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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.

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.

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.

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.

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.

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.

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.

Based on Tamaschke’s axioms for affine spaces a relationship between affine subspaces of an affine space and groups of translations is established. Further, it is shown that every non-desarguesian translation plane possesses a distinguished class of affine subspaces.

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.

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.

Matching a pair of affine invariant regions between images results in estimation of the affine transformation between the regions. However, the parameters of the affine transformations are rarely used directly for matching images, mainly due to the lack of an appropriate error metric of the distance between them. In this paper we derive a novel metric for measuring the distance between affine transformations: Given an image region, we show that minimization of this metric is equivalent to the minimization of the mean squared distance between affine transformations of a point, sampled uniformly on the image region. Moreover, the metric of the distance between affine transformations is equivalent to the l 2 norm of a linear transformation of the difference between the six parameters of the affine transformations. We employ the metric for estimating homographies and for estimating the fundamental matrix between images. We show that both homography estimation and fundamental matrix estimation methods, based on the proposed metric, are superior to current linear estimation methods as they provide better accuracy without increasing the computational complexity.

In this paper, a continuous method approach is adopted to study both the entire process and the limiting behaviors of the dual affine scaling continuous trajectories for linear programming. Our approach is different from the method presented by Adler and Monteiro (Adler and Monteiro, Math. Program. 50:29---51, 1991). Many new theoretical results on the trajectories resulting from the dual affine scaling continuous method model for linear programming are obtained.

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

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

The processing of the images simultaneously degraded by blur and affine transformation has become a key task in many applications and many novel methods are designed specifically for it in which the moment-based methods play an important role. However, the existing moment-based methods all resort to non-orthogonal moments invariants which have problem of information redundancy and are sensitive to noise. In this paper, we construct a new set of combined invariants of orthogonal Legendre moments which hold for blur and affine transformation together. The experimental results show that the proposed invariants have better discriminative power and robustness to noise with the comparison to other invariants.

Affine transform is widely used in the high speed image processing systems. This transform plays an important role in various high speed applications like Optical quadrature microscopy (OQM), image stabilisation in digital camera and image registration etc. In these applications, transformations of image consume most of the execution time. Hence, for high speed imaging systems, acceleration of Affine transform is very much sought for. In this paper, the pipelined architecture implementation of a proposed inherent parallel algorithm for Affine transform has been presented. The acceleration of the image transformation will help in reducing the processing time of high speed imaging systems. The architecture is mapped in Field programmable gate array (FPGA) and the result shows that the proposed algorithm is almost 4 times faster than the conventional algorithm while retaining the image quality. Using the proposed algorithm, an image of size 1,920 × 1,080 can be transformed with a frame rate of 540 frames per second and the multiplane image synthesis for image stabilisation on the same digital image can be performed with a frame rate of 65 fps.

In this work we study a class of affine maximal surfaces with singularities that we call affine maximal maps. Our main goal is the classification of the complete affine maximal maps with one or two embedded regular ends. In particular, we obtain a new characterization of the elliptic paraboloid and a large family of global examples.

This paper focuses on the proximal point regularization technique for a class of optimal control processes governed by affine switched systems. We consider switched control systems described by nonlinear ordinary differential equations which are affine in the input. The affine structure of the dynamical models under consideration makes it possible to establish some continuity/approximability properties and to specify these models as convex control systems. We show that, for some classes of cost functionals, the associated optimal control problem (OCP) corresponds to a conventional convex optimization problem in a suitable Hilbert space. The latter can be reliably solved using standard first-order optimization algorithms and consistent regularization schemes. In particular, we propose a conceptual numerical approach based on the gradient-type method and classic proximal point techniques.

