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A specific study of the orthographic, weak-perspective and paraperspective cameras.An algebraic procedure solving the affine correction problem for each camera model.A closed-form solution for each camera model.A full characterization of the affine corrections' generic ambiguities for each camera model.An experimental evaluation comparing the algebraic procedures to global polynomial optimization and an interior-point method. Given a general affine camera, we study the problem of finding the closest metric affine camera, where the latter is one of the orthographic, weak-perspective and paraperspective projection models. This problem typically arises in stratified Structure-from-Motion methods such as factorization-based methods. For each type of metric affine camera, we give a closed-form solution and its implementation through an algebraic procedure. Using our algebraic procedure, we can then provide a complete analysis of the problem's generic ambiguity space. This also gives the means to generate the other solutions if any.

We derive an explicit relation between local affine approximations resulting from matching of affine invariant regions and the epipolar geometry in the case of a two view geometry. Most methods that employ the affine relations do so indirectly by generating pointwise correspondences from the affine relations. In this paper we derive an explicit relation between the local affine approximations and the epipolar geometry. We show that each affine approximation between images is equivalent to 3 linear constraints on the fundamental matrix and that the linear conditions guarantee the existence of an homography, compatible with the fundamental matrix. We further show that two affine relations constrain the location of the epipole to a conic section. Therefore, the location of the epipole can be extracted from 3 regions by intersecting conics. The result is further employed to derive a procedure for estimating the fundamental matrix, based on the estimated location of the epipole. It is shown to be more accurate and to require less iterations in LO-RANSAC based estimation, than the current point based approaches that employ the affine relation to generate pointwise correspondences and then calculate the fundamental matrix from the pointwise relations.

Usually, state-of-the-art interest point detectors tend to over-represent the local image structures associating several interest points for each local image structure. This fact avoids interest points to clearly stand out against its neighborhood, losing the ability to clearly describe the global uniqueness of each local image structure. In order to solve this problem we propose a sparse affine invariant blob detector, which tries to describe each blob structure with a single interest point. The proposed detector is carried out in two stages: an initial stage, where a set of scale invariant interest points are located by means of the idea of blob movement and blob evolution (creation, annihilation and merging) along different scales by using a precise description of the image provided by the Gaussian curvature, providing a global bottom-up estimation of the image structure. During the second stage, the shape and location of each scale invariant interest point is refined by fitting an anisotropic Gaussian function, which minimizes the error with the underlying image and simultaneously estimates both the shape and location, by means of a non-linear least squares approach. A comparative evaluation of affine invariant detectors is presented, showing that our approach outperforms state-of-the-art affine invariant detectors in terms of precision and recall, and obtains a similar performance to that of the best ones in terms of repeatability and matching. In addition we demonstrate that our detector does not over-represent blob structures and provides a sparse detection that improves distinctiveness and reduces drastically the computational cost of matching tasks. In order to verify the accuracy and the reduction in the computational cost we have evaluated our detector in image registration tasks.

We propose a novel algorithm for affine registration of 2D point sets. The main idea is to treat the 2D points as complex numbers and from each point set, a polynomial with complex coefficients can be computed whose roots are the points in the given point set. The two-step algorithm first reduces the affine registration problem to a rigid registration problem, and the unknown rotation is then computed using the coefficients of these polynomials. The algorithm is entirely algebraic with clear underlying geometric motivation. The implementation is straightforward and it takes less than a second to compute the affine transformation for point sets containing hundreds of points. We validate the algorithm on a variety of synthetic 2D point sets as well as point sets extracted from real-world images.

The contribution of this paper is twofold: (1) it provides a thorough analysis of the frequency domain relationships relating two affine-warped images and (2) based on a fundamental equation between energy radial projections, it presents an original algorithm for estimating the global 2D affine transformation between the two images. It is well known that operating in the frequency domain allows one to separate the estimate of the affine matrix, related to the magnitudes of the Fourier transforms of the two images, from the estimate of the translation vector, related to their phases. Exploiting this property, our algorithm consists of two main steps: (1) the affine matrix is first estimated by solving, with a coarse-to-fine strategy, a suitable minimization problem formulated upon the radial projections of the image energies, and (2) after compensation for the contribution of the affine matrix, the translation vector is then recovered by means of phase correlation. The proposed method is very robust against perspective distortion and, with moderate translational displacements, it may also work when the two images differ along their peripheral areas. Experimental evidence of these characteristics is reported and discussed. The algorithm can be efficiently implemented via FFT and well suits applications requiring unsupervised and/or quasi-real-time estimation of global motion that can be described with 2D affine transformations.

In this paper, we will study the recognition problem for finite point configurations, in a statistical manner. We study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. Here, a general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. No approximations are made, resulting in an exact expression of the uncertainty region. In particular, we will concentrate on the affine shape, where often analytical computations is possible. In this case confidence intervals for invariants can be obtained from a priori assumptions on the densities of the detected points in the images. However, the theory is completely general and can be used to compute the density of any invariant (Euclidean, affine, similarity, projective, etc.) from arbitrary densities of the individual points. These confidence intervals can be used in such applications as geometrical hashing, recognition of ordered point configurations, and error analysis of reconstruction algorithms. Finally, an example will be given, illustrating the theory for the problem of recognizing planar point configurations from images taken by an affine camera. This case is of particular importance in applications, where details on a conveyor belt are captured by a camera, with image plane parallel to the conveyor belt and extracted feature points from the images are used to sort the objects.

In this paper the problem of computing the point correspondences in a sequence of time-varying images of a 3D object undergoing nonrigid (affine) motion is addressed. It is assumed that the images are obtained through affine projections. The correspondences are established only from the analysis of the unknown 3D affine structure of the object, without making use of any attributes of the feature points. It is shown that it is possible to establish the point correspondences uniquely (up to symmetry) in the sense that they yield a unique affine structure of the object and that the computation is possible in polynomial time. Two different algorithms for computing the point correspondences are presented. Results on various real image sequences, including a sequence containing independently moving objects, demonstrate the applicability of the structure based approach to motion correspondence.

One method for determining whether two curves may be different perspective views of the same curve is to use invariant signature functions. A significant difficulty with this approach is that projective invariants often require unwieldy combinations of point correspondences and high order derivatives. A special case of the projective viewing transformation is the affine transformation. Affine invariants require fewer point correspondences and/or lower order derivatives than projective invariants, and as such are faster to compute and less sensitive to noise. In this paper, affine invariants requiring one point correspondence and second order derivatives, and requiring two point correspondences and first order derivatives, are investigated. In both of these cases, the affine invariant is useful when the ratio of the sine of the angle of inclination of the plane of the transformed curve to the image plane to the distance of the transformed curve from the image plane is less than a curve-dependent constant. Examples are worked out in detail to illustrate our results.