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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.

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 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 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.

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

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.

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 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 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.

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.

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.

Moment has been widely used for contour matching. To use the moment to achieve contour matching under affine transformations, the affine-invariant support point set (SPS) should be constructed first. Then, a novel method of acquiring SPS based on the contour projection (SPS-CP) is proposed here. For an arbitrary selected contour point, the contour is projected onto the line vertical to the vector connecting the contour centroid and the selected point, and the contour points with the sampled projection values are picked up to form the SPS-CP of the point. SPS-CP which captures the global structure of the contour is stably affine-invariant. Experiments on synthetic and real data demonstrate that moments generated from SPS-CP outperform those generated from SPSs sampled by uniform spacing or affine length.

We study the reach control problem for affine systems on simplices, and the focus is on cases when it is known that the problem is not solvable by continuous state feedback. We examine from a geometric viewpoint the structural properties of the system which make continuous state feedbacks fail. This structure is encoded by so-called reach control indices, which are defined and developed in the paper. Based on these indices, we propose a subdivision algorithm and associated piecewise affine feedback. The method is shown to solve the reach control problem in all remaining cases, assuming it is solvable by open-loop controls.

For affine control problems, it is known that the optimal controller for these classes of problems can be expressed in terms of the associated co-state. A method to obtain the co-state vector in terms of the state vector is given in this article. Using this method, the optimal control can be written as a function of state variable. Further the method is illustrated using examples.

In this paper we classify convex compact ancient solutions to the affine curve shortening flow: namely, any convex compact ancient solution to the affine curve shortening flow must be a shrinking ellipse. The method combines a rescaling argument inspired by \cite{Wang}, affine invariance of the equation and monotonicity of the affine isoperimetric ratio. We will give two proofs. The essential ideas are related, but the first one uses level set represenation of the evolution. The second proof employs Schauder's estimates, and it also provides a new simple proof for the corresponding classification result to the higher dimensional affine normal flow., 5 pages, comments are welcome

We study improper affine spheres with some admissible singularities, called improper affine maps, and associated to the unimodular Hessian equation. In particular, we characterize when a curve of R 3 is the singular curve of some improper affine map with prescribed cuspidal edges and swallowtails. Also, we consider improper affine maps with isolated singularities and show some similarities and differences between the Hessian +1 equation and the Hessian −1 equation. As a consequence, we construct global examples with the desired singularities.

Common fixed point results for new classes of non-commuting self-maps and non-expansive. Within the class of all self-maps ff and TT of a ww −starshaped subset MM of XX where ff is affine or ww −affine. We apply them to obtain several invariant approximation results which unify, extend, and complement well-known results.

Fitting piecewise affine models to data points is a pervasive task in many scientific disciplines. In this work, we address the k-Piecewise Affine Model Fitting with Piecewise Linear Separability problem (k-PAMF-PLS) where, given a set of m points { a 1 , ź , a m } ź R n and the corresponding observations { b 1 , ź , b m } ź R , we have to partition the domain R n into k piecewise linearly (or affinely) separable subdomains and to determine an affine submodel (i.e., an affine function) for each of them so as to minimize the total linear fitting error w.r.t. the observations bi.To solve k-PAMF-PLS to optimality, we propose a Mixed-Integer Linear Programming (MILP) formulation where symmetries are broken by separating shifted column inequalities. For medium-to-large scale instances, we develop a four-step heuristic involving, at each iteration, a point reassignment step based on the identification of critical points and a domain partition step based on multicategory linear classification. Differently from traditional approaches proposed in the literature for similar fitting problems, in both our exact and heuristic methods the domain partitioning and submodel fitting aspects are taken into account simultaneously.Computational experiments on real-world and structured randomly generated instances show that, with our MILP formulation with symmetry breaking constraints, we can solve to proven optimality many small-size instances. Our four-step heuristic turns out to provide close-to-optimal solutions for the small-size instances, while allowing to tackle instances of much larger size. The experiments also show that the combined impact of the main features of our heuristic is quite substantial when compared to standard variants not including them.We conclude the paper with an application to the identification of dynamical piecewise affine systems, for which we obtain promising results of comparable quality with those achieved with state-of-the-art methods from the literature (hinging hyperplane models and sum-of-norms regularization) on benchmark data sets. HighlightsWe address the fundamental piecewise affine model fitting problem.We propose an MILP formulation, strengthened with symmetry breaking constraints.We present a heuristic with point-reassignment and domain partition at each iteration.We report extensive results on real-world and structured random instances.We conclude with an application to the identification of hybrid dynamical systems.

