  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.
