Showing papers on "Affine transformation published in 2014"

Canonical bases for cluster algebras

Abstract: In previous work, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for the ring of functions on Y extending to a basis for functions on U. Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SL_r, parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

Bounded biharmonic weights for real-time deformation

Abstract: Changing an object's shape is a basic operation in computer graphics, necessary for transforming raster images, vector graphics, geometric models, and animated characters. The fastest approaches for such object deformation involve linearly blending a small number of given affine transformations, typically each associated with bones of an internal skeleton, vertices of an enclosing cage, or a collection of loose point handles. Unfortunately, linear blending schemes are not always easy to use because they may require manually painting influence weights or modeling closed polyhedral cages around the input object. Our goal is to make the design and control of deformations simpler by allowing the user to work freely with the most convenient combination of handle types. We develop linear blending weights that produce smooth and intuitive deformations for points, bones, and cages of arbitrary topology. Our weights, called bounded biharmonic weights, minimize the Laplacian energy subject to bound constraints. Doing so spreads the influences of the handles in a shape-aware and localized manner, even for objects with complex and concave boundaries. The variational weight optimization also makes it possible to customize the weights so that they preserve the shape of specified essential object features. We demonstrate successful use of our blending weights for real-time deformation of 2D and 3D shapes.

A Novel Coarse-to-Fine Scheme for Automatic Image Registration Based on SIFT and Mutual Information

Abstract: Automatic image registration is a vital yet challenging task, particularly for remote sensing images. A fully automatic registration approach which is accurate, robust, and fast is required. For this purpose, a novel coarse-to-fine scheme for automatic image registration is proposed in this paper. This scheme consists of a preregistration process (coarse registration) and a fine-tuning process (fine registration). To begin with, the preregistration process is implemented by the scale-invariant feature transform approach equipped with a reliable outlier removal procedure. The coarse results provide a near-optimal initial solution for the optimizer in the fine-tuning process. Next, the fine-tuning process is implemented by the maximization of mutual information using a modified Marquardt-Levenberg search strategy in a multiresolution framework. The proposed algorithm is tested on various remote sensing optical and synthetic aperture radar images taken at different situations (multispectral, multisensor, and multitemporal) with the affine transformation model. The experimental results demonstrate the accuracy, robustness, and efficiency of the proposed algorithm.

Online adaptive policy learning algorithm for H∞ state feedback control of unknown affine nonlinear discrete-time systems.

Abstract: The problem of H∞ state feedback control of affine nonlinear discrete-time systems with unknown dynamics is investigated in this paper. An online adaptive policy learning algorithm (APLA) based on adaptive dynamic programming (ADP) is proposed for learning in real-time the solution to the Hamilton-Jacobi-Isaacs (HJI) equation, which appears in the H∞ control problem. In the proposed algorithm, three neural networks (NNs) are utilized to find suitable approximations of the optimal value function and the saddle point feedback control and disturbance policies. Novel weight updating laws are given to tune the critic, actor, and disturbance NNs simultaneously by using data generated in real-time along the system trajectories. Considering NN approximation errors, we provide the stability analysis of the proposed algorithm with Lyapunov approach. Moreover, the need of the system input dynamics for the proposed algorithm is relaxed by using a NN identification scheme. Finally, simulation examples show the effectiveness of the proposed algorithm.

(Hierarchical) Identity-Based Encryption from Affine Message Authentication

Abstract: We provide a generic transformation from any affine message authentication code (MAC) to an identity-based encryption (IBE) scheme over pairing groups of prime order. If the MAC satisfies a security notion related to unforgeability against chosen-message attacks and, for example, the k-Linear assumption holds, then the resulting IBE scheme is adaptively secure. Our security reduction is tightness preserving, i.e., if the MAC has a tight security reduction so has the IBE scheme. Furthermore, the transformation also extends to hierarchical identity-based encryption (HIBE). We also show how to construct affine MACs with a tight security reduction to standard assumptions. This, among other things, provides the first tightly secure HIBE in the standard model.

Sound compilation of reals

Abstract: Writing accurate numerical software is hard because of many sources of unavoidable uncertainties, including finite numerical precision of implementations. We present a programming model where the user writes a program in a real-valued implementation and specification language that explicitly includes different types of uncertainties. We then present a compilation algorithm that generates a finite-precision implementation that is guaranteed to meet the desired precision with respect to real numbers. Our compilation performs a number of verification steps for different candidate precisions. It generates verification conditions that treat all sources of uncertainties in a unified way and encode reasoning about finite-precision roundoff errors into reasoning about real numbers. Such verification conditions can be used as a standardized format for verifying the precision and the correctness of numerical programs. Due to their non-linear nature, precise reasoning about these verification conditions remains difficult and cannot be handled using state-of-the art SMT solvers alone. We therefore propose a new procedure that combines exact SMT solving over reals with approximate and sound affine and interval arithmetic. We show that this approach overcomes scalability limitations of SMT solvers while providing improved precision over affine and interval arithmetic. Our implementation gives promising results on several numerical models, including dynamical systems, transcendental functions, and controller implementations.

