Unknown Affine Transformation Research Articles

Affine Invariants of Vector Fields.

Vector fields are a special kind of multidimensional data, which are in a certain sense similar to digital color images, but are distinct from them in several aspects. In each pixel, the field is assigned to a vector that shows the direction and the magnitude of the quantity, which has been measured. To detect the patterns of interest in the field, special matching methods must be developed. In this paper, we propose a method for the description and matching of vector field patterns under an unknown affine transformation of the field. Unlike digital images, transformations of vector fields act not only on the spatial coordinates but also on the field values, which makes the detection different from the image case. To measure the similarity between the template and the field patch, we propose original invariants with respect to total affine transformation. They are designed from the vector field moments. It is demonstrated by experiments on real data from fluid mechanics that they perform significantly better than potential competitors.

Affine invariants of generalized polygons and matching under affine transformations

A generalized polygon is an ordered set of vertices. This notion generalizes the concept of the boundary of a polygonal shape because self-intersections are allowed. In this paper we study the problem of matching generalized polygons under affine transformations. Our approach is based on invariants. Firstly we associate an ordered set of complex numbers with each polygon and construct a collection of complex scalar functions on the space of plane polygons. These invariant functions are defined as quotients of the so-called Fourier descriptors, also known as discrete Fourier transforms.Each one of these functions is invariant under similarity transformations; that is, the function associates the same complex number to similar polygons. Moreover, if two polygons are affine related (one of them is the image of the other under an affine transformation), the pseudo-hyperbolic distance between their associated values is a constant that depends only on the affine transformation involved, but independent of the polygons.More formally, given a collection {Z1,Z2,…,Zm} of n-sided polygons in the plane and a query polygon W, we give algorithms to find all Zℓ such that f(Zℓ)=W+ΔW, where f is an unknown affine transformation and ΔW=(Δw1,…,Δwn) with |Δwk|≤ρ, where ρ is certain tolerance.

Robust automated multiple view inspection

Recently, Automated Multiple View Inspection (AMVI) has been developed for automated defect detection of manufactured objects, and the framework was successfully implemented for calibrated image sequences. However, it is not easy to be implemented in industrial environments because the calibration is a difficult and an unstable process. To overcome these disadvantages, the robust AMVI strategy, which assumes that an unknown affine transformation exists between each pair of uncalibrated images, is proposed. This transformation is estimated using two complementary robust procedures: a global approximation of the affine mapping is computed by creating candidate correspondences via B-splines and selecting those which better satisfy the epipolar constraint for uncalibrated images. Then, we use this approximation as initial estimate of a robust intensity-based matching approach, which is applied locally on each potential defect. The result is that false alarms are discarded, and the defects of an industrial object are actually tracked along the uncalibrated image sequence. The method is successful as shown in our experiments on aluminum die castings.

3-D Reconstruction of Urban Scenes from Image Sequences

In this paper, we address the problem of the recovery of a realistic textured model of a scene from a sequence of images, without any prior knowledge either about the parameters of the cameras or about their motion. We do not require any knowledge of the absolute coordinates of some control points in the scene to achieve this goal. First, using various computer vision tools, we establish correspondences between the images and recover the epipolar geometry, from which we show how to compute the complete set of perspective projection matrices for all camera positions. Then, we proceed to reconstruct the geometry of the scene. We show how to rely on information of the scene such as parallel lines or known angles in order to reconstruct the geometry of the scene up to, respectively, an unknown affine transformation or an unknown similitude. Alternatively, if this information is not available, we can still recover the Euclidean structure of the scene through the techniques of self-calibration. The scene geometry is modeled as a set of polyhedra. Textures to be mapped on the scene polygons are extracted automatically from the images. We show how several images can be combined through mosaicing in order to automatically remove visual artifacts such as pedestrians or trees from the textures.This vision system has been implemented as a vision server, which provides to a CAD-CAM modeler geometry or texture information extracted from the set of images. The whole system allows efficient and fast production of scene models of high quality for such applications as simulation, virtual, or augmented reality.

Part I: Modeling image curves using invariant 3-D object curve models-a path to 3-D recognition and shape estimation from image contours

This paper and its companion are concerned with the problems of 3-D object recognition and shape estimation from image curves using a 3-D object curve model that is invariant to affine transformation onto the image space, and a binocular stereo imaging system. The objects of interest here are the ones that have markings (e.g., characters, letters, special drawings and symbols, etc.) on their surfaces. The 3-D curves on the object are modeled as B-splines, which are characterized by a set of parameters (the control points) from which the 3-D curve can be totally generated. The B-splines are invariant under affine transformations. That means that the affine projected object curve onto the image space is a B-spline whose control points are related to the object control points through the affine transformation. Part I deals with issues relating to the curve modeling process. In particular, the authors address the problems of estimating the control points from the data curve, and of deciding on the best order B-spline and the best number of control points to be used to model the image or object curve(s). A minimum mean-square error (mmse) estimation technique which is invariant to affine transformations is presented as a noniterative, simple, and fast approach for control point estimation. The best B-spline is decided upon using a Bayesian selection rule. Finally, we present a matching algorithm that allocates a sample curve to one of p prototype curves when the sample curve is an a priori unknown affine transformation of one of the prototype curves stored in the data base. The approach is tried on a variety of images of real objects. >