Fast Bilateral-Space Stereo for Synthetic Defocus Supplemental Material

We will now dig deeper into the details of this matrix factorization, and discuss the two specific bilateral representations we use: the simplified bilateral grid, and the permutohedral lattice [1]. Filtering with both the permutohedral lattice and the simplified bilateral grid works by “splatting” a value at each pixel onto a small number of vertices, performing a separable blur in the space of vertices, and “slicing” out values at each pixel to get a filtered set of values. The difference between the two representations is in the arrangement of the vertices (the permutohedral lattice is tetrahedral, while the simplified bilateral grid is rectangular), and the nature of the splat interpolation (the lattice uses barycentric interpolation, and the simplified bilateral grid uses nearest-neighbor assignment). Intuitively, the permutohedral lattice approximates the Gaussian in the affinity function as the convolution of a tent filter (barycentric interpolation), a [1, 2, 1] blur kernel, and another tent filter, which is a good binomial approximation of a Gaussian function. The simplified bilateral grid approximates a Gaussian using a boxcar or “rect” filter (nearest-neighbor interpolation) and a narrow [1,2,1] blur kernel, which is a significantly less accurate but more efficient representation. In the factorization produced by the permutohedral lattice, assuming that our Gaussian affinity is in a D dimensional space, the splat matrix S has D + 1 non-zero elements per row, we haveD+1 blur matrices, and we approximate B̄ as an outer product of the blur matrices. In the simplified bilateral grid, we have 1 non-zero element per row of our splat matrix, we have D blur matrices, and we approximate B̄ as the sum of blur matrices.