A Bayesian Reflection on Surfaces

TL;DR

This paper introduces a Bayesian framework for continuous-basis field inference, enabling maximally informative, scalable, and efficient surface reconstruction from measurements like images, with theoretical and practical advancements over traditional methods.

Contribution

It presents the Generalized Kalman Filter, a novel Bayesian inference method for continuous fields, addressing information accuracy, memory tradeoffs, and multiscale updates.

Findings

Effective inference of continuous surfaces from image data

Theoretical justification for multigrid methods

Enhanced information preservation in probabilistic surface modeling

Abstract

The topic of this paper is a novel Bayesian continuous-basis field representation and inference framework. Within this paper several problems are solved: The maximally informative inference of continuous-basis fields, that is where the basis for the field is itself a continuous object and not representable in a finite manner; the tradeoff between accuracy of representation in terms of information learned, and memory or storage capacity in bits; the approximation of probability distributions so that a maximal amount of information about the object being inferred is preserved; an information theoretic justification for multigrid methodology. The maximally informative field inference framework is described in full generality and denoted the Generalized Kalman Filter. The Generalized Kalman Filter allows the update of field knowledge from previous knowledge at any scale, and new data, to new knowledge at any other scale. An application example instance, the inference of continuous surfaces from measurements (for example, camera image data), is presented.