Showing papers on "Affine transformation published in 2010"

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

Abstract: The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

Abstract: The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.

Ensemble samplers with affine invariance

Abstract: We propose a family of Markov chain Monte Carlo methods whose performance is unaffected by affine tranformations of space. These algorithms are easy to construct and require little or no additional computational overhead. They should be particularly useful for sampling badly scaled distributions. Computational tests show that the affine invariant methods can be significantly faster than standard MCMC methods on highly skewed distributions.

Robust Subspace Segmentation by Low-Rank Representation

Abstract: We propose low-rank representation (LRR) to segment data drawn from a union of multiple linear (or affine) subspaces. Given a set of data vectors, LRR seeks the lowest-rank representation among all the candidates that represent all vectors as the linear combination of the bases in a dictionary. Unlike the well-known sparse representation (SR), which computes the sparsest representation of each data vector individually, LRR aims at finding the lowest-rank representation of a collection of vectors jointly. LRR better captures the global structure of data, giving a more effective tool for robust subspace segmentation from corrupted data. Both theoretical and experimental results show that LRR is a promising tool for subspace segmentation.

Face recognition based on image sets

Abstract: We introduce a novel method for face recognition from image sets. In our setting each test and training example is a set of images of an individual's face, not just a single image, so recognition decisions need to be based on comparisons of image sets. Methods for this have two main aspects: the models used to represent the individual image sets; and the similarity metric used to compare the models. Here, we represent images as points in a linear or affine feature space and characterize each image set by a convex geometric region (the affine or convex hull) spanned by its feature points. Set dissimilarity is measured by geometric distances (distances of closest approach) between convex models. To reduce the influence of outliers we use robust methods to discard input points that are far from the fitted model. The kernel trick allows the approach to be extended to implicit feature mappings, thus handling complex and nonlinear manifolds of face images. Experiments on two public face datasets show that our proposed methods outperform a number of existing state-of-the-art ones.

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 under affine constraints (ARMP) and show that SVP recovers the minimum rank solution for affine constraints that satisfy a restricted isometry property (RIP). Our method guarantees geometric convergence rate even in the presence of noise and requires strictly weaker assumptions on the RIP constants than the existing methods. We also introduce a Newton-step for our SVP framework to speed-up the convergence with substantial empirical gains. Next, we address a practically important application of ARMP - the problem of low-rank matrix completion, for which the defining affine constraints do not directly obey RIP, hence the guarantees of SVP do not hold. However, we provide partial progress towards a proof of exact recovery for our algorithm by showing a more restricted isometry property and observe empirically that our algorithm recovers low-rank incoherent matrices from an almost optimal number of uniformly sampled entries. We also demonstrate empirically that our algorithms outperform existing methods, such as those of [5, 18, 14], for ARMP and the matrix completion problem by an order of magnitude and are also more robust to noise and sampling schemes. In particular, results show that our SVP-Newton method is significantly robust to noise and performs impressively on a more realistic power-law sampling scheme for the matrix completion problem.

Prescribed Performance Adaptive Control for Multi-Input Multi-Output Affine in the Control Nonlinear Systems

Abstract: We consider the tracking problem of unknown, robustly stabilizable, multi-input multi-output (MIMO), affine in the control, nonlinear systems with guaranteed prescribed performance. By prescribed performance we mean that the tracking error converges to a predefined arbitrarily small residual set, with convergence rate no less than a prespecified value, exhibiting maximum overshoot as well as undershoot less than some sufficiently small preassigned constants. Utilizing an output error transformation, we obtain a transformed system whose robust stabilization is proven necessary and sufficient to achieve prescribed performance guarantees for the output tracking error of the original system, provided that initially the transformed system is well defined. Consequently, a switching robust control Lyapunov function (RCLF)-based adaptive, state feedback controller is designed, to solve the stated problem. The proposed controller is continuous and successfully overcomes the problem of computing the control law when the approximation model becomes uncontrollable. Simulations illustrate the approach.

isl: an integer set library for the polyhedral model

Abstract: In compiler research, polytopes and related mathematical objects have been successfully used for several decades to represent and manipulate computer programs in an approach that has become known as the polyhedral model. The key insight is that the kernels of many compute-intensive applications are composed of loops with bounds that are affine combinations of symbolic constants and outer loop iterators. The iterations of a loop nest can then be represented as the integer points in a (parametric) polytope and manipulated as a whole, rather than as individual iterations. A similar reasoning holds for the elements of an array and for mappings between loop iterations and array elements.

