Showing papers on "Affine transformation published in 2005"

A Comparison of Affine Region Detectors

Abstract: The paper gives a snapshot of the state of the art in affine covariant region detectors, and compares their performance on a set of test images under varying imaging conditions. Six types of detectors are included: detectors based on affine normalization around Harris (Mikolajczyk and Schmid, 2002; Schaffalitzky and Zisserman, 2002) and Hessian points (Mikolajczyk and Schmid, 2002), a detector of `maximally stable extremal regions', proposed by Matas et al. (2002); an edge-based region detector (Tuytelaars and Van Gool, 1999) and a detector based on intensity extrema (Tuytelaars and Van Gool, 2000), and a detector of `salient regions', proposed by Kadir, Zisserman and Brady (2004). The performance is measured against changes in viewpoint, scale, illumination, defocus and image compression. The objective of this paper is also to establish a reference test set of images and performance software, so that future detectors can be evaluated in the same framework.

Vertex component analysis: a fast algorithm to unmix hyperspectral data

Abstract: Given a set of mixed spectral (multispectral or hyperspectral) vectors, linear spectral mixture analysis, or linear unmixing, aims at estimating the number of reference substances, also called endmembers, their spectral signatures, and their abundance fractions. This paper presents a new method for unsupervised endmember extraction from hyperspectral data, termed vertex component analysis (VCA). The algorithm exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. In a series of experiments using simulated and real data, the VCA algorithm competes with state-of-the-art methods, with a computational complexity between one and two orders of magnitude lower than the best available method.

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.

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.

A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations

Abstract: In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem - we shall "simply" treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe - Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter-time space - and develop a new greedy adaptive procedure to "optimally" construct the parameter-time sample set. Second, we propose error estimation and adjoint procedures that provide rigorous and sharp bounds for the error in specific outputs of interest: the estimates serve ap riorito construct our samples, and a posteriori to confirm fidelity. Third, based on the assumption of affine parameter dependence, we develop offline- online computational procedures: in the offline stage, we generate the reduced-basis space; in the online stage, given a new parameter value, we calculate the reduced-basis output and associated error bound. The operation count for the online stage depends only on N (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts.

A bounded-error approach to piecewise affine system identification

Abstract: This paper proposes a three-stage procedure for parametric identification of piecewise affine autoregressive exogenous (PWARX) models. The first stage simultaneously classifies the data points and estimates the number of submodels and the corresponding parameters by solving the partition into a minimum number of feasible subsystems (MIN PFS) problem for a suitable set of linear complementary inequalities derived from data. Second, a refinement procedure reduces misclassifications and improves parameter estimates. The third stage determines a polyhedral partition of the regressor set via two-class or multiclass linear separation techniques. As a main feature, the algorithm imposes that the identification error is bounded by a quantity /spl delta/. Such a bound is a useful tuning parameter to trade off between quality of fit and model complexity. The performance of the proposed PWA system identification procedure is demonstrated via numerical examples and on experimental data from an electronic component placement process in a pick-and-place machine.

Deformable registration of diffusion tensor MR images with explicit orientation optimization.

Abstract: In this paper we present a novel deformable registration algorithm for diffusion tensor (DT) MR images that enables explicit analytic optimization of tensor reorientation. The optimization seeks a piecewise affine transformation that divides the image domain into uniform regions and transforms each of them affinely. The objective function captures both the image similarity and the smoothness of the transformation across region boundaries. The image similarity enables explicit orientation optimization by incorporating tensor reorientation, which is necessary for warping DT images. The objective function is formulated in a way that allows explicit implementation of analytic derivatives to drive fast and accurate optimization using the conjugate gradient method. The optimal transformation is hierarchically refined in a subdivision framework. A comparison with affine registration for inter-subject normalization of 8 subjects shows that our algorithm improves the alignment of manually segmented white matter structures (corpus callosum and cortio-spinal tracts).

