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The affine projection algorithm (APA) attracts a wide attention for its advantages such as simple frame, fast convergence rate, and so on. This paper is based on the AP algorithm to introduce nonlinear update equation for the weight error vector. So the affine projection mixed-norm (APMN) algorithm is proposed. In this paper, we use a new approach to analyse the APMN algorithm. In this thesis, the stability and convergence rate of APMN algorithm are analyzed in theory and, simulation and verification of theoretical analysis is also performed.

In this paper we present a new panorama generation method based on affine model. The images used for panorama generation are captured by an Unmanned Aerial Vehicle (UAV). We focus our research on terrain data, which contains few high buildings. In our method a Best-First Affine Model is used to generate panorama, with the affine parameters solved by a locally optimized RANSAC. The process of our image stitching method is fully automatic. Compared with existing methods, the panorama generated by ours is a quasi-orthorectified one and free from visible distortions.

An efficient MIP-APSA (EMIP-APSA) is proposed via incorporating l 0-norm as a better measure of sparseness into a recently presented memory-improved proportionate affine projection sign algorithm (MIP-APSA) to enhance performance for sparse system identification. Also, to reduce computational complexity of EMIP-APSA, we achieve a simple implementation of the EMIP-APSA (SEMIP-APSA) while maintaining the consistent performance in terms of convergence rate and steady-state misalignment. Simulation results demonstrate that the proposed EMIP-APSA and SEMIP-APSA obtain a lower steady-state misalignment in comparison with the MIP-APSA for sparse system identification in the impulsive noise environment.

We develop an algorithm for the computation of a locally affine optical flow field as an extension of the Lucas-Kanade LK method. The classical LK method solves a system of linear equations assuming that the flow field is locally constant. Our method solves a collection of systems of linear equations assuming that the flow field is locally affine. Since our method combines the minimisation of the total variation and the decomposition of the region, the method is a local version of the $l_2^2$ -l 1 optical flow computation. Since the linearly diverging vector field from a point is locally affine, our method is suitable for optical flow computation for diverging image sequences such as front-view sequences observed by car-mounted cameras.

The Curvature Scale-Space (CSS) technique has been selected in MPEG-7 for shape similarity retrieval. While the technique is invariant to shape transformations with respect to four parameters namely zoom, rotation and translation (which needs two parameters to represent), our algorithm proposed in the paper, called Affine CSS (ACSS), treats the two left over parameters. Against any prognosis, simulating all views depend on these two parameters is feasible. The enriched algorithm is used to find similar shapes from a very large prototype database, and also a small classified database of marine creatures, which consists of original as well as affine transformed shapes. An improvement is observed over the conventional CSS algorithm.

In this paper we present a new result concerning description of asymptotics of the invariant measure of the affine recursion in the critical case. We discuss also relations of this model with the smoothing transform.

This paper proposes an accurate method for multiple key points tracking in long microscopic sequences. Tracking in normal-scale image sequences is proved to be a valuable fundamental technology in computer vision, while tracking in microscopic sequences is a more challenging work due to its poor image quality resulted from the complexity of microscopic imaging process. The micro stereo imaging process can be implemented in a tilting rotation of the stage which produces an affine geometric transformation on the projection of rigid spatial micro structure. This paper finds that the projection's affine invariance leads tracking of point templates to be a feasible solution, due to the fixed spatial relationship among the composed of simple fundamental components such as points, lines and planes. At the same time, we apply an adaptive particle filter (PF) of points tracking algorithm to sample and calculate the weights from those multiple point templates, which can resolve the visual distortion, illumination variability and irregular motion estimation. The experimental results are precise and robust for rigid multiple key points tracking in long micro image sequences.

