Showing papers on "Affine transformation published in 2009"

ASIFT: A New Framework for Fully Affine Invariant Image 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 consequence the solid object recognition problem has often been led back to the computation of affine invariant image local features. Such invariant features could be obtained by normalization methods, but no fully affine normalization method exists for the time being. Even scale invariance is dealt with rigorously only by the scale-invariant feature transform (SIFT) method. By simulating zooms out and normalizing translation and rotation, SIFT is invariant to four out of the six parameters of an affine transform. The method proposed in this paper, affine-SIFT (ASIFT), simulates all image views obtainable by varying the two camera axis orientation parameters, namely, the latitude and the longitude angles, left over by the SIFT method. Then it covers the other four parameters by using the SIFT method itself. The resulting method will be mathematically proved to be fully affine invariant. Against any prognosis, simulating all views depending on the two camera orientation parameters is feasible with no dramatic computational load. A two-resolution scheme further reduces the ASIFT complexity to about twice that of SIFT. A new notion, the transition tilt, measuring the amount of distortion from one view to another, is introduced. While an absolute tilt from a frontal to a slanted view exceeding 6 is rare, much higher transition tilts are common when two slanted views of an object are compared (see Figure hightransitiontiltsillustration). The attainable transition tilt is measured for each affine image comparison method. The new method permits one to reliably identify features that have undergone transition tilts of large magnitude, up to 36 and higher. This fact is substantiated by many experiments which show that ASIFT significantly outperforms the state-of-the-art methods SIFT, maximally stable extremal region (MSER), Harris-affine, and Hessian-affine.

Sparse subspace clustering

Abstract: We propose a method based on sparse representation (SR) to cluster data drawn from multiple low-dimensional linear or affine subspaces embedded in a high-dimensional space. Our method is based on the fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all other data points. In general, finding such a SR is NP hard. Our key contribution is to show that, under mild assumptions, the SR can be obtained `exactly' by using l1 optimization. The segmentation of the data is obtained by applying spectral clustering to a similarity matrix built from this SR. Our method can handle noise, outliers as well as missing data. We apply our subspace clustering algorithm to the problem of segmenting multiple motions in video. Experiments on 167 video sequences show that our approach significantly outperforms state-of-the-art methods.

Moments and Moment Invariants in Pattern Recognition

Abstract: Moments as projections of an images intensity onto a proper polynomial basis can be applied to many different aspects of image processing. These include invariant pattern recognition, image normalization, image registration, focus/ defocus measurement, and watermarking. This book presents a survey of both recent and traditional image analysis and pattern recognition methods, based on image moments, and offers new concepts of invariants to linear filtering and implicit invariants. In addition to the theory, attention is paid to efficient algorithms for moment computation in a discrete domain, and to computational aspects of orthogonal moments. The authors also illustrate the theory through practical examples, demonstrating moment invariants in real applications across computer vision, remote sensing and medical imaging. Key features: Presents a systematic review of the basic definitions and properties of moments covering geometric moments and complex moments. Considers invariants to traditional transforms translation, rotation, scaling, and affine transform - from a new point of view, which offers new possibilities of designing optimal sets of invariants. Reviews and extends a recent field of invariants with respect to convolution/blurring. Introduces implicit moment invariants as a tool for recognizing elastically deformed objects. Compares various classes of orthogonal moments (Legendre, Zernike, Fourier-Mellin, Chebyshev, among others) and demonstrates their application to image reconstruction from moments. Offers comprehensive advice on the construction of various invariants illustrated with practical examples. Includes an accompanying website providing efficient numerical algorithms for moment computation and for constructing invariants of various kinds, with about 250 slides suitable for a graduate university course. Moments and Moment Invariants in Pattern Recognition is ideal for researchers and engineers involved in pattern recognition in medical imaging, remote sensing, robotics and computer vision. Post graduate students in image processing and pattern recognition will also find the book of interest.

