Metric (mathematics)

About: Metric (mathematics) is a research topic. Over the lifetime, 42617 publications have been published within this topic receiving 836571 citations. The topic is also known as: distance function & metric.

Reconciling distance functions and level sets

Abstract: This paper is concerned with the simulation of the partial differential equation driven evolution of a closed surface by means of an implicit representation. In most applications, the natural choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian have proposed to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately the solution to this equation is not a distance function. As a consequence, the practical application of the level set method is plagued with such questions as when do we have to reinitialize the distance function? how do we reinitialize the distance function?, which reveal a disagreement between the theory and its implementation. This paper proposes an alternative to the use of Hamilton-Jacobi equations which eliminates this contradiction: in our method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory any more. This is achieved through the introduction of a new equation. Besides its theoretical advantages, the proposed method also has several practical advantages which we demonstrate in three applications: (i) the segmentation of the human cortex surfaces from MRI images using two coupled surfaces (X. Zeng, et al., in Proceedings of the International Conference on Computer Vision and Pattern Recognition, June 1998), (ii) the construction of a hierarchy of Euclidean skeletons of a 3D surface, (iii) the reconstruction of the surface of 3D objects through stereo (O. Faugeras and R. Keriven, Lecture Notes in Computer Science, Vol. 1252, pp. 272-283).

Coarse-to-fine low-rank structure-from-motion

Abstract: We address the problem of deformable shape and motion recovery from point correspondences in multiple perspective images. We use the low-rank shape model, i.e. the 3D shape is represented as a linear combination of unknown shape bases. We propose a new way of looking at the low-rank shape model. Instead of considering it as a whole, we assume a coarse-to-fine ordering of the deformation modes, which can be seen as a model prior. This has several advantages. First, the high level of ambiguity of the original low-rank shape model is drastically reduced since the shape bases can not anymore be arbitrarily re-combined. Second, this allows us to propose a coarse-to-fine reconstruction algorithm which starts by computing the mean shape and iteratively adds deformation modes. It directly gives the sought after metric model, thereby avoiding the difficult upgrading step required by most of the other methods. Third, this makes it possible to automatically select the number of deformation modes as the reconstruction algorithm proceeds. We propose to incorporate two other priors, accounting for temporal and spatial smoothness, which are shown to improve the quality of the recovered model parameters. The proposed model and reconstruction algorithm are successfully demonstrated on several videos and are shown to outperform the previously proposed algorithms.

A new sharpness metric based on local kurtosis, edge and energy information

Abstract: Sharpness metrics that use the whole frequency spectrum of the image cannot separate the sharpness information from the scene content. The sharpness metrics that use spatial gradients of the edges work only for comparisons among versions of the same image. We have developed a content independent, no-reference sharpness metric based on the local frequency spectrum around the image edges. In this approach, we create an edge profile by detecting edge pixels and assigning them to 8×8 pixel blocks. Then we compute sharpness using the average 2D kurtosis of the 8×8 DCT blocks. However, average kurtosis is highly sensitive to asymmetry in the DCT, e.g. different amounts of energy and edges in the x and y directions, therefore causing problems with different content and asymmetric sharpness enhancement. Thus we compensate the kurtosis using spatial edge extent information and the amount of vertical and horizontal energy in the DCT. The results show high correlation with subjective quality for sharpness-enhanced video and high potential to deal with asymmetric enhancement. For compressed, extremely sharpened and noisy video, the metric correlates with subjective scores up to the point where impairments become strongly noticeable in the subjective quality evaluation. The metric can be used by itself as a control variable for high-quality image capture and display systems, high-quality sharpness enhancement algorithms, and as a key component of a more general overall quality metric.

Mesh gradation control

Abstract: This paper gives a procedure to control the size variation in a mesh adaption scheme where the size specication (the so-called control space) is used to govern the mesh generation stage. The method consists in replacing the initial control space by a reduced one by means of size or metric. It allows to improve, a priori, the quality of the adapted mesh and to speed up the adaption procedure. Several numerical examples show the eciency of the reduction scheme. ? 1998 John Wiley & Sons, Ltd.