Showing papers on "Affine transformation published in 2011"

Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms

Abstract: This book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite dimension (algebraic systems) and in infinite dimension (ordinary and partial differential equations) Its focus is on local and global Newton methods for direct problems or Gauss-Newton methods for inverse problems The term 'affine invariance' means that the presented algorithms and their convergence analysis are invariant under one out of four subclasses of affine transformations of the problem to be solved Compared to traditional textbooks, the distinguishing affine invariance approach leads to shorter theorems and proofs and permits the construction of fully adaptive algorithms Lots of numerical illustrations, comparison tables, and exercises make the text useful in computational mathematics classes At the same time, the book opens many directions for possible future research

ASIFT: An Algorithm for Fully Affine Invariant Comparison

Abstract: If a physical object has a smooth or piecewise smooth boundary, its images obtained by cameras in varying positions undergo smooth apparent deformations. These deformations are locally well approximated by affine transforms of the image plane. In consequencethe solid object recognition problem has often been led back to the computation of affine invariant image local features. The similarity invariance (invariance to translation, rotation, and zoom) is dealt with rigorously by the SIFT method The method illustrated and demonstrated in this work, AffineSIFT (ASIFT), simulates a set of sample views of the initial images, obtainable by varying the two camera axis orientation parameters, namely the latitude and the longitude angles, which are not treated by the SIFT method. Then it applies the SIFT method itself to all images thus generated. Thus, ASIFT covers effectively all six parameters of the affine transform. Source Code The source code (ANSI C), its documentation, and the online demo are accessible at the IPOL web page of this article 1 .

Sparse approximated nearest points for image set classification

Abstract: Classification based on image sets has recently attracted great research interest as it holds more promise than single image based classification. In this paper, we propose an efficient and robust algorithm for image set classification. An image set is represented as a triplet: a number of image samples, their mean and an affine hull model. The affine hull model is used to account for unseen appearances in the form of affine combinations of sample images. We introduce a novel between-set distance called Sparse Approximated Nearest Point (SANP) distance. Unlike existing methods, the dissimilarity of two sets is measured as the distance between their nearest points, which can be sparsely approximated from the image samples of their respective set. Different from standard sparse modeling of a single image, this novel sparse formulation for the image set enforces sparsity on the sample coefficients rather than the model coefficients and jointly optimizes the nearest points as well as their sparse approximations. A convex formulation for searching the optimal SANP between two sets is proposed and the accelerated proximal gradient method is adapted to efficiently solve this optimization. Experimental evaluation was performed on the Honda, MoBo and Youtube datasets. Comparison with existing techniques shows that our method consistently achieves better results.

Local Affine Multidimensional Projection

Abstract: Multidimensional projection techniques have experienced many improvements lately, mainly regarding computational times and accuracy. However, existing methods do not yet provide flexible enough mechanisms for visualization-oriented fully interactive applications. This work presents a new multidimensional projection technique designed to be more flexible and versatile than other methods. This novel approach, called Local Affine Multidimensional Projection (LAMP), relies on orthogonal mapping theory to build accurate local transformations that can be dynamically modified according to user knowledge. The accuracy, flexibility and computational efficiency of LAMP is confirmed by a comprehensive set of comparisons. LAMP's versatility is exploited in an application which seeks to correlate data that, in principle, has no connection as well as in visual exploration of textual documents.

Smoothly varying affine stitching

Abstract: Traditional image stitching using parametric transforms such as homography, only produces perceptually correct composites for planar scenes or parallax free camera motion between source frames. This limits mosaicing to source images taken from the same physical location. In this paper, we introduce a smoothly varying affine stitching field which is flexible enough to handle parallax while retaining the good extrapolation and occlusion handling properties of parametric transforms. Our algorithm which jointly estimates both the stitching field and correspondence, permits the stitching of general motion source images, provided the scenes do not contain abrupt protrusions.

SpaRCS: Recovering low-rank and sparse matrices from compressive measurements

Abstract: We consider the problem of recovering a matrix M that is the sum of a low-rank matrix L and a sparse matrix S from a small set of linear measurements of the form y = A(M)= A(L + S). This model subsumes three important classes of signal recovery problems: compressive sensing, affine rank minimization, and robust principal component analysis. We propose a natural optimization problem for signal recovery under this model and develop a new greedy algorithm called SpaRCS to solve it. Empirically, SpaRCS inherits a number of desirable properties from the state-of-the-art CoSaMP and ADMiRA algorithms, including exponential convergence and efficient implementation. Simulation results with video compressive sensing, hyperspectral imaging, and robust matrix completion data sets demonstrate both the accuracy and efficacy of the algorithm.

