Start Date: 2018-05-01 End Date: 2021-11-23
In illumination optics free-form optical designs are now frequently used. To design these optical components Optimal Transport (OT) based methods are becoming more and more popular. But in these methods it is assumed that the source has an infinitesimal size, which is not realistic and is solved in practice with tedious iterative methods.
Significant progress has been recently achieved on the numerical resolution of OT problems. Free-form (FF) Reflectors for idealized collimated source of illumination can be computed rapidly with resolutions of millions of points. More realistic point source and extended illumination resolution, which are important for applications, are still open as the available OT solvers are unable to deal efficiently with the more complicated structure of these problems. It is well known that these more complicated problems can be relaxed into huge linear program. A numerical approach called Sinkhorn iterations and popularized recently by Cuturi for OT relies on the “entropic” regularization of these linear program and alternate projection solver. Using GPU parallelization this again allows to solve regularized OT problems with millions of points.
Within the research project the following objectives will be addressed:
– the development of numerical algorithms for the finite source OT problem
– the evaluation of the Sinkhorn iterations method
– the implementation of the algorithms on GPU
We expect that the research results in new methods to iteratively move from a point source to an extended source efficiently together with error estimations on the achieved target distribution compared to the desired target distribution. All obtained results will be verified with available commercial software tools.
In this work, we address the “freeform optics” inverse problem of designing a reflector surface mapping a prescribed source distribution of light to a prescribed target far-field distribution, for the point light source and the extended light source.
When the source is a point source, the light distribution has support only on the optics ray directions. In this setting, the inverse problem is well-posed for arbitrary source and target probability distributions. It can be recast as an optimal transport problem and is a classic example of an optimal transport problem with a non-euclidean displacement cost. We explore the use of entropic Optimal Transport and the associated Sinkhorn algorithm to solve it numerically. As the reflector modeling is based on the Kantorovich potentials, several questions arise. First, on the convergence of the discrete entropic approximation and here we follow the recent work of Berman and in particular the imposed discretization requirements therein. Secondly, the correction of the bias induced by the entropic Optimal Transport using the recent notion of Sinkhorn divergences is shown to be necessary to achieve satisfactory results.
For the point source problem, we discuss the necessary mathematical and numerical tools needed to produce and analyze the obtained numerical results. We find that Sinkhorn algorithm may be adapted to the resolution of the point source to far-field reflector problem.
We are not aware of any similar mathematical formulation in the extended source case: i.e. the source has an “ ´etendue” with support in the product space: physical domain-ray directions. We propose to leverage the well-posed variational formulation of the point source problem to build a smooth parameterization of the reflector and the map modeling the reflection. Under this parametrization, we can construct a smooth cost function to optimize for the best solution in this class of reflectors. Both steps, the parameterization and the cost function, are related to entropic optimal transport distances.
We also take advantage of recent progress in the optimization techniques and the efficient implementations of Sinkhorn algorithm to perform a numerical study