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.

近似理论在图像处理，尤其是图像解卷积和分解中起着重要作用。对于分段平滑图像，过去三十年发展了许多方法。本研究的目标是设计类似且实用的方法来处理纹理图像。这个问题是由法医成像引起的，因为指纹、鞋印和子弹弹道证据都是有纹理的图像。特别是，已知纹理信息几乎被模糊算子破坏，例如从低成本显微镜捕获的模糊弹道图像。这项工作的贡献是双重的：首先，我们提出了一个数学模型，用于将纹理图像解卷积和分解为四个有意义的分量，使用基于方向平均曲率的高阶偏微分方程方法。其次，我们揭示了泛函分析和多尺度采样理论之间的联系，例如谐波分析和滤波器组。提供了理论结果和自然图像的例子来说明所提出模型的性能。