Asynchronous Output-Feedback Control of Networked Nonlinear Systems With Multiple Packet Dropouts: T–S Fuzzy Affine Model-Based Approach

Abstract: This paper investigates the problem of robust output-feedback control for a class of networked nonlinear systems with multiple packet dropouts. The nonlinear plant is represented by Takagi-Sugeno (T-S) fuzzy affine dynamic models with norm-bounded uncertainties, and stochastic variables that satisfy the Bernoulli random binary distribution are adopted to characterize the data-missing phenomenon. The objective is to design an admissible output-feedback controller that guarantees the stochastic stability of the resulting closed-loop system with a prescribed disturbance attenuation level. It is assumed that the plant premise variables, which are often the state variables or their functions, are not measurable so that the controller implementation with state-space partition may not be synchronous with the state trajectories of the plant. Based on a piecewise quadratic Lyapunov function combined with an S-procedure and some matrix inequality convexifying techniques, two different approaches to robust output-feedback controller design are developed for the underlying T-S fuzzy affine systems with unreliable communication links. The solutions to the problem are formulated in the form of linear matrix inequalities (LMIs). Finally, simulation examples are provided to illustrate the effectiveness of the proposed approaches.

Computing Smooth Time Trajectories for Camera and Deformable Shape in Structure from Motion with Occlusion

Abstract: We address the classical computer vision problems of rigid and nonrigid structure from motion (SFM) with occlusion. We assume that the columns of the input observation matrix W describe smooth 2D point trajectories over time. We then derive a family of efficient methods that estimate the column space of W using compact parameterizations in the Discrete Cosine Transform (DCT) domain. Our methods tolerate high percentages of missing data and incorporate new models for the smooth time trajectories of 2D-points, affine and weak-perspective cameras, and 3D deformable shape. We solve a rigid SFM problem by estimating the smooth time trajectory of a single camera moving around the structure of interest. By considering a weak-perspective camera model from the outset, we directly compute euclidean 3D shape reconstructions without requiring postprocessing steps such as euclidean upgrade and bundle adjustment. Our results on real SFM data sets with high percentages of missing data compared positively to those in the literature. In nonrigid SFM, we propose a novel 3D shape trajectory approach that solves for the deformable structure as the smooth time trajectory of a single point in a linear shape space. A key result shows that, compared to state-of-the-art algorithms, our nonrigid SFM method can better model complex articulated deformation with higher frequency DCT components while still maintaining the low-rank factorization constraint. Finally, we also offer an approach for nonrigid SFM when W is presented with missing data.

Representations of Hecke Algebras at Roots of Unity

Abstract: Generic Iwahori-Hecke algebras- Kazhdan-Lusztig cells and cellular bases- Specialisations and decomposition maps- Hecke algebras and finite groups of Lie type- Representation theory of Ariki-Koike algebras- Canonical bases in affine type A and Ariki's theorem- Decomposition numbers for exceptional types

From contours to 3D object detection and pose estimation

Abstract: This paper addresses view-invariant object detection and pose estimation from a single image. While recent work focuses on object-centered representations of point-based object features, we revisit the viewer-centered framework, and use image contours as basic features. Given training examples of arbitrary views of an object, we learn a sparse object model in terms of a few view-dependent shape templates. The shape templates are jointly used for detecting object occurrences and estimating their 3D poses in a new image. Instrumental to this is our new mid-level feature, called bag of boundaries (BOB), aimed at lifting from individual edges toward their more informative summaries for identifying object boundaries amidst the background clutter. In inference, BOBs are placed on deformable grids both in the image and the shape templates, and then matched. This is formulated as a convex optimization problem that accommodates invariance to non-rigid, locally affine shape deformations. Evaluation on benchmark datasets demonstrates our competitive results relative to the state of the art.

Sampling and Reconstructing Signals From a Union of Linear Subspaces

Abstract: In this paper, we study the problem of sampling and reconstructing signals which are assumed to lie on or close to one of several subspaces of a Hilbert space. Importantly, we here consider a very general setting in which we allow infinitely many subspaces in infinite dimensional Hilbert spaces. This general approach allows us to unify many results derived recently in areas such as compressed sensing, affine rank minimization, analog compressed sensing and structured matrix decompositions.

Loop transformations: convexity, pruning and optimization

Abstract: High-level loop transformations are a key instrument in mapping computational kernels to effectively exploit the resources in modern processor architectures. Nevertheless, selecting required compositions of loop transformations to achieve this remains a significantly challenging task; current compilers may be off by orders of magnitude in performance compared to hand-optimized programs. To address this fundamental challenge, we first present a convex characterization of all distinct, semantics-preserving, multidimensional affine transformations. We then bring together algebraic, algorithmic, and performance analysis results to design a tractable optimization algorithm over this highly expressive space. Our framework has been implemented and validated experimentally on a representative set of benchmarks running on state-of-the-art multi-core platforms.

