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Mixed Oblique Extension Principles (MOEP) provide an important method to construct affine dual frames from refinable functions. This paper addresses MOEP under the setting of reducing subspaces of L 2 ( R d ) . We obtain an MOEP for (non)homogeneous affine dual frames and (non)homogeneous affine Parseval frames.

Despite extensive research on content-based image retrieval, challenges such as low accuracy, incapability to handle complex queries and high time consumption persist. Initially, a preprocessing technique is introduced in this study, a technique that uses a median filter to remove noise to achieve improved accuracy and reliability. Then, Fourier and circularity descriptors are extract in an effective manner correspondent to the texture and affine shape adaptation features. In addition, various descriptors, such as color histogram, color moment, color autocorrelogram and color coherency vector, are extracted as the invariant color features. The multiple ant colony optimization (MACOBTC) approach is implemented with whole features to find relevant features. Finally, the relevant features are utilized for the greedy learning of deep Boltzmann machine classifier (GDBM). The proposed approach obtains effective performance and accurate results on four datasets and is analyzed with various parameters such as accuracy, precision, recall, Jaccard, Dice, and Kappa coefficients. The GDBM provides a 25% increase in accuracy compared with existing techniques, such as the a priori classification algorithm.

The performance analysis of the improved proportionate affine projection (IPAP) algorithm is performed in this brief. Based on the energy-conservation arguments, the steady-state analysis of the IPAP algorithm is performed, which provides a general expression of steady-state excess mean-square error for the proportionate-type affine projection algorithms. The tracking behavior is also studied and the step size that optimizes the tracking performance is provided. Simulation results confirm the accuracy of the proposed expressions under different operating scenarios.

We consider the person re-identification problem, assuming the availability of a sequence of images for each person, commonly referred to as video-based or multi-shot re-identification. We approach this problem from the perspective of learning discriminative distance metric functions. While existing distance metric learning methods typically employ the average feature vector as the data exemplar, this discards the inherent structure of the data. To overcome this issue, we describe the image sequence data using affine hulls. We show that directly computing the distance between the closest points on these affine hulls as in existing recognition algorithms is not sufficiently discriminative in the context of person re-identification. To this end, we incorporate affine hull data modeling into the traditional distance metric learning framework, learning discriminative feature representations directly using affine hulls. We perform extensive experiments on several publicly available data sets to show that the proposed approach improves the performance of existing metric learning algorithms irrespective of the feature space employed to perform metric learning. Furthermore, we advance the state of the art on iLIDS-VID, PRID, and SAIVT, with absolute rank-1 performance improvements of 6.0%, 11.4%, and 6.0% respectively.

In this paper, we study a simplified affine motion model-based coding framework to overcome the limitation of a translational motion model and maintain low-computational complexity. The proposed framework mainly has three key contributions. First, we propose to reduce the number of affine motion parameters from 6 to 4. The proposed four-parameter affine motion model can not only handle most of the complex motions in natural videos, but also save the bits for two parameters. Second, to efficiently encode the affine motion parameters, we propose two motion prediction modes, i.e., an advanced affine motion vector prediction scheme combined with a gradient-based fast affine motion estimation algorithm and an affine model merge scheme, where the latter attempts to reuse the affine motion parameters (instead of the motion vectors) of neighboring blocks. Third, we propose two fast affine motion compensation algorithms. One is the one-step sub-pixel interpolation that reduces the computations of each interpolation. The other is the interpolation-precision-based adaptive block size motion compensation that performs motion compensation at the block level rather than the pixel level to reduce the number of interpolation. Our proposed techniques have been implemented based on the state-of-the-art high-efficiency video coding standard, and the experimental results show that the proposed techniques altogether achieve, on average, 11.1% and 19.3% bits saving for random access and low-delay configurations, respectively, on typical video sequences that have rich rotation or zooming motions. Meanwhile, the computational complexity increases of both the encoder and the decoder are within an acceptable range.