Matrix description of general motion correction applied to multishot images

Abstract: Motion of an object degrades MR images, as the acquisition is time-dependent, and thus k-space is inconsistently sampled. This causes ghosts. Current motion correction methods make restrictive assumptions on the type of motions, for example, that it is a translation or rotation, and use special properties of k-space for these transformations. Such methods, however, cannot be generalized easily to nonrigid types of motions, and even rotations in multiple shots can be a problem. Here, a method is presented that can handle general nonrigid motion models. A general matrix equation gives the corrupted image from the ideal object. Thus, inversion of this system allows us to get the ideal image from the corrupted one. This inversion is possible by efficient methods mixing Fourier transforms with the conjugate gradient method. A faster but empirical inversion is discussed as well as methods to determine the motion. Simulated three-dimensional affine data and two-dimensional pulsation data and in vivo nonrigid data are used for demonstration. All examples are multishot images where the object moves between shots. The results indicate that it is now possible to correct for nonrigid types of motion that are representative of many types of patient motion, although computation times remain an issue.

A maximum entropy framework for part-based texture and object recognition

Abstract: This paper presents a probabilistic part-based approach for texture and object recognition. Textures are represented using a part dictionary found by quantizing the appearance of scale- or affine- invariant keypoints. Object classes are represented using a dictionary of composite semi-local parts, or groups of neighboring keypoints with stable and distinctive appearance and geometric layout. A discriminative maximum entropy framework is used to learn the posterior distribution of the class label given the occurrences of parts from the dictionary in the training set. Experiments on two texture and two object databases demonstrate the effectiveness of this framework for visual classification.

Deformable registration of diffusion tensor MR images with explicit orientation optimization

Abstract: In this paper we present a novel deformable registration algorithm for diffusion tensor (DT) MR images that enables explicit analytic optimization of tensor reorientation. The optimization seeks a piecewise affine transformation that divides the image domain into uniform regions and transforms each of them affinely. The objective function captures both the image similarity and the smoothness of the transformation across region boundaries. The image similarity enables explicit orientation optimization by incorporating tensor reorientation, which is necessary for warping DT images. The objective function is formulated in a way that allows explicit implementation of analytic derivatives to drive fast and accurate optimization using the conjugate gradient method. The optimal transformation is hierarchically refined in a subdivision framework. A comparison with affine registration for inter-subject normalization of 8 subjects shows that our algorithm improves the alignment of manually segmented white matter structures (corpus callosum and cortio-spinal tracts).

Formulae for Arithmetic on Genus 2 Hyperelliptic Curves

Abstract: The ideal class group of hyperelliptic curves can be used in cryptosystems based on the discrete logarithm problem. In this article we present explicit formulae to perform the group operations for genus 2 curves. The formulae are completely general but to achieve the lowest number of operations we treat odd and even characteristic separately. We present 3 different coordinate systems which are suitable for different environments, e. g. on a smart card we should avoid inversions while in software a limited number is acceptable. The presented formulae render genus two hyperelliptic curves very useful in practice. The first system are affine coordinates where each group operation needs one inversion. Then we consider projective coordinates avoiding inversions on the cost of more multiplications and a further coordinate. Finally, we introduce a new system of coordinates and state algorithms showing that doublings are comparably cheap and no inversions are needed. A comparison between the systems concludes the paper.

Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration

Abstract: We investigate the projective properties of the feature consisting of two concentric circles. We demonstrate there exist geometric and algebraic constraints on its projection. We show how these constraints greatly simplify the recoveries of the affine and Euclidean structures of a 3D plane. As an application, we assess the performances of two camera calibration algorithms.

Deformation invariant image matching

Abstract: We propose a novel framework to build descriptors of local intensity that are invariant to general deformations. In this framework, an image is embedded as a 2D surface in 3D space, with intensity weighted relative to distance in x-y. We show that as this weight increases, geodesic distances on the embedded surface are less affected by image deformations. In the limit, distances are deformation invariant. We use geodesic sampling to get neighborhood samples for interest points, and then use a geodesic-intensity histogram (GIH) as a deformation invariant local descriptor. In addition to its invariance, the new descriptor automatically finds its support region. This means it can safely gather information from a large neighborhood to improve discriminability. Furthermore, we propose a matching method for this descriptor that is invariant to affine lighting changes. We have tested this new descriptor on interest point matching for two data sets, one with synthetic deformation and lighting change, and another with real non-affine deformations. Our method shows promising matching results compared to several other approaches

Efficient population registration of 3d data

Abstract: We present a population registration framework that acts on large collections or populations of data volumes. The data alignment procedure runs in a simultaneous fashion, with every member of the population approaching the central tendency of the collection at the same time. Such a mechanism eliminates the need for selecting a particular reference frame a priori, resulting in a non-biased estimate of a digital atlas. Our algorithm adopts an affine congealing framework with an information theoretic objective function and is optimized via a gradient-based stochastic approximation process embedded in a multi-resolution setting. We present experimental results on both synthetic and real images.

