Showing papers on "Affine involution published in 2007"
TL;DR: This paper shows how to embed a subset of size n of the Euclidean space into a O(ε-2 log n)-dimensional Euclidan space, so that no set of size s ≤ k changes its volume by more than (1 + εs-1).
Abstract: Let X be a subset of n points of the Euclidean space, and let 0 < e < 1. A classical result of Johnson and Lindenstrauss [JL] states that there is a projection of X onto a subspace of dimension O(e-2 log n) with distortion ≤ 1+ e. We show a natural extension of the above result to a stronger preservation of the geometry of finite spaces. By a k-fold increase of the number of dimensions used compared with [JL], a good preservation of volumes and of distances between points and affine spaces is achieved. Specifically, we show how to embed a subset of size n of the Euclidean space into a O(e-2 log n)-dimensional Euclidean space, so that no set of size s ≤ k changes its volume by more than (1 + es-1. Moreover, distances of points from affine hulls of sets of at most k - 1 points in the space do not change by more than a factor of 1 + e. A consequence of the above with k = 3 is that angles can be preserved using asymptotically the same number of dimensions as the one used in [JL]. Our method can be applied to many problems with high-dimensional nature such as Projective Clustering and Approximated Nearest Affine Neighbour Search. In particular, it shows a first polylogarithmic query time approximation algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced by Feige [F], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph.
39 citations
TL;DR: In this paper, an affine analogue and generalization of the geometry of special Lagrangian submanifolds of Cn is proposed, which is called affine affine Lagrangians.
Abstract: In this paper we propose an affine analogue and generalization of the geometry of special Lagrangian submanifolds of Cn.
7 citations
5 citations
15 Apr 2007
TL;DR: The problem of shape-based recognition of objects under affine transformations is considered and a robust and highly discriminative affine invariant function that can be used for within-class object recognition applications is focused on.
Abstract: The problem of shape-based recognition of objects under affine transformations is considered. We focus on the construction of a robust and highly discriminative affine invariant function that can be used for within-class object recognition applications. Using the boundaries of the objects of interest, a training scheme, based on principal component analysis (PCA), is proposed to derive a set of basis functions with desired properties. The derived bases are then used for the construction of a novel affine invariant function. The proposed invariant function is evaluated for the problem of aircraft silhouette identification and appears to achieve comparable performance to a popular wavelet-based affine invariant function. At the same time, the proposed framework is much simpler than that based on wavelet analysis.
2 citations
28 Jan 2007
TL;DR: In this article, Euclidean integral invariants for 3D curves in 3D with respect to the special and the full affine groups were derived using an inductive approach.
Abstract: In this paper we obtain, for the first time, explicit formulae for integral invariants for curves in 3D with respect
to the special and the full affine groups. Using an inductive approach we first compute Euclidean integral
invariants and use them to build the affine invariants. The motivation comes from problems in computer vision.
Since integration diminishes the effects of noise, integral invariants have advantage in such applications. We use
integral invariants to construct signatures that characterize curves up to the special affine transformations.
2 citations
1 Jan 2007
TL;DR: In this paper, a scale and affine invariant detector was proposed for matching cartographic images over detecting interest points that are invariant to scale, rotation and translation and robust to illumination changes and limited changes of viewpoint.
Abstract: In this paper we propose a novel approach for matching cartographic images over detecting interest points invariant to scale and affine transformations. Our scale and affine invariant detectors are based on the following recent results: Interest points extracted with the SIFT detector which is adapted to affine transformations and give repeatable results (geometrically stable). This provides a set of distinctive points which are invariant to scale, rotation and translation as well as robust to illumination changes and limited changes of viewpoint. The characteristic scale determines a scale invariant region for each point. The characteristic scale and the affine shape of neighbourhood determine an affine invariant region for each point. We apply an unsupervised classification to reduce the space of sets of interest points by using weighted bipartite graph matching in solving the point correspondence. Diffusion map: projection of the bipartite graph in a reduce space on which we apply K-means to classify the representatives clusters. The performance of our approach detector is also confirmed by excellent matching results. Keywords: Interest points, Local features, Scale invariance, Affine invariance, Matching and Recognition
1 citations
Posted Content•
14 Oct 2007TL;DR: In this article, the equivalence problem was used to classify space curves up to affine transformations with affine transformation invariants rather than the other methods for finding the invariants of space curves.
Abstract: In this paper, we will classify space curves up to affine transformations with the equivalence problem rather than the other methods for finding the invariants of space curves. As a generalization, we will study the shapes of space curves as the geodesics of affine geometry.
1 citations
TL;DR: In this article, the infinitesimal affine transformations of the Berwald connection of a spray are discussed, and the relation between the projective transformations and the affine transformation of its Berwald-Thomas-Whitehead connection is discussed.
Abstract: We discuss the infinitesimal affine transformations of the Berwald connection of a spray, and the relation between the projective transformations of a spray and the affine transformations of its Berwald–Thomas–Whitehead connection.
TL;DR: In this paper, the Centre Symmetry Set (CSS) of a smooth surface in real affine 3-space is studied and the singularities of the envelope of these chords are classified.
Abstract: Let M be a smooth surface in real affine 3-space. Consider the pairs of points of this surface at which the tangent planes are parallel and in particular the chords (infinite lines) joining these pairs. We study in detail and classify the singularities of the envelope of these chords, that is a (singular) surface tangent to all of them. This is called the Centre Symmetry Set (CSS) of M. The study is local in character and is based upon a more general investigation by the authors of the n-dimensional case. The construction of the CSS is affinely invariant and generalises the focal set of a surface in euclidean space and the affine focal set of a surface in affine space. Many standard and some unusual singularities occur in a natural way as the singularities of the CSS. There are illustrations of the various cases by means of concrete examples.