Shih_Image_Processing_and_Mathematical_Morpholo.pdf

22 Image Processing and Mathematical Morphology
Example 2.4: Let A and B be as shown below. Draw diagrams for A ° B and
A • B.
2.6 Hit-or-Miss Transformation
The hit-or-miss transformation (HMT) on a binary image A is defi ned as
follows. The structuring element B is a pair of binary images B 1 and B 2, where
B 1 Õ A and B 2 Õ A – (complement of A). The HMT of A by (B 1, B 2), denoted by
A * s (B1, B2), is defi ned as
A * s (B1, B2) = (A � B1) « (A – � B2) (2.23)
Note that erosion is a special case of HMT, where B2 is an empty set. Because
A – ______
� B2 = A � Bˆ 2 , we have
It is equivalent to set difference:
A * s (B ______
1, B2) = (A � B1) « ( A � Bˆ 2 ) (2.24)
A * s (B1, B2) = (A � B1) - (A � Bˆ 2). (2.25)
Morphological HMT is a natural and powerful morphological tool for shape
recognition and the processing of binary images. The generalization of HMT
to process grayscale images has been proposed based on grayscale erosion
[Bloomberg and Maragos 1990; Naegel et al. 2007a]. Naegel et al. [2007b]
applied gray-level HMTs to angiographic image processing. Using grayscale
HMT, Khosravi and Schafer [1996] developed a class of rank-based templatematching
criteria that are multiplier-free and independent of the dc variations
of the image. Schaefer and Casasent [1995] presented a modifi ed version of
morphological HMT for object detection. Raducanu and Grana [2000] proposed
that HMT be based on level sets, calling it the level set HMT (LSHMT),
to obtain a translation invariant recognition tool, with some robustness