Numerical Conditioning Problems and Solutions for Nonparametric i.i.d. Statistical Active Contours

In this paper, we propose an active contour model based on nonparametric independent and identically distributed (i.i.d.) statistics of the image that can segment an image without any a priori information about the intensity distributions of the region of interest or the background. This is not, however, the first active contour model proposed to solve the segmentation problem under these same assumptions. In contrast to prior active contour models based on nonparametric i.i.d. statistics, we do not formulate our optimization criterion according to any distance measure between estimated probability densities inside and outside the active contour. Instead, treating the segmentation problem as a pixel-wise classification problem, we formulate an active contour to minimize the unbiased pixel-wise average misclassification probability (AMP). This not only simplifies the problem by avoiding the need to arbitrarily select among many sensible distance measures to measure the difference between the probability densities estimated inside and outside the active contour, but it also solves a numerical conditioning problem that arises with such prior active contour models. As a result, the AMP model exhibits faster convergence with higher accuracy and robustness when compared to active contour models previously formulated to solve the same nonparametric i.i.d. statistical segmentation problem via probability distances. To discuss this improved numerical behavior more precisely, we introduce the notion of “conditioning ratio” and demonstrate that the proposed AMP active contour is numerically better conditioned (i.e., exhibits a much smaller conditioning ratio) than prior probability distance-based active contours.