Affine Grassmannians and the geometric Satake in mixed characteristic

Abstract: We endow the set of lattices in Q_p^n with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake correspondence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.

Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility

Abstract: The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved nonconvex relaxations. In this work we consider elementary methods based on projections for solving a sparse feasibility problem without employing convex heuristics. It has been shown recently that, locally, the fundamental method of alternating projections must converge linearly to a solution to the sparse feasibility problem with an affine constraint. In this paper we apply different analytical tools that allow us to show global linear convergence of alternating projections under familiar constraint qualifications. These analytical tools can also be applied to other algorithms. This is demonstrated with the prominent Douglas-Rachford algorithm where we establish local linear convergence of this method applied to the sparse affine feasibility problem.

Online approximate optimal control for affine non-linear systems with unknown internal dynamics using adaptive dynamic programming

Abstract: In this study, a novel online adaptive dynamic programming (ADP)-based algorithm is developed for solving the optimal control problem of affine non-linear continuous-time systems with unknown internal dynamics. The present algorithm employs an observer-critic architecture to approximate the Hamilton-Jacobi-Bellman equation. Two neural networks (NNs) are used in this architecture: an NN state observer is constructed to estimate the unknown system dynamics and a critic NN is designed to derive the optimal control instead of typical action-critic dual networks employed in traditional ADP algorithms. Based on the developed architecture, the observer NN and the critic NN are tuned simultaneously. Meanwhile, unlike existing tuning laws for the critic, the newly developed critic update rule not only ensures convergence of the critic to the optimal control but also guarantees stability of the closed-loop system. No initial stabilising control is required, and by using recorded and instantaneous data simultaneously for the adaptation of the critic, the restrictive persistence of excitation condition is relaxed. In addition, Lyapunov direct method is utilised to demonstrate the uniform ultimate boundedness of the weights of the observer NN and the critic NN. Finally, an example is provided to verify the effectiveness of the present approach.

A Cyclic Douglas–Rachford Iteration Scheme

Abstract: In this paper, we present two Douglas–Rachford inspired iteration schemes which can be applied directly to N-set convex feasibility problems in Hilbert space. Our main results are weak convergence of the methods to a point whose nearest point projections onto each of the N sets coincide. For affine subspaces, convergence is in norm. Initial results from numerical experiments, comparing our methods to the classical (product-space) Douglas–Rachford scheme, are promising.

A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model

Abstract: Compared with a probability model, a non-probabilistic convex model only requires a small number of experimental samples to discern the uncertainty parameter bounds instead of the exact probability distribution. Therefore, it can be used for uncertainty analysis of many complex structures lacking experimental samples. Based on the multidimensional parallelepiped convex model, we propose a new method for non-probabilistic structural reliability analysis in which marginal intervals are used to express scattering levels for the parameters, and relevant angles are used to express the correlations between uncertain variables. Using an affine coordinate transformation, the multidimensional parallelepiped uncertainty domain and the limit-state function are transformed to a standard parameter space, and a non-probabilistic reliability index is used to measure the structural reliability. Finally, the method proposed herein was applied to several numerical examples.

Minimal length elements of extended affine Weyl groups

Abstract: Let . We also introduce the class polynomials, which play a crucial role in the study of affine Deligne–Lusztig varieties He [Geometric and cohomological properties of affine Deligne–Lusztig varieties, Ann. of Math. (2) 179 (2014), 367–404].

A locally convergent rotationally invariant particle swarm optimization algorithm

Abstract: Several well-studied issues in the particle swarm optimization algorithm are outlined and some earlier methods that address these issues are investigated from the theoretical and experimental points of view. These issues are the: stagnation of particles in some points in the search space, inability to change the value of one or more decision variables, poor performance when the swarm size is small, lack of guarantee to converge even to a local optimum (local optimizer), poor performance when the number of dimensions grows, and sensitivity of the algorithm to the rotation of the search space. The significance of each of these issues is discussed and it is argued that none of the particle swarm optimizers we are aware of can address all of these issues at the same time. To address all of these issues at the same time, a new general form of velocity update rule for the particle swarm optimization algorithm that contains a user-definable function $$f$$ is proposed. It is proven that the proposed velocity update rule guarantees to address all of these issues if the function $$f$$ satisfies the following two conditions: (i) the function $$f$$ is designed in such a way that for any input vector $$\vec {y}$$ in the search space, there exists a region $$A$$ which contains $$\vec {y}$$ and $$ f\!\left( {\vec {y}} \right) $$ can be located anywhere in $$A$$ , and (ii) $$f$$ is invariant under any affine transformation. An example of function $$f$$ is provided that satisfies these conditions and its performance is examined through some experiments. The experiments confirm that the proposed algorithm (with an appropriate function $$f)$$ can effectively address all of these issues at the same time. Also, comparisons with earlier methods show that the overall ability of the proposed method for solving benchmark functions is significantly better.

