Optimal reconstructive tau-openings for disjoint and statistically modeled nondisjoint grains

When a binary granular image is sieved by a reconstructive tau-opening, some grains are entirely removed while the remaining grains are fully passed. If desired grains are viewed as composing the signal and undesired grains are viewed as noise (or clutter), then the reconstructive tau-opening acts as a filter and a basic task is to find an optimal filter. The present paper proceeds parametrically by optimally selecting a tau-opening from a Euclidean granulometry {Psi(t)}. The optimal value of t results in minimization of the expected area of the symmetric difference between the pure signal and the output of the reconstructed tau-opening applied to the noisy signal. There is no demand of similarity between filter and image generators. The general theory does not require grains to be nonintersecting. When there is grain intersection, overlapping is statistically modeled in a manner compatible with the assumption that the binary image has resulted from thresholding an image of touching or partially overlapping blobs. The watershed is postulated as a canonical segmentation procedure and filtering is performed relative to the segmented image. Error minimization is more difficult when there is grain intersection because the random geometry of the segmented grains must be described as a function of input geometry, the intersection model, and the effect of the segmentation. (C) 1997 Elsevier Science B.V.