Optimal morphological representation and restoration of binary images: theory and applications
Dan Schonfeld · 1991
In many image analysis applications there is a need to develop an image representation scheme which contains various important aspects of the image in a compressed form. In this thesis, we introduce the morphological image representation, a general class of morphological representations of binary images, and establish its relationship with some important shape analysis tools (e.g., pattern-spectrum, morphological skeleton, discrete medial axis transform). Thus, we develop a unified theory for the geometrical representation of binary images. The applicability of the morphological image representation, in tasks such as image coding and pattern recognition, depends upon its ability to handle noise in the image and representation domains. The morphological representation, however, is extremely sensitive to the presence of various noise degradations, in both domains. In this thesis, we present a study of the effect of noise degradation, both in the image and the representation domains. The effectiveness of morphological filters in the restoration of noisy, binary images is studied. We prove that the class of alternating sequential filters is a set of parametric, smoothing morphological filters which best preserve the crucial structure of input images, in the least mean difference sense. A minimax estimation procedure is also proposed which allows us to obtain the optimal alternating sequential filter. Subsequently, we demonstrate that the morphological representation of the optimally restored binary image closely resembles the morphological representation of the original image. Finally, we study the effect of the noise degradation in the representation domain. We prove that the generalized reduced morphological skeleton is the optimal morphological representation among a class of invertible morphological representations. The generalized reduced morphological skeleton results in the minimization of an upper-bound on the probability of error in reconstruction of the original image from its degraded morphological representation.