In the last fifty years, approximately, advances in computers and the availability of images in digital form have made it possible to process and to analyze them in automatic (or semi-automatic) ways. Alongside with general signal processing, the discipline of image processing has acquired a great importance for practical applications as well as for theoretical investigations. Some general image processing references are (Castleman, 1979) (Rosenfeld & Kak, 1982) (Jain, 1989) (Pratt, 1991) (Haralick & Shapiro, 1992) (Russ, 2002) (Gonzalez & Woods, 2006). Mathematical Morphology, which was founded by Serra and Matheron in the 1960s, has distinguished itself from other types of image processing in the sense that, among other aspects, has focused on the importance of shapes. The principles of Mathematical Morphology can be found in numerous references such as (Serra, 1982) (Serra, 1988) (Giardina & Dougherty, 1988) (Schmitt & Mattioli, 1993) (Maragos & Schafer, 1990) (Heijmans, 1994) (Soille, 2003) (Dougherty & Lotufo, 2003) (Ronse, 2005).
Underlying Algebraic Structure And Basic Operations
In morphological processing, the underlying algebraic structure is a complete lattice (Serra, 1988). A complete lattice is a set of elements with a partial ordering relationship, which will be denoted as ≤, and with two operations defined called supremum (sup) and infimum (inf):
The sup operation computes the smallest element that is larger than or equal to the operands. Thus, if a, b are two elements of a lattice, “a sup b” is the element of the lattice that is larger than both a and b, and there is no smaller element that is so.
The inf operation computes the greatest element that is smaller than or equal to the operands.
Moreover, every subset of a lattice has an infimum element and a supremum element.
For sets and gray-level images, these operations are:
Key Terms in this Chapter
Idempotence: A transformation ? is said to be idempotent if, when sequentially applied twice, it does not change the output of the first application, i.e.,? ? = ?.
Multi-Scale Transformation: A transformation that displays some characteristics controllable by means of (at least) a parameter, which is called the size or scale parameter.
Extensivitity: A transformation is extensive when its output is larger than or equal to the input. Anti-extensivity is the opposite concept: a transformation is anti-extensive when its output is smaller than or equal to the input.
Increasingness: A transformation is increasing when it preserves ordering. If ? is increasing, then a = b ? ?(a) = ?(b).
Morphological Filter: An increasing and idempotent transformation.
Duality: The duality principle states that, for each morphological operator, there exists a dual one. In sets, the duality is established with respect to the set complementation operation (see further details in the text).
Image Transformation: An operation that processes an input image and produces an output image.
Lattice: A complete lattice is a set of elements with a partial ordering relationship and two operations called supremum and infimum.