Қысымдалған жарық сәулелерінің геометриясы немесе каустикалар

Envelope of light rays reflected or refracted by a curved surface/object

In optics, a caustic or caustic network is the envelope of light rays which have been reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. The caustic is a curve or surface to which each of the light rays is tangent, defining a boundary of an envelope of rays as a curve of concentrated light. Rippling caustics are commonly formed when light shines through waves on a body of water. Another familiar caustic is the rainbow. Scattering of light by raindrops causes different wavelengths of light to be refracted into arcs of differing radius, producing the bow.

In computer graphics, most modern rendering systems support caustics. Some of them even support volumetric caustics. This is accomplished by raytracing the possible paths of a light beam, accounting for the refraction and reflection. Photon mapping is one implementation of this. Volumetric caustics can also be achieved by volumetric path tracing. Some computer graphic systems work by "forward ray tracing" wherein photons are modeled as coming from a light source and bouncing around the environment according to rules. Caustics are formed in the regions where sufficient photons strike a surface causing it to be brighter than the average area in the scene. “Backward ray tracing” works in the reverse manner beginning at the surface and determining if there is a direct path to the light source. Some examples of 3D ray traced caustics can be found here. The focus of most computer graphics systems is aesthetics rather than physical accuracy. This is especially true when it comes to real time graphics in computer games where generic pre calculated textures are mostly used instead of physically correct calculations.

Caustic engineering describes the process of solving the inverse problem to computer graphics. That is, given a specific image, to determine a surface whose refracted or reflected light forms this image. In the discrete version of this problem, the surface is divided into several micro surfaces which are assumed smooth, i. e. the light reflected/refracted by each micro surface forms a Gaussian caustic. Gaussian caustic means that each micro surface obey Gaussian distribution. The position and orientation of each of the micro surfaces are then obtained using a combination of Poisson integration and simulated annealing. There have been many different approaches to address the continuous problem. One approach uses an idea from transportation theory called optimal transport to find a mapping between incoming light rays and the target surface. After obtaining such a mapping, the surface is optimized by adapting it iteratively using Snell's law of refraction.

Controlling caustic pattern is rather a challenging problem as very minor changes of the surface will significantly affect the quality of the pattern since light ray directions might be interfered by other light rays as they intersect with and refract through the material. This will lead to a scattered, discontinuous pattern. To tackle this problem, optimal transport based is one of the existing proposed methods to control caustic pattern by redirecting light's directions as it propagates through the surface of a certain transparent material. This is done by solving an inverse optimization problem based on optimal transport. Given a reference image of an object/pattern, the target is to formulate the mathematical description of the material surface through which light refracts and converges to the similar pattern of the reference image. This is done by rearranging/recomputing the initial light intensity until the minimum of the optimization problem is reached.

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.

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.

Introduce U as an intermediate variable indicating 2D projected vertex coordinate positions. The gradient of these properties can be derived by chain rule indirectly. After applying the stochastic gradient descent, the optimal , and could be achieved. Subsequently, these quantities are used to carve or mill the material to generate the target pattern.

One common approach is to utilize the ability to perform differential operations in various deep learning auto differentiation frameworks/libraries such as: Tensorflow, PyTorch, Theano. One more approach is to make use of the OpenDR framework to build a forward graphics model and to automatically obtain derivatives with respect to the model parameters for optimization. As optimization properties are obtained, the target image can be generated. OpenDR provides a local optimization method that can be incorporated into probabilistic programming frameworks. This can be used to solve the problem of caustic.