A High-Speed CGH Calculation Method for Mirror Images on Bézier Surfaces using Optical Path Length Minimization

TL;DR

This paper tackles the challenge of rendering reflections from free-form Bézier mirrors in CGH, where existing methods either cost excessive computation or fail to handle curved surfaces. It introduces a Fermat's-principle–based formulation that minimizes the optical-path length $L(s,t)$ on the Bézier surface and solves for the reflection point with Newton's method, using proximity-based initialization and GPU-accelerated ray tracing. The approach yields substantial speedups (over 100× versus mirror-subdivision and over 2,800× versus Bézier clipping) while maintaining high fidelity, and it naturally extends to multiple reflections. Experimental validations with cylindrical and parabolic mirrors, motion parallax, and multi-mirror scenarios demonstrate accurate image depth, parallax, and scalable computation, enabling potential real-time holographic rendering of complex scenes.

Abstract

Rendering reflections in curved mirrors is crucial for enhancing the realism in computer-generated hologram (CGH), yet it poses a fundamental challenge due to the unique computational principles of CGH. Conventional methods using Bézier clipping are computationally prohibitive, and a previously proposed mirror surface subdivision method suffered from the computation time increasing with mirror curvature. To address these limitations, this paper proposes a novel calculation method based on Fermat's principle that directly and efficiently determines the reflection point by minimizing the optical path length from a point light source to a hologram pixel via the mirror surface, using Newton's method for optimization. Experimental results demonstrate that this method significantly reduces computation time compared to previous approaches. Furthermore, it enables the rendering of multiple reflections from several mirrors, a capability that was challenging for conventional techniques.

Figures (24)

• Figure 1: Conceptual diagram of CGH calculation using the point-based method. The light waves here propagate from a point light source $(x_i, y_i, z_i)$ on the object surface to each pixel $(x, y)$ on the hologram plane.
• Figure 2: Conceptual diagram of hidden surface removal in the point-based method that applies ray tracing. A ray is traced from a point light source placed on the object in the background (Object 1) towards a pixel on the hologram plane. A determination is made as to whether this ray intersects with the occluding object in front (Object 2), and the visibility coefficient $c_i$ is determined. If the ray intersects with the occluding object, $c_i=0$ and light wave propagation calculations from that point light source are not carried out. In contrast, if the ray is not occluded and reaches the pixel, $c_i=1$ and light wave propagation calculations are carried out.
• Figure 3: Conceptual diagram of specular reflection calculations on a planar mirror. First, a point light source on the real object is replicated as a specular image point, at a position symmetrical to the planar mirror. Next, a determination is made as to whether the reflection point calculated from the line connecting this specular image point and the pixel is within the range of the mirror. If this point is outside the range of the mirror, no calculations to propagate light waves to that pixel are carried out. Furthermore, if the light path from the point light source via the reflection point on the mirror surface to the pixel is blocked by another object, no calculations are carried out, thereby accurately rendering only physically valid mirror images.
• Figure 4: Example of a quadratic $\times$ quadratic Bézier surface constructed using $3 \times 3$ control points $F_{ij}$. A Bézier surface is constructed as an $(n-1) \times (m-1)$ degree surface depending on the number of control points $n$ and $m$. Any point $P(s,t)$ on the surface is uniquely determined by two parameters $s$ and $t$ (usually $0\le s,t \le 1$) along the directions indicated by the arrows.
• Figure 5: Conceptual diagram of specular reflection calculation on a curved mirror using Bézier clipping. First, the error between the angle of incidence and the angle of reflection of the light ray from the point light source $P_L$ at the candidate point is calculated. Then, Bézier clipping is used to modify the candidate point in a direction that reduces the error, and the angle error is calculated again. This process is repeated to find a highly accurate reflection point $P_M$. After that, the distance from the point light source $P_L$ to the reflection point $P_M$ is calculated, and the specular image point $P^{'}_L$ is calculated by extending the vector from the viewpoint $P_H$ to the reflection point $P_M$ by that distance.