The curve indicator random field
Jonas August, Steven W. Zucker · 2001
Can the organization of local image measurements into curves be directly related to natural image structure? By viewing curve enhancement as a statistical estimation problem, we suggest that it can. In particular, the classical Gestalt perceptual organization cues of proximity and good continuation—the basis of many current curve enhancement systems—can be statistically measured in images. As a prior for our estimation approach we introduce 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 artists pen. The explicit formulation of the CIRF allows the calculation of tractable formulas for its cumulants and moment generating functional. A novel aspect of the CIRF is that contour intersections can be explicitly incorporated. More fundamentally, the CIRF is a model of an ideal edge/line map, and therefore provides a basis for rigorously understanding real (noisy, blurry) edge/line measurements as an observation of the CIRF. This model therefore allows us to derive nonlinear filters for enhancing contour structure in noisy images. In particular, we first introduce linear, quadratic, and cubic Volterra filters for rapidly estimating the CIRF embedded in large amounts of noise. Then we derive coupled integro-elliptic reaction-diffusion-convection partial differential equations for estimating the posterior mean of the CIRF. Example computations illustrate a striking degree of noisy cleaning and contour enhancement. But the framework also suggests we seek in natural images those correlations that were exploited in deriving filters. We present the results of some edge correlation measurements that not only confirm the presence of Gestalt cues, but also suggest that curvature has a role in curve enhancement. Markov process model for contour curvature is therefore introduced, where it is shown that its most probable realizations include the Euler spiral, a curve minimizing changes in curvature. Contour computations with curvature highlight how our filters are curvature-selective, even when curvature is not explicitly measured in the input.