CV-SLAM: a new ceiling vision-based SLAM technique

Abstract: We propose a fast and robust CV-SLAM (ceiling vision-based simultaneous localization and mapping) technique using a single ceiling vision sensor. The proposed algorithm is suitable for system that demands very high localization accuracy such as an intelligent robot vacuum cleaner. A single camera looking upward direction (called ceiling vision system) is mounted on the robot, and salient image features are detected and tracked through the image sequence. Compared with the conventional frontal view systems, the ceiling vision has advantage in tracking, since it involves only rotation and affine transform without scale change. And, in this paper, we solve the rotation and affine transform problems using 3D gradient orientation estimation method and multi-view description of landmarks. By applying these methods to the solution for data association, we can reconstruct the 3D landmark map in real-time through the extend Kalman filter based SLAM framework. Furthermore, relocation problem is solved efficiently by using a wide base line matching between the reconstructed 3D map and a 2D ceiling image. Experimental results demonstrate the accuracy and robustness of the proposed algorithm in real environments.

Motion layer extraction in the presence of occlusion using graph cuts

Abstract: Extracting layers from video is very important for video representation, analysis, compression, and synthesis. Assuming that a scene can be approximately described by multiple planar regions, this paper describes a robust and novel approach to automatically extract a set of affine or projective transformations induced by these regions, detect the occlusion pixels over multiple consecutive frames, and segment the scene into several motion layers. First, after determining a number of seed regions using correspondences in two frames, we expand the seed regions and reject the outliers employing the graph cuts method integrated with level set representation. Next, these initial regions are merged into several initial layers according to the motion similarity. Third, an occlusion order constraint on multiple frames is explored, which enforces that the occlusion area increases with the temporal order in a short period and effectively maintains segmentation consistency over multiple consecutive frames. Then, the correct layer segmentation is obtained by using a graph cuts algorithm and the occlusions between the overlapping layers are explicitly determined. Several experimental results are demonstrated to show that our approach is effective and robust.

Affine Structure From Sound

Abstract: We consider the problem of localizing a set of microphones together with a set of external acoustic events (e.g., hand claps), emitted at unknown times and unknown locations. We propose a solution that approximates this problem under a far field approximation defined in the calculus of affine geometry, and that relies on singular value decomposition (SVD) to recover the affine structure of the problem. We then define low-dimensional optimization techniques for embedding the solution into Euclidean geometry, and further techniques for recovering the locations and emission times of the acoustic events. The approach is useful for the calibration of ad-hoc microphone arrays and sensor networks.

Metric 3D reconstruction and texture acquisition of surfaces of revolution from a single uncalibrated view

Abstract: Image analysis and computer vision can be effectively employed to recover the three-dimensional structure of imaged objects, together with their surface properties. In this paper, we address the problem of metric reconstruction and texture acquisition from a single uncalibrated view of a surface of revolution (SOR). Geometric constraints induced in the image by the symmetry properties of the SOR structure are exploited to perform self-calibration of a natural camera, 3D metric reconstruction, and texture acquisition. By exploiting the analogy with the geometry of single axis motion, we demonstrate that the imaged apparent contour and the visible segments of two imaged cross sections in a single SOR view provide enough information for these tasks. Original contributions of the paper are: single view self-calibration and reconstruction based on planar rectification, previously developed for planar surfaces, has been extended to deal also with the SOR class of curved surfaces; self-calibration is obtained by estimating both camera focal length (one parameter) and principal point (two parameters) from three independent linear constraints for the SOR fixed entities; the invariant-based description of the SOR scaling function has been extended from affine to perspective projection. The solution proposed exploits both the geometric and topological properties of the transformation that relates the apparent contour to the SOR scaling function. Therefore, with this method, a metric localization of the SOR occluded parts can be made, so as to cope with them correctly. For the reconstruction of textured SORs, texture acquisition is performed without requiring the estimation of external camera calibration parameters, but only using internal camera parameters obtained from self-calibration.