Motion Segmentation in the Presence of Outlying, Incomplete, or Corrupted Trajectories

Abstract: In this paper, we study the problem of segmenting tracked feature point trajectories of multiple moving objects in an image sequence. Using the affine camera model, this problem can be cast as the problem of segmenting samples drawn from multiple linear subspaces. In practice, due to limitations of the tracker, occlusions, and the presence of nonrigid objects in the scene, the obtained motion trajectories may contain grossly mistracked features, missing entries, or corrupted entries. In this paper, we develop a robust subspace separation scheme that deals with these practical issues in a unified mathematical framework. Our methods draw strong connections between lossy compression, rank minimization, and sparse representation. We test our methods extensively on the Hopkins155 motion segmentation database and other motion sequences with outliers and missing data. We compare the performance of our methods to state-of-the-art motion segmentation methods based on expectation-maximization and spectral clustering. For data without outliers or missing information, the results of our methods are on par with the state-of-the-art results and, in many cases, exceed them. In addition, our methods give surprisingly good performance in the presence of the three types of pathological trajectories mentioned above. All code and results are publicly available at http://perception.csl.uiuc.edu/coding/motion/.

Affine Point Processes and Portfolio Credit Risk

Abstract: This paper analyzes a family of multivariate point process models of correlated event timing whose arrival intensity is driven by an affine jump diffusion. The components of an affine point process are self- and cross-exciting and facilitate the description of complex event dependence structures. ODEs characterize the transform of an affine point process and the probability distribution of an integer-valued affine point process. The moments of an affine point process take a closed form. This guarantees a high degree of computational tractability in applications. We illustrate this in the context of portfolio credit risk, where the correlation of corporate defaults is the main issue. We consider the valuation of securities exposed to correlated default risk and demonstrate the significance of our results through market calibration experiments. We show that a simple model variant can capture the default clustering implied by index and tranche market prices during September 2008, a month that witnessed significant volatility.

Affine Point Processes and Portfolio Credit Risk

Abstract: This paper analyzes a family of multivariate point process models of correlated event timing whose arrival intensity is driven by an affine jump diffusion. The components of an affine point process are self- and cross-exciting, and facilitate the description of complex event dependence structures. Ordinary differential equations characterize the transform of an affine point process and the probability distribution of an integer-valued affine point process. The moments of an affine point process take a closed form. This guarantees a high degree of computational tractability in applications. We illustrate this in the context of portfolio credit risk, where the correlation of corporate defaults is the main issue. We consider the valuation of securities exposed to correlated default risk, and demonstrate the significance of our results through market calibration experiments. We show that a simple model variant can capture the default clustering implied by index and tranche market prices during September 2008, a month that witnessed significant volatility.

A Partial Intensity Invariant Feature Descriptor for Multimodal Retinal Image Registration

Abstract: Detection of vascular bifurcations is a challenging task in multimodal retinal image registration. Existing algorithms based on bifurcations usually fail in correctly aligning poor quality retinal image pairs. To solve this problem, we propose a novel highly distinctive local feature descriptor named partial intensity invariant feature descriptor (PIIFD) and describe a robust automatic retinal image registration framework named Harris-PIIFD. PIIFD is invariant to image rotation, partially invariant to image intensity, affine transformation, and viewpoint/perspective change. Our Harris-PIIFD framework consists of four steps. First, corner points are used as control point candidates instead of bifurcations since corner points are sufficient and uniformly distributed across the image domain. Second, PIIFDs are extracted for all corner points, and a bilateral matching technique is applied to identify corresponding PIIFDs matches between image pairs. Third, incorrect matches are removed and inaccurate matches are refined. Finally, an adaptive transformation is used to register the image pairs. PIIFD is so distinctive that it can be correctly identified even in nonvascular areas. When tested on 168 pairs of multimodal retinal images, the Harris-PIIFD far outperforms existing algorithms in terms of robustness, accuracy, and computational efficiency.