Robust and reliable features with noise immunity, rotation-invariance, and low-dimensionality are the challenging aspects of pattern recognition. In this study, the authors presented a novel low-dimensional binary feature descriptor local diagonal Laplacian pattern (LDLP) for medical image registration. LDLP method is developed by defining the local relationship between a centre pixel and its diagonal neighbours and encoding it to a binary feature vector. The idea of centre-diagonal pixel correlation has drastically reduced the length of the feature vector without compromising the quality of local texture analysis. In the proposed work, first, the LDLP feature histograms of computed tomography (CT), magnetic resonance (MR), and ultrasound images are obtained. Further, these LDLP features of individual medical images are considered as target/fixed objects while their corresponding rotated and noisy features are considered as moving/floating objects to perform mono-modal rigid registration using an improved Procrustes analysis-based affine transform. The registration quality is examined by calculating the squared intensity error and the results are compared with the existing binary patterns such as local binary patterns, local tetra patterns, and local diagonal extrema patterns. The proposed LDLP feature descriptor-based rigid registration has attained relatively better performance in terms of registration accuracy and computational complexity.

Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal . In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces.

We study the regularity properties of the value function associated with an affine optimal control problem with quadratic cost plus a potential, for a fixed final time and initial point. Without assuming any condition on singular minimizers, we prove that the value function is continuous on an open and dense subset of the interior of the attainable set. As a byproduct we obtain that it is actually smooth on a possibly smaller set, still open and dense.

In affine differential geometry of hypersurface, C. Scharlach et al. found an inequality involving intrinsic and extrinsic curvatures, and classified elliptic and hyperbolic affine hyperspheres realizing the equality if an affine invariant 2-dimensional distribution D 2 is integrable. In this paper, we continue to study affine hyperspheres realizing the equality, including parabolic affine hyperspheres. As main results, firstly we classify parabolic affine hyperspheres realizing the equality if its scalar curvature is constant, or D 2 is integrable. Next, by introducing a well-defined 3-dimensional distribution D 3 when D 2 is not integrable, we complete the classification of locally strongly convex affine hyperspheres realizing the equality if D 3 is integrable. Finally, we pose a conjecture and a problem in order to determine all affine hyperspheres attaining the equality.

This paper presents a fast and accurate affine canonicalization method for planar shapes. This method improves on previous ones based on iterative optimization that produce multiple canonical versions. Canonicalization provides a common reference frame for shape comparison without the loss of discrimination ability often caused by invariant features. It also gives for free the alignment transformation between any pair of shapes. The proposed method is based on the properties of the joint angular distribution of marginal skewness and kurtosis, the so-called SK signature, which can be efficiently computed in closed form from the raw image moments. The experiments demonstrate that the method is robust to the non-affine distortions caused by natural perspective image conditions. Thus, it can be used as an automatic preprocessing step to add affine invariance in statistical pattern recognition applications.

In this letter, we propose a novel cross-range scaling algorithm to estimate the effective rotational velocity (RV) of non-uniform rotating targets by using affine invariant normalization and weigh...

Video compression marks its necessity when a huge sized video needs to be transmitted. The process starts with the identification of GoP (group of pictures), which depends on I- (intra), B- (bidirectional) and P- (predicted) frames determination. GoP is fixed, where consecutive frames are placed in an orderly manner based on the GoP size. Conventionally, B-frames lead to buffering of memory within the past and future frames consuming more computational time. Such issues are handled by an adaptive framework for determining frames based on matching criteria rather than fixed GoP. NSEW (North–South–East–West) affine translation (NAT) is proposed for replacing B with either I- or P-frame. The proposed framework involves video compression using affine motion-based free-form transformation and video decompression using warping methodologies for the purpose of compressing and decompressing the video sequence, based on the resulted I- and P-frames. B-spline transformation was also initiated at local level along with global affine transformation to improve the subjective quality of the decompressed video sequence. The methodology was investigated for the file size, computational time, peak-signal-to-noise ratio (PSNR) and Structural Similarity index (SSIM), which proved the superiority of the proposed technique. Further, the methodology was also investigated with optimizing the affine motion parameters (AMP) using nonlinear least squares, Broyden–Fletcher–Goldfarb–Shanno (BFGS) and limited-memory BFGS which yet again proved to be far more superior to conventional techniques with an average PSNR of 38.98 dB with LBFGS. To further improve the subjective quality, affine B-spline-based motion estimation using LBFGS was implemented and observed the average PSNR gain to be 42.03 dB.