Bundling features for large scale partial-duplicate web image search

Abstract: In state-of-the-art image retrieval systems, an image is represented by a bag of visual words obtained by quantizing high-dimensional local image descriptors, and scalable schemes inspired by text retrieval are then applied for large scale image indexing and retrieval. Bag-of-words representations, however: 1) reduce the discriminative power of image features due to feature quantization; and 2) ignore geometric relationships among visual words. Exploiting such geometric constraints, by estimating a 2D affine transformation between a query image and each candidate image, has been shown to greatly improve retrieval precision but at high computational cost. In this paper we present a novel scheme where image features are bundled into local groups. Each group of bundled features becomes much more discriminative than a single feature, and within each group simple and robust geometric constraints can be efficiently enforced. Experiments in Web image search, with a database of more than one million images, show that our scheme achieves a 49% improvement in average precision over the baseline bag-of-words approach. Retrieval performance is comparable to existing full geometric verification approaches while being much less computationally expensive. When combined with full geometric verification we achieve a 77% precision improvement over the baseline bag-of-words approach, and a 24% improvement over full geometric verification alone.

Spectral Curvature Clustering (SCC)

Abstract: This paper presents novel techniques for improving the performance of a multi-way spectral clustering framework (Govindu in Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), vol. 1, pp. 1150---1157, 2005; Chen and Lerman, 2007, preprint in the supplementary webpage) for segmenting affine subspaces. Specifically, it suggests an iterative sampling procedure to improve the uniform sampling strategy, an automatic scheme of inferring the tuning parameter from the data, a precise initialization procedure for K-means, as well as a simple strategy for isolating outliers. The resulting algorithm, Spectral Curvature Clustering (SCC), requires only linear storage and takes linear running time in the size of the data. It is supported by theory which both justifies its successful performance and guides our practical choices. We compare it with other existing methods on a few artificial instances of affine subspaces. Application of the algorithm to several real-world problems is also discussed.

Super-Resolution Without Explicit Subpixel Motion Estimation

Abstract: The need for precise (subpixel accuracy) motion estimates in conventional super-resolution has limited its applicability to only video sequences with relatively simple motions such as global translational or affine displacements. In this paper, we introduce a novel framework for adaptive enhancement and spatiotemporal upscaling of videos containing complex activities without explicit need for accurate motion estimation. Our approach is based on multidimensional kernel regression, where each pixel in the video sequence is approximated with a 3-D local (Taylor) series, capturing the essential local behavior of its spatiotemporal neighborhood. The coefficients of this series are estimated by solving a local weighted least-squares problem, where the weights are a function of the 3-D space-time orientation in the neighborhood. As this framework is fundamentally based upon the comparison of neighboring pixels in both space and time, it implicitly contains information about the local motion of the pixels across time, therefore rendering unnecessary an explicit computation of motions of modest size. The proposed approach not only significantly widens the applicability of super-resolution methods to a broad variety of video sequences containing complex motions, but also yields improved overall performance. Using several examples, we illustrate that the developed algorithm has super-resolution capabilities that provide improved optical resolution in the output, while being able to work on general input video with essentially arbitrary motion.

Guaranteed Rank Minimization via Singular Value Projection

Abstract: Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization with affine constraints (ARMP) and show that SVP recovers the minimum rank solution for affine constraints that satisfy the "restricted isometry property" and show robustness of our method to noise. Our results improve upon a recent breakthrough by Recht, Fazel and Parillo (RFP07) and Lee and Bresler (LB09) in three significant ways: 1) our method (SVP) is significantly simpler to analyze and easier to implement, 2) we give recovery guarantees under strictly weaker isometry assumptions 3) we give geometric convergence guarantees for SVP even in presense of noise and, as demonstrated empirically, SVP is significantly faster on real-world and synthetic problems. In addition, we address the practically important problem of low-rank matrix completion (MCP), which can be seen as a special case of ARMP. We empirically demonstrate that our algorithm recovers low-rank incoherent matrices from an almost optimal number of uniformly sampled entries. We make partial progress towards proving exact recovery and provide some intuition for the strong performance of SVP applied to matrix completion by showing a more restricted isometry property. Our algorithm outperforms existing methods, such as those of \cite{RFP07,CR08,CT09,CCS08,KOM09,LB09}, for ARMP and the matrix-completion problem by an order of magnitude and is also significantly more robust to noise.

