Approximation theory plays an important role in image processing, especially image deconvolution and decomposition. For piecewise smooth images, there are many methods that have been developed over the past thirty years. The goal of this study is to devise similar and practical methodology for handling textured images. This problem is motivated by forensic imaging, since fingerprints, shoeprints and bullet ballistic evidence are textured images. In particular, it is known that texture information is almost destroyed by a blur operator, such as a blurred ballistic image captured from a low-cost microscope. The contribution of this work is twofold: first, we propose a mathematical model for textured image deconvolution and decomposition into four meaningful components, using a high-order partial differential equation approach based on the directional mean curvature. Second, we uncover a link between functional analysis and multiscale sampling theory, e.g., harmonic analysis and filter banks. Both theoretical results and examples with natural images are provided to illustrate the performance of the proposed model.
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