Clara Cuciurean-Zapan

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

Abstract When a binary granular image is sieved by a reconstructive τ-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 τ-opening acts as a filter and a basic task is to find an optimal filter. The present paper proceeds parametrically by optimally selecting a τ-opening from a Euclidean granulometry {Ψt}. 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 τ-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.

Optimal openings for overlapping signal and noise grains

Optimal openings are considered for extraction of signal from noise in the random binary union-noise model. Disjointness of signal and noise is not assumed, nor are grains within the signal or within the noise assumed to be disjoint. There is a constraint on the overlapping, but this reflects the manner in which binary granular images are derived from gray-scale images of touching objects. The method assumes that the degraded image is segmented by the binary watershed algorithm and that an optimal opening by reconstruction must be found to remove segmented noise grains while passing segmented signal grains.

Behavior of adaptive digital erosions

Design of statistically optimal erosion-based filters is problematic due to the complexity of the search process. Specialized search techniques and constraints on optimality are used to mitigate the full search. Adaptation of structuring elements has also ben employed. The present paper looks at the behavior of an adaptive filter relative to the actual optimal filter for a single erosion in two models, signal-union-noise and dilation. It does so in the context of state transitions, where filter states are stacks that determine the structuring element upon thresholding.

Computational morphology and representation of operators between complete lattices

The representations of translation-invariant mappings in the context of computational morphology are in terms of elementary binary erosions and dilations. The general representations in the context of complete lattices are in terms of abstract dilations and antidilations. The present paper considers the relationship between the specialized computational and general lattice representations. The computational and lattice-based representations are directly demonstrated to be equivalent in the computational setting.