LS galleries, the path model, and MV cycles

Abstract: We give an interpretation of the path model of a representation [18] of a complex semisimple algebraic group G in terms of the geometry of its affine Grassmannian. In this setting, the paths are replaced by LS–galleries in the affine Coxeter complex associated to the Weyl group of G. To explain the connection with geometry, consider a Demazure–Hansen–Bott–Samelson desingularization ˆ Σ(λ) of the closure of an orbit G(C[[t]]).λ in the affine Grassmannian. The homology of ˆ Σ(λ) has a basis given by Bia lynicki–Birula cell’s, which are indexed by the T –fixed points in ˆ Σ(λ). Now the points of ˆ Σ(λ) can be identified with galleries of a fixed type in the affine Tits building associated to G, and the T –fixed points correspond in this language to combinatorial galleries of a fixed type in the affine Coxeter complex. We determine those galleries such that the associated cell has a non-empty intersection with G(C[[t]]).λ (identified with an open subset of ˆ Σ(λ)), and we show that the closures of the strata associated to LS-galleries are exactly the MV–cycles [24], which form a basis of the representation V (λ) for the Langland’s dual group G ∨ .

Non Mean Reverting Affine Processes for Stochastic Mortality

Abstract: In this paper we use doubly stochastic processes (or Cox processes) in order to model the random evolution of mortality of an individual. These processes have been widely used in the credit risk literature in modelling default arrival, and in this context have proved to be quite flexible, especially when the intensity process is of the affine class. We investigate the applicability of affine processes in describing the individual's intensity of mortality, and provide a calibration to the Italian and UK populations. Results from the calibration seem to suggest that, in spite of their popularity in the financial context, mean reverting processes are not suitable for describing the death intensity of individuals. On the contrary, affine processes whose deterministic part increases exponentially seem to be appropriate. As for the stochastic part, negative jumps seem to do a better job than diffusive components. Stress analysis and analytical results indicate that increasing the randomness of the intensity process results in improvements in survivorship.

Toric degenerations of spherical varieties

Abstract: We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.

A tensor decomposition for geometric grouping and segmentation

Abstract: While spectral clustering has been applied successfully to problems in computer vision, their applicability is limited to pairwise similarity measures that form a probability matrix. However many geometric problems with parametric forms require more than two observations to estimate a similarity measure, e.g. epipolar geometry. In such cases we can only define the probability of belonging to the same cluster for an n-tuple of points and not just a pair, leading to an n-dimensional probability tensor. However spectral clustering methods are not available for tensors. In this paper we present an algorithm to infer a similarity matrix by decomposing the n-dimensional probability tensor. Our method exploits the super-symmetry of the probability tensor to provide a randomised scheme that does not require the explicit computation of the probability tensor. Our approach is fast and accurate and its applicability is illustrated on two significant problems, namely perceptually salient geometric grouping and parametric motion segmentation (like affine, epipolar etc).

Otolith shape contour analysis using affine transformation invariant wavelet transforms and curvature scale space representation

Abstract: Fish otolith morphology has been closely related to landmark selection in order to establish the most discriminating points that can help to differentiate or find common characteristics in sets of otolith images. Fourier analysis has traditionally been used to represent otolith images, since it can reconstruct a version of the contour that is close to the original by choosing a reduced set of harmonic terms. However, it is difficult to locate the contour's singularities from this spectrum. As an alternative, wavelet transform and curvature scale space representation allow us to quantify the irregularities of the contour and determine its precise position. These properties make these techniques suitable for pattern recognition purposes, ageing, stock determination and species identification studies. In the present study both techniques are applied and used in an otolith classification system that shows robustness against affine image transformations, shears and the presence of noise. The results are interpreted and discussed in relation to traditional morphology studies.