Cicero Mota

Analysis of Superimposed Oriented Patterns

Estimation of local orientation in images may be posed as the problem of finding the minimum gray-level variance axis in a local neighborhood. In bivariate images, the solution is given by the eigenvector corresponding to the smaller eigenvalue of a 2times2 tensor. For an ideal single orientation, the tensor is rank-deficient, i.e., the smaller eigenvalue vanishes. A large minimal eigenvalue signals the presence of more than one local orientation, what may be caused by non-opaque additive or opaque occluding objects, crossings, bifurcations, or corners. We describe a framework for estimating such superimposed orientations. Our analysis is based on the eigensystem analysis of suitably extended tensors for both additive and occluding superpositions. Unlike in the single-orientation case, the eigensystem analysis does not directly yield the orientations, rather, it provides so-called mixed-orientation parameters (MOPs). We, therefore, show how to decompose the MOPs into the individual orientations. We also show how to use tensor invariants to increase efficiency, and derive a new feature for describing local neighborhoods which is invariant to rigid transformations. Applications are, e.g., in texture analysis, directional filtering and interpolation, feature extraction for corners and crossings, tracking, and signal separation

Analytic solutions for multiple motions

A novel framework for single and multiple motion estimation is presented. It is based on a generalized structure tensor that contains blurred products of directional derivatives. The order of differentiation increases with the number of motions but more general linear filters can be used instead of derivatives. From the general framework, a hierarchical algorithm for motion estimation is derived and its performance is demonstrated on a synthetic sequence.

Estimation of multiple local orientations in image signals

Local orientation estimation can be posed as the problem of finding the minimum grey level variance axis within a local neighbourhood. In 2D image signals, this corresponds to the eigensystem analysis of a 2 /spl times/ 2-tensor, which yields valid results for single orientations. We describe extensions to multiple overlaid orientations, which may be caused by transparent objects, crossings, bifurcations, corners etc. Multiple orientation detection is based on the eigensystem analysis of an appropriately extended tensor, yielding so-called mixed orientation parameters. These mixed orientation parameters can be regarded as another tensor built from the sought individual orientation parameters. We show how the mixed orientation tensor can be decomposed into the individual orientations by finding the roots of a polynomial. Applications are, e.g., in directional filtering and interpolation, feature extraction for corners or crossings, and signal separation.

Analysing superimposed oriented patterns

Estimation of local orientation in images is often posed as the task of finding the minimum variance axis in a local neighborhood. The solution is given as the eigenvector belonging to the smaller eigenvalue of a 2/spl times/2 tensor. Ideally, the tensor is rank-deficient, i.e., the smaller eigenvalue is zero. A large minimal eigenvalue signals the presence of more than one local orientation. We describe a framework for estimating such superimposed orientations. Our analysis of superimposed orientations is based on the eigensystem analysis of a suitably extended tensor. We show how to carry out the eigensystem analysis efficiently using tensor invariants. Unlike in the single orientation case, the eigensystem analysis does not directly yield the orientations, rather, it provides so-called mixed orientation parameters. We therefore show how to decompose the mixed orientation parameters into the individual orientations. These, in turn, allow the superimposed patterns to be separated.

Estimation of multiple orientations in multidimensional signals

We present a solution to the general problem of estimating multiple orientations in multidimensional signals. The solution is divided in a linear part that provides the mixed-orientation space (MOS) and a nonlinear part that gives the actual orientation spaces. We show that the angle between two overlaid orientations is an invariant that can be derived from the MOS without solving the nonlinear part and that all other invariants are generated by this angle. Results obtained for synthetic images illustrate that the above invariant is a useful image feature for various applications such as pattern recognition and texture segmentation.

Estimation of multiple motions: regularization and performance evaluation

This paper deals with the problem of estimating multiple transparent motions that can occur in computer vision applications, e.g. in the case of semi-transparencies and occlusions, and also in medical imaging when different layers of tissue move independently. Methods based on the known optical-flow equation for two motions are extended in three ways. Firstly, we include a regularization term to cope with sparse flow fields. We obtain an Euler-Lagrange system of differential equations that becomes linear due to the use of the mixed motion parameters. The system of equations is solved for the mixed-motion parameters in analogy to the case of only one motion. To extract the motion parameters, the velocity vectors are treated as complex numbers and are obtained as the roots of a complex polynomial of a degree that is equal to the number of overlaid motions. Secondly, we extend a Fourier-Transform based method proposed by Vernon such as to obtain analytic solutions for more than two motions. Thirdly, we not only solve for the overlaid motions but also separate the moving layers. Performance is demonstrated by using synthetic and real sequences.

Analysis of motion and curvature in image sequences

We briefly review a recent development in the area of computer vision and multidimensional signal processing. Image sequences are regarded as hypersurfaces and useful properties are derived from the geometry of that hypersurface. Besides demonstrating the uniqueness of curvature, new methods for the analysis of single and multiple motions are presented including the case of occluded motions.

Estimation of Multiple Orientations at Corners and Junctions

Features like junctions and corners are a rich source of information for image understanding. We present a novel theoretical framework for the analysis of such 2D features in scalar and multispectral images. We model the features as occluding superpositions of two different orientations and derive a new constraint equation based on the tensor product of two directional derivatives. The eigensystem analysis of a 3 × 3-tensor then provides the so-called mixed-orientation parameters (MOP) vector that encodes the two orientations uniquely, but only implicitly. We then show how to separate the MOP vector into the two orientations by .nding the roots of a second-order polynomial. Based on the orientations, the occluding boundary and the center of the junction are easily determined. The results con.rm the validity, robustness, and accuracy of the approach.

Spatial and spectral analysis of occluded motions

We present a spatio-temporal analysis of motion at occluding boundaries. The main result is an analytical description of the motions and the distortions that occur at the occluding boundary. Based on this result we analyze occluding motions in the Fourier domain and show that the distortion term has an hyperbolic decay independent of the shape of the occluding boundary. Moreover, we derive the exact expression for the distortion term for the case of straight boundaries. The results are illustrated by using simulations with synthetic movies.

Linear and regularized solutions for multiple motions

We extend a novel framework for the estimation of multiple transparent motions to include regularization. We use mixed-motion parameters to obtain linear Euler-Lagrange equations with a regularization term. The equations are solved iteratively for the mixed-motion parameters based on an update rule that is similar to the case of only one motion. The motion parameters are then obtained as the roots of a complex polynomial of a degree that is equal to the number of overlaid motions. An experimental error analysis is performed and reported.