Switched affine systems control design with application to DCߝDC converters

Abstract: This study presents a novel procedure for switched affine systems control design specially developed to deal with switched converters where the main goal is to attain a set of equilibrium points. The main contribution is on the determination of a switching function, which assures global stability and minimises a guaranteed quadratic cost. The implementation of the switching function taking into account only partial information is analysed and discussed with particular interest. The theoretical results are applied to buck, boost and buck-boost converters control design. Several simulations show the usefulness of the methodology and its favourable impact in a class of real-world control design problems.

TILT: Transform Invariant Low-rank Textures

Abstract: In this paper, we show how to efficiently and effectively extract a class of "low-rank textures" in a 3D scene from 2D images despite significant corruptions and warping. The low-rank textures capture geometrically meaningful structures in an image, which encompass conventional local features such as edges and corners as well as all kinds of regular, symmetric patterns ubiquitous in urban environments and man-made objects. Our approach to finding these low-rank textures leverages the recent breakthroughs in convex optimization that enable robust recovery of a high-dimensional low-rank matrix despite gross sparse errors. In the case of planar regions with significant affine or projective deformation, our method can accurately recover both the intrinsic low-rank texture and the precise domain transformation, and hence the 3D geometry and appearance of the planar regions. Extensive experimental results demonstrate that this new technique works effectively for many regular and near-regular patterns or objects that are approximately low-rank, such as symmetrical patterns, building facades, printed texts, and human faces.

Affine SL(2) conformal blocks from 4d gauge theories

Abstract: We study Nekrasov’s instanton partition function of four-dimensional $${\mathcal{N}=2}$$ gauge theories in the presence of surface operators. This can be computed order by order in the instanton expansion by using results available in the mathematical literature. Focusing in the case of SU(2) quiver gauge theories, we find that the results agree with a modified version of the conformal blocks of affine SL(2) algebra. These conformal blocks provide, in the critical limit, the eigenfunctions of the corresponding quantized Hitchin Hamiltonians.

High order discretization schemes for the cir process: application to affine term structure and heston models

Abstract: This paper presents weak second and third order schemes for the Cox-Ingersoll-Ross (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method to get weak second-order schemes that extends the one introduced by Ninomiya and Victoir~\cite{NV}. Combining these both results, this allows to propose a second-order scheme for more general affine diffusions. Simulation examples are given to illustrate the convergence of these schemes on CIR and Heston models. Algorithms are stated in a pseudocode language.

Affine SL(2) conformal blocks from 4d gauge theories

Abstract: We study Nekrasov's instanton partition function of four-dimensional N=2 gauge theories in the presence of surface operators. This can be computed order by order in the instanton expansion by using results available in the mathematical literature. Focusing in the case of SU(2) quiver gauge theories, we find that the results agree with a modified version of the conformal blocks of affine SL(2) Lie algebra. These conformal blocks provide, in the critical limit, the eigenfunctions of the corresponding quantized Hitchin Hamiltonians.

Passive Actuators' Fault-Tolerant Control for Affine Nonlinear Systems

Abstract: In this brief, the problem of passive fault-tolerant control (FTC) for nonlinear affine systems with actuator faults is considered. Two types of faults, additive and loss-of-effectiveness faults, are treated. In each case, a Lyapunov-based feedback controller is proposed, which ensures the local uniform asymptotic (exponential) stability of the faulty system, if the safe nominal system is locally uniformly asymptotically (exponentially) stable. The effectiveness of the FT controllers is shown on the autonomous helicopter numerical example.

Optimal control of affine nonlinear continuous-time systems

Abstract: In this paper, the optimal regulation and tracking control of affine nonlinear continuous-time systems with known dynamics is undertaken using a novel single online approximator (SOL)-based scheme. The SOLA-based adaptive approach is designed to learn the infinite horizon continuous-time Hamilton-Jacobi-Bellman (HJB) equation and its corresponding optimal control input. A novel parameter tuning algorithm is derived which not only ensures the optimal cost (HJB) function and control input are achieved, but also ensures the system states remain bounded during the online learning process. Lyapunov techniques show that all signals are uniformly ultimately bounded (UUB) and the approximated control signal approaches the optimal control input with small bounded error. In the absence of OLA reconstruction errors, asymptotic convergence to the optimal control is shown. Simulation results illustrate the effectiveness of the approach.

