Morphological pseudo bandpass image decompositions

A morphological bandpass filter would, ideally, strictly limit the sizes of all features in an image to lie between the sizes of two similarly shaped but differently scaled structuring elements. A morphological bandpass decomposition of an image would be a disjoint set of morphological bandpass images with features of increasing size such that the set sums to the original image. Such strict bandpass limitations in size are not possible in general for arbitrary structuring element families. Hence, a true bandpass decomposition is not generally possible. Pseudo bandpass decompositions, in which intraband size limitations are relaxed, are possible and can be useful image analysis tools. This paper describes four pseudo bandpass image decompositions, one of which, the opening spectrum, is relatively well known and three of which are new. They are a decomposition derived from iteration of the tophat transform, a morphological reconstruction of a Euclidean (quasi) granulometry, and a reconstruction of the opening spectrum. Properties of the opening spectrum and the tophat transform are reviewed. The tophat spectrum is defined, some of its properties are deduced, and it is compared to the opening spectrum. The reconstruction-based decompositions are defined and compared to the others. Comparative examples are given and a practical use described.