Perturbation Methods for Image Synthesis

Abstract

This thesis presents new mathematical and computational tools for applying perturbation methods to image synthesis. The key idea is to characterize families of closely related optical paths by expanding a given path into a high-dimensional Taylor series. Our methods are based on the closed-form expressions for linear and higher-order approximations of specularly reflected rays of light, which are derived from the Implicit Function Theorem and Fermat's Variation Principle. The expressions hold for general multiple-bounce specular paths and provide a mathematical foundation for exploiting path coherence in incremental rendering. To illustrate their use, new algorithms based on the perturbation formulas are developed for several applications, including fast approximation of specular reflections on curved surface and direct caustic contour generation. First, path Jacobians are introduced to formulate the linear perturbation of a ray path connecting a fixed point and a perturbed point through any number of intermediate reflection points. These Jacobian matrices are the derivatives of the reflection points with respect to the perturbed endpoint. A recurrence relation is derived to compute the closed-form expressions of path Jacobians for a general multiple-bounce reflection path, even when the new reflection path cannot be expressed in a closed form. Next, the concept of path Jacobian is generalized to tensors of third order, which we call path Hessians. A simiar recurrence formula for path Hessians is derived from tensor calculus. The expressions for path Jacobians and path Hessians hold for any reflection path involving implicitly-defined reflective surfaces. Based on the close-form expressions for the path Jacobian and the path Hessian, a reflection path can be expanded as a Taylor series up to the second order. This perturbation formula gives rise to a fast and accurate interpolation scheme for the unknown reflection paths nearby. Finally, new algorithms are presented for some applications that will benefit from our perturbation method. In one application, a new approach is presented for interactively approximating specular reflections in arbitrary curved surfaces. The technique is quite general as it employs local perturbations to interpolate point samples and is applicable to any smooth implicitly-defined reflecting surface that is equipped with a ray-intersection procedure. After ray tracing a sparse set of reflection paths with respect to a given vantage point and static reflecting surfaces, the algorithm rapidly approximates reflections of arbitrary points in 3-space by expressing them as perturbations of nearby points with known reflections. The reflection of each new point is approximated to second-order accuracy using the Taylor expansions of one or more nearby reflection paths. After preprocessing, the approach is fast and accurate enough to compute specular reflections of tessellated objects in arbitrary curved surfaces at interactive rates using standard graphics hardware. In another application, the path Jacobian formula is used to directly compute caustic contours on a plane formed by a given point light source and an implicitly-defined specular surface. Caustic irradiance for a given point on the plane is computed using wavefront tracing and some classical results from differential geometry. Using the closed-form expression for path Jacobian, an analytic formula for the gradient of caustic irradiance is also derived. The level curves with constant caustic irradiance on the plane, caustic contours, are traced by numerically solving an isolux ordinary differential equation.

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