A topological sampling theorem for Robust boundary reconstruction and image segmentation

AbstractExisting 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 α-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

3D Object Digitization: Topology Preserving Reconstruction

In this paper we derive a sampling theorem, which is the first one to guarantee topology preservation during digitization of 3D objects. This new theorem is applicable to several reconstruction methods, e.g. a unionof-balls reconstruction and the trilinear interpolation.

DIY heat exchanger to the dew point ventilation system

Wer falsch lüftet, holt sich oft Kälte oder Feuchtigkeit ins Haus. Gegen die Feuchtigkeit hilft ein Taupunkt-Lüftungssystem, wie es in Heft 1/2022 vorgestellt wurde. Doch gegen ein Auskühlen des Kellers durch das Lüften nützt dies nichts. Abhilfe schafft die Kombination des Lüftungssystems mit einem Wärmetauscher. Unser Projekt zeigt, wie man diesen günstig und mit wenig Aufwand selber bauen kann

3D Object Digitization: Majority Interpolation and Marching Cubes

In a previous paper we showed that a 3D object can be digitized without changing the topology if the object is r-regular and if the reconstruction method fulfills certain requirements. In this paper we give two important examples for such reconstruction methods. First, we introduce Majority Interpolation, an algorithm to interpolate sampling points at doubled resolution such that topological ambiguities are resolved. Second, we show how the well-known Marching Cubes algorithm has to be modified such that it is topology preserving. This is the first approach of digitizing 3D objects which guarantees topology preservation for voxel-based or polygonal surface-based reconstructions.

3D Object Digitization: Volume and Surface Area Estimation

Measuring volume and surface area of objects given its digitizations are important problems in 3D image analysis. Good estimators should be multigrid convergent, i.e. the error goes to zero with increasing sampling density. We will give such estimators both for volume and for surface area estimation based on simple counting of voxels.

Grid-Independent Necessary Criterions for Shape Preserving Digitization

The question, which shapes can be digitized without any change in the fundamental geometric and topological properties is of great importance for the reliability of image analysis algorithms, but is nevertheless unanswered for a lot of digitization schemes. While r-regularity is a sufficient criterion for shapes to be reconstructed correctly by using any regular or irregular sampling grid of certain density, necessary criteria are up to now unknown. The author proves such a necessary criterion: If you choose some sampling grid and you want a shape to be digitized correctly with any alignment of this grid, then the shape has to be a bordered 2D-manifold, i.e. its boundary has to have no junctions. This implies that any correct digitization is an extended well-composed set and thus the well known problems of defining connectivity in 2D are always due to wrong sampling or improper original shapes. This is of great importance, since extended well-composed sets have many nice topological properties, for example the Jordan curve theorem holds and the Euler characteristic is locally computable. Moreover the author proves a second necessary criterion: In case of a correct digitization with a grid of a certain density, shapes are not allowed to have corners with an angle smaller than 60 degrees. In case of common square grids the smallest possible angle is even 90 degrees. If some shape has some corner with a too small angle, the shape can not be digitized topologically correctly with every alignment of some sampling grid, if this grid exceeds a certain density. Thus the intuitive assumption that a finer grid would lead to a better digitization (in a topological sense) is simply wrong

Shape Preserving Sampling and Reconstruction of Grayscale Images

Abstract. The expressiveness of a lot of image analysis algorithms depends on the question whether shape information is preserved during digitization. Most existing approaches to answer this are restricted to binary images and only consider nearest neighbor reconstruction. This paper generalizes this to grayscale images and to several reconstruction methods. It is shown that a certain class of images can be sampled with regular and even irregular grids and reconstructed with different interpolation methods without any change in the topology of the level sets of interest.

Affichage de base pour un système d'affichage autostéréoscopique

A basic display (10) for an autostereoscopic display arrangement (30, 32) having pixels (12) arranged in a periodic raster, in which at least one of the two diagonals (f1, f2) that form an angle of 45° with the display horizontal (x) has the property that, for two arbitrary pixels (12) that fulfill the condition that a straight line (g) passing through the center points (P) of the two pixels forms an angle of between −2° and 2° with the diagonal (f1, f2), the center distance (d1, d2) of the pixels is greater than 1.5 times, preferably greater than 1.8 times, more preferably greater than twice a basic distance (e) that is defined as the minimum of the center distances of all pixel pairs