Accurate and robust localization of duplicated region in copy---move image forgery

Abstract: Copy---move image forgery detection has recently become a very active research topic in blind image forensics. In copy---move image forgery, a region from some image location is copied and pasted to a different location of the same image. Typically, post-processing is applied to better hide the forgery. Using keypoint-based features, such as SIFT features, for detecting copy---move image forgeries has produced promising results. The main idea is detecting duplicated regions in an image by exploiting the similarity between keypoint-based features in these regions. In this paper, we have adopted keypoint-based features for copy---move image forgery detection; however, our emphasis is on accurate and robust localization of duplicated regions. In this context, we are interested in estimating the transformation (e.g., affine) between the copied and pasted regions more accurately as well as extracting these regions as robustly by reducing the number of false positives and negatives. To address these issues, we propose using a more powerful set of keypoint-based features, called MIFT, which shares the properties of SIFT features but also are invariant to mirror reflection transformations. Moreover, we propose refining the affine transformation using an iterative scheme which improves the estimation of the affine transformation parameters by incrementally finding additional keypoint matches. To reduce false positives and negatives when extracting the copied and pasted regions, we propose using "dense" MIFT features, instead of standard pixel correlation, along with hysteresis thresholding and morphological operations. The proposed approach has been evaluated and compared with competitive approaches through a comprehensive set of experiments using a large dataset of real images (i.e., CASIA v2.0). Our results indicate that our method can detect duplicated regions in copy---move image forgery with higher accuracy, especially when the size of the duplicated region is small.

Structural control of elastic moduli in ferrogels and the importance of non-affine deformations

Abstract: One of the central appealing properties of magnetic gels and elastomers is that their elastic moduli can reversibly be adjusted from outside by applying magnetic fields. The impact of the internal magnetic particle distribution on this effect has been outlined and analyzed theoretically. In most cases, however, affine sample deformations are studied and often regular particle arrangements are considered. Here we challenge these two major simplifications by a systematic approach using a minimal dipole-spring model. Starting from different regular lattices, we take into account increasingly randomized structures, until we finally investigate an irregular texture taken from a real experimental sample. On the one hand, we find that the elastic tunability qualitatively depends on the structural properties, here in two spatial dimensions. On the other hand, we demonstrate that the assumption of affine deformations leads to increasingly erroneous results the more realistic the particle distribution becomes. Understanding the consequences of the assumptions made in the modeling process is important on our way to support an improved design of these fascinating materials.

Affine cartesian codes

Abstract: We compute the basic parameters (dimension, length, minimum distance) of affine evaluation codes defined on a cartesian product of finite sets. Given a sequence of positive integers, we construct an evaluation code, over a degenerate torus, with prescribed parameters of a certain type. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation codes.

A Novel Affine Arithmetic Method to Solve Optimal Power Flow Problems With Uncertainties

Abstract: An affine arithmetic (AA) method is proposed in this paper to solve the optimal power flow (OPF) problem with uncertain generation sources. In the AA-based OPF problem, all the state and control variables are treated in affine form, comprising a center value and the corresponding noise magnitudes, to represent forecast, model error, and other sources of uncertainty without the need to assume a probability density function (pdf). The proposed AA-based OPF problem is used to determine the operating margins of the thermal generators in systems with uncertain wind and solar generation dispatch. The AA-based approach is benchmarked against Monte Carlo simulation (MCS) intervals in order to determine its effectiveness. The proposed technique is tested and demonstrated on the IEEE 30-bus system and also a real 1211-bus European system.

A cohomological classification of vector bundles on smooth affine threefolds

Abstract: We give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2. As a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties. The proofs of these results involve three main ingredients. First, we give a description of the first nonstable A1-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel’s A1-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in A1-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.

