Peer Stelldinger

Towards a general sampling theory for shape preservation

Computerized image analysis makes statements about the continuous world by looking at a discrete representation. Therefore, it is important to know precisely which information is preserved during digitization. We analyze this question in the context of shape recognition. Existing results in this area are based on very restricted models and thus not applicable to real imaging situations. We present generalizations in several directions: first, we introduce a new shape similarity measure that approximates human perception better. Second, we prove a geometric sampling theorem for arbitrary dimensional spaces. Third, we extend our sampling theorem to two-dimensional images that are subjected to blurring by a disk point spread function. Our findings are steps towards a general sampling theory for shapes that shall ultimately describe the behavior of real optical systems.

A topological sampling theorem for Robust boundary reconstruction and image segmentation

Existing theories on shape digitization impose strong constraints on admissible shapes, and require error-free data. Consequently, these theories are not applicable to most real-world situations. In this paper, we propose a new approach that overcomes many of these limitations. It assumes that segmentation algorithms represent the detected boundary by a set of points whose deviation from the true contours is bounded. Given these error bounds, we reconstruct boundary connectivity by means of Delaunay triangulation and @a-shapes. We prove that this procedure is guaranteed to result in topologically correct image segmentations under certain realistic conditions. Experiments on real and synthetic images demonstrate the good performance of the new method and confirm the predictions of our theory.

Topology Preserving Marching Cubes-like Algorithms on the Face-Centered Cubic Grid

The well-known marching cubes algorithm is modified to apply to the face-centered cubic (fee) grid. Thus, the local configurations that are considered when extracting the local surface patches are not cubic anymore. This paper presents three different partitionings of the fee grid to be used for the local configurations. The three candidates are evaluated theoretically and experimentally and compared with the original marching cubes algorithm. It is proved that the reconstructed surface is topologically equivalent to the surface of the original object when the surface of the original object that is digitized is smooth and a sufficiently dense fee grid is used.

Topologically correct image segmentation using alpha shapes

Existing theories on shape digitization impose strong constraints on feasible shapes and require error-free measurements We use Delaunay triangulation and α-shapes to prove that topologically correct segmentations can be obtained under much more realistic conditions Our key assumption is that sampling points represent object boundaries with a certain maximum error Experiments on real and generated images demonstrate the good performance and correctness of the new method.

Shape Preservation during Digitization: Tight Bounds Based on the Morphing Distance

We define strong r-similarity and the morphing distance to bound geometric distortions between shapes of equal topology. We then derive a necessary and sufficient condition for a set and its digitizations to be r-similar, regardless of the sampling grid. We also extend these results to certain gray scale images. Our findings are steps towards a theory of shape digitization for real optical systems.

Provably correct reconstruction of surfaces from sparse noisy samples

Automated three-dimensional surface reconstruction is a very large and still fast growing area of applied computer vision and there exists a huge number of heuristic algorithms. Nevertheless, the number of algorithms which give formal guarantees about the correctness of the reconstructed surface is quite limited. Moreover such theoretical approaches are proven to be correct only for objects with smooth surfaces and extremely dense samplings with no or very few noise. We define an alternative surface reconstruction method and prove that it preserves the topological structure of multi-region objects under much weaker constraints and thus under much more realistic conditions. We derive the necessary error bounds for some digitization methods often used in discrete geometry, i.e. supercover and m-cell intersection sampling. We also give a detailed analysis of the behavior of our algorithm and compare it with other approaches.

Fast and Accurate 3D Edge Detection for Surface Reconstruction

Although edge detection is a well investigated topic, 3D edge detectors mostly lack either accuracy or speed. We will show, how to build a highly accurate subvoxel edge detector, which is fast enough for practical applications. In contrast to other approaches we use a spline interpolation in order to have an efficient approximation of the theoretically ideal sinc interpolator. We give theoretical bounds for the accuracy and show experimentally that our approach reaches these bounds while the often-used subpixel-accurate parabola fit leads to much higher edge displacements.

Topologically correct surface reconstruction using alpha shapes and relations to ball-pivoting

The problem to reconstruct a surface given a finite set of boundary points is of growing interest, e.g. in the context of laser range images. While a lot of heuristic methods have been published in this context (e.g. the ball-pivoting algorithm), there exist only a few algorithms which guarantee the reconstruction to be homeomorphic to the original surface if a certain sampling density is reached. However, the sampling density mentioned is in most cases much higher than what seems to be sufficient on real data. In this paper we show how recently proved results about homology extraction from surface samples can be adopted to surface reconstruction and we significantly improve the bounds on the sampling density in case of noise-free samplings. This allows us to prove for the first time that the ball-pivoting algorithm reconstructs certain object surfaces without any topological changes and we can give bounds on the reconstruction error regarding both position and normal direction of the boundary.

Shape Preserving Digitization of Ideal and Blurred Binary Images

In order to make image analysis methods more reliable it is important to analyse to what extend shape information is preserved during image digitization. Most existing approaches to this problem consider topology preservation and are restricted to ideal binary images. We extend these results in two ways. First, we characterize the set of binary images which can be correctly digitized by both regular and irregular sampling grids, such that not only topology is preserved but also the Hausdorff distance between the original image and the reconstruction is bounded. Second, we prove an analogous theorem for gray scale images that arise from blurring of binary images with a certain filter type. These results are steps towards a theory of shape digitization applicable to real optical systems.

On covering a digital disc with concentric circles in Z2

Digital circles and digital discs satisfy many bizarre anisotropic properties, understanding of which is essential for solving various problems in image analysis and computer graphics. In this paper we study the underlying properties of absentee pixels that appear while covering a digital disc with concentric digital circles. We present, for the first time, a mathematical characterization of these pixels based on number theory and digital geometry. Interestingly, the absentees occur in multitude, and we show that their count varies quadratically with the radius. The notion of infimum parabola and supremum parabola has been used to derive the count of these absentees. Using this parabolic characterization, we derive a necessary and sufficient condition for a pixel to be a disc absentee, and obtain the geometric properties of the absentees. An algorithm to locate the absentees is presented. We show that the ratio of the absentee pixels to the total number of disc pixels approaches a constant with increasing radius. Test results have been furnished to substantiate our theoretical findings.