Static analysis of finite precision computations

Abstract: We define several abstract semantics for the static analysis of finite precision computations, that bound not only the ranges of values taken by numerical variables of a program, but also the difference with the result of the same sequence of operations in an idealized real number semantics. These domains point out with more or less detail (control point, block, function for instance) sources of numerical errors in the program and the way they were propagated by further computations, thus allowing to evaluate not only the rounding error, but also sensitivity to inputs or parameters of the program. We describe two classes of abstractions, a non relational one based on intervals, and a weakly relational one based on parametrized zonotopic abstract domains called affine sets, especially well suited for sensitivity analysis and test generation. These abstract domains are implemented in the Fluctuat static analyzer, and we finally present some experiments.

Affine Legendre Moment Invariants for Image Watermarking Robust to Geometric Distortions

Abstract: Geometric distortions are generally simple and effective attacks for many watermarking methods. They can make detection and extraction of the embedded watermark difficult or even impossible by destroying the synchronization between the watermark reader and the embedded watermark. In this paper, we propose a new watermarking approach which allows watermark detection and extraction under affine transformation attacks. The novelty of our approach stands on a set of affine invariants we derived from Legendre moments. Watermark embedding and detection are directly performed on this set of invariants. We also show how these moments can be exploited for estimating the geometric distortion parameters in order to permit watermark extraction. Experimental results show that the proposed watermarking scheme is robust to a wide range of attacks: geometric distortion, filtering, compression, and additive noise.

Generalized Gaussian Scale-Space Axiomatics Comprising Linear Scale-Space, Affine Scale-Space and Spatio-Temporal Scale-Space

Abstract: This paper describes a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space and linear spatio-temporal scale-space using a similar set of assumptions (scale-space axioms). The notion of non-enhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general non-isotropic kernels over both spatial and spatio-temporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of shape adaptation and velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal spatio-temporal scale-spaces can be derived from similar assumptions. The mathematical structure of these scale-spaces is analyzed in detail concerning transformation properties over space and time, the temporal cascade structure they satisfy over time as well as properties of the resulting multi-scale spatio-temporal derivative operators. It is also shown how temporal derivatives with respect to transformed time can be defined, leading to the formulation of a novel analogue of scale normalized derivatives for time-causal scale-spaces. The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatio-temporal scale-space as well as the time-causal spatio-temporal scale-space relate to spatio-temporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to space-time separable cells in the LGN as well as to both space-time separable and non-separable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasi-quadrature measures expressed in terms of scale normalized spatio-temporal derivatives. The theories presented as well as their relations to biological vision show that it is possible to describe a general set of Gaussian and/or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area.

Affine processes on positive semidefinite matrices

Abstract: This article provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices This analysis has been motivated by a large and growing use of matrix-valued affine processes in finance, including multi-asset option pricing with stochastic volatility and correlation structures, and fixed-income models with stochastically correlated risk factors and default intensities

Equivalences Between GIT Quotients of Landau-Ginzburg B-Models

Abstract: We define the category of B-branes in a (not necessarily affine) Landau-Ginzburg B-model, incorporating the notion of R-charge. Our definition is a direct generalization of the category of perfect complexes. We then consider pairs of Landau-Ginzburg B-models that arise as different GIT quotients of a vector space by a one-dimensional torus, and show that for each such pair the two categories of B-branes are quasi-equivalent. In fact we produce a whole set of quasi-equivalences indexed by the integers, and show that the resulting auto-equivalences are all spherical twists.

Simplest Color Balance

Abstract: In this paper we present the simplest possible color balance algorithm. The assumption underlying this algorithm is that the highest values of R, G, B observed in the image must correspond to white, and the lowest values to obscurity. The algorithm simply stretches, as much as it can, the values of the three channels Red, Green, Blue (R, G, B), so that they occupy the maximal possible range [0, 255] by applying an affine transform ax+b to each channel. Since many images contain a few aberrant pixels that already occupy the 0 and 255 values, the proposed method saturates a small percentage of the pixels with the highest values to 255 and a small percentage of the pixels with the lowest values to 0, before applying the affine transform. Source Code The source code (ANSI C), its documentation, and the online demo are accessible at the IPOL web page of this article.

In-vehicle camera traffic sign detection and recognition

Abstract: In this paper, we discuss theoretical foundations and a practical realization of a real-time traffic sign detection, tracking and recognition system operating on board of a vehicle. In the proposed framework, a generic detector refinement procedure based on mean shift clustering is introduced. This technique is shown to improve the detection accuracy and reduce the number of false positives for a broad class of object detectors for which a soft response’s confidence can be sensibly estimated. The track of an already established candidate is maintained over time using an instance-specific tracking function that encodes the relationship between a unique feature representation of the target object and the affine distortions it is subject to. We show that this function can be learned on-the-fly via regression from random transformations applied to the image of the object in known pose. Secondly, we demonstrate its capability of reconstructing the full-face view of a sign from substantial view angles. In the recognition stage, a concept of class similarity measure learned from image pairs is discussed and its realization using SimBoost, a novel version of AdaBoost algorithm, is analyzed. Suitability of the proposed method for solving multi-class traffic sign classification problems is shown experimentally for different feature representations of an image. Overall performance of our system is evaluated based on a prototype C++ implementation. Illustrative output generated by this demo application is provided as a supplementary material attached to this paper.

