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The normalized least-mean-squares (NLMS) algorithm has a problem that the convergence slows down for correlated input signals. The reason for this phenomenon is explained by looking at the algorithm from a geometrical point of view. This observation motivates the affine projection algorithm (APA) as a natural generalization of the NLMS algorithm. The APA exploits most recent multiple regressors, while the NLMS algorithm uses only the current, single regressor. In the APA, the current coefficient vector is orthogonally projected onto the affine subspace defined by the regressors for updating the coefficient vector. By increasing the number of regressors, which is called the projection order, the convergence rate of the APA is improved especially for correlated input signals. The role of the step-size is made clear. Investigations from the affine projection point of view give us a deep insight into the properties of the APA. We also see that alternative approaches are possible to derive the update equation for the APA. To stabilize the numerical inversion of a matrix in the update equation, a regularization term is often added. This variant of the APA is called the regularized APA (R-APA), whereas the original APA is called the basic APA (B-APA). This chapter also explains that the B-APA with unity step-size has a decorrelating property, and that there are formal similarities between the recursive least-squares (RLS) algorithm and the R-APA.

After the birth of the basic affine projection algorithm (B-APA) in the middle of 1980s, several adaptive filtering algorithms that also exploit multiple regressors were put forward independently. They are now recognized as variants of the B-APA, forming a family of affine projection algorithms. In the family, there are such algorithms as the regularized affine projection algorithm (R-APA), the decorrelation affine projection algorithm (D-APA), the partial-rank algorithm (PRA), the normalized least-mean-squares algorithm with orthogonal correction factors (NLMS-OCF), the binormalized data-reusing least-mean-squares algorithm (BNDR-LMS). The R-APA and the D-APA are discussed in the preceding chapter. If we update the coefficient vector every p samples instead of every sample, where p is the projection order, then we obtain the PRA. By applying the Gram–Schmidt orthogonalization to the regressors, the B-APA is transformed into the NLMS-OCF. The BNDR-LMS is just a special case of the B-APA where the projection order equals 2. We can also use sparse regressors in the APAs. These algorithms are formulated by a single update equation with several parameters. By setting those parameters at appropriate values, the update equation for each of the algorithms in the APA family is expressed.

The paper presents an original method for Finite Time Response (FTR) control of the affine systems. The FTR property is specific to linear systems only, known in the literature as dead-beat algorithms. In this work, it is developed as a new procedure for the affine systems FTR synthesis, called the Equivalent Input Method (EIM). For this purpose it calculates an equivalent input which will determine, according to a quadratic criterion, the best approximation of the affine component. This way the system is approximated by an affine system with an input variable equal to the sum of the original input and the equivalent input, but having only a residual affine component. This residual affine component has a smaller norm than the initial affine component. Considering zero the residual affine component, a FTR linear system synthesis procedure is applied. In the real system, controlled by a FTR control law, the residual affine component creates at each step a disturbance that FTR algorithm seeks to cancel. This approach is justified by the fact that the disturbance residual affine component is much smaller in norm than the original affine component. Under certain circumstances, this residual affine component can be zero. The controllability and algorithm convergence is analyzed. The proposed EIM method can be applied also for nonlinear systems approximated by Piecewise Affine Subsystems (PWAS). An experimental platform has been designed in Matlab environment allowing implementation of various affine systems and their control algorithms. Simulation results are included to support the method presented in the paper.

Several pattern-matching techniques have focused on affine invariant pattern matching, mainly because rotation, scale, translation, and shear are common image transformations. In some situations, other transformations may be modeled as a small deformation on top of an affine transformation. This work presents an algorithm which aims at improving existing Fourier Transform (FT)-based pattern matching techniques in such a situation. The pattern is first decomposed into non-overlapping concentric circular rings, which are centered in middle of the pattern. Then, the FT of each ring is computed. Essentially, adding the individual complex-valued FTs provides the overall FT of the pattern. Past techniques used the overall FT to identify the parameters of the affine transformation between two patterns. In this work, it is assumed that the rings may be rotated with respect to each other, thus, parameters of transformations beyond the affine ones can be computed. The proposed method determines this variable angle of rotation starting from the FT of the outermost ring and moving inwards to the FT of the innermost ring. The variable angle of rotation provides information about the directional properties of a pattern. Two methods are investigated, namely a dynamic programming algorithm and a greedy algorithm, in order to determine the variable angle of rotation. The intuition behind this approach is that since the rings are not necessarily aligned in the same manner for different patterns, their ring FTs may also be rotated with respect to each other. Simulations demonstrate the effectiveness of the proposed technique.