The decision on whether a pair of closed contours is derived from different views of the same object, a task commonly known as affine invariant matching, can be encapsulated as the search for the existence of an affine transform between them. Past research has demonstrated that such search process can be effectively and swiftly accomplished with the use of genetic algorithms. On this basis, a successful attempt was developed for the heavily broken contour situation. In essence, a distance image and a correspondence map are utilized to recover a closed boundary from a fragmented scene contour. However, the pre-processing task involved in generating the distance image and the correspondence map consumes large amount of computation. This paper proposes a solution to overcome this problem with a fast algorithm, namely labelled chamfer distance transform. In our method, the generation of the distance image and the correspondence map is integrated into a single process which only involves small amount of arithmetic operations. Evaluation reveals that the time taken to match a pair of object shapes is about 10 to 30 times faster than the parent method.

We introduce an (equi-)affine invariant geometric structure by which surfaces that go through squeeze and shear transformations can still be properly analyzed. The definition of an affine invariant metric enables us to evaluate a new form of geodesic distances and to construct an invariant Laplacian from which local and global diffusion geometry is constructed. Applications of the proposed framework demonstrate its power in generalizing and enriching the existing set of tools for shape analysis.

A Bayesian online object tracking method is proposed in this paper. Within the inference framework, estimation of the dynamical transition model and observation model is computed sequentially based on object appearance warping and KD-tree-based matching. Once the object to be tracked is located in the previous frame, the proposed method randomly samples structured local image patches in the current frame via Gaussian particle filtering within and around the previous target region to form the candidates. Then, a random KD-tree forest is established to organize the sampling data, and find the nearest neighbor (NN) to the object region in the previous frame. The information provided by the matching output is interpreted as the tracking result for the current frame. The trees update online until the tracking procedure is finished. Experiments demonstrate the efficiency and competitive performance of the proposed algorithm compared with some state-of-the-art works.

Visual tracking remains a challenging problem because of intrinsic appearance variability of object and extrinsic disturbance. Many algorithms have been recently proposed to capture the varying appearance of targets. Most existing tracking methods, however, fail to estimate the scale and orientation of the target. To deal with this problem, we model the second-order statistics of image regions using a kernel function and perform covariance matching under the Log-Euclidean Riemannian metric. Applying kernel-based covariance matrix as image region descriptor, we construct a region similarity measure that describes the relationship between the candidate object region and a given appearance template. After that, tracking is implemented by minimizing this similarity measure, in which gradient descent method is utilized to iteratively search the best matched object region. The corresponding optimization problem can be derived by calculating the first derivative of the similarity measure with respect to the affine transformation parameters and setting them to be zero. Experimental results compared with several methods demonstrate the robust performance of the proposed algorithm under challenging conditions.

Tchebichef moments are successfully used in the field of image analysis because of their polynomial properties of discrete and orthogonal. In this paper, two new affine invariant sets are introduced for object recognition using discrete orthogonal Tchebichef moments. The current study constructs affine Tchebichef invariants by normalization method. Firstly, image is normalized to a standard form using Tchebichef moments as normalization constraints. Then, the affine invariants can be obtained at the standard form. The experimental results are presented to illustrate the performance of the invariants for affine deformed images.

Affine projection algorithms (APAs) are very good candidates for echo cancellation. The two main reasons for that are: they may converge and track much faster than the NLMS algorithm and they can be efficient from an arithmetic complexity viewpoint. In this chapter, we derive some useful APAs for SAEC with the WL model.

A wavelet transform based affine projection algorithm is proposed, with a variable step-size scheme. The numerical simulations and engineering application verify the performance of the algorithm. On the same conditions, the proposed algorithm has a faster convergence and the same steady state MSE compare to normal affine projection algorithm. That means to achieve the same rate of convergence less order is needed by this algorithm, which counteract the computing expenses induced by wavelet transform. By means of variable step-size scheme, the proposed algorithm has a faster convergence rate and a lower steady state MSE or misadjustment than the corresponding normal variable step-size affine projection algorithms.