Efficient algorithm for low-rank matrix factorization with missing components and performance comparison of latest algorithms

Abstract: This paper examines numerical algorithms for factorization of a low-rank matrix with missing components. We first propose a new method that incorporates a damping factor into the Wiberg method to solve the problem. The new method is characterized by the way it constrains the ambiguity of the matrix factorization, which helps improve both the global convergence ability and the local convergence speed. We then present experimental comparisons with the latest methods used to solve the problem. No comprehensive comparison of the methods that have been proposed recently has yet been reported in literature. In our experiments, we prioritize the assessment of the global convergence performance of each method, that is, how often and how fast the method can reach the global optimum starting from random initial values. Our conclusion is that top performance is achieved by a group of methods based on Newton-family minimization with damping factor that reduce the problem by eliminating either of the two factored matrices. Our method, which belongs to this group, consistently shows a 100% global convergence rate for different types of affine structure from motion data with a very high population of missing components.

Density Approximations for Multivariate Affine Jump-Diffusion Processes

Abstract: We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existence and differentiability of transition densities of affine models and show how they naturally fit into the approximation framework. Empirical applications in option pricing, credit risk, and likelihood inference highlight the usefulness of our expansions. The approximations are extremely fast to evaluate, and they perform very accurately and numerically stable.

Real-Time Affine Global Motion Estimation Using Phase Correlation and its Application for Digital Image Stabilization

Abstract: We propose a fast and robust 2D-affine global motion estimation algorithm based on phase-correlation in the Fourier-Mellin domain and robust least square model fitting of sparse motion vector field and its application for digital image stabilization Rotation-scale-translation (RST) approximation of affine parameters is obtained at the coarsest level of the image pyramid, thus ensuring convergence for a much larger range of motions Despite working at the coarsest resolution level, using subpixel-accurate phase correlation provides sufficiently accurate coarse estimates for the subsequent refinement stage of the algorithm The refinement stage consists of RANSAC based robust least-square model fitting for sparse motion vector field, estimated using block-based subpixel-accurate phase correlation at randomly selected high activity regions in finest level of image pyramid Resulting algorithm is very robust to outliers such as foreground objects and flat regions We investigate the robustness of the proposed method for digital image stabilization application Experimental results show that the proposed algorithm is capable of estimating larger range of motions as compared to another phase correlation method and optical flow algorithm

Efficient subspace segmentation via quadratic programming

Abstract: We explore in this paper efficient algorithmic solutions to robust subspace segmentation We propose the SSQP, namely Subspace Segmentation via Quadratic Programming, to partition data drawn from multiple subspaces into multiple clusters The basic idea of SSQP is to express each datum as the linear combination of other data regularized by an overall term targeting zero reconstruction coefficients over vectors from different subspaces The derived coefficient matrix by solving a quadratic programming problem is taken as an affinity matrix, upon which spectral clustering is applied to obtain the ultimate segmentation result Similar to sparse subspace clustering (SCC) and low-rank representation (LRR), SSQP is robust to data noises as validated by experiments on toy data Experiments on Hopkins 155 database show that SSQP can achieve competitive accuracy as SCC and LRR in segmenting affine subspaces, while experimental results on the Extended Yale Face Database B demonstrate SSQP's superiority over SCC and LRR Beyond segmentation accuracy, all experiments show that SSQP is much faster than both SSC and LRR in the practice of subspace segmentation

A New Robust Watermarking Scheme Based on RDWT-SVD

Abstract: In this paper, a new method for non-blind image watermarking that is robust against affine transformation and ordinary image manipulation is presented. The suggested method presents a watermarking scheme based on redundant discrete wavelet transform and Singular Value Decomposition. After applying RDWT to both cover and watermark images, we apply SVD to the LL subbands of them. We then modify singular values of the cover image using singular values of the visual watermark. The advantage of the proposed technique is its robustness against most common attacks. Analysis and experimental results show higher performance of the proposed method in comparison with the DWT-SVD method.

Yangians and cohomology rings of Laumon spaces

Abstract: Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL n. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspon- dences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of sln(s ±1 , t)) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of coho- mology. This basis is an affine analog of the Gelfand-Tsetlin basis. The affine analog of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space Mn,d of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is