Homological properties of finite type Khovanov-Lauda-Rouquier algebras

Abstract: We give an algebraic construction of standard modules-infinite-dimensional modules categorifying the Poincare-Birkhoff-Witt basis of the underlying quantized enveloping algebra-for Khovanov-Lauda-Rouquier algebras in all finite types. This allows us to prove in an elementary way that these algebras satisfy the homological properties of an "affine quasihereditary algebra." In simply laced types these properties were established originally by Kato via a geometric approach. We also construct some Koszul-like projective resolutions of standard modules corresponding to multiplicity-free positive roots.

Non-affine Extensions to Polyhedral Code Generation

Abstract: This paper describes a loop transformation framework that extends a polyhedral representation of loop nests to represent and transform computations with non-affine index arrays in loop bounds and subscripts via a new interface between compile-time and run-time abstractions. Polyhedra scanning code generation, which historically applies an affine mapping to the subscript expressions of the statements in a loop nest, is modified to apply non-affine mappings involving index arrays that are represented at compile time by uninterpreted functions; non-affine loop bounds involving index arrays are also represented. When appropriate, an inspector is utilized to capture the non-affine subscript mappings, and a generalized loop coalescing transformation is introduced as a non-affine transformation to support non-affine loop bounds. With this support, complex sequences of new and existing transformations can then be composed. We demonstrate the effectiveness of this framework by optimizing sparse matrix vector multiplication operations targeting GPUs for different matrix structures and parallelization strategies. This approach achieves performance that is comparable to or greater than the hand-tuned CUSP library; for two of the implementations it achieves an average 1.14× improvement over CUSP across a collection of sparse matrices, while the third performs on average within 8% of CUSP.

Evaluation of respiratory and cardiac motion correction schemes in dual gated PET/CT cardiac imaging

Abstract: Purpose: Cardiac imaging suffers from both respiratory and cardiac motion. One of the proposed solutions involves double gated acquisitions. Although such an approach may lead to both respiratory and cardiac motion compensation there are issues associated with (a) the combination of data from cardiac and respiratory motion bins, and (b) poor statistical quality images as a result of using only part of the acquired data. The main objective of this work was to evaluate different schemes of combining binned data in order to identify the best strategy to reconstruct motion free cardiac images from dual gated positron emission tomography (PET) acquisitions. Methods: A digital phantom study as well as seven human studies were used in this evaluation. PET data were acquired in list mode (LM). A real-time position management system and an electrocardiogram device were used to provide the respiratory and cardiac motion triggers registered within the LM file. Acquired data were subsequently binned considering four and six cardiac gates, or the diastole only in combination with eight respiratory amplitude gates. PET images were corrected for attenuation, but no randoms nor scatter corrections were included. Reconstructed images from each of the bins considered above were subsequently used in combination with an affine or an elastic registration algorithm to derive transformation parameters allowing the combination of all acquired data in a particular position in the cardiac and respiratory cycles. Images were assessed in terms of signal-to-noise ratio (SNR), contrast, image profile, coefficient-of-variation (COV), and relative difference of the recovered activity concentration. Results: Regardless of the considered motion compensation strategy, the nonrigid motion model performed better than the affine model, leading to higher SNR and contrast combined with a lower COV. Nevertheless, when compensating for respiration only, no statistically significant differences were observed in the performance of the two motion models considered. Superior image SNR and contrast were seen using the affine respiratory motion model in combination with the diastole cardiac bin in comparison to the use of the whole cardiac cycle. In contrast, when simultaneously correcting for cardiac beating and respiration, the elastic respiratory motion model outperformed the affine model. In this context, four cardiac bins associated with eight respiratory amplitude bins seemed to be adequate. Conclusions: Considering the compensation of respiratory motion effects only, both affine and elastic based approaches led to an accurate resizing and positioning of the myocardium. The use of the diastolic phase combined with an affine model based respiratory motion correction may therefore be a simple approach leading to significant quality improvements in cardiac PET imaging. However, the best performance was obtained with the combined correction for both cardiac and respiratory movements considering all the dual-gated bins independently through the use of an elastic model based motion compensation.

Nonlinear Kalman Filtering in Affine Term Structure Models

Abstract: The extended Kalman filter, which linearizes the relationship between security prices and state variables, is widely used in fixed-income applications. We investigate whether the unscented Kalman filter should be used to capture nonlinearities and compare the performance of the Kalman filter with that of the particle filter. We analyze the cross section of swap rates, which are mildly nonlinear in the states, and cap prices, which are highly nonlinear. When caps are used to filter the states, the unscented Kalman filter significantly outperforms its extended counterpart. The unscented Kalman filter also performs well when compared with the much more computationally intensive particle filter. These findings suggest that the unscented Kalman filter may be a good approach for a variety of problems in fixed-income pricing. This paper was accepted by Wei Xiong, finance.