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 2 x 2 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.

No abstract

No abstract

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.

This paper deals with the problem of estimating multiple motions at points where these motions are overlaid. We present a new approach that is based on block-matching and can deal with both transparent motions and occlusions. We derive a block-matching constraint for an arbitrary number of moving layers. We use this constraint to design a hierarchical algorithm that can distinguish between the occurrence of single, transparent, and occluded motions and can thus select the appropriate local motion model. The algorithm adapts to the amount of noise in the image sequence by use of a statistical confidence test. The algorithm is further extended to deal with very noisy images by using a regularization based on Markov Random Fields. Performance is demonstrated on image sequences synthesized from natural textures with high levels of additive dynamic noise.