Measuring rail profile in the presence of multiple degrees of freedom vibration is a very challenging task. This paper presents a novel method based on the local affine invariant feature descriptor to calibrate distorted profiles, which are obtained by traditional rail measurement system. It has three major modules: local affine invariant (LAI) feature descriptor, affine transformation estimation and parameters refinement. LAI feature descriptor is based on the affine geometry invariant and generated by calculating the proportions of different areas. Using the proposed LAI descriptor, we implement a three-stage profile calibration including matching, estimation, and refinement based on grouping and fast iterative closest point (FICP) algorithm. The performance of proposed LAI descriptor and calibrating method is tested by performing extensive experiments. The experimental results show that our LAI descriptor is highly descriptive and robust with respect to varying resolution and noise, and the LAI descriptor based calibration is effective and repeatable.

Robust image matching is crucial for many applications of remote sensing and photogrammetry, such as image fusion, image registration, and change detection. In this paper, we propose a robust feature matching method based on support-line voting and affine-invariant ratios. We first use popular feature matching algorithms, such as SIFT, to obtain a set of initial matches. A support-line descriptor based on multiple adaptive binning gradient histograms is subsequently applied in the support-line voting stage to filter outliers. In addition, we use affine-invariant ratios computed by a two-line structure to refine the matching results and estimate the local affine transformation. The local affine model is more robust to distortions caused by elevation differences than the global affine transformation, especially for high-resolution remote sensing images and UAV images. Thus, the proposed method is suitable for both rigid and non-rigid image matching problems. Finally, we extract as many high-precision correspondences as possible based on the local affine extension and build a grid-wise affine model for remote sensing image registration. We compare the proposed method with six state-of-the-art algorithms on several data sets and show that our method significantly outperforms the other methods. The proposed method achieves 94.46% average precision on 15 challenging remote sensing image pairs, while the second-best method, RANSAC, only achieves 70.3%. In addition, the number of detected correct matches of the proposed method is approximately four times the number of initial SIFT matches.

In this paper, we introduce an affine scaling cubic regularization algorithm for solving optimization problem without available derivatives subject to bound constraints employing a polynomial interpolation approach to handle the unavailable derivatives of the original objective function. We first define an affine scaling cubic model of the approximate objective function which is obtained by the polynomial interpolation approach with an affine scaling method. At each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique. The global convergence and local superlinear convergence of the proposed algorithm are established under some mild conditions. Preliminary numerical results are reported to show the effectiveness of the proposed algorithm.

Calculating the geometry relationship by the positions of the correspondences from stereo images is a fundamental method to obtain the depth information. Such a method was quite widespread and popular thanks to its efficiency and easily accessed implementation. General speaking, the more density of the correspondences are, the more precisely the depth information can be calculated. Theoretically, a sufficient strengthened correspondences match algorithm can be utilized for depth information calculation under any circumstances. Unfortunately, the updated image feature detectors are all sensitive to the view point changes: with the rising of the stereo images' view angle, the number of matched features drastically reduced, resulting in the number of matched features not adequate to cover every details of the stereo images. This disadvantages of feature detections in practice hampers its application for the depth calibration. To tackle the sensitive of view point to stereo images, in this paper, we will propose an affine invariant affine scale space structure, which is more robust to detect the correspondences from stereo images. The purpose of affine scale space is to create a more general approach to the affine invariant image scale representation by modifying the corresponding Gaussian filters in order to cope with the specific change of view point. The affine adaptation of the scale space is to retain a linear relationship with the transiting of the view point. With linear relationship, the affine scale space can be established as a more general approach for the detection of correspondences from stereo images. With a better correspondences detection, a more precise depth information can be made.

