Bayesian modeling of uncertainty in low-level vision

The need for error modeling, multisensor fusion, and robust algorithms is becoming increasingly recognized in computer vision. Bayesian modeling is a powerful, practical, and general framework for meeting these requirements. This article develops a Bayesian model for describing and manipulating the dense fields, such as depth maps, associated with low-level computer vision. Our model consists of three components: a prior model, a sensor model, and a posterior model. The prior model captures a priori information about the structure of the field. We construct this model using the smoothness constraints from regularization to define a Markov Random Field. The sensor model describes the behavior and noise characteristics of our measurement system. We develop a number of sensor models for both sparse and dense measurements. The posterior model combines the information from the prior and sensor models using Bayes' rule. We show how to compute optimal estimates from the posterior model and also how to compute the uncertainty (variance) in these estimates. To demonstrate the utility of our Bayesian framework, we present three examples of its application to real vision problems. The first application is the on-line extraction of depth from motion. Using a two-dimensional generalization of the Kalman filter, we develop an incremental algorithm that provides a dense on-line estimate of depth whose accuracy improves over time. In the second application, we use a Bayesian model to determine observer motion from sparse depth (range) measurements. In the third application, we use the Bayesian interpretation of regularization to choose the optimal smoothing parameter for interpolation. The uncertainty modeling techniques that we develop, and the utility of these techniques in various applications, support our claim that Bayesian modeling is a powerful and practical framework for low-level vision.