This paper presents a matrix projection method for matching a pair of shapes, and estimating the affine transformation that aligns the two shapes. First, shapes are considered as 2D disordered point sets, and their normalized forms are given. Then, we prove that only a rotational transform exists between the normalized forms of the two shapes under affine distortions. Second, correspondences are found by minimizing the inner product between one matrix and projection of the other. Finally, the affine transformation for shape registration is estimated by the correspondences. Experimental results show that our approach compares favorably to other methods under affine distortions.

Local feature descriptors are widely used in many computer vision applications. Over the past couple of decades, several local feature descriptors have been proposed which are robust to challenging conditions. Since they show different characteristics in different environment, it is necessary to evaluate their performance in an intensive and consistent manner. However, there has been no relevant work that addresses this problem, especially for the affine invariant region detectors which are popularly used in object recognition and classification. In this paper, we present a useful and rigorous performance evaluation of local descriptors for affine invariant region detector, in which MSER (maximally stable extremal regions) detector is employed. We intensively evaluate local patch based descriptors as well as binary descriptors, including SIFT (scale invariant feature transform), SURF (speeded up robust features), BRIEF (binary robust independent elementary features), FREAK (fast retina keypoint), Shape descriptor, and LIOP (local intensity order pattern). Intensive evaluation on standard dataset shows that LIOP outperforms the other descriptors in terms of precision and recall metric.

In the multivariate one-sample location model, only L-estimators are affine equivariant; we propose a class of flexible robust, affine equivariant L-estimators of location, for distributions invoking affine-invariance of Mahalanobis distances of individual observations. An involved iteration process for their computation is numerically illustrated.

The main goal of this paper is to compute a class of polynomial vector fields, whose associated dynamical system has a given affine variety as attractive and invariant set, a given point in such an affine variety as invariant and attractive and another given affine variety as invariant set, solving the application of this technique in the robotic area. This objective is reached by using some tools taken from algebraic geometry. Practical examples of how these vector fields can be computed are reported. Moreover, by using these techniques, two feedback control laws, respectively, for a unicycle-like mobile robot and for a car-like mobile robot, which make them move, within the workspace, approaching to a selected algebraic curve, are given.

In this chapter, I outline the GATSM framework, explain how GATSMs may be estimated from yield curve data, and use estimation results to detail the practical pitfalls of applying them in ZLB environments. The outline first serves to establish the present standard for term structure modeling, given that GATSMs are used extensively by researchers and practitioners, as already discussed in chapter 2. I also use GATSMs to represent the shadow term structure in the shadow/ZLB-GATSMs in the remainder of the book, so it is important to establish the precise notation and estimation methods for GATSMs that will carry over to those shadow/ZLB-GATSMs. Related to that point, I can then present the shadow/ZLB-GATSM framework and its estimation in chapter 4 as a relatively tractable modification to the GATSM class.

This paper presents a novel and reliable tracking-by detection method for image regions that undergo affine transformations such as translation, rotation, scale, dilatation and shear deformations, which span the six degrees of freedom of motion. Our method takes advantage of the intrinsic Lie group structure of the 2D affine motion matrices and imposes this motion structure on a kernelized structured output SVM classifier that provides an appearance based prediction function to directly estimate the object transformation between frames using geodesic distances on manifolds unlike the existing methods proceeding by linearizing the motion. We demonstrate that these combined motion and appearance model structures greatly improve the tracking performance while an incorporated particle filter on the motion hypothesis space keeps the computational load feasible. Experimentally, we show that our algorithm is able to outperform state-of-the-art affine trackers in various scenarios.