In this paper, we introduce the notion of partial affine system that is a subset of an affine system. It has potential applications in signal analysis. A general affine system has been extensively studied; however, the partial one has not. The main focus of this paper is on partial affine system–based frames and dual frames. We obtain a necessary condition and a sufficient condition for a partial affine system to be a frame and present a characterization of partial affine system–based dual frames. Some examples are also provided.

Estimating homography using only one affine correspondence.The proposed theory makes multi-homography estimation less ambiguous.Stochastic sampling can be omitted from robust homography estimation.Affine-covariant detectors are compared w.r.t. quality of estimated homographies.Equivalence of affine and perspective-invariances for known epipolar geometry. We propose a method, called HAF, to estimate planar homography from an affine correspondence satisfying the epipolar constraint in an image pair. An affine correspondence consists of a point pair and the related local affine transformation mapping the pixels infinitely close to the point locations from the first to the second images. As a minimal solver, it estimates the homography from only one correspondence, however, it is generalized for the over-determined case as well. The required local affinities are obtained by affine-covariant feature detectors accurately. As a side-effect of the tests, the state-of-the-art affine-covariant detectors are compared to each other w.r.t. the accuracy of the estimated point-wise homographies. The proposed method is validated both on the publicly available AdelaideRMF dataset and in a synthetic testing environment.

A novel target detection method based on affine invariant interest point detection, feature encoding, and large-margin dimensionality reduction (LDR) is proposed for optical remote sensing images. First, four types of interest point detectors are introduced, and their performance in extracting low-level affine invariant descriptors using affine shape estimation is compared. Such a description can deal with significant affine transformations, including viewpoints. Second, feature encoding, which extends bag-of-words (BOW) by encoding high-order statistics, is selected to generate mid-level representation. Finally, LDR based on the large-margin constraint and stochastic subgradient is introduced to make the high-dimensional mid-level representation applicable for target detection. The experiments on aircraft and vehicle detections illustrate the effectiveness of the affine invariant description and LDR (compared with principal component analysis) in improving the detection performance. The experiments also demonstrate the effectiveness of the proposed method compared with popular approaches including Gabor, HOG, LBP, BOW, and R-CNN.

In this paper we introduce a notion of a para-complex affine hypersphere. We give a complete local classification of such hypersurfaces and give several examples. It turns out that every para-complex affine hypersphere can be constructed from (real) affine hyperspheres. As an application, we classify all 2-dimensional para-complex affine hyperspheres.

Mismatch removal is a key step in many computer vision problems. In this paper, we handle the mismatch removal problem by adopting shape interaction matrix (SIM). Given the homogeneous coordinates of the two corresponding point sets, we first compute the SIMs of the two point sets. Then, we detect the mismatches by picking out the most different entries between the two SIMs. Even under strong affine transformations, outliers, noises, and burstiness, our method can still work well. Actually, this paper is the first non-iterative mismatch removal method that achieves affine invariance. Extensive results on synthetic 2D points matching data sets and real image matching data sets verify the effectiveness, efficiency, and robustness of our method in removing mismatches. Moreover, when applied to partial-duplicate image search, our method reaches higher retrieval precisions with shorter time cost compared with the state-of-the-art geometric verification methods.

