Gaurav Garg,
*Ph.D. Dissertation*, Stanford University, September 2006

**Abstract**

The use of the reflectance fields of real world objects to render realistic looking images is rapidly increasing. The reflectance field describes the transport of light between the light incident on an object and the light exitant from it. This has numerous applications in areas that include entertainment, cultural heritage, digital libraries and space exploration. The central problem with this approach is the lack of fast methods to acquire the reflectance field data. This dissertation addresses this problem and describes a system for acquiring the reflectance field of real world objects that performs many orders of magnitude faster than the previous approaches.

The system models the 8D reflectance field as a transport matrix between the 4D incident light field and the 4D exitant light field. It is a challenging task to measure this matrix because of its large size. However, in some cases the matrix is sparse, e.g. in scenes with little or no inter-reflections. To measure such matrices, this thesis describes a hierarchical technique called dual photography which exploits this sparseness to parallelize the acquisition process. This technique, however, performs poorly for scenes with significant diffuse inter-reflections because in such cases the matrix is dense. Fortunately, in these cases the matrix is often data-sparse. Data-sparseness refers to the fact that sub-blocks of the matrix can be well approximated using low-rank representations. Additionally, the transport matrix is symmetric. Symmetry enables simultaneous measurements from both sides, rows and columns, of the transport matrix. These measurements are used to develop a hierarchical acquisition algorithm that can exploit the data-sparseness by a local rank-1 approximation. This technique, called symmetric photography, parallelizes the acquisition for dense but data-sparse transport matrices.

In the process, this thesis introduces the use of hierarchical tensors as the underlying data structure to represent data-sparse matrices, specifically through local rank-1 factorizations of the transport matrix. Besides providing an efficient representation for storage, it enables fast acquisition of the approximated transport matrix and fast rendering of the images from the captured matrix. The prototype acquisition system consists of an array of mirrors and a pair of coaxial projector and camera controlled by a computer. The effectiveness of the system is demonstrated with scenes rendered from reflectance fields that were captured by this system. In these renderings one can change the viewpoint as well as relight objects using arbitrary incident light fields.

**Dissertation**