Numerical experiments indicate that images, in general, possess a considerable degree of affine self-similarity, that is, blocks are well approximated in root mean square error (RMSE) by a number of other blocks when affine greyscale transformations are employed. This has led to a simple L2-based model of affine image self-similarity which includes the method of fractal image coding (cross-scale, affine greyscale similarity) and the nonlocal means denoising method (same-scale, translational similarity). We revisit this model in terms of the structural similarity (SSIM) image quality measure, first deriving the optimal affine coefficients for SSIM-based approximations, and then applying them to various test images. We show that the SSIM-based model of self-similarity removes the "unfair advantage" of low-variance blocks exhibited in L2- based approximations. We also demonstrate experimentally that the local variance is the principal factor for self-similarity in natural images both in RMSE and in SSIM-based models.

In this paper, the well-known SIFT detector is extended with a bivariate feature localization. This is done by using function models that assume a Gaussian feature shape for the detected features. As function models we propose (a) a bivariate Gaussian and (b) a Difference of Gaussians. The proposed detector has all properties of SIFT, but provides invariance to affine transformations and blurring. It shows superior performance for strong viewpoint changes compared to the original SIFT. Compared to the most accurate affine invariant detectors, it provides competitive results for the standard test scenarios while performing superior in case of motion blur in video sequences.

This paper is addressing the problem of realigning broken objects without correspondences. We consider linear transformations between the object fragments and present the method through 2D and 3D affine transformations. The basic idea is to construct and solve a polynomial system of equations which provides the unknown parameters of the alignment. We have quantitatively evaluated the proposed algorithm on a large synthetic dataset containing 2D and 3D images. The results show that the method performs well and robust against segmentation errors. We also present experiments on 2D real images as well as on volumetric medical images applied to surgical planning.

The matching of point sets that are characterized only by their geometric configuration is a challenging problem. In this paper, we present a novel point registration algorithm for robustly identifying objects represented by two dimensional point clouds under affine distortions. We make no assumptions about the initial orientation of the point clouds and only incorporate the geometric configuration of the points to recover the affine transformation that aligns the parts that originate from the same locally planar surface of the three dimensional object. Our algorithm can deal well with noise and outliers and is inherently robust against partial occlusions. It is in essence a GOODSAC approach based on geometric hashing to guess a good initial affine transformation that is iteratively refined in order to retrieve a characteristic common point set with minimal squared error. We successfully apply it for the biometric identification of the bluespotted ribbontail ray Taeniura lymma.

This paper presents a novel topic model named Affine Invariant Topic Model(AITM) for generic object recognition Abandoning the “bag of words” assumption in traditional topic models, AITM incorporates spatial structure into traditional LDA AITM extends LDA by modeling visual words with latent affine transformations as well as latent topics, treating topics as different parts of objects and assuming a common affine transformation of visual words given a certain topic MCMC is employed to make inference for latent variables, MCMC-EM algorithm is used to parameter estimation, and Bayesian decision rule is used to perform classification Experiments on two challenging data sets demonstrate the efficiency of AITM.

In digital image watermark applications, geometric affine transform attacks (including rotation, scaling, change of aspect ratio, translation, shearing etc) can prevent detection of watermarks. In order to solve this problem, we propose an affine resilient watermark algorithm, which uses a trace transform to normalize the image before watermark embedding and detection. Because the normalization is invariant to affine transforms, the watermark vector, which is embedded in the normalized image, is invariant to affine transforms too. What's more, the original host image is not required in watermark detection. Experimental results verify that the false positive rate is very low, and the proposed watermark algorithm is resistant to affine transforms, cropping, noising, JPEG and other image processing attacks.

In this paper we extend a standard affine optical flow model to 4D and present how affine parameters can be used for estimation of 3D object structure, 3D motion and rotation using a 1D camera grid. Local changes of the projected motion vector field are modelled not only on the image plane as usual for affine optical flow, but also in camera displacement direction, and in time. We identify all parameters of this 4D fully affine model with terms depending on scene structure, scene motion, and camera displacement. We model the scene by planar, translating, and rotating surface patches and project them with a pinhole camera grid model. Imaged intensities of the projected surface points are then modelled by a brightness change model handling illumination changes. Experiments demonstrate the accuracy of the new model. It outperforms not only 2D affine optical flow models but range flow for varying illumination. Moreover we are able to estimate surface normals and rotation parameters. Experiments on real data of a plant physiology experiment confirm the applicability of our model.

