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In the area of combination of adaptive filters, two main approaches, namely convex and affine combinations have been introduced. In this article, the relation between these two approaches is investigated. First, the problem of obtaining optimal convex combination coefficients is formulated as the projection of the optimal affine combination weights to the unit simplex in a weighted inner product space. Based on this formulation the closed form expressions for optimal combination weights and target MSE levels are obtained for two and three branch cases.

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.

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.

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.

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 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.

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.

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.

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.

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.

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 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.

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.

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.

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...

Piecewise affine models often provide a good approximation to describe continuous systems, but may involve a high degree of simplification. To compare solutions of the continuous and piecewise affine models, it is important to quantify the differences between solutions in each region of the state space. As an approach, we will use enveloping "bands" to characterize continuous activation or inhibition functions, and then describe the differences between continuous and piecewise affine solutions in terms of the width δ of these bands. As a case study, we will consider the negative feedback loop, a classical motif in two dimensions which results in oscillating behaviour. For this example, it is shown that the two types of models may differ only on a compact invariant set (the interior of a limit cycle), whose diameter is a function of the band width δ.

The behavior of affine processes, which are ubiquitous in a wide range of applications, depends crucially on the choice of state space. We study the case where the state space is compact, and prove in particular that (i) no diffusion is possible; (ii) jumps are possible and enforce a grid-like structure of the state space; (iii) jump components can feed into drift components, but not vice versa. Using our main structural theorem, we classify all bivariate affine processes with compact state space. Unlike the classical case, the characteristic function of an affine process with compact state space may vanish, even in very simple cases., Electronic Journal of Probability, 23, ISSN:1083-6489

In this work, the affine point set matching is formulated under a variational Bayesian framework and the model points are projected forward into the scene space by a linear transformation. A directed acyclic graph is presented to represent the relationship between the parameters, latent variables, model and scene point sets and an iterative approximate algorithm is proposed for the estimation of the posterior distributions over parameters. Furthermore, the anisotropic covariance is assumed on the transition variable and one Gaussian component is provided for the inference of outlier points. Experimental results demonstrate that the proposed algorithm achieves good performance in terms of both robustness and accuracy.

A specific study of the orthographic, weak-perspective and paraperspective cameras.An algebraic procedure solving the affine correction problem for each camera model.A closed-form solution for each camera model.A full characterization of the affine corrections' generic ambiguities for each camera model.An experimental evaluation comparing the algebraic procedures to global polynomial optimization and an interior-point method. Given a general affine camera, we study the problem of finding the closest metric affine camera, where the latter is one of the orthographic, weak-perspective and paraperspective projection models. This problem typically arises in stratified Structure-from-Motion methods such as factorization-based methods. For each type of metric affine camera, we give a closed-form solution and its implementation through an algebraic procedure. Using our algebraic procedure, we can then provide a complete analysis of the problem's generic ambiguity space. This also gives the means to generate the other solutions if any.

In this paper we introduce an affine invariant distance definition from a $$2D$$2D point to the boundary of a bounded shape using morphological multiscale analysis. We study the mathematical behavior of this distance by examining separately the cases of convex and non-convex shapes. We prove that the proposed distance is bounded in the convex hull of the shape and infinite otherwise. A numerical scheme is given as well as experiments illustrating the behavior of the affine invariant distance.