Viewpoint Invariant Texture Description Using Fractal Analysis

Abstract: Image texture provides a rich visual description of the surfaces in the scene. Many texture signatures based on various statistical descriptions and various local measurements have been developed. Existing signatures, in general, are not invariant to 3D geometric transformations, which is a serious limitation for many applications. In this paper we introduce a new texture signature, called the multifractal spectrum (MFS). The MFS is invariant under the bi-Lipschitz map, which includes view-point changes and non-rigid deformations of the texture surface, as well as local affine illumination changes. It provides an efficient framework combining global spatial invariance and local robust measurements. Intuitively, the MFS could be viewed as a "better histogram" with greater robustness to various environmental changes and the advantage of capturing some geometrical distribution information encoded in the texture. Experiments demonstrate that the MFS codes the essential structure of textures with very low dimension, and thus represents an useful tool for texture classification.

McCormick-Based Relaxations of Algorithms

Abstract: Theory and implementation for the global optimization of a wide class of algorithms is presented via convex/affine relaxations. The basis for the proposed relaxations is the systematic construction of subgradients for the convex relaxations of factorable functions by McCormick [Math. Prog., 10 (1976), pp. 147-175]. Similar to the convex relaxation, the subgradient propagation relies on the recursive application of a few rules, namely, the calculation of subgradients for addition, multiplication, and composition operations. Subgradients at interior points can be calculated for any factorable function for which a McCormick relaxation exists, provided that subgradients are known for the relaxations of the univariate intrinsic functions. For boundary points, additional assumptions are necessary. An automated implementation based on operator overloading is presented, and the calculation of bounds based on affine relaxation is demonstrated for illustrative examples. Two numerical examples for the global optimization of algorithms are presented. In both examples a parameter estimation problem with embedded differential equations is considered. The solution of the differential equations is approximated by algorithms with a fixed number of iterations.

Affine Moser–Trudinger and Morrey–Sobolev inequalities

Abstract: An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L n energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L n Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L p energy, with p > n, is replaced by the affine energy.

A fully affine invariant image comparison method

Abstract: A fully affine invariant image comparison method, Affine-SIFT (ASIFT) is introduced. While SIFT is fully invariant with respect to only four parameters namely zoom, rotation and translation, the new method treats the two left over parameters : the angles defining the camera axis orientation. Against any prognosis, simulating all views depending on these two parameters is feasible. The method permits to reliably identify features that have undergone very large affine distortions measured by a new parameter, the transition tilt. State-of-the-art methods hardly exceed transition tilts of 2 (SIFT), 2.5 (Harris-Affine and Hessian-Affine) and 10 (MSER). ASIFT can handle transition tilts up 36 and higher (see Fig. 1).

Robust Video Stabilization Based on Particle Filter Tracking of Projected Camera Motion

Abstract: Video stabilization is an important technique in digital cameras Its impact increases rapidly with the rising popularity of handheld cameras and cameras mounted on moving platforms (eg, cars) Stabilization of two images can be viewed as an image registration problem However, to ensure the visual quality of the whole video, video stabilization has a particular emphasis on the accuracy and robustness over long image sequences In this paper, we propose a novel technique for video stabilization based on the particle filtering framework We extend the traditional use of particle filters in object tracking to tracking of the projected affine model of the camera motions We rely on the inverse of the resulting image transform to obtain a stable video sequence The correspondence between scale-invariant feature transform points is used to obtain a crude estimate of the projected camera motion We subsequently postprocess the crude estimate with particle filters to obtain a smooth estimate It is shown both theoretically and experimentally that particle filtering can reduce the error variance compared to estimation without particle filtering The superior performance of our algorithm over other methods for video stabilization is demonstrated through computer simulated experiments

A Fast and Log-Euclidean Polyaffine Framework for Locally Linear Registration

Abstract: In this article, we focus on the parameterization of non-rigid geometrical deformations with a small number of flexible degrees of freedom. In previous work, we proposed a general framework called polyaffine to parameterize deformations with a finite number of rigid or affine components, while guaranteeing the invertibility of global deformations. However, this framework lacks some important properties: the inverse of a polyaffine transformation is not polyaffine in general, and the polyaffine fusion of affine components is not invariant with respect to a change of coordinate system. We present here a novel general framework, called Log-Euclidean polyaffine, which overcomes these defects. We also detail a simple algorithm, the Fast Polyaffine Transform, which allows to compute very efficiently Log-Euclidean polyaffine transformations and their inverses on regular grids. The results presented here on real 3D locally affine registration suggest that our novel framework provides a general and efficient way of fusing local rigid or affine deformations into a global invertible transformation without introducing artifacts, independently of the way local deformations are first estimated.