Dimensional reduction is a primary way to analyze and work with complex and large amount of multidimensional data by avoiding the effect of curse of dimensionality. This problem of constructing low dimensional embedding gains importance in number of fields like artificial intelligence, image processing, geographical research and lot more. In this paper, we introduce a modified locally linear embedding, an unsupervised learning algorithm that computes low dimensional data from complex high dimensional data using affine transformation and neighborhood preserving embedding. Unlike novel locally linear embedding, our method is affine invariant where each point is being represented by an affine combination of its neighboring points. At the end, we conduct the experiment to evaluate our proposed method and compare its performance with existing methods. Results show that our proposed method is unaffected by affine transformation, specifically shear while existing methods fail to produce correct results in case of shear.

The problem of aligning of 3D point data is the known registration task. The most popular registration algorithm is the Iterative Closest Point (ICP). This paper proposes a new algorithm for affine registration of point clouds by incorporating the affine transformation into the point-to-plane ICP algorithm. At each iterative step of the algorithm, a closed-form solution for the affine transformation is derived.

The modern era of information technology suffers a serious loss due to the lack of a cutting edge methodology for storing mega sized videos. It is at this juncture, video compression makes a mark for its necessity. There have been several research outcomes where almost all researchers have followed a particular methodology of adopting GoP (Group of pictures) for video compression, focusing on I (Intra), B (Bi-directional) & P (Predicted) frame determination. These frames remain fixed throughout the process of GoP regardless of the camera motion. Moreover, it also leads to buffering of memory within the past and future thereby consuming more computational time for B-frames. These vital issues are handled by an adaptive framework of determining frames based on a matching criteria rather than utilizing fixed GoP pattern. NSEW affine translation (NAT) is introduced for replacing B-frames with either I or P frame. The framework involves VCAME (Video Compression using Affine Motion Estimation) & VDAW (Video Decompression using Affine Warping) methodologies for compressing and decompressing a video sequence, based on the resulted I & P frames. The methodology was investigated over four vital parameters, the file size, computational time, SSIM (Structural Similarity Index) & PSNR (Peak Signal to Noise ratio), which proved the superiority of the proposed technique. Further, the methodology was also investigated with optimizing the affine motion parameters (AMP) using nonlinear least squares, BFGS (Broyden–Fletcher–Goldfarb–Shanno) and Limited-memory BFGS which yet again proved to be far more superior than conventional techniques yielding an average PSNR gain of 2.52 dB .

In this study, partial registration problem with outliers and missing data in the affine case is discussed. To solve this problem, a novel objective function is proposed based on bidirectional distance and trimmed strategy, and then a new affine trimmed iterative closest point algorithm is given. First, when bidirectional distance measurement is applied, the ill-posed partial registration problem in the affine case is prevented. Second, the overlapping percentage is solved by using trimmed strategy which uses as many correct overlapping points as possible. The authors' method computes the affine transformation, correspondence and overlapping percentage automatically at each iterative step. In this way, it handles partially overlapping registration with outliers and missing data in the affine case well. Experimental results demonstrate that their method is more robust and precise than the state-of-the-art algorithms. It also has good convergence and similar running time with traditional algorithms.

Correlation filter achieves promising performance with high speed in visual tracking. However, conventional correlation filter based trackers cannot tackle affine transformation issues such as scale variation, rotation and skew. To address this problem, in this paper, we propose a part-based representation tracker via kernelized correlation filter (KCF) for visual tracking. A Spatial-Temporal Angle Matrix (STAM), severed as confidence metric, is proposed to select reliable patches from parts via multiple correlation filters. These stable patches are used to estimate a 2D affine transformation matrix of the target in a geometric method. Specially, the whole combination scheme for these stable patches is proposed to exploit sampling space in order to obtain numerous affine matrices and their corresponding candidates. The diversiform candidates would help to seek for the optimal candidate to represent the object's accurate affine transformation in a higher probability. Both qualitative and quantitative evaluations on VOT2014 challenge and Object Tracking Benchmark (OTB) show that the proposed tracking method achieves favorable performance compared with other state-of-the-art methods.