This paper proposes a novel method to address the registration of images with affine transformation. Firstly, the Maximally Stable Extremal Region (MSER) detection method is performed on the reference image and the image to be registered, respectively. And the coarse affine transformation matrix between the two images is estimated by the matched MSER pairs. Two circular regions containing roughly the same image content are also obtained by fitting and normalizing the centroids of the matched MSERs from the two images. Secondly, a scale invariant and approximate affine transformation invariant feature point detection algorithm based on the Gabor filter decomposition and phase congruency is performed on the two coarsely aligned regions, and two feature point sets are achieved, respectively. Finally, the affine transformation matrix between the two feature point sets is obtained by using a probabilistic point set registration algorithm, and the final affine transformation matrix between the reference image and the image to be registered is achieved according to the coarse affine transformation matrix and the affine transformation matrix between the two feature point sets. Several sets of experiments demonstrate that our proposed method performs competitively with the classical scale-invariant feature transform (SIFT) method for images with scale changes, and performs better than the traditional MSER and Affine-SIFT (ASIFT) methods for images with affine distortions. Moreover, the proposed method shows higher computation efficiency and robustness to illumination change than some existing area-based or feature-based methods do. A combination of coarse alignment and refinement is employed in the method.Input images are coarsely aligned by fitting and normalizing the matched MSERs.A phase congruency and point set registration based refinement step is used.The method accurately and efficiently aligns images with affine distortions.The method is robust to illumination changes.

In this paper, we propose a variational approach for the affine point set matching problems under the Bayesian probabilistic framework. A directed acyclic graph is provided for the representation of the joint probability over affine transformation, random variables and the point sets. Based on the directed graph, a variational iterative algorithm is derived to approximate the posteriors of the random variables and the anisotropic Gaussian mixtures are used for the estimation of the spurious outliers instead of the frequently-used uniform distribution. Experimental results demonstrate that our method achieves good performance in terms of both robustness and accuracy and is comparable to other state-of-the-art point registration algorithms especially in the case of complicated outliers.

For vision-based mobile robot navigation, images of the same scene may undergo a general affine transformation in the case of significant viewpoint changes. So, a novel method for detecting affine invariant interest points is proposed to obtain the invariant local features, which is coined polynomial local orientation tensor (PLOT). The new detector is based on image local orientation tensor that is constructed from the polynomial expansion of image signal. Firstly, the properties of local orientation tensor of PLOT are analyzed, and a suitable tuning parameter of local orientation tensor is chosen so as to extract invariant features. The initial interest points are detected by local maxima search for the smaller eigenvalues of the orientation tensor. Then, an iterative procedure is used to allow the initial interest points to converge to affine invariant interest points and regions. The performances of this detector are evaluated on the repeatability criteria and recall versus 1-precision graphs, and then are compared with other existing approaches. Experimental results for PLOT show strong performance under affine transformation in the real-world conditions.

This paper investigates the structure of general affine subspaces of L2(Rd) . For a d × d expansive matrix A, it shows that every affine subspace can be decomposed as an orthogonal sum of spaces each of which is generated by dilating some shift invariant space in this affine subspace, and every non-zero and non-reducing affine subspace is the orthogonal direct sum of a reducing subspace and a purely non-reducing subspace, and every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces when |detA| = 2.

In this paper, a new affine projection sign algorithm with a variable step-size, robust in impulsive noise environments, is proposed. For that purpose, l1-norm of the a posteriori error vector is minimized under a box constraint on the step-size. Since the proposed l1-norm minimization problem is non-differentiable convex one, an efficient numerical procedure is developed for the affine projection structure algorithm. Finally, it is demonstrated that the proposed algorithm, not requiring any a priori knowledge, yields more robust convergence in various impulsive noise environments than conventional algorithms with robustness to outliers and recent variable step-size algorithms with a priori information.

The problem of aligning of 3D point data is the known registration task. The most popular registration algorithm is the Iterative Closest Point (ICP) algorithm. The traditional ICP algorithm is a fast and accurate approach for rigid registration between two point clouds but it is unable to handle affine case. Recently, extension of the ICP algorithm for composition of scaling, rotation, and translation is proposed. A generalized ICP version for an arbitrary affine transformation is also suggested. In this paper, a new iterative algorithm for registration of point clouds based on the point-to-plane ICP algorithm with affine transformations is proposed. At each iteration, a closed-form solution to the affine transformation is derived. This approach allows us to get a precise solution for transformations such as rotation, translation, and scaling. With the help of computer simulation, the proposed algorithm is compared with common registration algorithms.