In this chapter, the family of kernel affine projection algorithms with coherence criterion is presented. The proportionality principle is translated to the kernel-based version. A new algorithm called Kernel Proportionate Affine Projection Algorithm (KPAPA) is proposed. It is proved that the additional computational increase burden is independent of the order of the algorithm, being dependent only on the order of the kernel expansion. The Dichotomous Coordinate Descent (DCD) method and an example of an efficient implementation of KAPA using DCD are presented. This chapter also discusses the influence of the coherence value, the step size value, and the dictionary size on the performance of KAPA and KPAPA algorithms. The effectiveness of the proposed algorithms and the effect of different parameters are confirmed by computer simulations for nonlinear system identification application.

We solve the problem of finding all indefinite improper affine spheres passing through a given regular curve of R3R3 with a prescribed affine co-normal vector field along this curve. We prove the problem is well-posed when the initial data are non-characteristic and show that uniqueness of the solution can fail at characteristic directions. As application we classify the indefinite improper affine spheres admitting a geodesic planar curve., Ministerio de Educación y Ciencia Grant No. MTM2013-43970-P and Junta de Andalucía Grant No. FQM-325.

The research of affine scale space is to create a more general approach to the affine invariant image scale representation by modifying the corresponding Gaussian filters in order to cope with the specific change of view point. It has the purpose to retain a linear relationship with the transiting of the view point. With this linear relationship, the affine scale space could be established as a more general approach for the affine invariant image retrieval, including affine feature detection and affine feature descriptor. The scope of this paper is to discuss the accessible to the affine scale space, its performance and a practical implementation to construct it in order to cope with the high complexity brought in by the scale space and the affine adaptation.

We present a new approach for aligning families of 2D gels. Instead of choosing one of the gels as reference and performing a pairwise alignment, we construct an ideal gel that is representative of the entire family and obtain a set of piecewise affine transformations that optimally align each gel of the family to the ideal gel. The coefficients defining the transformations as well as the ideal landmarks are obtained as the solution of a large-scale quadratic programming problem that can be solved efficiently by interior-point methods.

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.

The paper introduces modern concepts and tools from affine geometry into power system analysis. It is shown that such an approach allows: i) a new non linear formulation of such classical problems as load flow and state estimation, ii) a more efficient way of solving such problems through non iterative methods. The new approach is illustrated for a small but representative example of a load flow for a two-bus network.

This note shows how the fixed points of an affine transformation in the plane can be constructed by an elementary geometric method. The approach presented here also shows how the affinities of the plane can be classified with the help of their fixed points.

In this paper, a modeling method of high dimensional piecewise affine models is proposed. Because the model is expressed by radial basis function networks, whose RBFs are located at the grid points of the orthogonal coordinate in the input space, the shape of the piecewise affine model becomes easily understood. The more the number of RBFs is, the higher the approximation capability of the model is. The algorithm to increase the number of RBFs was developed in focusing on the distribution of estimation errors. By approximating nonlinear processes with piecewise affine models, control theories using mixed logical dynamical systems can be applied.

In this paper, we enhance a recently proposed method for adaptive identification of piecewise affine systems by the use of concurrent learning. It is shown that the concurrent use of recorded and instantaneous data leads to exponential convergence of all subsystem parameters under verifiable conditions on the recorded data. A key advantage of the proposed method is that linear independence of the recorded data is sufficient, compared to the persistence of excitation assumed by previous adaptive parameter identifiers. Furthermore, the procedure tremendously improves the performance of adaptive identification for piecewise affine systems that previously suffered from slow convergence.

In this paper, we propose an affine parameter estimation algorithm from block motion vectors for extracting accurate motion information with the assumption that the undergoing motion can be characterized by an affine model. The motion may be caused either by a moving camera or a moving object. The proposed method first extracts motion vectors from a sequence of images by using size-variable block matching and then processes them by adaptive robust estimation to estimate affine parameters. Typically, a robust estimation filters out outliers (velocity vectors that do not fit into the model) by fitting velocity vectors to a predefined model. To filter out potential outliers, our adaptive robust estimation defines a continuous weight function based on a Sigmoid function. During the estimation process, we tune the Sigmoid function gradually to its hard-limit as the errors between the model and input data are decreased, so that we can effectively separate non-outliers from outliers with the help of the finally tuned hard-limit form of the weight function. Experimental results show that the suggested approach is very effective in estimating affine parameters reliably.

