Industrial Geometry

S09207 Pattern and 3D Shape Recognition

Summary

In this project pattern and shape recognition have been investigated from partial differential equations (PDEs) and inverse problems points of view.

In the fundamental and inspiring book of Y. Meyer, patterns are characterized as oscillating functions, which in turn are considered elements of dual Sobolev spaces. The concept of oscillating patterns can be used for pattern recognition and edge enhancing techniques with regularization methods exploiting the Bregman distance, a concept which has been established by Burger et al. Recently, we have extended this approach to texture enhancing.

Manay et al. have introduced the research area of signatures a few years ago. Since then integral invariants and according signatures have been identified to be useful for shape classification, which is an important research topic in computer vision, artificial intelligence and pattern recognition. Integral invariants and signatures are transformations of shapes. In general, the invariants are constructed in such a way that they are invariant under geometric transformations and allow for a compact representation of shapes. Our joint collaboration with H. Pottmann was to identify the inverse problems point of view of integral invariants, which is a core research area of the Infmath Imaging group.

Filtering of high-dimensional data, such as color data, is an active research area in PDEs. In the computer vision areas research is heavily driven by modeling of such differential equations for high-dimensional data. We emphasize that there are much less publications concerned with variational filtering methods. Our work is concerned with generalizing morphological partial differential equations for analysis of intensity data to high-dimensional data. The challenging task is to establish a solution concept with a rigorous mathematical analysis. The difficulty is that the concept of viscosity solutions, which applies to morphological differential equations, does not generalize to high-dimensional data.

Left: One slice of artificially distorted tensor data $\vec{u}:\mathbb{R}^2\rightarrow \mathbb{R}^9\cong \mathbb{R}^{3\times 3}$ of the human brain. Right: Filtered data.

Results have specifically been obtained in the areas of Integral Invariants and of Vector Valued Denoising. A detailed report of the first funding period is available here. See also the list of Publications.