A general purpose sampling algorithm for continuous distributions (the t-walk)

Abstract: We develop a new general purpose MCMC sampler for arbitrary continuous distributions that requires no tuning. We call this MCMC the t-walk. The t-walk maintains two independent points in the sample space, and all moves are based on proposals that are then accepted with a standard Metropolis-Hastings acceptance probability on the product space. Hence the t-walk is provably convergent under the usual mild requirements. We restrict proposal distributions, or `moves', to those that produce an algorithm that is invariant to scale, and approximately invariant to affine transformations of the state space. Hence scaling of proposals, and effectively also coordinate transformations, that might be used to increase efficiency of the sampler, are not needed since the t-walk's operation is identical on any scaled version of the target distribution. Four moves are given that result in an effective sampling algorithm. We use the simple device of updating only a random subset of coordinates at each step to allow application of the t-walk to high-dimensional problems. In a series of test problems across dimensions we find that the t-walk is only a small factor less efficient than optimally tuned algorithms, but significantly outperforms general random-walk M-H samplers that are not tuned for specific problems. Further, the t-walk remains effective for target distributions for which no optimal affine transformation exists such as those where correlation structure is very different in differing regions of state space. Several examples are presented showing good mixing and convergence characteristics, varying in dimensions from 1 to 200 and with radically different scale and correlation structure, using exactly the same sampler. The t-walk is available for R, Python, MatLab and C++ at http://www.cimat.mx/~jac/twalk/

An Affine Arithmetic-Based Methodology for Reliable Power Flow Analysis in the Presence of Data Uncertainty

Abstract: Power flow studies are typically used to determine the steady state or operating conditions of power systems for specified sets of load and generation values, and is one of the most intensely used tools in power engineering. When the input conditions are uncertain, numerous scenarios need to be analyzed to cover the required range of uncertainty. Under such conditions, reliable solution algorithms that incorporate the effect of data uncertainty into the power flow analysis are required. To address this problem, this paper proposes a new solution methodology based on the use of affine arithmetic, which is an enhanced model for self-validated numerical analysis in which the quantities of interest are represented as affine combinations of certain primitive variables representing the sources of uncertainty in the data or approximations made during the computation. The application of this technique to the power flow problem is explained in detail, and several numerical results are presented and discussed, demonstrating the effectiveness of the proposed methodology, especially in comparison to previously proposed interval arithmetic's techniques.

Distributed Estimation Over an Adaptive Incremental Network Based on the Affine Projection Algorithm

Abstract: We study the problem of distributed estimation based on the affine projection algorithm (APA), which is developed from Newton's method for minimizing a cost function. The proposed solution is formulated to ameliorate the limited convergence properties of least-mean-square (LMS) type distributed adaptive filters with colored inputs. The analysis of transient and steady-state performances at each individual node within the network is developed by using a weighted spatial-temporal energy conservation relation and confirmed by computer simulations. The simulation results also verify that the proposed algorithm provides not only a faster convergence rate but also an improved steady-state performance as compared to an LMS-based scheme. In addition, the new approach attains an acceptable misadjustment performance with lower computational and memory cost, provided the number of regressor vectors and filter length parameters are appropriately chosen, as compared to a distributed recursive-least-squares (RLS) based method.