In this work we adapt the theory of piecewise differential systems to the world of piecewise affine maps. Through this adaptation it is possible to define a sliding map in the sense of Filippov. We prove that there exists a choice for the pieces of the piecewise affine map, such that the associate sliding map possesses chaotic dynamics.

© 2016 Elsevier B.V. It is well-known that Vrancken–Li–Simon classified locally strongly convex affine hyperspheres in Rn+1 whose affine metric are of constant sectional curvatures, but on the other side it is still a difficult problem to classify n-dimensional locally strongly convex affine hyperspheres whose affine metrics are Einstein. In this paper, we have solved the problem in case n=4. publisher: Elsevier articletitle: On four-dimensional Einstein affine hyperspheres journaltitle: Differential Geometry and its Applications articlelink: http://dx.doi.org/10.1016/j.difgeo.2016.10.003 content_type: article copyright: © 2016 Elsevier B.V. All rights reserved. ispartof: Differential Geometry and its Applications vol:50 pages:20-33 status: published

We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the transversal vector field on M we introduce a new affine normal plane and the familly of affine distance functions on M. We show that the singularities of the family of affine distance functions appear at points on the affine normal plane and the affine focal points correspond to degenerate singularities of this family. Moreover we show that if M is immersed in a locally strictly convex hypersurface, then the affine normal plane contains the affine normal vector to the hypersurface and conclude that any surface immersed in a locally strictly convex hypersphere is affine semiumbilical.

A new kind of geometric invariants is proposed in this paper, which is called affine weighted moment invariant (AWMI). By combination of affine differential invariants and the framework of global integral, they can more effectively extract features of images, increase the number of low-order invariants. The experimental results show that AWMIs have good stability and distinguishability. Also, we find that the accuracy of image retrieval and classification has been improved by combining AWMIs with other traditional moment invariants.

An affine equivalence problem is to find affine mappings A and B such that \(F=B\circ S\circ A\) for given two permutations F and S, which was first studied by Biryukov et al. Their algorithm for solving an affine equivalence problem is quite efficient and has been used in the cryptanalytic toolbox for many cryptographic schemes. Recently, Baek et al. presented a specialized affine equivalence algorithm (SAEA), which solves an affine equivalence problem in the case that S is a concatenation of several smaller S-boxes. The SAEA is more efficient than the affine equivalence algorithm for special cases, but its complexity mainly depends on the entire input size of F.

We proposed a kind of naturally combined shape-color affine moment invariants (SCAMI), which consider both shape and color affine transformations simultaneously in one single system. In the real scene, color and shape deformations always exist in images simultaneously. Simple shape invariants or color invariants cannot be qualified for this situation. The conventional method is just to make a simple linear combination of the two factors. Meanwhile, the manual selection of weights is a complex issue. Our construction method is based on the multiple integration framework. The integral kernel is assigned as the continued product of the shape and color invariant cores. It is the first time to directly derive an invariant to dual affine transformations of shape and color. The manual selection of weights is no longer necessary, and both the shape and color transformations are extended to affine transformation group. With the various of invariant cores, a set of lower-order invariants are constructed and the completeness and independence are discussed detailed. A set of SCAMIs, which called SCAMI24, are recommended, and the effectiveness and robustness have been evaluated on both synthetic and real datasets.

Refinable function-based affine frames and affine bi-frames have been extensively studied in the literature. All these works are based on some restrictions on refinable functions. This paper addresses what are expected from two general refinable functions. We introduce the notion of weak (quasi-) affine bi-frame; present a refinable function-based construction of weak (quasi-) affine bi-frames; and obtain a fast algorithm associated with weak affine bi-frames. An example is also given to show that our construction is optimal in some sense.