Features for recognition of affinely distorted objects are of great demand. The affine moment invariants can be generated by a few methods, namely the graph method, the tensor method and the direct solution of the Cayley-Aronhold differential equation. The proof of their equivalence is complicated; it can be derived from the Gurevich's proof for affine tensor invariants. The theme of this paper is this derivation.

In this chapter, we will discuss algebraically approximate and noisy realization problems of affine dynamical systems, which realize any input response map, equivalently, as an input/output map with causality. Affine dynamical systems were proposed and the realization problems of the systems were solved in the reference [Matsuo & Hasegawa, 2003]. We characterized the finite-dimensionality of affine dynamical systems. We obtained the same results as ones established in linear system theory.

In this paper, a new algorithm for 2D shape characterization is proposed. This method characterizes a planar object using a triangle-area representation obtained from its closed contour. As main novelty with respect to previous approaches, in our approach the triangle side lengths at each contour point are adapted to the local variations of the shape, removing noise from the contour without missing relevant points. This representation is invariant to affine transformations, and robust against noise. The performance of our proposal is demonstrated using a standard test on the well-known MPEG-7 CE-shape-1 data set.

This paper presents a detailed evaluation of a new approach that uses affine invariants for wide baseline image matching. Previously published work presented a new approach to classify tentative feature matches as inliers or outliers during wide baseline image matching. After typical feature matching algorithms are run and tentative matches are created, the approach is used to classify matches as inliers or outliers to a transformation model. The approach uses the affine invariant property that ratios of areas of shapes are constant under an affine transformation. Thus, by randomly sampling corresponding shapes in the image pair a histogram of ratios of areas can be generated. The matches that contribute to the maximum histogram value are then candidate inliers. This paper evaluates the robustness of the approach under varying degrees of incorrect matches, localization error and perspective rotation often encountered during wide baseline matching. The evaluation shows the affine invariant approach provides similar accuracy as RANSAC under a wide range of conditions while maintaining an order of magnitude increase in efficiency.

A new global method for image registration in the presence of affine and radiometric deformations is introduced. The method proposed utilizes kernel operators in order to find corresponding regions without using local features. Application of polynomial type kernel functions results in a low complexity algorithm, allowing estimation of the radiometric deformation regardless of the affine geometric transformation. Preliminary experimentation shows high registration accuracy for the joint task, given real images with varying illuminations.

A new type of affine moment invariants for color images is proposed in this paper. The traditional affine moment invariants can be computed on each color channel separately, yet when the channels are transformed together, by the same affine transform, additional invariants can be computed. They have low order and therefore high robustness to noise. The new invariants are compared with another set of invariants for color images using second powers of the image function. The basic properties of the new features are tested on real images in a numerical experiment.

This paper deals with the computational issues encountered in the construction of invariant sets for LTI (Linear Time Invariant) systems subject to linear constraints. Three algorithms to compute or approximate the invariant set are presented. Two of theme are based on expansive and contractive strategy, while the third one uses the transition graph over the partition of the closed loop piecewise affine system.

We present a novel image transform called Scale Manipulation Features (SMF). The transform calculates affine invariant features of objects in a global manner and avoids using any sort of edge detection. The transform can be used for registration of affine transformed images in the presence of non homogenous illumination changes and for estimation of the illumination changes. The computational load of the method is relatively low since it is linear in the data size. In this paper we introduce the transform and demonstrate its applications for illumination compensation and for object registration in the presence of an affine geometric transformation and varying illumination.

The paper proposes a unified direct approach to a number of problems arising in image processing. In particular, the areas of image registration, and object or pattern recognition are addressed when the images of interest display significant geometric distortion due to some physical or geometrical conditions. The proposed method performs a direct multi-objective search in image response space for an optimal piece-wise affine transformation of the images using a hybrid evolutionary algorithm. In its most general form, the entire algorithm works in two relatively independent passes. First, the global search attempts to find the optimal solution for the principal affine transformation. During the second pass, the correction procedure seeks for the optimal piecewise approximation of the actual image transformation using the result of the first pass as the initial approximation.