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.

The known factorisation algorithm for the maximum likelihood affine reconstruction requires that all the feature points used must be visible in all views. We derive here a conditional solution for the 3D coordinates of the feature points and translation vectors given the inhomogeneous affine projection matrices, but no single feature point needs to be visible in all views. The solution represents closed-form maximum likelihood, generalised affine triangulation under missing data and unknown translations, and it implies two iterative algorithms for the maximum likelihood affine reconstruction where all the measured data may be used. For the case where at least four non-coplanar points are visible in all views, a non-iterative solution for affine reconstruction with missing data is also proposed. As reported, the closed-form-expression additionally has applications in affine bundle adjustment and identifying degenerate configurations.

This paper proposes a new scheme based on independent component analysis (ICA) for object recognition with affine transformation and for affine motion estimation between video frames. For different affine shapes of a recognized object, an invariant descriptor can be extracted by ICA, and it can solve some object recognition problems. This method also can be used to estimate the affine motion between two frames, which is important in high compression rate coding such as MPEG4 or MPEG7 standard. Simulation results show that the proposed method has a better performance than other traditional methods in pattern recognition and affine motion estimation.

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.

Piecewise affine inverse problems form a general class of nonlinear inverse problems. In particular inverse problems obeying certain variational structures, such as Fermat's principle in travel time tomography, are of this type. In a piecewise affine inverse problem a parameter is to be reconstructed when its mapping through a piecewise affine operator is observed, possibly with errors. A piecewise affine operator is defined by partitioning the parameter space and assigning a specific affine operator to each part. A Bayesian approach with a Gaussian random field prior on the parameter space is used. Both problems with a discrete finite partition and a continuous partition of the parameter space are considered. The main result is that the posterior distribution is decomposed into a mixture of truncated Gaussian distributions, and the expression for the mixing distribution is partially analytically tractable. The general framework has, to the authors' knowledge, not previously been published, although the r...

Piecewise affine systems are systems with a piecewise defined state equation. The continuous dynamics of these systems are affine but switch at the boundaries of certain subsets of the state-space. At these boundaries sliding modes may occur. A method will be presented to obtain a discrete-time system from a given continuous-time piecewise affine system. This method takes into account sliding modes as well as changes of the dynamics which occur between two sampling instances. Even for low sampling rates the resulting model shows a high accuracy.

We investigate necessary and sufficient conditions under which a general nonlinear affine control system with outputs can be written as a gradient control system corresponding to some pseudo-Riemannian metric defined on the state space. The results rely on a suitable notion of compatibility of the system with respect to a given affine connection, and on the output behavior of the prolonged system and the gradient extension. The symmetric product associated with an affine connection plays a key role in the discussion., Comment: 21 pages, no figures

In this paper, we consider minimization of a real-valued function $f$ over $\bold R\sp {n+1}$ and study the choice of the affine normal of the level set hypersurfaces of $f$ as a direction for minimization. The affine normal vector arises in affine differential geometry when answering the question of what hypersurfaces are invariant under unimodular affine transformations. It can be computed at points of a hypersurface from local geometry or, in an alternate description, centers of gravity of slices. In the case where $f$ is quadratic, the line passing through any chosen point parallel to its affine normal will pass through the critical point of $f$. We study numerical techniques for calculating affine normal directions of level set surfaces of convex $f$ for minimization algorithms.

In this paper we propose a new approach for the control of discrete-time piecewise affine hybrid systems on full-dimensional polytopes. The new strategy is divided into a local and a supervisory control problem. The local problem is to reach and cross one facet of a polytope ensuring that the next sample of the time-discrete trajectory is picked up in the adjacent polytope. Therefore, a local piecewise affine control law is obtained by solving an optimization problem, minimizing the retention period of the trajectory in the polytope or the quadratic sum of the input signal. This procedure is based on system inherent bounds. The supervisory control problem is to find a suitable combination of polytopes and local control strategies that transfers the trajectory to the operating point and keeps it there.