Affine invariant measures are powerful tools to develop robust shape descriptors that can be applied, for example, to shape matching, shape retrieval, or symmetry detection problems. In this work we introduce estimators for the affine structure of surfaces represented by triangle meshes, i.e. affine co-normal and normal vectors, affine curvature tensors, affine mean and Gaussian curvatures, and affine principal directions and curvatures. The proposed method estimates the affine normal using a finite differences scheme together with a least-squares approximation, followed by a weighted average strategy to approach discrete affine curvature tensors. When compared to the exact geometric measures of analytic models, experiments on regular meshes obtain small error, which decreases for finer meshes, and outperforms the state-of-the-art method in some cases. Experiments to evaluate affine invariance show that the difference between measures before and after equi-affine transformations remains small even after large deformations.

A generalized polygon is an ordered set of vertices. This notion generalizes the concept of the boundary of a polygonal shape because self-intersections are allowed. In this paper we study the problem of matching generalized polygons under affine transformations. Our approach is based on invariants. Firstly we associate an ordered set of complex numbers with each polygon and construct a collection of complex scalar functions on the space of plane polygons. These invariant functions are defined as quotients of the so-called Fourier descriptors, also known as discrete Fourier transforms.Each one of these functions is invariant under similarity transformations; that is, the function associates the same complex number to similar polygons. Moreover, if two polygons are affine related (one of them is the image of the other under an affine transformation), the pseudo-hyperbolic distance between their associated values is a constant that depends only on the affine transformation involved, but independent of the polygons.More formally, given a collection { Z 1 , Z 2 , ź , Z m } of n-sided polygons in the plane and a query polygon W, we give algorithms to find all Z ź such that f ( Z ź ) = W + Δ W , where f is an unknown affine transformation and Δ W = ( Δ w 1 , ź , Δ w n ) with | Δ w k | ź ź , where ź is certain tolerance.

Rolling shutter distortions degrade the quality of videos captured by hand-held cameras. This letter proposes an affine motion model for removing rolling shutter distortions. The model represents the image motion during image capture as a sequence of affine transformations and computes the composition of these affine transformations precisely. Because an affine transformation can be represented by a neural network with one layer of linear neurons, the motion model can be represented by a multilayer neural network of linear neurons. Thus, the backpropagation algorithm can be used to improve the efficiency of the optimization process that estimates the model parameters. The proposed model is calibration-free. It is more general than other rolling shutter motion models because it only assumes that the velocity of the image during image acquisition is piecewise constant. Experimental results demonstrate that the model is more accurate than two state-of-the-art models and that the model parameters can be estimated efficiently.

In this work we give a systemic study of affine translation surfaces in affine 3-dimensional space. Specifically, we obtain the complete classification of minimal affine translation surfaces. Moreover, we consider affine translation surfaces with some natural geometric conditions, such as constant affine mean curvature and constant Gauss–Kronecker curvature. Some characterization results with these geometric conditions are also obtained.

In this study, an affine registration algorithm with reference-omitted scheme and soft correspondence is proposed. It is an iterative method with two-step matching process at each iteration, named as forward matching and backward matching. Due to the introduction of backward matching, two sets of points are alternately to be reference set, such that the selection of reference set is omitted. Failure caused by different reference sets can be corrected with the reference-omitted scheme, and even there is obvious difference in scale. Additionally, soft correspondence is applied to avoid estimating the initial transformations. The simulation and real experimental results show that the proposed method substantially outperforms the current affine registration methods, especially when the scale difference between two sets of points is obvious or there are outliers in one set.

