistics of patches extracted from natural images. The goal of these investigations was never the construction of a natural images transform''. We nevertheless advocate that the methodology used in these experiments can lead us to think about images with a fresh eye.
In particular, inspired by the work of Lee and Mumford, who showed experimentally that 3x3 patches of natural images organize themselves around nonlinear low dimensional manifolds, we propose to construct basis functions to efficiently parametrize the manifolds of image patches. Thinking about an image as a parameterization of a set of patches alleviates the need for a universal transform, and opens the door to an entirely new way of processing images. In this work, we describe how we can remove the noise from an image by iteratively reconstructing and denoising the set of patches. This approach outperforms the most successful denoising techniques.

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