A fundamental extension of the standard regularization technique is proposed for making data approximations using multivalued functions which are essential for solving the transparency problems in computational vision. Conventional standard regularization techniques can approximate scattered data by using a single-valued function which is smooth everywhere in the domain. However, to incorporate discontinuities of the functions, it is necessary to introduce the line process or an equivalent technique to break the coherence or smoothness of the approximating functions. Multilayer representations have been used in reconstruction of multiple overlapping surfaces. However this technique should incorporate auxiliary fields for segmenting given data. Furthermore, these two different approaches both have the difficulty implementing optimizations of their energy functionals since they always become nonquadratic, nonconvex minimization problems with respect to an unknown surface and auxiliary field parameters. This paper shows that by using a direct representation of multivalued functions, data approximation made using a multivalued function can be reduced to minimizations of a single quadratic convex functional. Therefore, since the Euler-Lagrange equation of the functional becomes linear in this case, it is possible to benefit from simple relaxation techniques of guaranteed convergence to the optimal solution.

A computational framework is provided for representing and detecting multiple orientation fields from a set of local differentiation filters such as multiscale Gaussian derivatives. The representation is direct and closed-form, i.e., it is not necessary to steer the filters in order to detect multiple orientations. They can be estimated in a single-shot manner by solving algebraic equations. The filter does not need to be strongly tuned to orientations, since the derived algorithm does not suffer from the problem of interference between signal components of the multiple orientations. The capability of extracting the characteristic image structures of different scales is demonstrated by simulation. These advantages are accomplished by using the principle of superposition.>

A closed-form single-shot stereo disparity estimation algorithm is proposed that can compute multiple disparities due to transparency directly from signal differences and variations on epipolar lines of a binocular image pair. The transparent stereo constraint equations have been derived by using a novel mathematical technique, the principle of superposition. A computationally tractable single-shot algorithm is derived by using the first-order approximation of the constraint equations with respect to disparities. The algorithm can compute multiple disparities from only two images, in contrast to the previous algorithms for motion transparency, which needed at least n+1 frames for n simultaneous motion estimates. The derived algorithm can be viewed as the SSD (sum of squared differences) for signal matching extended to deal with multiple disparities. However, the constraints are not dedicated solely to the SSD method and several other implementations are possible.>

The proposed one-shot algorithm requires only first- and second-order image derivatives in local neighborhoods to detect multiple orientations and to classify L-, T-, X-junctions. This algorithm has two stages. The first stage detects multiple orientations with a parametrized filter based on the principle of superposition, and the second stage classifies L-, T-, X- junctions with first-order image derivatives in local areas in order to evaluate the borders. The computational cost is therefore low, and the method is robust against noise. The algorithm provided successful results for artificial images and real images.

A unified theoretical framework for motion transparency and motion boundaries by devising fundamental constraint equations of multiple optical flow is proposed. This framework can handle flow discontinuities at motion boundaries as well as flow multiplicities due to transparency of objects in a unified manner. The constraint equations are formulated by a composition of homogeneously parametrized differential operators on the space-time image. Fitting algorithms for the constraints which result in eigensystem analyses are described. To determine the number of flows, the authors use the margin energy, a measure of goodness of fit which is the difference between the first and the second lower eigenenergy of the eigensystem. They also hypothesize a criterion for multiplicity. The measure and the criterion are derived from the analogy of quantum mechanics. It is demonstrated that the margin energy can determine the transparency and discontinuities of the flow field as regions of more than one flow.>

The theory and algorithm of a general method that constructs one-to-one mappings on an n-dimensional digital lattice are presented. The mapping is constructed so that any given equivolume affine transformation can be approximated. Equivolume affine transformations include translation, reflection, and skew. It is shown that (n/sup 2/-1) fundamental skew transformations, n fundamental translations, and some reflective transformations are sufficient to represent arbitrary equivolume affine transformation. One-to-one integer approximation of the fundamental transformations and approximation error propagation rules are described. Minimum error decomposition algorithms for the equivolume affine transformation in n-dimensional space and two-dimensional space are proposed.>

The authors propose a simultaneous closed-form estimation method for multiple optical flow from image sequences in which each image point has multiple motions. This method only requires convolution for space-time filtering and low-dimensional eigensystem analysis as an optimization process. The authors propose a mixture flow model of a multiple flow and energy integral minimization as a model fitting method. It is shown that symmetry between component flows of the mixture flow can reduce the dimension of the eigensystem and make the optimization unimodal and stable. Successful experiments on double-flow estimation of random texture patterns and natural scene images are reported.>