integration energy that aligns the vertex normals obtained from the Optimal Transport with the target normals obtained from the Snell's law computation above. as mesh generated in step Solving Optimal Transport cannot adapt to the sharp instances from the discontinuities, this energy is to penalize the vertices to not change significantly from the incoming light ray. is the energy measuring the flux over the triangle in the mesh. is the energy that regularizes the shape of the triangles to maintain its well-shapedness. is barrier energy to ensure that surface does not deform beyond a certain distance threshold . Differentiable inverse rendering caustic pattern design Basic principle Inverse graphics is a method of observing the data from an image and inferring all possible properties including 3D geometry, lighting, materials, and motion in order to generate a realistic image. In conventional computer graphics, to render an image with desired appearance and effects, it is given all the relevant properties/characteristics. This could be described as the forward method. On the contrary, in caustic design, the properties and characteristics of objects (especially the material surface) are not trivial. The given constraint is the target image to obtain. Therefore, the goal is to obtain its properties and characteristics by observing and inferring the target image. This can be considered the inverse/backward method. The following is the basic loss function explaining how to optimize the parameters: where, : loss function, mean square error of the rendered image and the target : contains elements which can influence the generated image : target image Designed pipeline At first, the target pattern is designed and the forward pass computed to get the synthetic pattern. It's compared to the target pattern and get the loss. The objection is to let the synthetic pattern is similar to the target pattern as much as possible. And then do the back propagation to get the optimized properties need to use in caustic manufacturing. Elements contributing to generated image Appearance (): per-pixel surface appearance is modeled as product of mipmapped texture and per-pixel brightness. Geometry (): assume a 3D scene to be approximated by triangles, parameterized by vertices . Camera (): focal length, the point of view, the center of the camera. There could be more elements, for example albedo and refraction coefficient. General differentiable framework Introduce U as an intermediate variable indicating 2D projected vertex coordinate positions. The gradient of these properties can