Efficient Continuous Relaxations for Dense CRF Supplementary Materials

In this paper, the filter based method that we use for our experiments is the one by Adams et al. [1]. In this method, the original computation is approximated by a convolution in a higher dimensional space. The original points are associated to a set of vertices on which the convolution is performed. The considered vertices are the one from the permutohedral lattice. Krähenbühl and Koltun [2] provided an implementation of this method. In their implementation, they added a pixelwise normalisation of the output of the permutohedral lattice and say that it performs well in practice. We observe that for the variances considered in this paper and without using the normalisation by Krähenbühl and Koltun, the results given by the permutohedral lattice is a constant factor away from the value computed by brute force in most cases. As can be seen in Figure 1, in the case where we compute ∑ a,bKa,b1, the left graph, the ratio between the value obtained by brute force and the value obtained using the permutohedral lattice is 0.6 for large enough images. On the other hand, for a different value of the input points where we compute ∑ b>aKa,b− ∑ b<aKa,b, the right graph, we get a ratio of 0.48 between the two results. The case where we consider a variance of 50 is special. We know that the highest the variance value, the worst the approximation of the permutohedral is. If the experience on the full computation is conducted on an image of size 320× 213, the ratio between the brute force approach and the permutohedral lattice is 0.633. At the same time is also worth noting that in all these results, if we consider the outputs as vectors, as is done when computing our gradients, the vectors given by the brute force and the ones given by the permutohedral lattice are collinear for all image size and all variances. We can thus expect that for other input values, the direction of gradient provided by the permutohedral lattice is correct, but the norm of this vector may be incorrect.