In this paper, we introduce a novel second-order regularizer, the Affine Total-Variation term, to capture the geometry of piecewise affine functions. The approach can be characterized by two convex decompositions of a given image into piecewise affine structure and texture and noise, respectively. A convergent multiplier-based method is presented for computing a global optimum by computationally cheap iterative steps. Experiments with images and vector fields validate our approach and illustrate the difference to classical TV denoising and decomposition.

This chapter applies anisotropic Gaussian scale space theory for modeling affine shape modifications of moving blobs in the context of vision-based driver assistance systems. First, affine blobs are detected in an image sequence and tracked; second, their scale ratios are used for the derivation of 3D motion characteristics. For example, this also allows to estimate the navigation angles of a moving camera in 3D space. The theoretical concept is explained in detail, and illustrated by a few experiments, including an indoor experiment and also the estimation of navigation angles of a car (i.e., of the ego-vehicle) in provided test sequences. The numerical evaluations indicate the validity of the idea and advantages to vehicle vision.

In this paper, we consider Voronoi diagrams from the view point of affine differential geometry. A main object of affine differential geometry is to study hypersurfaces in an affine space that are invariant under the action of the group of affine transformations. Since incidence relations (configurations of vertexes, edges, etc.) in computational geometry are invariant under affine transformations, we may say that affine differential geometry gives a new sight in computational geometry. The Euclidean distance function can be generalized by a divergence function in affine differential geometry. For such divergence functions, we show that Voronoi diagrams on statistical manifolds are invariant under ( *** 1)-conformal transformations. We then give some typical figures of Voronoi diagrams on a manifold. These figures may give good intuition for Voronoi diagrams on a manifold because the figures or constructing algorithms on a manifold strongly depend on the realization or on the choice of local coordinate systems. We also consider the upper envelope type theorems on statistical manifolds, and give a constructing algorithm of Voronoi diagrams on ( *** 1)-conformally flat statistical manifolds.

A series of extensive numerical experiments indicates that images, in general, possess a considerable degree of affine self-similarity, that is, blocks are well approximated by a number of other blocks --- at the same or different scales --- when affine greyscale transformations are employed. We introduce a simple model of affine image self-similarity which includes the method of fractal image coding (cross-scale, affine greyscale similarity) and the nonlocal means denoising method (same-scale, translational similarity) as special cases.

Exemplar-based inpainting techniques demonstrate some advantages in filling both textural and structural images, based upon the simple process of patch sampling and copying. An implicit assumption of such methods is that an exemplar patch could be found for copying. But the assumption might be false for images with complex structures, resulting in inpainting artefact. To inpaint structure and texture more efficiently, this paper adopts a weak assumption that affine transformed patches could be found for filling unknown regions. The algorithm based on this assumption firstly establishes the affine relation between the unknown region and a source patch, then transform the source pixels to fill the unknowns. This algorithm generates more satisfying inpainting results, as illustrated by the provided experiments.

In this chapter, we will discuss approximate and noisy realization problems of affine dynamical systems, which realize any input response map, equivalently, as an input/output map with causality. Affine dynamical systems were proposed and the realization problem of the systems were solved in the reference [Matsuo & Hasegawa, 2003]. We characterized the finite-dimensionality of affine dynamical systems. We obtained the same results as ones established in linear system theory.

Affine parameter estimation technique applied to image registration is found useful in obtaining reliable fusion of same object's images taken from different modalities, into single image with strong features. Usually, the minimization in affine parameter estimation technique can be done by least squares in a quadratic way. However, this will be sensitive to the presence of outliers. Therefore, affine parameter estimation technique for image registration calls for methods that are robust enough to withstand the influence of outliers. Progressively, some robust estimation techniques demanding non-quadratic and non-convex potentials adopted from statistical literature have been used for solving these. Addressing the minimization of error function in a factual framework for finding the global optimal solution, the minimization can begin with the convex estimator at the coarser level and gradually introduce non-convexity i.e., from soft to hard redescending non-convex estimators when the iteration reaches finer level of multiresolution pyramid. Comparison has been made to find the performance results of proposed method with the registration results found using different robust estimators.

