In this work, we recover the 3D shape of mirrors, sunglasses, and stainless steel implements. A computer monitor displays several images of parallel stripes, each image at a different angle. Reflections of these stripes in a mirroring surface are captured by the camera. For every image point, the direction of the displayed stripes and their reflections in the image are related by a 1D homography matrix, computed with a robust version of the statistically accurate heteroscedastic approach. By focusing on a sparse set of image points for which monitor-image correspondence is computed, the depth and the local shape may be estimated from these homographies. The depth estimation relies on statistically correct minimization and provides accurate, reliable results. Even for the image points where the depth estimation process is inherently unstable, we are able to characterize this instability and develop an algorithm to detect and correct it. After correcting the instability, dense surface recovery of mirroring objects is performed using constrained interpolation, which does not simply interpolate the surface depth values but also uses the locally computed 1D homographies to solve for the depth, the correspondence, and the local surface shape. The method was implemented and the shape of several objects was densely recovered at submillimeter accuracy.

In this work we recover the 3D shape of mirroring objects such as mirrors, sunglasses, and stainless steel objects. A computer monitor displays several images of parallel stripes, each image at a different angle. Reflections of these stripes in a mirroring surface are captured by the camera. For every image point, the directions of the displayed stripes and their reflections in the image are related by a 1D homography which can be computed robustly and using the statistically accurate heteroscedastic model, without monitor-image correspondence, which is generally required by other techniques. Focusing on a small set of image points for which monitor-image correspondence is computed, the depth and the local shape may be calculated relying on this homography. This is done by an optimization process which is related to the one proposed by Savarese, Chen and Perona, but is different and more stable. Then dense surface recovery is performed using constrained interpolation, which does not simply interpolate the surface depth values, but rather solves for the depth, the correspondence, and the local surface shape, simultaneously at each interpolated point. Consistency with the ID homography is thus required. The proposed method as well as the method described in [A. Ripsman and M. Jenkin, IEEE Int. Symposium on Computational Intelligence (2001)] are inherently unstable on a small part of the surface. We propose a method to detect these instabilities and correct them. The method was implemented and the shapes of a mirror, sunglasses, and a stainless steel ashtray were recovered at sub-millimeter accuracy.

Abstract In this article we study the group Steiner network problem, which is defined in the following way. Given a graph G = (V,E), a partition of its vertices into K groups and connectivity requirements between the different groups, the aim is to find simultaneously a set of representatives, one for each group, and a minimum cost connected subgraph that satisfies the connectivity requirements between the groups (representatives). This problem is a generalization of the Steiner network problem and the group Steiner tree problem, two known NP‐complete problems. We present an approximation algorithm for a special case of the group Steiner network problem with an approximation ratio of min {2(1 + 2x),2I}, where I is the cardinality of the largest group and x is a parameter that depends on the cost function. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 49(2), 160–167 2007

Robust regression techniques are used today in many computer vision algorithms. Chen and Meer recently presented a new robust regression technique named the projection based M-estimator. Unlike other methods in the RANSAC family of techniques, where performance depends on a user supplied scale parameter, in the pbM-estimator technique this scale parameter is estimated automatically from the data using kernel smoothing density estimation. In this work we improve the performance of the pbM-estimator by changing its cost function. Replacing the cost function of the pbM-estimator with the changed one yields the modified pbM-estimator. The cost function of the modified pbM-estimator is more stable relative to the scale parameter and is also a better classifier. Thus we get a more robust and effective technique. A new general method to estimate the runtime of robust regression algorithms is proposed. Using it we show, that the modified pbM-estimator runs 2 -3 times faster than the pbM-estimator. Experimental results of fundamental matrix estimation are presented demonstrating the correctness of the proposed analysis method and the advantages of the modified pbM-estimator.