Optimal Transport Problem Compute Target Light Distribution Compute Mapping from Initial Distribution to Target Distribution Optimizing Target Surface Compute Normal Representation of Surface Surface Refinement Solving optimal transport problem As the case refraction occurs through a transparent surface, for instance the patterns appearing under clear water surface, 3 main phenomena can be observed: Very bright (condensed light intensity) points (so-called singularity) Curve-like objects that connect the points Regions with low light intensity To perform computation, the following 3 quantities are being respectively introduced to describe the geometric characteristics of the pattern: point singularity (measuring light intensity at certain highly concentrated light-point), curve singularity (measuring light intensity at/around a light-curve), and irradiance measure (measuring intensity in a certain poorly concentrated light-area). Putting them altogether, the following function defines the total radiant flux measure at a certain section Ω on the target surface: After this step, there are two existing measures of the radiant flux measures of the source (uniform distribution, by initialization) and the target (computed in previous step). What remains to compute is the mapping from the source to target. In order to do this, there are several quantities to be defined. Firstly, two light intensities evaluated by probabilities: (light intensity evaluated by dividing by the flux of the union region between and ), (light intensity evaluated by dividing by the flux of the union region between and ) are defined. Secondly, the source mesh is generated as multiple sites , which is later being deformed. Next, a power diagram (a set of power cells) is defined on this set of sites weighted by a weight vector . Finally, the goal is to decide whether which power cells are going to be move. Considering all vertices on the surface, finding the minimizer of the following convex function will produce the matched power diagram for the target: Optimizing target surface After solving optimal transport problem, the vertices are achieved. However, this gives no information about what the final surface should look like. To achieve the desired target surface given the incoming light ray , outgoing light ray and power diagram from the step above, the surface normals representation can be computed according to Snell's law as: where, : refraction coefficient : target position obtained from solving above optimal transport problem As the normal representation is obtained, surface refinement is then achieved by minimizing the following compound energy function: where, is the