Affine registration has a long and venerable history in computer vision literature, and extensive work have been done for affine registrations in ℝ2 and ℝ3. In this paper, we study affine registrations in ℝ m for m > 3, and to justify breaking this dimension barrier, we show two interesting types of matching problems that can be formulated and solved as affine registration problems in dimensions higher than three: stereo correspondence under motion and image set matching. More specifically, for an object undergoing non-rigid motion that can be linearly modelled using a small number of shape basis vectors, the stereo correspondence problem can be solved by affine registering points in ℝ3n . And given two collections of images related by an unknown linear transformation of the image space, the correspondences between images in the two collections can be recovered by solving an affine registration problem in ℝm, where m is the dimension of a PCA subspace. The algorithm proposed in this paper estimates the affine transformation between two point sets in ℝm. It does not require continuous optimization, and our analysis shows that, in the absence of data noise, the algorithm will recover the exact affine transformation for almost all point sets with the worst-case time complexity of O(mk 2), k the size of the point set. We validate the proposed algorithm on a variety of synthetic point sets in different dimensions with varying degrees of deformation and noise, and we also show experimentally that the two types of matching problems can indeed be solved satisfactorily using the proposed affine registration algorithm.

in this study, we propose a simple, yet general and powerful framework for constructing accurate affine invariant regions and use it for near-duplicate detection problem. In our framework, a method for extracting reliable seed points is first proposed. Then, regions which are invariant to most common affine transformations are extracted from seed points by a new method named the Thresholding Seeded Growing Region (TSGR). After that, an improved ellipse fitting method based on the Direct Least Square Fitting (DLSF) is used to fit the irregularly-shaped contours of TSGRs to obtain ellipse regions as the final invariant regions. At last, SIFT-PCA descriptors are computed on the obtained regions. In the experiment, our framework is evaluated by retrieving near-duplicate in an image database containing 1000 images. It gives a satisfying result of 96.8% precision at 100% recall.

In this paper, an alternative approach to the computation of control invariant sets for piecewise affine systems is presented. Based on two approximation operators, two algorithms that provide outer and inner approximations of the maximal robust control invariant set are presented. These algorithms can be used to obtain a robust control invariant set for the system. An illustrative example is presented.

In this paper, we approach the problem of determining the basic components from repetitive textured surfaces undergoing free-form deformations. Traditional methods for texture modeling are usually based on measurements performed on fronto-parallel planar surfaces. Recently, affine invariant descriptors have been proposed as an effective way to extract local information from non-planar texture surfaces. However, affine transformations are unable to model local image distortions caused by changes in surface curvature. Here, we propose a method for selecting the most representative candidates for the basic texture elements of a texture field while preserving the descriptors' affine invariance requirement. Our contribution in this paper is twofold. First, we investigate the distribution of extracted affine invariant descriptors on a nonlinear manifold embedding. Secondly, we describe a learning procedure that allows us to group repetitive texture elements while removing candidates presenting high levels of curvature-induced distortion. We demonstrate the effectiveness of our method on a set of images obtained from man-made texture surfaces undergoing a range of non-rigid deformations.

In this paper we introduce affine invariants based on a multiscale framework combined with nonlinear comparison operations. The resulting descriptors are histograms, which are computed from a set of comparison results using binary coding. The new constructions are analogous to other multiscale affine invariants, but the use of highly nonlinear operations yields clear advantages in discriminability. This is also demonstrated by the experiments, where comparable recognition rates are achieved with only a fraction of the computational load. The new methods are straightforward to implement and fast to evaluate from given image patches.

