Hole-spectrum representation and model-based optimal morphological restoration of binary images degraded by subtractive noise

A shape-based image representation grounded on the distribution of holes within an image is developed, and the manner in which this representation can be used to design optimal morphological filters to restore images suffering from subtractive-noise degradation is investigated. The image and noise models are predicated on the existence of some class of shape primitives into which both image and noise can be decomposed (relative to union), and this decomposition is developed within the framework of a general algebraic paradigm for component-based filtering that does not depend on the linear-space structure typically used in spectral representations. Both deterministic and nondeterministic models are considered, and in each case the necessary model constraints are fully explored. Moreover, the type of filters that are naturally compatible with the image-noise models are analyzed. Specifically, optimal morphological filter design is studied in terms of the shape-based hole spectrum (as linear filter design is studied in terms of the frequency spectrum). Various forms of a design algorithm are discussed, the particulars depending on a symmetric-difference error analysis yielding approximate error expressions in terms of the spectral decomposition and the geometry of the underlying shape primitives. Finally, the statistical estimation procedures required for practical implementation of the entire spectrum-filter paradigm are explained.