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Co-occurrence matrix is traditionally used for the representation of texture information. In this paper, the co-occurrence matrix is combined with Gaussian curvature for 3D shape representation and a novel 3D shape description approach named Gaussian curvature co-occurrence matrix is proposed. Normalization process to Gaussian curvature co-occurrence matrix and the invariants independence of the translation, scaling and rotation transforms are demonstrated. Experiments indicate a better classification rate and running complexity to objects with slight different shape characteristic compared with traditional methods. [ABSTRACT FROM AUTHOR]

Discrete distortion for two- and three-dimensional combinatorial manifolds is a discrete alternative to Ricci curvature known for differentiable manifolds. Here, we show that distortion can be successfully used to estimate mean curvature at any point of a surface. We compare our approach with the continuous case and with a common discrete approximation of mean curvature, which depends on the area of the star of each vertex in the triangulated surface. This provides a new, area-independent, tool for curvature estimation and for morphological shape analysis. We illustrate our approach through experimental results showing the behavior of discrete distortion. [ABSTRACT FROM AUTHOR]

In this paper we present a novel scale invariant interest point detector of blobs which incorporates the idea of blob movement along the scales. This trajectory of the blobs through the scale space is shown to be valuable information in order to estimate the most stable locations and scales of the interest points. Our detector evaluates interest points in terms of their self trajectory along the scales and its evolution obtaining non-redundant and discriminant features. Moreover, in this paper we present a differential geometry view to understand how interest points can be detected. We propose to analyze the gaussian curvature to classify image regions as blobs, edges or corners. Our interest point detector has been compared with some of the most important scale invariant detectors on infrared (IR) images, outperforming their results in terms of: number of interest points detected and discrimination of the interest points. [ABSTRACT FROM AUTHOR]

This paper describes a new approach for regularising triangulated graphs. We commence by embedding the graph onto a manifold using the heat-kernel embedding. Under the embedding, each first-order cycle of the graph becomes a triangle. Our aim is to use curvature information associated with the edges of the graph to effect regularisation. Using the difference in Euclidean and geodesic distances between nodes under the embedding, we compute sectional curvatures associated with the edges of the graph. Using the Gauss Bonnet Theorem we compute the Gaussian curvature associated with each node from the sectional curvatures and through the angular excess of the geodesic triangles. Using the curvature information we perform regularisation with the advantage of not requiring the solution of a partial differential equation. We experiment with the resulting regularization process, and explore its effect on both graph matching and graph clustering. [ABSTRACT FROM AUTHOR]

Chapter 27 of the book "Math Amazements: Astounding Investigations Uncover Math in Your World" is presented. It discusses the skills of the German mathematician Carl Friedrich Gauss, which were obvious even when he was still very young. Mathematical areas explored by Gauss are described, including his development of the Gaussian curve, a mathematical equation for the normal distribution curve. Classroom activities are also presented on math puzzles and equations.