Supplementary Document for Patches, Planes and Probabilities: A Non-local Prior for Volumetric 3D Reconstruction

This supplementary document presents derivations for the proposed sum-product belief propagation algorithm, pseudocode of the inference algorithm, the derivations of our depth-map prediction method as well as additional experiments. First, we present the message derivations of the sum-product belief propagation algorithm which were omitted in the original document. We then present the pseudocode of our inference algorithm, and in particular the message passing scheme. Besides, we show how Bayes optimal depth predictions can be obtained under our probabilistic model. Finally, we present a number of additional experiments. In particular, we present results by varying the parameters for the model with pairwise smoothness potentials. Next, we present an evaluation for the BARUS&HOLLEY dataset which excludes the tree regions where the LIDAR ground truth is not accurate and show that our algorithm outperforms previous algorithms. Finally, we present an experiment using a uniform prior over plane orientations as opposed to the Manhattan world prior that we utilize in the paper. 1. Message Equations for Sum-product Belief Propagation This section presents the message equations and their derivations that were omitted in the main submission due to lack of space. We refer the reader to the submission file for the notation and the probabilistic model. The general form of the message equation for sum-product belief propagation on factor graphs is given by μf→x(x) = ∑ Xf\x φf (Xf ) ∏ y∈Xf\x μy→f (y) (1) μx→f (x) = ∏ g∈Fx\f μg→x(x) (2) where f denotes a factor, x is a random variable, Xf denotes all variables associated with factor f and Fx is the set of factors to which variable x is connected. Below, we repeat the factor graph of our model for completeness.