Shape recognition deals with the study geometric structures. Modern surface processing methods can cope with non-rigidity--by measuring the lack of isometry, deal with similarity or scaling--by multiplying the Euclidean arc-length by the Gaussian curvature, and manage equi-affine transformations--by resorting to the special affine arc-length definition in classical equi-affine differential geometry. Here, we propose a computational framework that is invariant to the full affine group of transformations (similarity and equi-affine). Thus, by construction, it can handle non-rigid shapes. Technically, we add the similarity invariant property to an equi-affine invariant one and establish an affine invariant pseudo-metric. As an example, we show how diffusion geometry can encapsulate the proposed measure to provide robust signatures and other analysis tools for affine invariant surface matching and comparison.

Feature-based image matching algorithms play an indispensable role in automatic target recognition (ATR). In this work, a fast image matching algorithm (FIMA) is proposed which utilizes the geometry feature of extended centroid (EC) to build affine invariants. Based on affine invariants of the length ratio of two parallel line segments, FIMA overcomes the invalidation problem of the state-of-the-art algorithms based on affine geometry features, and increases the feature diversity of different targets, thus reducing misjudgment rate during recognizing targets. However, it is found that FIMA suffers from the parallelogram contour problem and the coincidence invalidation. An advanced FIMA is designed to cope with these problems. Experiments prove that the proposed algorithms have better robustness for Gaussian noise, gray-scale change, contrast change, illumination and small three-dimensional rotation. Compared with the latest fast image matching algorithms based on geometry features, FIMA reaches the speedup of approximate 1.75 times. Thus, FIMA would be more suitable for actual ATR applications.

We derive an explicit relation between local affine approximations resulting from matching of affine invariant regions and the epipolar geometry in the case of a two view geometry. Most methods that employ the affine relations do so indirectly by generating pointwise correspondences from the affine relations. In this paper we derive an explicit relation between the local affine approximations and the epipolar geometry. We show that each affine approximation between images is equivalent to 3 linear constraints on the fundamental matrix and that the linear conditions guarantee the existence of an homography, compatible with the fundamental matrix. We further show that two affine relations constrain the location of the epipole to a conic section. Therefore, the location of the epipole can be extracted from 3 regions by intersecting conics. The result is further employed to derive a procedure for estimating the fundamental matrix, based on the estimated location of the epipole. It is shown to be more accurate and to require less iterations in LO-RANSAC based estimation, than the current point based approaches that employ the affine relation to generate pointwise correspondences and then calculate the fundamental matrix from the pointwise relations.

Current techniques for shape analysis tend to seek invariance to similarity transformations (rotation, translation, and scale), but certain imaging situations require invariance to larger groups, such as affine or projective groups. Here we present a general Riemannian framework for shape analysis of planar objects where metrics and related quantities are invariant to affine and projective groups. Highlighting two possibilities for representing object boundaries-ordered points (or landmarks) and parameterized curves-we study different combinations of these representations (points and curves) and transformations (affine and projective). Specifically, we provide solutions to three out of four situations and develop algorithms for computing geodesics and intrinsic sample statistics, leading up to Gaussian-type statistical models, and classifying test shapes using such models learned from training data. In the case of parameterized curves, we also achieve the desired goal of invariance to re-parameterizations. The geodesics are constructed by particularizing the path-straightening algorithm to geometries of current manifolds and are used, in turn, to compute shape statistics and Gaussian-type shape models. We demonstrate these ideas using a number of examples from shape and activity recognition.

This paper studies the elliptical statistical affine shape theory under certain particular conditions on the evenness or oddness of the number of landmarks. In such a case, the related distributions are polynomials, and the inference is easily performed; as an example, a landmark data is studied, and the performance of the polynomial density versus the usual series density is compared.

In order to describe the shape of the trees in nature better and simulate the trees realistically, This method that by changing the parameters of affine transformation matrix the different types of three dimensional fractal images are drawn is discussed.and the model of generating three dimensional plant is built based on iterated function system.Finally take an example like a set of images through contractive affine transformation and in the chaosPro environment,the graphics are generated and the different types of natural scenery of trees are simulated,using the established model and translation, rotation and scaling of affine transformation.