Optimal and Adaptive Design of Reconstructive Granulometric Filters

Abstract

The present paper provides a brief overview of the optimization and adaptive theories of reconstructive binary granulometric filters. When the generating sets of a Euclidean granulometry are compact and convex, the granulometry represents a parameterized family of-openings, each being a union of openings with structuring elements scalar multiplied by t > 0. The concept is generalized to a multivariate granulometry by allowing each structuring element to be parameterized by a vector, thereby generating a family of-openings parameterized by the vectors. Each such multivariate granulometry induces a reconstructive multivariate granulometry defined by passing each connected component of an image that is not eliminated by the original granulometry and not passing each connected component that is fully eliminated. Given ideal and observed (corrupted) random sets, the optimization problem for image restoration by multivariate reconstructive gran-ulometries is find to a value of the parameter vector that minimizes the expected error (expected symmetric-difference area) between the ideal and the filtered observation. The corresponding adaptive filter problem is to allow the parameter vector to transition during a scanning process in such a way as to produce desired results and to study the steady-state properties of the multi-dimensional Markov chain defined by the state space of the now random parameter vector. A second approach to granulometric filtering is to decompose an image by means of the spectral components produced by the granulometry and then pass some subfamily of the spectral components. Given ideal and observed image processes, the optimization for these bandpass filters is to pass spectral components in a manner that minimizes restoration error. One can also proceed adaptively by allowing the parameters defining the passbands to adapt, thereby producing a multi-dimensional state space for a Markov chain whose steady-state properties characterize adaptivity. These four optimization and adaptation problems are discussed in the present paper, with emphasis how they yield filters for restoration in the signal-union-noise model.