Efficient freeform lens optimization for computational caustic displays

Abstract

Phase-only light modulation shows great promise for many imaging applications, including future projection displays. While images can be formed efficiently by avoiding per-pixel attenuation of light most projection efforts utilizing phase-only modulators are based on holographic principles which rely on interference of coherent laser light and a Fourier lens. Limitations of this type of an approach include scaling to higher power as well as visible artifacts such as speckle and image noise. We propose an alternative approach: operating the spatial phase modulator with broadband illumination by treating it as a programmable freeform lens. We describe a simple optimization approach for generating phase modulation patterns or freeform lenses that, when illuminated by a collimated, broadband light source, will project a pre-defined caustic image on a designated image plane. The optimization procedure is based on a simple geometric optics image formation model and can be implemented computationally efficient. We perform simulations and show early experimental results that suggest that the implementation on a phase-only modulator can create structured light fields suitable, for example, for efficient illumination of a spatial light modulator (SLM) within a traditional projector. In an alternative application, the algorithm provides a fast way to compute geometries for static, freeform lens manufacturing.

High Brightness HDR Projection Using Dynamic Freeform Lensing

“…Typical per-frame computation times were previously on the order of 20min or more [Damberg and Heidrich 2015], while the Fourier version in Algorithm 1 takes approximately 0.6s at the same resolution (256 × 128) on a Core i5 desktop computer, a speedup of approximately 2,000 times. The conversion to Fourier domain solves also results in operations that are more friendly for parallel GPU implementation.…”

“…In our previous work Damberg and Heidrich [2015], a simplified new formulation is derived in which we optimize directly for the phase function (i.e., the shape of the wavefront in the lens plane) or, equivalently, the lens shape, without a need for a subsequent integration step. This is made possible by a new parameterization of the problem that allows expression of the optimization directly in the lens plane rather than the image plane.…”

“…In this section, we briefly summarize the basic freeform lensing algorithm [Damberg and Heidrich 2015]. The new algorithmic contributions will then be presented in Section 4.…”

“…Our final algorithm is shown in Algorithm 1 and is identical to the proximal point method except that the b image used by the proximal operator is updated at every iteration using the warping procedure from our previous work [Damberg and Heidrich 2015]. After precomputing the Fourier transforms of f ∇ 2 , each iteration of the algorithm can be implemented with an image warping, some componentwise operations, and a forward/inverse Fourier transform.…”

“…This work is based on our earlier work on generating freeform lenses or goal-driven caustics using common approximations in optics to directly optimize the phase modulation pattern or lens shape of a freeform lens Damberg and Heidrich [2015]. The method is briefly reviewed in Section 3.2.…”