## 1. Introduction

Caustics, or the family of light transport paths that contain a *S*^{+}*D* sub-path, are responsible for some of the most complex and distinctive visual phenomena found in nature. As a component of the rendering equation, however, they often prove challenging to solve owing to the relatively tiny proportion which carry a significant contribution of energy between the emitter and the eye. Consequently, caustics require the use of bi-directional and multi-pass rendering techniques which simulate energy flow both forwards from the light source and backwards from the eye. Among the most popular of these is photon mapping [Jen96b]: a two-pass approach that has seen widespread adoption thanks to its speed, robustness and extensibility.

A typical photon map stores packets of radiant flux at diffuse surfaces which have first undergone specular reflection and refraction. The exitant radiance from the caustic component at any given point can be reconstructed from the photon distribution using a density estimation kernel. This approach capitalises on the correlation between the integral of incident illumination at each point on a surface, transforming the sparse photon point set into a continuous function of illumination. The principal drawback of the photon mapping method is the relatively low fidelity of the cached dataset which generally results in error being introduced into the radiance reconstruction.

Error reduction strategies for density estimation algorithms have been rigorously researched and a large body of work now exists which addresses nuances of the problem specific to the photon mapping framework. Photon relaxation [SJ09] represents a recent contribution that aims to reduce error by directly diffusing the underlying point distribution. Though shown to be effective at smoothing out multi-frequency noise, this approach can significantly degrade intricate, subtle or high frequency detail.

After covering the background literature in Section 2 and introducing the problem in Section 3, this paper:

- • Discusses the advantages of augmenting the standard photon map with information about each photon's initial trajectory. After motivation in Section 4, the chosen parameters are proposed in Section 4.1.
- • Explores the new k-NN query enabled by this approach (Section 4.2) along with the idea that the photon kd-tree can be extended to a higher-dimensional space thereby allowing efficient, parameter-aware querying. These combine to effectively isolate overlapping or interfering illumination.
- • Introduces a new method of identifying anisotropic structure within the photon distribution using principal components analysis (Section 4.3).
- • Presents and discusses the resulting denoised images in Section 5. Due to the low kernel bandwidths made practical by this approach, low render times and low bias are achievable whilst still yielding low noise images.