Conformal Geometry and Its Applications on 3D Shape Matching, Recognition, and Stitching

Three-dimensional shape matching is a fundamental issue in computer vision with many applications such as shape registration, 3D object recognition, and classification. However, shape matching with noise, occlusion, and clutter is a challenging problem. In this paper, we analyze a family of quasi-conformal maps including harmonic maps, conformal maps, and least-squares conformal maps with regards to 3D shape matching. As a result, we propose a novel and computationally efficient shape matching framework by using least-squares conformal maps. According to conformal geometry theory, each 3D surface with disk topology can be mapped to a 2D domain through a global optimization and the resulting map is a diffeomorphism, i.e., one-to-one and onto. This allows us to simplify the 3D shape-matching problem to a 2D image-matching problem, by comparing the resulting 2D parametric maps, which are stable, insensitive to resolution changes and robust to occlusion, and noise. Therefore, highly accurate and efficient 3D shape matching algorithms can be achieved by using the above three parametric maps. Finally, the robustness of least-squares conformal maps is evaluated and analyzed comprehensively in 3D shape matching with occlusion, noise, and resolution variation. In order to further demonstrate the performance of our proposed method, we also conduct a series of experiments on two computer vision applications, i.e., 3D face recognition and 3D nonrigid surface alignment and stitching.

1 Introduction and Previous Work

Three-dimensional shape matching is a fundamental issue in computer vision with many applications, such as shape registration, partial scan alignment, and 3D object recognition and classification [8], [50], [37], [23]. As 3D scanning technologies improve, large databases of 3D scans require automated methods for matching. However, matching 3D shapes in noisy and cluttered scenes is a challenging task. Moreover, since most 3D shape scanners can only capture 2.5D data of the target surfaces, aligning and stitching partial 3D surfaces is a fundamental problem in many research areas, such as computer vision, mechanical engineering, and molecular biology.