The goal of this paper is to clarify when a stochastic partial differential equation with an affine realization admits affine state processes. This includes a characterization of the set of initial points of the realization. Several examples, as the HJMM equation from mathematical finance, illustrate our results., 27 pages

Since few years ago it has been generally accepted without any dispute that the 3D affine transformation applied to high-resolution satellite imageries (HRSI), produces results as accurate as those obtained by the RPCs derived from rigorous photogrammetric model. However, as the higher order terms are absent in the affine transformation, the degree of success of this model obviously hinges upon the geometric nature of the imagery to be geo-rectified. In authors view, there are a latent confusion and misunderstanding in the minds of the photogrammetric practitioners as regards the potential of the 3D affine transformation as a replacement model for the geometric correction of the HRSI. The main intention of this paper is, therefore, to analyse the 3D affine transformation by concentrating more on its limitations. To obtain deeper insight into the nature of the 3D affine model, it is applied to images with larger field of view as well as the images of highly mountainous terrains. The geo-coding success of the affine model is then evaluated by comparing the object coordinates of a dense cloud of homologous points derived by the affine model with the object coordinates of the same points obtained by the standard terrain-independent rational functions. Extensive tests conducted over excessively mountainous as well as the hilly terrains indicate that there are clear distortion trends in the residual ground coordinates that cannot be fully absorbed into the 3D affine coefficients. The sources of these non-linear trends such as the satellite attitude and position variations, the terrain relief, the earth curvature and their impact on the final accuracy are analysed using the scatter patterns of the residual errors.

Since unity step size could guarantee the fastest convergence and more detailed analysis for the affine projection (AP) algorithm, a statistical tracking behavior of AP algorithm is discussed in this paper. Deterministic recursive equations are derived for the mean weight error and mean-square error. All the possible correlations between the adaptive filtering coefficients and the past measurement noise are considered as well.

For a non Archimedean local field which is not of characteristic $2$, nor an extension of $\mathbb Q_2$, we construct a pseudo-differential calculus covariant under a unimodular subgroup of the affine group of the field. Our phase space is a quotient group of the covariance group. Our main result is a generalisation on that context of the Calder\'on-Vaillancourt estimate. Our construction can be thought as the non Archimedean version of Unterberger's Fuchs calculus and our methods are mainly based on Wigner functions and on coherent states transform., Comment: to appear in JFG

Quantum circuits of the generalized affine transform are devised based on the novel enhanced quantum representation of digital images. A novel quantum image encryption algorithm combining the generalized affine transform with logistic map is suggested. The gray-level information of the quantum image is encrypted by the XOR operation with a key generator controlled by the logistic map, while the position information of the quantum image is encoded by the generalized affine transform. The encryption keys include the independent control parameters used in the generalized affine transform and the logistic map. Thus, the key space is large enough to frustrate the possible brute-force attack. Numerical simulations and analyses indicate that the proposed algorithm is realizable, robust and has a better performance than its classical counterpart in terms of computational complexity.

This paper deals with the following problem: What can be said about the shape of an object if a certain invariant of it is known? Such a, herein called, shape invariant interpretation problem has not been studied/solved for the most of invariants, but also it is not known to which extent the shape interpretation of certain invariants does exist. In this paper, we consider a well-known second-order affine moment invariant. This invariant has been expressed recently Xu and Li (2008) as the average square area of triangles whose one vertex is the shape centroid while the remaining two vertices vary through the shape considered. The main results of the paper are (i) the ellipses are shapes which minimize such an average square triangle area, i.e., which minimize the affine invariant considered; (ii) this minimum is $$1/(16\pi ^2)$$1/(16?2) and is reached by the ellipses only. As by-products, we obtain several results including the expression of the second Hu moment invariant in terms of one shape compactness measure and one shape ellipticity measure. This expression further leads to the shape interpretation of the second Hu moment invariant, which is also given in the paper.

Shape is considered to be one of the most promising tools to represent and recognise an object. In this study, an effective and rigorous shape matching algorithm is developed based on a new descriptor and relaxation labelling technique. For each contour point, the descriptor captures the distribution of all points within the shape region along the vector perpendicular to that from the centroid to the point. In addition to stable affine invariance, the descriptor is robust to noise since it makes use of all points in the shape region. The descriptor distance is used to initialise the contour point matching probability, and relaxation labelling technique is utilised to update the matching probability using a new compatibility coefficient function, which is defined based on the shape projection preserving characteristic. The experiments on synthetic and real remote sensing data are provided to test the performance of the authors’ proposed algorithm. Compared to other four state-of-the-art contour-based shape matching algorithms, their algorithm is more robust and capable of shape matching under affine transformations and noise.