In many cases, feature-matching problems can boil down to the computation of affine invariant local image features. However, many methods which are used to obtain these image features are based on affine not perspective transformation. They typically fail to get enough matching points at extreme viewpoints. In this paper, a novel method based on perspective projection to simulate all image views by sampling a camera pose in 3D space is presented. Only four variables—three angles (pan, tilt and roll) and a scale factor which are used to describe the camera pose—are involved to represent an affine transformation matrix. All these samples generated by our method are more similar to the real images captured by camera than those generated by traditional methods. We demonstrate the performance and robustness of our method by comparing it with popular SIFT, ASIFT descriptors and randomized trees method. Experimental results show that a large range of viewpoints by learning the behavior of key points patterns can be handled even when the camera is placed at some extreme positions.

We present in this article inductive schemes that allow to classify all those affine hypersurfaces with affine normal parallel second fundamental (cubic) form, for every possible dimensional case greater or equal than two. The solution proposed is a follow up to previous works on the same topic by this author and it uses, firstly, the reduction of the problem, eminently geometric, to the classification of a certain class of solutions to an equation of Monge-Ampere type. Then it is applied the so-called “method of algorithmic sequence of coordinate changes”, in order to achieve the latter.

A surface immersed in R 4 is called a proper affine sphere if the position vector belongs to the affine normal plane. We classify proper affine spheres with ∇⊥ g ⊥=0 whose affine mean curvature vector has constant length. Moreover, we find some concrete examples of affine spheres which are not affine umbilical.

We describe a new approach to the problem of visual pattern recognition. This approach is based on figure encoding. The encoding used is invariant under affine transformations of discrete sets of points (discrete figures).

Set-membership (SM) adaptive filtering is appealing in many practical situations, particularly those with inherent power and computational constraints. The main feature of the SM algorithms is their data-selective coefficient update leading to lower computational complexity and power consumption. The set-membership affine projection (SM-AP) algorithm does not trade convergence speed with misadjustment and computation complexity as many existing adaptive filtering algorithms. In this work analytical results related to the SM-AP algorithm are presented for the first time, providing tools to setup its parameters as well as some interpretation to its desirable features. The analysis results in expressions for the excess mean square error (MSE) in stationary environments and the transient behavior of the learning curves. Simulation results confirm the accuracy of the analysis and the good features of the SM-AP algorithms.

Landmark point-based registration is an important tool in medical image analysis and applications. An efficient and robust new method is sought that does not require optimization and is less susceptible to noise.A new quaternion-based affine registration algorithm for matching 3D point sets, a generalization of a previously reported method for 2D point sets, was developed. The new algorithm computes the exact affine transformation and the unknown correspondence between two 3D point sets but does not require any optimization. The method performs robustly in the presence of noise for non-degenerate cases. The method performs a reduction of general affine case to an orthogonal case, and then computes the unknown rotation using the quaternion representation of the 3D points. The method assumes no prior knowledge of point-wise correspondence between the two point sets. The algebraic and geometric concepts underlying the method are shown to be both clear and intuitive.Experimental evaluation of the method was performed using both randomly generated synthetic 3D point sets and Stanford Bunny dataset. The algorithm performed well even for noisy data. A feasibility test was performed with medical MR scans showing promising results.The algorithm for point-based correspondence registration demonstrated robust results, even in noisy cases, and was shown to be feasible for use with medical images.

In this paper, we introduce the pseudo-affine transformations, a new class of maps, and obtain their characterization in terms of g-triangle-reflections.

We consider the problem of estimating the geometric deformation of an object, with respect to some reference observation on it. Existing solutions, set in the standard coordinate system imposed by the measurement system, lead to high-dimensional, non-convex optimization problems. We propose a novel framework that employs a set of non-linear functionals to replace this originally high dimensional problem by an equivalent problem that is linear in the unknown transformation parameters. The proposed solution includes the case where the deformation relating the observed signature of the object and the reference template is composed both of the geometric deformation due to the affine transformation of the coordinate system and a constant amplitude gain. The proposed solution is unique and exact and is applicable to any affine transformation regardless of its magnitude.

We extend the concept of Codazzi-equivalence from Riemannian metrics in [14] to affine connections. Applications to relative hypersurface theory show that this concept simplifies the investigation of pairs of hypersurfaces with parallel normalization, moreover we get a better understanding of the affine Gaus maps. We give a new proof of Calabi’s global affine Minkowski problem; see [1, 10].