Matching techniques are part-and-parcel of shape recognition. A coarse-to-fine method is presented which finds point correspondence between open or closed curves and is invariant to various image deformations, including affine transformation, perspective distortion, non-rigid motion and so forth. The method is inspired by the idea to use point correspondences established at one level to generate a priori information, which is either topological or geometric, to match features at finer levels. This has all been achieved through an analysis of the curve topology and a synthesis of the B-spline interpolation techniques. This is in contrast to existing multi-scale methods for curve matching that use pure feature correlation or 3D structure recovery at a fixed scale. The presented method proves to be robust and accurate and can serve as a powerful aid to measure similarity of shape, as demonstrated in various experiments on real images.

In order to reconstruct 3-D Euclidean shape by the Tomasi-Kanade factorization, one needs to specify an affine camera model such as orthographic, weak perspective, and paraperspective. We present a new method that does not require any such specific models. We show that a minimal requirement for an affine camera to mimic perspective projection leads to a unique camera model, called symmetric affine camera, which has two free functions. We determine their values from input images by linear computation and demonstrate by experiments that an appropriate camera model is automatically selected.

In this paper a new method for matching contours called CTFDP is presented. It is invariant to affine transformations and can provide robust and accurate estimation of point correspondence between closed curves. This has all been achieved by exploiting the dynamic programming techniques in a coarse-to-fine framework. By normalizing the shape into a standard point distribution, the new method can compare different shapes despite the shearing and scaling effect of affine transformation. Using the coarse-to-fine dynamic programming technique, the shapes are aligned to each other by iteratively seeking for correspondences and estimating relative transformations so as to prune the start points in the dynamic programming stage in turn. Experiments on artificial and real images have validated the robustness and accuracy of the presented method.

In this paper, a registration method based on Harris corners is proposed. It is composed of three steps. First, corner extraction and matching. We use the gray level information around the corner to setup the correspondences, then use the affine invariant of Mahalannobis distance to remove the mis-matched corner points. From this correspondence of the corner points, the affine matrix between two different images can be determined. Finally, map all points in the sensed image to the reference using the estimated transformation matrix and assign the corresponding gray level by re-sampling the image in the sensed image. Experiments with different types of multi-sensor images demonstrated the feasibility of our method.

We show how to recover the affine parts of the secret key for a certain class of HFE-Cryptosystems. Further we will show that any system with branches can be decomposed in its single branches in polynomial time on average. The attack on the affine parts generalizes the results from [1, 11] to a bigger class of systems and is achieved by a different approach. Despite the fact that systems with branches are not used anymore (see [11, 6]), our second attack is a still of interest, as it shows that branches belong to the list of algebraic properties, which cannot be hidden by composition with secret affine transformations. We derived both algorithms by considering the cryptosystem as objects from the theory of nonassociative algebras and applying classical techniques from this theory. This general framework might be a useful tool for future investigations of HFE-Cryptosystems, e.g. to detect further invariants, which are not hidden by composition with affine transformations.

Recognition of planar objects from their images taken from different viewpoints requires affine invariants calculated from the object boundaries. The equivalence between affine transformation and the source mixture model simplifies the recognition problem. The rotation, scaling, skewing, and translation effects of the affine transformation can be undone by using blind source separation (BSS) techniques to go back to a canonical object view. Then, the problem is reduced from a more involved affine invariant search to a simple shape matching task. Point correspondence between two curves initially related by an affine transformation is obtained with further processing.

This paper presents a method for computing the reach set of affine systems for sets of initial states given as low-dimensional polytopes. An affine representation for polytopes is introduced to improve the efficiency of set representations. Using the affine representation, we present a procedure to compute conservative over-approximations of the reach set, which uses the Krylov subspace approximation method to handle large-scale affine systems (systems of order over 100).

In this paper, we propose a Bayesian approach for affine auto-calibration. By the Bayesian approach, a posterior distribution for the affine camera parameters can be constructed, where also the prior knowledge can be taken into account. Moreover, due to the linearity of the affine camera model, the structure and translations can be analytically marginalised out from the posterior distribution, if certain prior distributions are assumed. The marginalisation reduces the dimensionality of the problem substantially that makes the MCMC methods better suitable for exploring the posterior of the intrinsic camera parameters. The experiments verify that the proposed approach is a versatile, statistically sound alternative for the existing affine auto-calibration methods.