Bayesian decision theory, the maximum local mass estimate, and color constancy

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Abstract
Vision algorithms are often developed in a Bayesian framework. Two estimators are commonly used: maximum a posteriori (MAP), and minimum mean squared error (MMSE). We argue that neither is appropriate for perception problems. The MAP estimator makes insufficient use of structure in the posterior probability. The squared error penalty of the MMSE estimator does not reflect typical penalties. We describe a new estimator, which we call maximum local mass (MLM) [10, 26, 65], which integrates the local probability density. The MLM method is sensitive to local structure of the posterior probability, which MAP is not. The new method uses an optimality criterion that is appropriate for perception tasks: it finds the most probable approximately correct answer. For the case of low observation noise, we provide an efficient approximation. We apply this new estimator to color constancy. An unknown illuminant falls on surfaces of unknown colors. We seek to estimate both the illuminant spectrum and the surface spectra from photosensor responses which depend on the product of the unknown spectra. In simulations, we show that the MLM method performs better than the MAP estimator, and better than two standard color constancy algorithms. The MLM method may prove useful in other vision problems as well.

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