| Fundamentally, computer graphics deals with generating a 2D image given the information about a scene. The information consists of the geometry of the scene, lighting conditions, material properties (BRDF) of the objects and viewpoint (camera position and orientation). The generation of the image requires interplay between these components and includes global effects like shadows and inter-reflections referred to as light transport. With the growth of the video game, animation and special effects industries (for example, advertising in sports) the focus has shifted to photo realistic rendering in real time. However, this makes the light transport very complex and relighting intractable in real time.;First, we develop a number of new mathematical results based on the spherical convolution framework. We derive a number of novel identities in the frequency domain. They apply in a number of canonical cases, including single and multiple images of objects under the same and different lighting conditions. We develop more general algorithms for inverse rendering problems, which can directly relight and change material properties by transferring the BRDF or lighting from another object or illumination. We demonstrate this practically with real objects of known geometry in complex lighting. The new identities derived can be used to check the consistency of an image, in order to detect tampering or image splicing.;We then perform a theoretical analysis of locally low dimensional light transport. Blockwise or Clustered Principal Component Analysis (CPCA) is commonly used to achieve real-time rendering of shadows and glossy reflections with precomputed radiance transfer (PRT). We carry out a theoretical analysis of how light transport dimensionality increases with local patch size. We show mathematically that for symmetric patches of area A, the number of basis functions for glossy reflections increases linearly with A, while for simple cast shadows, it often increases as A . These results are confirmed numerically on a number of test scenes. We also carry out an analysis of the cost of rendering, trading off local dimensionality and the number of patches, deriving an optimal block size.;We also conduct a theoretical and empirical analysis of the BRDF in-out factorization. For Phong BRDFs, we obtain analytic results, showing that the number of terms needed grows linearly with the Phong exponent, while the factors correspond closely to spherical harmonic basis functions. More generally, we show that the number of terms is quadratic in the frequency content of the BRDF along the reflected or half-angle direction. This analysis gives clear practical guidance on setting the appropriate number of terms for different materials in the PRT system.;This dissertation performs the theoretical analysis of light transport in different domains such as spherical harmonics and blockwise PCA. In particular, we focus on the efficient compression and acquisition of light transport. The analysis in turn allows us to develop novel algorithms for relighting and inverse rendering, forgery detection, efficient compression and real time rendering of 6D light transport with changing viewpoint and lighting and efficient acquisition of light transport.;Finally, we present an efficient light transport acquisition system. Light transport acquisition requires generating a large number of images of the scene under different lighting and viewpoint conditions. We develop a new framework for capturing light transport data of a real scene, based on the recently developed theory of compressive sensing thus reducing the acquisition and storage costs by more than two orders of magnitude. We develop a novel hierarchical decoding algorithm that improves reconstruction quality by exploiting inter-pixel coherency relations.;We believe that the theoretical analysis presented above plays a critical role in the better understanding of light transport. |
