Foundations of semi-differential invariants

This paper is concerned with the problem of feature point registration and scene recognition from images under weak perspective transformations which are well approximated by affine transformations and under possible occlusion and/or appearance of new objects. It presents a set of local absolute affine invariants derived from the convex hull of scattered feature points (e.g., fiducial or marking points, corner points, inflection points, etc.) extracted from the image. The affine invariants are constructed from the areas of the triangles formed by connecting three vertices among a set of four consecutive vertices (quadruplets) of the convex hull, and hence do make direct use of the area invariance property associated with the affine transformation. Because they are locally constructed, they are very well suited to handle the occlusion and/or appearance of new objects. These invariants are used to establish the correspondences between the convex hull vertices of a test image with a reference image in order to undo the affine transformation between them. A point matching approach for recognition follows this. The time complexity for registering L feature points on the test image with N feature points of the reference image is of order O(N x L). The method has been tested on real indoor and outdoor images and performs well.

We propose a mathematical method to analyze the numerous algorithms performing Image Matching by Affine Simulation (IMAS). To become affine invariant they apply a discrete set of affine transforms to the images, previous to the comparison of all images by a Scale Invariant Image Matching (SIIM), like SIFT. Obviously this multiplication of images to be compared increases the image matching complexity. Three questions arise: a) what is the best set of affine transforms to apply to each image to gain full practical affine invariance? b) what is the lowest attainable complexity for the resulting method? c) how to choose the underlying SIIM method? We provide an explicit answer and a mathematical proof of quasi-optimality of the solution to the first question. As an answer to b) we find that the near-optimal complexity ratio between full affine matching and scale invariant matching is more than halved, compared to the current IMAS methods. This means that the number of key points necessary for affine matching can be halved, and that the matching complexity is divided by four for exactly the same performance. This also means that an affine invariant set of descriptors can be associated with any image. The price to pay for full affine invariance is that the cardinality of this set is 6.34 times larger than for a SIIM.

This paper outlines a new geometric parameterization of 2D curves where parameterization is in terms of geometric invariants and parameters that determine intrinsic coordinate systems. This new approach handles two fundamental problems: single-computation alignment, and recognition of 2D shapes under Euclidean or affine transformations. The approach is model-based: every shape is first fitted by a quartic represented by a fourth degree 2D polynomial. Based on the decomposition of this equation into three covariant conics, we are able, in both the Euclidean and the affine cases, to define a unique intrinsic coordinate system for non-singular bounded quartics that incorporates usable alignment information contained in the polynomial representation, a complete set of geometric invariants, and thus an associated canonical form for a quartic. This representation permits shape recognition based on 11 Euclidean invariants, or 8 affine invariants. This is illustrated in experiments with real data sets.

We construct an algebraic-combinatorial model of the SOS compartment of the EGFR biochemical network. A Petri net is used to construct an initial representation of the biochemical decision making network, which in turn defines a hyperdigraph. We observe that the linear algebraic structure of each hyperdigraph admits a canonical set of algebraic-combinatorial invariants that correspond to the information flow conservation laws governing a molecular kinetic reaction network. The linear algebraic structure of the hyperdigraph and its sets of invariants can be generalized to define a discrete algebraic-geometric structure, which is referred to as an oriented matroid. Oriented matroids define a polyhedral optimization geometry that is used to determine optimal subpaths that span the nullspace of a set of kinetic chemical reaction equations. Sets of constrained submodular path optimizations on the hyperdigraph are objectively obtained as a spanning tree of minimum cycle paths. This complete set of subcircuits is used to identify the network pinch points and invariant flow subpaths. We demonstrate that this family of minimal circuits also characteristically identifies additional significant biochemical reaction pattern features. We use the SOS Compartment A of the EGFR biochemical pathway to develop and demonstrate the application of our algebraic-combinatorial mathematical modeling methodology.

Asymptotic Stability of Uncalibrated Eye-in-Hand Visual Servoing: An Affine Invariance Perspective

Affine moment invariants are widely used in pattern recognition, most of the existing affine moment invariants are based on geometric central moments or continuous orthogonal moments. This paper proposes an approach to construct a set of discrete Tchebichef moment invariants which are simultaneously invariant to affine transformation and to convolution with circularly symmetric point spread function. Some important properties of proposed invariants are presented and proved. These invariants are used for image recognition. The experimental results show that the proposed method has better performance in pattern recognition compared to existing method.

In this paper a method for distant object recognition is proposed. We propose an iterative structure for distant object recognition using an active-zooming camera like PTZ camera. We adopt Graphcut-based segmentation algorithm, Grabcut, to our structure in order to select candidate regions for zooming and gazing. Grabcut is utilized through the modified strategy for our purpose. And we also propose two failure detection methods based on contour points and homography for our object recognition system. We show that our candidate region selection method is available for distance change from the camera, and also validate our recognition method through the real experiment.