The study of affine differential geometry is partially motivated by the fact that the direction of the affine normal coincides with the tangent direction of the gravity curve at the base point. A way to prove this is to compute the Taylor expansion of the hypersurface up to third order derivatives. By computing fourth and fifth order coefficients we characterize the second order behavior of the gravity curve at its base point in terms of affine invariants.

We propose a robust classification algorithm for curves in 2D and 3D, under the special and full groups of affine transformations. To each plane or spatial curve we assign a plane signature curve. Curves, equivalent under an affine transformation, have the same signature. The signatures introduced in this paper are based on integral invariants, which behave much better on noisy images than classically known differential invariants. The comparison with other types of invariants is given in the introduction. Though the integral invariants for planar curves were known before, the affine integral invariants for spatial curves are proposed here for the first time. Using the inductive variation of the moving frame method we compute affine invariants in terms of Euclidean invariants. We present two types of signatures, the global signature and the local signature. Both signatures are independent of parameterization (curve sampling). The global signature depends on the choice of the initial point and does not allow us to compare fragments of curves, and is therefore sensitive to occlusions. The local signature, although is slightly more sensitive to noise, is independent of the choice of the initial point and is not sensitive to occlusions in an image. It helps establish local equivalence of curves. The robustness of these invariants and signatures in their application to the problem of classification of noisy spatial curves extracted from a 3D object is analyzed., Comment: 30 pages, 16 figures

Image and geometry processing applications estimate the local geometry of objects using information localized at points. They usually consider information about the tangents as a side product of the points coordinates. This work proposes parabolic polygons as a model for discrete curves, which intrinsically combines points and tangents. This model is naturally affine invariant, which makes it particularly adapted to computer vision applications. As a direct application of this affine invariance, this paper introduces an affine curvature estimator that has a great potential to improve computer vision tasks such as matching and registering. As a proof-of-concept, this work also proposes an affine invariant curve reconstruction from point and tangent data.

In this paper, we study n-dimensional improper affine spheres of which the center map is congruent to the original immersion. We show how to construct such hypersurfaces starting from (n − 1)-dimensional proper elliptic affine hyperspheres. We also show that they correspond to solutions of the Monge-Ampere equations which are homogeneous of degree 2.

We present the study of a data-driven motion synthesis approach based on a 1D affine image-matching equation. We start by deriving the relevant properties of the exact matching operator, such as the existence of a singular point. Next, we approximate such operator by the Green’s function of a second-order differential equation, finding that it leads to a more compelling motion impression, due to the incorporation of blur. We then proceed to show that, by judicious choice of the matching parameters, the 1D affine Green’s filter allows the simulation of a broad class of effects, such as zoom-in and zoom-out, and of complex nonrigid motions such as that of a pulsating heart.

Purpose This paper is to propose a new framework for medical image registration with large nonrigid deformations, which still remains one of the biggest challenges for image fusion and further analysis in many medical applications.

We present a general geometric description and Euler-Poincare characteristics of middle-sectioned simplices in a four-dimensional affine space. We demonstrate the relationship between similar geometric objects and four-dimensional analogs of the triangular Sierpinski napkin.

The Margulis invariant of an affine hyperbolic element measures the signed Lorentzian displacement along the unique closed geodesic in its class. Given a group of affine transformations of Minkowski spacetime whose linear part is Schottky, the Margulis invariant is a useful tool in determining properness of its action. This paper, based on a presentation given at CGG IV Oostende 2005, describes the affine deformation space of a rank two Schottky group. Several illustrations are included, contrasting the difference between the case of a three-holed sphere and that of a one-holed torus.

A method is presented for ray casting implicit surfaces defined by fractal combinations of procedural noise functions. The method is robust and uses affine arithmetic to bound the variation of the implicit function along a ray. The method is also efficient due to a modification in the affine arithmetic representation that introduces a condensation step at the end of every non-affine operation. We show that our method is able to retain the tight estimation capabilities of affine arithmetic for ray casting implicit surfaces made from procedural noise functions while being faster to compute and more efficient to store.