This paper describes methods for tracking using point and line features in affine views to provide fundamental invariance to changes of focal length. It first demonstrates how an earlier method of transfer-based tracking using spatio-temporal matching of point features in a stereo active head is indeed zoom-invariant. In order to also make use of lines the paper then illustrates how the affine tri- and quadrifocal tensors may be applied to tracking with zoom in monocular and stereo systems respectively. The usefulness of the tensors is evident from their ability to transfer a fixation point in two uncalibrated images into novel views. Whereas the trifocal tensor is already familiar and in common use for matching and reconstruction, we believe this to be the first practical application of a quadrifocal tensor. We develop expressions for affine tri- and quadrifocal tensors, and using novel affine specializations of existing projective algorithms we show how computation of the tensors is faster, simpler and more stable. Experiments on real images are presented.

This paper investigates the identification of continuous piecewise affine systems in state space form with jointly unknown partition and subsystem matrices. The partition of the system is generated by the so-called centers. By representing continuous piecewise affine systems in the max-form and using a recursive Gauss-Newton algorithm for a suitable cost function, we derive adaptive laws to online estimate parameters including both subsystem matrices and centers. The effectiveness of the proposed approach is demonstrated with a numerical example.

In this paper we consider convex improper affine maps of the 3-dimensional affine space and classify their singularities. The main tool developed is a generating family with properties that closely resembles the area function for non-convex improper affine maps., 11 pages, 2 figures

An approach of the edge feature matching of remote sensing images based on parameter decomposition of the affine transformation model is presented. First, the adaptability of affine transformation model used in image matching is analyzed, and parameters of the affine transformation model are decomposed. Second, the sum of squared gradient (SSG) similarity is constructed based on the edge feature in the images, and the parameters of the affine transformation model between the images are solved with the Genetic Algorithms (GA) which is used to obtain the global optimum solution of the similarity. Finally, the performance of the approach is analyzed in theory and validated with remote sensing image matching experiments.

Affine Differential Geometry provides a set of measures invariant under a larger set of transformations compared to rigid motions. This leads to several applications using robust shape descriptors. Although affine-invariant operations are already used for surfaces, they do not intend to approximate the definitions of Affine Differential Geometry, which are the basis for further differential invariants. In this work we propose estimators for the local affine structure of an implicit surface, i.e. the affine metric, the co-normal and normal vectors, and the affine Gaussian and mean curvatures. The direct derivation of the formulae from the implicit function theorem lead to very intensive computations and numerical instabilities. This work further proposes a geometrical reduction allowing a much simpler and more stable formulae, and compares the results by incorporating the proposed estimators in Marching Cubes based algorithms.

In this paper we investigate analytic affine control systems = X + uY , u ∈ [ a,b ] , where X,Y is an orthonormal frame for a generalized Martinet sub-Lorentzian structure of order k of Hamiltonian type. We construct normal forms for such systems and, among other things, we study the connection between the presence of the singular trajectory starting at q 0 on the boundary of the reachable set from q 0 with the minimal number of analytic functions needed for describing the reachable set from q 0 .

The size of 3D data stored around the web has become bigger. Therefore the development of recognition applications and retrieval systems of 3D models is important. This paper deals with invariants for 3D models recognition. Thus under general affine transform we propose in a first time determinants of three points to realize invariance under affinity. To solve starting point problem we needs Fourier Series (FS) to extract affine invariant descriptors, called Fourier Series Descriptor (FSD). The difference between first and second approaches: in first approach determinants are computed on cartesian coordinates directly while in the second one determinants are computed from FS coefficients to eliminate dependency on starting point. The FS are also applied on 2D slices to generate affine invariants for 3D volume. FS can be computed based on hole points of volume, but this technique. The principal advantages of proposed approaches are the possibility to handle affine transform and 3D volume. Two types of 3D objects are used in the experimentations: mesh and volume, the Princeton Shape Benchmarek (PSB) is also used to test our descriptor based on FSD.

Affine invariant functions are constructed in spatial domain. Unlike the previous affine representation functions in transform domain, these functions are constructed directly on the object contour without any transformation. To eliminate the effect of the choice of points on the contour, an affine invariant function using seven points on the contour is constructed. For objects with several separable components, a closed curve is derived to construct the affine invariant functions. Several experiments have been conducted to evaluate the performance of the proposed method. Experimental results show that the constructed affine invariant functions can be used for object classification.

In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes, we show how to solve discrete Cauchy problems analogous to the Cauchy problems for smooth analytic improper affine spheres and smooth analytic affine minimal surfaces.