Affine transformation

About: Affine transformation is a research topic. Over the lifetime, 23531 publications have been published within this topic receiving 434668 citations. The topic is also known as: Affine map.

A sparse texture representation using local affine regions

Abstract: This paper introduces a texture representation suitable for recognizing images of textured surfaces under a wide range of transformations, including viewpoint changes and nonrigid deformations. At the feature extraction stage, a sparse set of affine Harris and Laplacian regions is found in the image. Each of these regions can be thought of as a texture element having a characteristic elliptic shape and a distinctive appearance pattern. This pattern is captured in an affine-invariant fashion via a process of shape normalization followed by the computation of two novel descriptors, the spin image and the RIFT descriptor. When affine invariance is not required, the original elliptical shape serves as an additional discriminative feature for texture recognition. The proposed approach is evaluated in retrieval and classification tasks using the entire Brodatz database and a publicly available collection of 1,000 photographs of textured surfaces taken from different viewpoints.

Indexing based on scale invariant interest points

Abstract: This paper presents a new method for detecting scale invariant interest points. The method is based on two recent results on scale space: (1) Interest points can be adapted to scale and give repeatable results (geometrically stable). (2) Local extrema over scale of normalized derivatives indicate the presence of characteristic local structures. Our method first computes a multi-scale representation for the Harris interest point detector. We then select points at which a local measure (the Laplacian) is maximal over scales. This allows a selection of distinctive points for which the characteristic scale is known. These points are invariant to scale, rotation and translation as well as robust to illumination changes and limited changes of viewpoint. For indexing, the image is characterized by a set of scale invariant points; the scale associated with each point allows the computation of a scale invariant descriptor. Our descriptors are, in addition, invariant to image rotation, of affine illumination changes and robust to small perspective deformations. Experimental results for indexing show an excellent performance up to a scale factor of 4 for a database with more than 5000 images.

Generalized principal component analysis (GPCA)

Abstract: This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point. When the number of subspaces is known, we show that these polynomials can be estimated linearly from data; hence, subspace segmentation is reduced to classifying one point per subspace. We select these points optimally from the data set by minimizing certain distance function, thus dealing automatically with moderate noise in the data. A basis for the complement of each subspace is then recovered by applying standard PCA to the collection of derivatives (normal vectors). Extensions of GPCA that deal with data in a high-dimensional space and with an unknown number of subspaces are also presented. Our experiments on low-dimensional data show that GPCA outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as k-subspaces and expectation maximization. We also present applications of GPCA to computer vision problems such as face clustering, temporal video segmentation, and 3D motion segmentation from point correspondences in multiple affine views.

Affine Processes and Applications in Finance

Abstract: We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and Ornstein-Uhlenbeck type processes. We show, and provide foundations for, a wide range of financial applications for regular affine processes.

Affine parameter-dependent Lyapunov functions and real parametric uncertainty

Abstract: This paper presents new tests to analyze the robust stability and/or performance of linear systems with uncertain real parameters. These tests are extensions of the notions of quadratic stability and performance where the fixed quadratic Lyapunov function is replaced by a Lyapunov function with affine dependence on the uncertain parameters. Admittedly with some conservatism, the construction of such parameter-dependent Lyapunov functions can be reduced to a linear matrix inequality (LMI) problem and hence is numerically tractable. These LMI-based tests are applicable to constant or time-varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations. They also avoid the frequency sweep needed in real-/spl mu/ analysis, and numerical experiments indicate that they often compare favorably with /spl mu/ analysis for time-invariant parameter uncertainty.