Useful for human visual perception, edge detection remains a crucial stage in numerous image processing applications. One of the most challenging goals in contour detection is to operate algorithms that can process visual information as humans require. To ensure that an edge detection technique is reliable, it needs to be rigorously assessed before being used in a computer vision tool. This assessment corresponds to a supervised evaluation process to quantify differences between a reference edge map and a candidate, computed by a performance measure/criterion. To achieve this task, a supervised evaluation computes a score between a ground truth edge map and a candidate image. This paper presents a survey of supervised edge detection evaluation methods. Considering a ground truth edge map, various methods have been developed to assess a desired contour. Several techniques are based on the number of false positive, false negative, true positive and/or true negative points. Other methods strongly penalize misplaced points when they are outside a window centered on a true or false point. In addition, many approaches compute the distance from the position where a contour point should be located. Most of these edge detection assessment methods will be detailed, highlighting their drawbacks using several examples. In this study, a new supervised edge map quality measure is proposed. The new measure provides an overall evaluation of the quality of a contour map by taking into account the number of false positives and false negatives, and the degrees of shifting. Numerous examples and experiments show the importance of penalizing false negative points differently than false positive pixels because some false points may not necessarily disturb the visibility of desired objects, whereas false negative points can significantly change the aspect of an object. Finally, an objective assessment is performed by varying the hysteresis thresholds on contours of real images obtained by filtering techniques. Theoretically, by varying the hysteresis thresholds of the thin edges obtained by filtering gradient computations, the minimum score of the measure corresponds to the best edge map, compared to the ground truth. Twenty-eight measures are compared using different edge detectors that are robust or not robust regarding noise. The scores of the different measures and different edge detectors are recorded and plotted as a function of the noise level in the original image. The plotted curve of a reliable edge detection measure must increase monotonously with the noise level and a reliable edge detector must be less penalized than a poor detector. In addition, the obtained edge map tied to the minimum score of a considered measure exposes the reliability of an edge detection evaluation measure if the edge map obtained is visually closer to the ground truth or not. Hence, experiments illustrate that the desired objects are not always completely visible using ill-suited evaluation measure.

Effective classification of planar shapes based on curve segment properties

The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on ℝd and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications.

Feasibility issues of Generalized Predictive Control structures are analysed in case of constraints on the input, output or other auxiliary variables with linear dependence on the system variables. The constraints are formulated as sets of linear equality or inequality relations on the input variables that will represent the optimisation domain for a quadratic cost function. Looking as the main goal at the prediction of the feasibility of this optimisation problem, sufficient conditions are presented in this direction. The main contribution is the description of the optimisation domain as an invariant set with respect to the predictive control law dynamics. The main advantage is that all the results are based on off-line calculations offering qualitative information prior to the effective implementation.

A new method is presented for solving equivalence problems for the extended jet of finite order of the scale-space corresponding to a 2-dimensional input image and the groups of spatially homogeneous affine and orthogonal transformations of local cartesian coordinate frames. By means of this method complete and irreducible sets of algebraic invariants are found that may describe any local and (or) multilocal affine or orthogonal invariant of scale-space. Consequently these sets may form bases for topological descriptions of scale-space.

The systems considered in the paper are m-input m-output, linear, time invariant decouplable systems S = (A, B, C) which can be described by the set of differential equations $ where A∊R n×n , B∊R n×m and C∊R m×n . In § 1 the decoupling problem is stated and some results relevant to our discussion are reviewed. In § 2 we analyse the effect of an integrator decoupling feedback law (IDF) upon the controllability and the observability of the system under consideration, showing that observability is preserved under IDF if and only if $. Furthermore it is shown that if a decouplable system satisfies this condition, then it is controllable and observable. In § 3 we give our main result : for the class Ω of decouplable systems which preserve observability under IDF, the set of integers {di } ; i = 1, 2,…, m is a complete set of independent invariants under feedback and similarity transformations. Finally a canonical form (unique) is given for the elements of Ω, completely determined by the set {di } ; i = 1,…, m.

The article is devoted to use wavelet transform, that have properties of frequency-spatial selectivity, and detect information elements of object (for example, contour) with required detailed elaboration. The morphological operator on wavelet transform base was developed. It allows receiving the contour description at different hierarchy levels. The algorithm of getting contour points on a base of iterative quasi Newton method of extremum search of function at wavelet transform space was developed. The algorithm of image contour analysis based on iterative search algorithms in wavelet transform space was offered.

The Ectocarpus genome sequence: insights into brown algal biology and the evolutionary diversity of the eukaryotes.