Regularization Uses Fractal Priors

Richard Szeliski

Many of the processing tasks arising in early vision involve the solution of ill-posed inverse problems. Two techniques that are often used to solve these inverse problems are regularization and Bayesian modeling. Regularization is used to find a solution that both fits the data and is also sufficiently smooth. Bayesian modeling uses a statistical prior model of the field being estimated to determine an optimal solution. One convenient way of specifying the prior model is to associate an energy function with each possible solution, and to use a Boltzmann distribution to relate the solution energy to its probability. This paper shows that regularization is an example of Bayesian modeling, and that using the regularization energy function for the surface interpolation problem results in a prior model that is fractal (self-affine over a range of scales). We derive an algorithm for generating typical (fractal) estimates from the posterior distribution. We also show how this algorithm can be used to estimate the uncertainty associated with a regularized solution, and how this uncertainty can be used at later stages of processing.

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