Theory and applications of steerable functions
David J. Heeger, Patrick C. Teo · 1998
A function is called steerable if transformed versions of the function can be expressed using linear combinations of a fixed set of basis functions. Steerability is a very general and often desirable property. As a result, steerable functions have recently been applied to an assortment of problems in image processing, computer vision, and computer graphics. In this dissertation, we propose a framework, based on Lie group theory, for studying and constructing functions steerable under any smooth transformation group. We argue that Lie theory is the appropriate mathematical tool for analyzing the properties of these steerable functions. This position is supported by the fact that all existing analytical approaches to steerability can be consistently explained within the framework. The design of a suitable set of basis functions given any arbitrary steerable function is one of the main problems concerning steerable functions. To this end, we have developed two very different algorithms. The first algorithm is a symbolic method that can be implemented in any symbolic package. This algorithm derives the minimal set of basis functions automatically given an arbitrary steerable function. The second algorithm addresses two practical considerations: approximate steerability and local steerability. In practice, functions that need to be steered might not be steerable with a finite number of basis functions. Moreover, it is often the case that only a small subset of transformations within the group of transformations needs to be considered. In response to these two concerns, the second algorithm computes the optimal set of k basis functions to steer an arbitrary function under a subset of the group of transformations. Lastly, we demonstrate the usefulness of steerable functions in a variety of applications. In particular, we present five applications that use steerable functions: (1) continuum approximation in modeling human vision (a method of approximating an infinite number of interacting mechanisms in a model), (2) the design of optimal steerable filters for gradient-based motion estimation, (3) efficient linear re-rendering of synthetic scenes under changes in illumination, (4) the construction of invariants from steerable filters, and (5) the application of steerable functions to discrete sets of points and lines.