The earth moverpsilas distance (EMD) is an important perceptually meaningful metric for comparing histograms, but it suffers from high (O(N3logN)) computational complexity. We present a novel linear time algorithm for approximating the EMD for low dimensional histograms using the sum of absolute values of the weighted wavelet coefficients of the difference histogram. EMD computation is a special case of the Kantorovich-Rubinstein transshipment problem, and we exploit the Holder continuity constraint in its dual form to convert it into a simple optimization problem with an explicit solution in the wavelet domain. We prove that the resulting wavelet EMD metric is equivalent to EMD, i.e. the ratio of the two is bounded. We also provide estimates for the bounds. The weighted wavelet transform can be computed in time linear in the number of histogram bins, while the comparison is about as fast as for normal Euclidean distance or chi2statistic. We experimentally show that wavelet EMD is a good approximation to EMD, has similar performance, but requires much less computation.

Recognition of specular objects is particularly difficult because their appearance is much more sensitive to lighting changes than that of Lambertian objects. We consider an approach in which we use a 3D model to deduce the lighting that best matches the model to the image. In this case, an important constraint is that incident lighting should be non-negative everywhere. In this paper, we propose a new method to enforce this constraint and explore its usefulness in specular object recognition, using the spherical harmonic representation of lighting. The method follows from a novel extension of Szego's eigenvalue distribution theorem to spherical harmonics, and uses semidefinite programming to perform a constrained optimization. The new method is faster as well as more accurate than previous methods. Experiments on both synthetic and real data indicate that the constraint can improve recognition of specular objects by better separating the correct and incorrect models