We present a batch method for recovering Euclidian camera motion from sparse image data. The main purpose of the algorithm is to recover the motion parameters using as much of the available information and as few computational steps as possible. The algorithm thus places itself in the gap between factorisation schemes, which make use of all available information in the initial recovery step, and sequential approaches which are able to handle sparseness in the image data. Euclidian camera matrices are approximated via the affine camera model, thus making the recovery direct in the sense that no intermediate projective reconstruction is made. Using a little known closure constraint, the FA-closure, we are able to formulate the camera coefficients linearly in the entries of the affine fundamental matrices. The novelty of the presented work is twofold: Firstly the presented formulation allows for a particularly good conditioning of the estimation of the initial motion parameters but also for an unprecedented diversity in the choice of possible regularisation terms. Secondly, the new autocalibration scheme presented here is in practice guaranteed to yield a Least Squares Estimate of the calibration parameters. As a bi-product, the affine camera model is rehabilitated as a useful model for most cameras and scene configurations, e.g. wide angle lenses observing a scene at close range. Experiments on real and synthetic data demonstrate the ability to reconstruct scenes which are very problematic for previous structure from motion techniques due to local ambiguities and error accumulation.

This paper proposes a new scheme based on two-source independent component analysis (ICA) for object recognition with affine transformation and for affine motion estimation between video frames. The idea is that, in two-source case, the indeterminate permutation matrix can be fixed. So an affine invariant description will be extracted by ICA algorithm. Simulation results show that ICA method has better performance comparing with traditional Fourier methods on object recognition and affine motion estimation.

A new method for obtaining affine transformations of loops for the localization (fast reuse) of program data is proposed. A technique of multidimensional scheduling is used with the following strategy of locality improvement: derive affine transformations allowing one to quickly reuse as much data as possible; if the amount of localized data is insufficient, apply a blocking. The method can easily be automated; the dependence on external parameters of the loops is explicitly taken into account.

The purpose of this article is the study of warped product manifolds which can be realized either as centroaffine or graph hypersurfaces in some affine space. First, we show that there exist many such realizations. Then we establish general optimal inequalities in terms of the warping function and the Tchebychev vector field for such affine hypersurfaces. We also investigate warped product affine hypersurfaces which verify the equality case of the inequalities. Several applications are also presented.

Both parametric and implicit representations are used in a variety of computer vision applications such as object modeling, recognition and pose estimation. In this paper, we present a new algebraic curve fitting technique based on the implicitization of affine invariant Fourier descriptors that can be used to model free-form objects captured from different viewpoints. Implicitization can be carried out quite efficiently using a numerical procedure rather than computing determinants of eliminant matrices, symbolically. Affine invariance of the proposed fitting technique is experimentally shown on a database of 2D free-form objects. Experimental results are provided to assess the robustness of our fitting method under data perturbations. Some invariant recognition examples are also presented.

In this paper, illumination-affine invariant methods are presented based on affine moment normalization techniques, Zernike moments, and multiband correlation functions. The methods are suitable for the illumination invariant recognition of 3D color texture. Complex valued moments (i.e., Zernike moments) and affine moment normalization are used in the derivation of illumination affine invariants where the real valued affine moment invariants fail to provide affine invariants that are independent of illumination changes. Three different moment normalization methods have been used, two of which are based on affine moment normalization technique and the third is based on reducing the affine transformation to a Euclidian transform. It is shown that for a change of illumination and orientation, the affinely normalized Zernike moment matrices are related by a linear transform. Experimental results are obtained in two tests: the first is used with textures of outdoor scenes while the second is performed on the well-known CUReT texture database. Both tests show high recognition efficiency of the proposed recognition methods.