We provide a formal proof of equivalence between the class of fractals created by recursive-turtle programs (RTP) and iterated affine transformations (IAT). We begin by reviewing RTP (a geometric interpretation of non-bracketed L-systems with a single production rule) and IAT (iterated function systems restricted to affine transformations). Next, we provide a simple extension to RTP that generalizes RTP from conformal transformations to arbitrary affine transformations. We then present constructive proofs of equivalence between the fractal geometry generated by RTP and IAT that yield conversion algorithms between these two methods. We conclude with possible extensions and a few open questions for future research.

In this paper, we consider the generative model for affine transformations on point sets and show how a priori information on the noise and the transformation can be incorporated into the model resulting in more accurate algorithms. While invariants have been widely used, the existing literature fails to fully account for the uncertainties introduced by both noise and the transformation. We show how using such priors leads to algorithms for Bayesian estimation and a probabilistic interpretation of invariants which addresses the limitations of current methods. We present synthetic and real results for object recognition, image registration and determining object planarity to demonstrate the power of using priors for image comparison.

In this paper, we describe an adaptation method for speech recognition systems that is based on a nonlinear transformation of the feature space. In contrast to most existing adaptation methods which assume some form of affine transformation of either the feature vectors or the acoustic models that model the feature vectors, our proposed method composes a general nonlinear transformation from two transformations, one of these being an affine transformation that combines the dimensions of the original feature space, and the other being a nonlinear transformation that is applied independently to each dimension of the previously transformed feature space leading to a general multidimensional nonlinear transformation of the original feature space. This method also differs from other affine techniques in the way the parameters of the transform are shared. In most previous methods, the parameters of the transformation are shared on the basis of the phonetic class, in our method, the parameters of the nonlinear transformation are shared not on the basis of the phonetic class, but rather on the location in the feature space. Experimental results show that the method outperforms affine methods providing up to a 25% relative improvement in the word error rate in an in-car speech recognition task.

Fractal sets can be generated by the iterated function system (IFS) codes using the contractive affine transformation. This paper presents various geometric affine transformations and their composition effects for two-dimensional (2-D) fractal sets. Here, the geometric transformations include translation, rotation, shearing, dilation/contraction, and reflection. First, a hierarchical fixed point-searching algorithm is proposed to determine the original coordinates of a 2-D fractal set directly from its IFS code. Then, the IFS code is modified according to the desired transformation. Instead of post processing the generated result, arbitrary affine transformation on the original fractal set can be directly obtained. On the other hand, the composite geometric transformations for 2-D fractal sets are also available. Finally, a complicated image frame can be synthesized by multiple 2-D fractal sets.

The notion of linear moment vector is introduced as an efficient approach for matching and normalization of affine deformed image. Least square error technique is introduced to improve the robustness and accuracy of the matching. By normalizing the linear moment vectors, the affine deformed image can be normalized. Our approaches are based only on regular moments that can be obtained easily from the image without extracting any feature point. The techniques can be generalized to n-dimensional case. The experiment results show that the proposed method gives good performance.

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

For a class of hybrid systems a method will be presented to obtain a time-discrete model from a given model in continuous time. The discretization method maps a continuous piecewise affine system to a time-discrete system. Difficulties arise in the procedure from the boundaries of the domains in which the affine systems are defined. The proposed procedure results in a precise time-discrete model even for a low sampling frequency.

This paper proposes a method for matching images based on their higher order moments without knowing the point correspondences. It is assumed that the disparity between the images can be explained by an affine transformation. The second-order statistics is used to transform the image points into canonical form, which reduces the affine matching problem for determining an orthonormal transformation matrix between the two point sets. Next, higher order complex moments are used to solve the remaining transformation. These affine moment descriptors are expressed in terms of the central moments of the original data. It is also shown that the resulting descriptors can be converted into affine moment invariants. A general framework for deriving affine moment descriptors as well as moment invariants is described. The experiments carried out with simulated data and real images indicate that the proposed method utilizing the second- and third-order statistics can provide good alignment results from noisy and spurious observations.