Face recognition (FR) is one of the most effervescent fields of research with extensive applications that span numerous domains, and it stands resolutely as one of the most challenging problems in computer vision. The accuracy of FR systems is severely affected when two images under consideration for a match, vary in their scale and/or affine angles. The prevalent affine and scale invariant recognition systems have been predominantly developed only for objects, and hence in this study, the authors propose a novel approach for faces based on the affine-SIFT (ASIFT) and two-dimensional principal component analysis (2DPCA) techniques, to accomplish the formidable task of facial image recognition, invariant of scale and affine angles, i.e. the ability to simulate with enough accuracy, all the distortions caused by the differences in resolution and the variation of the camera optical axis direction. In the formulation of ASIFT-2DPCA, they investigate three different variants of 2DPCA: classical 2DPCA, quaternion 2DPCA and sparse 2DPCA to gauge as to which is more effective. The authors'experimentations will demonstrate that the proposed approach can robustly handle affine and scale variations, and hence provide better accuracy and matching performance than the state-of-the-art methodologies.

The development of a virtual environments is desirable in many engineering applications for several reasons. For example, it can reduce the number of dangerous operations during the virtual inspection of mines, pipes for gas, water and sewer to inspect its current condition and identify possible failures. The proposal in this paper is an alternative solution to this problem obtaining a virtual image through the reconstruction of the inner pipe for assessment and fault detection. The proposed method employs regions with affine transformations used as dense correspondence, which are calculated through local correspondences, such as the detector/descriptor SIFT (Scale-invariant feature transform). The SIFT is required to obtain the position and orientation of the camera in the captured positions to get the structure of the scene through correspondences and to create a 3D reconstruction.

We introduce a new family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric planes coincide and contain the affine normal of the hyperquadric. In particular, any surface immersed in a locally strictly convex hyperquadric is affine semiumbilical with respect to the symmetric or antisymmetric equiaffine planes.

The aim in this paper is to construct an affine transformation using the classical physics analogy between the fields of optics and mechanics. Since optics and mechanics both have symplectic structures, the concept of optics can be replaced by that of mechanics and vice versa. We list the four types of eikonal (generating functions). We also introduce a unitary operator for the affine transformation. Using the unitary operator, the kernel (propagator) is calculated and the wavization (quantization) of the Gabor function is discussed. The dynamic properties of the affine transformed Wigner function are also discussed.

The affine rank minimization problem is to find a low-rank matrix satisfying a set of linear equations, which includes the well-known matrix completion problem as a special case and draws much attention in recent years. In this paper, a new model for affine rank minimization problem is proposed. The new model not only enhances the robustness of affine rank minimization problem, but also leads to high nonconvexity. We show that if the classical projected gradient method is applied to solve our new model, the linear convergence rate can be established under some conditions. Some preliminary experiments have been conducted to show the efficiency and effectiveness of our method.

In this paper, we discuss the digitization and applications of smooth affine shear tight frames, a recently developed new class of directional multiscale representation systems. An affine wavelet tight frame is generated by isotropic dilations and translations of directional wavelet generators, while an affine shear tight frame is generated by anisotropic dilations, shears, and translations of shearlet generators. These two tight frames are actually connected in the sense that an affine shear tight frame can be obtained from an affine wavelet tight frame through subsampling. Consequently, an affine shear tight frame has an underlying filter bank from the MRA structure of its associated affine wavelet tight frame. We discuss the digitization of digital affine shear filter banks associated with the affine shear tight frames. Moreover, we provide the detailed algorithmic steps for both the forward and backward digital affine shear transforms. Analysis of the redundancy rate and computational complexity shows...