Wide-baseline image matching using Line Signatures

Abstract: We present a wide-baseline image matching approach based on line segments Line segments are clustered into local groups according to spatial proximity Each group is treated as a feature called a Line Signature Similar to local features, line signatures are robust to occlusion, image clutter, and viewpoint changes The descriptor and similarity measure of line signatures are presented Under our framework, the feature matching is not only robust against affine distortion but also a considerable range of 3D viewpoint changes for non-planar surfaces When compared to matching approaches based on existing local features, our method shows improved results with low-texture scenes Moreover, extensive experiments validate that our method has advantages in matching structured non-planar scenes under large viewpoint changes and illumination variations

Affine Processes on Positive Semidefinite Matrices

Abstract: This paper provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. These matrix-valued affine processes have arisen from a large and growing range of useful applications 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.

A Total Variation-Based Algorithm for Pixel-Level Image Fusion

Abstract: In this paper, a total variation (TV) based approach is proposed for pixel-level fusion to fuse images acquired using multiple sensors. In this approach, fusion is posed as an inverse problem and a locally affine model is used as the forward model. A TV semi-norm based approach in conjunction with principal component analysis is used iteratively to estimate the fused image. The feasibility of the proposed algorithm is demonstrated on images from computed tomography (CT) and magnetic resonance imaging (MRI) as well as visible-band and infrared sensors. The results clearly indicate the feasibility of the proposed approach.

Optimality of Affine Policies in Multi-stage Robust Optimization

Abstract: In this paper, we show the optimality of a certain class of disturbance-affine control policies in the context of one-dimensional, constrained, multi-stage robust optimization. Our results cover the finite horizon case, with minimax (worst-case) objective, and convex state costs plus linear control costs. We develop a new proof methodology, which explores the relationship between the geometrical properties of the feasible set of solutions and the structure of the objective function. Apart from providing an elegant and conceptually simple proof technique, the approach also entails very fast algorithms for the case of piecewise affine state costs, which we explore in connection with a classical inventory management application.

Sampling and reconstructing signals from a union of linear subspaces

Abstract: In this note 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 minimisation and analog compressed sensing. Our main contribution is to show that a conceptually simple iterative projection algorithms is able to recover signals from a union of subspaces whenever the sampling operator satisfies a bi-Lipschitz embedding condition. Importantly, this result holds for all Hilbert spaces and unions of subspaces, as long as the sampling procedure satisfies the condition for the set of subspaces considered. In addition to recent results for finite unions of finite dimensional subspaces and infinite unions of subspaces in finite dimensional spaces, we also show that this bi-Lipschitz property can hold in an analog compressed sensing setting in which we have an infinite union of infinite dimensional subspaces living in infinite dimensional space.

A novel feature descriptor invariant to complex brightness changes

Abstract: We describe a novel and robust feature descriptor called ordinal spatial intensity distribution (OSID) which is invariant to any monotonically increasing brightness changes. Many traditional features are invariant to intensity shift or affine brightness changes but cannot handle more complex nonlinear brightness changes, which often occur due to the nonlinear camera response, variations in capture device parameters, temporal changes in the illumination, and viewpoint-dependent illumination and shadowing. A configuration of spatial patch sub-divisions is defined, and the descriptor is obtained by computing a 2-D histogram in the intensity ordering and spatial sub-division spaces. Extensive experiments show that the proposed descriptor significantly outperforms many state-of-the-art descriptors such as SIFT, GLOH, and PCA-SIFT under complex brightness changes. Moreover, the experiments demonstrate the proposed descriptor's superior performance even in the presence of image blur, viewpoint changes, and JPEG compression. The proposed descriptor has far reaching implications for many applications in computer vision including motion estimation, object tracking/recognition, image classification/retrieval, 3D reconstruction, and stereo.

A survey on evaluation methods for image interpolation

Abstract: Image interpolation is applied to Euclidean, affine and projective transformations in numerous imaging applications. However, due to the unique characteristics and wide applications of image interpolation, a separate study of their evaluation methods is crucial. The paper studies different existing methods for the evaluation of image interpolation techniques. Furthermore, an evaluation method utilizing ground truth images for the comparisons is proposed. Two main classes of analysis are proposed as the basis for the assessments: performance evaluation and cost evaluation. The presented methods are briefly described, followed by comparative discussions. This survey provides information for the appropriate use of the existing evaluation methods and their improvement, assisting also in the designing of new evaluation methods and techniques.