This paper is concerned with the identification of a class of piecewise affine systems called a piecewise affine autoregressive exogenous (PWARX) model. The PWARX model is composed of ARX sub-models each of which corresponds to a polyhedral region of the regression space. Under the temporary assumption that the number of sub-models is known a priori, the input-output data are collected into several clusters by using a statistical clustering algorithm. We utilize support vector classifiers to estimate the boundary hyperplane between two adjacent regions in the regression space. In each cluster, the parameter vector of the sub-model is obtained by the least squares method. It turns out that the present statistical clustering approach enables us to estimate the number of sub-models based on the information criteria such as CAIC and MDL. The estimate of the number of sub-models is performed by applying the identification procedure several times to the same data set, after having fixed the number of sub-models to different values. Finally, we verify the applicability of the present identification method through a numerical example of a Hammerstein model.

Affine (wavelet) frames related with subdivision scheme provide tools for building curves and surfaces in CAGD/CAM. At first we show that scaling filters can be parameterized by decomposed elementary matrices with the form Im−P+Pz−k, where k is a nonnegative integer, and P is a one rank constant-valued idempotent matrix P2=Im (Theorem 2.2). Then we use the method to construct m band affine (wavelet) frames with the given scaling filter and the dual frames. General solutions for affine (wavelet) frames and its dual affine (wavelet) frames are obtained with pre-assigned scaling filter (Theorems 3.5 and 3.6). At last a design example is given.

Affine projection algorithms are useful adaptive filters whose main purpose is to speed the convergence of LMS-type filters. Most analytical results on affine projection algorithms assume special regression models or Gaussian regression data. The available analysis also treat different affine projection filters separately. This paper provides a unified treatment of the mean-square error, tracking, and transient performances of a family of affine projection algorithms. The treatment relies on energy conservation arguments and does not restrict the regressors to specific models or to a Gaussian distribution. Simulation results illustrate the analysis and the derived performance expressions.

This paper deals with a trajectory tracking problem for a class of bimodal piecewise affine systems, which is inherently difficult because of the discontinuous changes of their vector fields. First, we introduce an error variable and an error system as a generalization of the tracking error and its system. As an error variable, a function switched by the mode of a piecewise affine system is adopted to overcome the inherent difficulty in trajectory tracking control of piecewise affine systems. Next, we design a tracking controller which stabilizes the error system using a Lyapunov-like function, which can be applied to systems including state jumps. Furthermore, the feasibility condition of tracking for SISO piecewise linear systems is simplified. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

This paper addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes and Wiener piecewise affine autoregressive exogenous models, in which the regressor space is partitioned into polyhedra with affine submodels for each polyhedron. In particular, we provide algorithms based on mixed-integer linear or quadratic programming which are guaranteed to converge to a global optimum. For the special case where the estimation data only seldom switches between the different submodels, we also suggest a way of trading off between optimality and complexity by using a change detection approach.

We give a classification of affine rotation surfaces in affine 3-space with flat affine metric.

In this paper we use Fourier series to produce normalized coefficients for color images which are invariant under any affine transformation. In the reference [Pattern Recogn. Lett., 23 (2002) 1109], the proposed algorithm is developed to be applied to shape extracted from binary images, thus it is used to discriminate objects based only on their shape similarity. That means that it is unable to discriminate two objects with the same shape but different colors and textures. It is known also that the shape extraction is not an easy and well-solved problem. The present paper addresses the problem of constructing invariants and motion estimates using both shape and color informations. The quantitative evaluation of the proposed approach shows promising results.

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.

This paper calculates the parameters of image position and orientation, proposes a mathematical model and adopts a new method with three steps of transformations based on parallel ray projection. Every step of the model is strict, and the map function of each transformation is the first order polynomials and other simple function. The final calculation of the parameters is for the linear equations with good status. As a result, the problem of the relativity of image parameter calculation is solved completely. Some experiments are carried out.