Photon mapping is the most widely used algorithm for generating high quality, global illumination images. It is based on simulating the light transport of photons (formally using a MonteCarlo method). However, as it is commonly implemented, it has a very small overestimation bias, of 1/k. The bias was discovered by modelling the photon mapping algorithm as a stochastic process, and noticing that the search for the nearest impacts can be modelled using Order Statistics [Garcia12] [Garcia14]. The influence of various stratification strategies in the bias was also modelled [Hartmann20], as well as the variance of photon mapping with common filtering kernels [Gan18].

The student will work within the theoretical framework described in the references above, and study the variance of photon mapping in the presence of simultaneous stratification and filtering kernel strategies.

**Description of the photon mapping algorithm**

The algorithm can be divided in the following phases

- Photon tracing from the light sources.
- The photons interact with the scene and are reflected, refracted or absorbed.
- The photon interaction points are stored for later use in a data structure called "photon map".
- Raytracing from the eye.
- For each pixel, the visible point in the scene is found.
- Multiple rays from that point are created, and the contribution of each (see below) is integrated in a final colour
- Photon map query
- Each ray from the previous step will possibly intersect the scene geometry at a point.
- A photon map query searches the photon map data structure for k photon impacts near this point, and calculates the flux density at the point.

**Literature:**

- [Garcia12]: Description and solution of an unreported intrinsic bias in photon mapping density estimation with constant kernel. R. García, C. Ureña, M. Sbert. Computer Graphics Forum Volume 31 (2012), number 1, pp 33-41
- [Garcia14]: Overestimation and underestimation biases in photon mapping with non-constant kernels. Garcia-Hernandez, R, Ureña, C, Poch, J, Sbert, M. IEEE Transactions on Visualization and Computer Graphics, Volume 20, Issue 10. Pp. 1441-1450
- [Gan18]: Variance of photon mapping density estimation. Z. Gan, Bachelor Thesis, LMU, 2018
- [Hartmann20]: Theoretical study of photon mapping with stratification. K. Hartmann, Bachelor Thesis, LMU, 2020.

- Study of the state of the art in the following areas: photon mapping, order statistics, montecarlo stratified algorithms.
- Extension of the codes which calculate bias and variance for the usecases in the references above to include the new usecases.
- Extension of the mathematical framework in the references above to take into account the new usecases.
- Study of the 1D, 2D (surface illumination) and 3D (volumetric effects) cases

- Strong background in Mathematics / Statistics

**Aufgabensteller:**

Prof. Dr. D. Kranzlmüller

**Dauer der Bachelor-Arbeit:**
3 Monate

**Anzahl Bearbeiter:**
1

**Betreuer:**