Volterra Filtering of Noisy Images of Curves

Abstract

How should one filter very noisy images of curves? While blurring with a Gaussian reduces noise, it also reduces contour contrast. Both non-homogeneous and anisotropic diffusion smooth images while preserving contours, but these methods assume a single local orientation and therefore they can merge or distort nearby or crossing contours. To avoid these difficulties, we view curve enhancement as a statistical estimation problem in the three-dimensional ( x, y, θ )-space of positions and directions, where our prior is a probabilistic model of an ideal edge/line map known as the curve indicator random field ( cirf ). Technically, this random field is a superposition of local times of Markov processes that model the individual curves; intuitively, it is an idealized artist’s sketch, where the value of the field is the amount of ink deposited by the artist’s pen. After reviewing the cirf framework and our earlier formulas for the CIRF cumulants, we compute the minimum mean squared error ( mmse ) estimate of the cirf embedded in large amounts of Gaussian white noise. The derivation involves a perturbation expansion in an infinite noise limit, and results in linear, quadratic, and cubic (Volterra) cirf filters for enhancing images of contours. The self-avoidingness of smooth curves in ( x, y, θ ) simplified our analysis and the resulting algorithms, which run in O ( n log n ) time, where n is the size of the input. This suggests that the Gestalt principle of good continuation may not only express the likely smoothness of contours, but it may have a computational basis as well.