A Zernike Moment Phase-Based Descriptor for Local Image Representation and Matching

Abstract: A local image descriptor robust to the common photometric transformations (blur, illumination, noise, and JPEG compression) and geometric transformations (rotation, scaling, translation, and viewpoint) is crucial to many image understanding and computer vision applications. In this paper, the representation and matching power of region descriptors are to be evaluated. A common set of elliptical interest regions is used to evaluate the performance. The elliptical regions are further normalized to be circular with a fixed size. The normalized circular regions will become affine invariant up to a rotational ambiguity. Here, a new distinctive image descriptor to represent the normalized region is proposed, which primarily comprises the Zernike moment (ZM) phase information. An accurate and robust estimation of the rotation angle between a pair of normalized regions is then described and used to measure the similarity between two matching regions. The discriminative power of the new ZM phase descriptor is compared with five major existing region descriptors (SIFT, GLOH, PCA-SIFT, complex moments, and steerable filters) based on the precision-recall criterion. The experimental results, involving more than 15 million region pairs, indicate the proposed ZM phase descriptor has, generally speaking, the best performance under the common photometric and geometric transformations. Both quantitative and qualitative analyses on the descriptor performances are given to account for the performance discrepancy. First, the key factor for its striking performance is due to the fact that the ZM phase has accurate estimation accuracy of the rotation angle between two matching regions. Second, the feature dimensionality and feature orthogonality also affect the descriptor performance. Third, the ZM phase is more robust under the nonuniform image intensity fluctuation. Finally, a time complexity analysis is provided.

Unified Real-Time Tracking and Recognition with Rotation-Invariant Fast Features

Abstract: We present a method that unifies tracking and video content recognition with applications to Mobile Augmented Reality (MAR). We introduce the Radial Gradient Transform (RGT) and an approximate RGT, yielding the Rotation-Invariant, Fast Feature (RIFF) descriptor. We demonstrate that RIFF is fast enough for real-time tracking, while robust enough for large scale retrieval tasks. At 26× the speed, our tracking-scheme obtains a more accurate global affine motionmodel than the Kanade Lucas Tomasi (KLT) tracker. The same descriptors can achieve 94% retrieval accuracy from a database of 104 images.

Correcting eddy current and motion effects by affine whole-brain registrations: evaluation of three-dimensional distortions and comparison with slicewise correction.

Abstract: Eddy-current (EC) and motion effects in diffusion-tensor imaging (DTI) bias the estimation of quantitative diffusion indices, such as the fractional anisotropy. Both effects can be retrospectively corrected by registering the strongly distorted diffusion-weighted images to less-distorted T2-weighted images acquired without diffusion weighting. Two different affine spatial transformations are usually employed for this correction: slicewise and whole-brain transformations. However, a relation between estimated transformation parameters and EC distortions has not been established yet for the latter approach. In this study, a novel diffusion-gradient-direction-independent estimation of the EC field is proposed based solely on affine whole-brain registration parameters. Using this model, it is demonstrated that a more distinct evaluation of the whole-brain EC effects is possible if the through-plane distortion was considered in addition to the well-known in-plane distortions. Moreover, a comparison of different whole-brain registrations relative to a slicewise approach is performed, in terms of the relative tensor error. Our findings suggest that for appropriate intersubject comparison of DTI data, a whole-brain registration containing nine affine parameters provides comparable performance (between 0 and 3%) to slicewise methods and can be performed in a fraction of the time.

Geometric Distortion Insensitive Image Watermarking in Affine Covariant Regions

Abstract: Feature-based image watermarking schemes, which aim to survive various geometric distortions, have attracted great attention in recent years. Existing schemes have shown robustness against rotation, scaling, and translation, but few are resistant to cropping, nonisotropic scaling, random bending attacks (RBAs), and affine transformations. Seo and Yoo present a geometrically invariant image watermarking based on affine covariant regions (ACRs) that provide a certain degree of robustness. To further enhance the robustness, we propose a new image watermarking scheme on the basis of Seo's work, which is insensitive to geometric distortions as well as common image processing operations. Our scheme is mainly composed of three components: 1) feature selection procedure based on graph theoretical clustering algorithm is applied to obtain a set of stable and nonoverlapped ACRs; 2) for each chosen ACR, local normalization, and orientation alignment are performed to generate a geometrically invariant region, which can obviously improve the robustness of the proposed watermarking scheme; and 3) in order to prevent the degradation in image quality caused by the normalization and inverse normalization, indirect inverse normalization is adopted to achieve a good compromise between the imperceptibility and robustness. Experiments are carried out on an image set of 100 images collected from Internet, and the preliminary results demonstrate that the developed method improves the performance over some representative image watermarking approaches in terms of robustness.