Optimization in the wavelet domain has been a very prominent research topic both for denoising, as well as compression, reflected in its use in the JPEG-2000 standard. Its performance depends to a great extent on the wavelet ψ itself, represented in the form of a filter in the case of the discrete wavelet transform. While other works solely optimize the coefficients in the wavelet domain, we will use a combined approach, optimizing the wavelet ψ and the coefficients simultaneously in order to adapt both to a given image, resulting in a better reconstruction of an image from less coefficients. We will use several orthonormal wavelet bases as a starting point, but we will also demonstrate that we can create wavelets from white Gaussian noise with our approach, which are in some cases even better in terms of performance. Experiments will be conducted on several images, demonstrating how the optimization algorithm adapts to textured, as well as more homogeneous images.
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