Optimal tracking control of affine nonlinear discrete-time systems with unknown internal dynamics

Abstract: In this paper, direct dynamic programming techniques are utilized to solve the Hamilton Jacobi-Bellman equation forward-in-time for the optimal tracking control of general affine nonlinear discrete-time systems using online approximators (OLA's). The proposed approach, referred as adaptive dynamic programming (ADP), is utilized to solve the infinite horizon optimal tracking control of affine nonlinear discrete-time systems in the presence of unknown internal dynamics and a known control coefficient matrix. The design is implemented using OLA's to realize the optimal feedback control signal and the associated cost function. The feedforward portion of the control input is derived and approximated using an additional OLA for steady state conditions. Novel tuning laws for the OLA's are derived, and all parameters are tuned online. Lyapunov techniques are used to show that all signals are uniformly ultimately bounded (UUB) and that the approximated control signal approaches the optimal control input with small bounded error. In the ideal case when there are no approximation errors, the approximated control converges to the optimal value asymptotically. Simulation results are included to show the effectiveness of the approach.

Optimality of affine policies in multi-stage robust optimization

Abstract: In this paper, we prove the optimality of disturbance-affine control policies in the context of onedimensional, box-constrained, multi-stage robust optimization. Our results cover the finite horizon case, with minimax (worstcase) objective, and convex state costs plus linear control costs. Our proof methodology, based on techniques from polyhedral geometry, is elegant and conceptually simple, and entails efficient algorithms for the case of piecewise affine state costs, when computing the optimal affine policies can be done by solving a single linear program.

Movement timing and invariance arise from several geometries.

Abstract: Human movements show several prominent features; movement duration is nearly independent of movement size (the isochrony principle), instantaneous speed depends on movement curvature (captured by the 2/3 power law), and complex movements are composed of simpler elements (movement compositionality). No existing theory can successfully account for all of these features, and the nature of the underlying motion primitives is still unknown. Also unknown is how the brain selects movement duration. Here we present a new theory of movement timing based on geometrical invariance. We propose that movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full affine geometries. Each geometry posses a canonical measure of distance along curves, an invariant arc-length parameter. We suggest that for continuous movements, the actual movement duration reflects a particular tensorial mixture of these canonical parameters. Near geometrical singularities, specific combinations are selected to compensate for time expansion or compression in individual parameters. The theory was mathematically formulated using Cartan's moving frame method. Its predictions were tested on three data sets: drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms (cloverleaves, lemniscates and limacons, with varying ratios between the sizes of the large versus the small loops). Our theory accounted well for the kinematic and temporal features of these movements, in most cases better than the constrained Minimum Jerk model, even when taking into account the number of estimated free parameters. During both drawing and locomotion equi-affine geometry was the most dominant geometry, with affine geometry second most important during drawing; Euclidian geometry was second most important during locomotion. We further discuss the implications of this theory: the origin of the dominance of equi-affine geometry, the possibility that the brain uses different mixtures of these geometries to encode movement duration and speed, and the ontogeny of such representations.

Extractors for Low-Weight Affine Sources

Abstract: We give polynomial time computable extractors for \emph{low-weight affince sources}. A distribution is affine if it samples a random points from some unknown low dimensional subspace of $\mathbb{F}_2^n$. A distribution is low weight affine if the corresponding linear space has a basis of low-weight vectors. Low-weight affine sources are thus a generalization of the well studied models of bit-fixing sources (which are just weight $1$ affine sources). For universal constants $c,\epsilon$, our extractors can extract almost all the entropy from weight $k^{\epsilon}$ affine sources of dimension $k$, as long as $k ≫ \log ^c n$, with error $2^{-k^{\Omega(1)}}$. In particular, our results give new extractors for low entropy bit-fixing sources, with exponentially small error, a parameter that is important for the application of these extractors to cryptography. Our techniques involve constructing new \emph{condensers} for \emph{affine somewhere random sources}.

Optimal registration of multiple deformed images using a physical model of the imaging distortion

Abstract: Methods and systems for image registration implementing a feature-based strategy that uses a retinal vessel network to identify features, uses an affine registration model estimated using feature correspondences, and corrects radial distortion to minimize the overall registration error. Also provided are methods and systems for retinal atlas generation. Further provided are methods and systems for testing registration methods.