In this paper, we examine the complexities involved in retrieving images from a database comprised of objects of very similar appearance. Such an operation requires a process that can discriminate among images at a very fine level, such as distinguishing among various species of fish. Furthermore, incidental environmental factors such as change in viewpoints and slight, nonessential shape deformation must be excluded from the similarity criteria. To this end, we propose a new method for content-based image retrieval and indexing, one that is well suited for discriminating among objects within the same class in a way that is insensitive to incidental environmental changes. The scheme comprises a global alignment and a local matching process. Affine transform is used to model the different viewpoints associated with positioning the camera, while multi-dimensional indexing techniques are used to make the global alignment scheme efficient. A local matching process based on dynamic programming allows the optimal matching of local structures using cost metrics that may ignore nonessential local shape deformation. Results show the method's ability to cancel out visual distortions caused by a changing viewpoint, and its tolerance to noise, occlusion, and slight deformations of the object.

International audience; In this paper we propose a novel approach for detecting interest points invariant to scale and affine transformations. Our scale and affine invariant detectors are based on the following recent results: (1) Interest points extracted with the Harris detector can be adapted to affine transformations and give repeatable results (geometrically stable). (2) The characteristic scale of a local structure is indicated by a local extremum over scale of normalized derivatives (the Laplacian). (3) The affine shape of a point neighborhood is estimated based on the second moment matrix. Our scale invariant detector computes a multi-scale representation for the Harris interest point detector and then selects points at which a local measure (the Laplacian) is maximal over scales. This provides a set of distinctive points which are invariant to scale, rotation and translation as well as robust to illumination changes and limited changes of viewpoint. The characteristic scale determines a scale invariant region for each point. We extend the scale invariant detector to affine invariance by estimating the affine shape of a point neighborhood. An iterative algorithm modifies location, scale and neighborhood of each point and converges to affine invariant points. This method can deal with significant affine transformations including large scale changes. The characteristic scale and the affine shape of neighborhood determine an affine invariant region for each point. We present a comparative evaluation of different detectors and show that our approach provides better results than existing methods. The performance of our detector is also confirmed by excellent matching results; the image is described by a set of scale/affine invariant descriptors computed on the regions associated with our points

This paper presents algorithms for tracking unknown objects in the presence of zoom. Since prior models are unavailable, point and line matches in affine views are used to characterize the structure and to transfer a fixation point into new images in a sequence. Because any affine projection matrix is permitted, the intrinsic camera parameters such as focal length may change freely. Also, since the techniques do not require long feature tracks, a further desirable property is insensitivity to partial occlusion caused, for instance, by part of the object falling off the image plane while zooming in. If only point matches are available, a previous method based on factorization is applied. When also incorporating lines, the affine trifocal and quadrifocal tensors are used for tracking in monocular and stereo systems respectively. Methods for computing the tensors, minimizing algebraic error, are developed. In comparison with their projective counterparts, the affine tensors offer significant advantages in terms of computation time and convenience of parameterization, and the relations between the different tensors are shown to be much simpler. Successful tracking is demonstrated on several real image sequences.

In this paper, we present a new geometric invariant shape representation using morphological multiscale analysis. The geometric invariant is based on the area and perimeter evolution of the shape under the action of a morphological multiscale analysis. First, we present some theoretical results on the perimeter and area evolution across the scales of a shape. In the case of similarity transformations, the proposed geometric invariant is based on a scale-normalized evolution of the isoperimetric ratio of the shape. In the case of general affine geometric transformations the proposed geometric invariant is based on a scale-normalized evolution of the area. We present some numerical experiments to evaluate the performance of the proposed models. We present an application of this technique to the problem of shape classification on a real shape database and we study the well-posedness of the proposed models in the framework of viscosity solution theory.

In this paper, a new set of affine moment invariants is proposed in the frequency domain. By thresholding the magnitude of the Discrete Fourier Transform (DFT) of affine transformation-related images, new images which are also related by affine transformation are constructed. Then some affine invariant features in the frequency domain are obtained from the reconstructed images. Experimental results are given to show that the new affine invariants are less sensitive to noise and the recognition rate is increased when using both the available spatial domain and the proposed frequency domain affine moment invariants.