COMPLEX LINEAR MORPHOLOGICAL TRANSFORMATIONS

Historical Notes from 1997:

"Morphological Transformation" represents the mapping of one shape into another. The mapping can be subtle or gross in terms of the difference between the original image and the transformed image. For example: An area of study known as "mathematical morphology" uses a structuring element to logically grow or shrink binary objects. These operations are logically defined and can be combined sequentially into sophisticated and useful image processing operations. Typically, structuring elements have been 3x3 binary elements. However, if larger structuring elements with more complicated shapes are used, one step operations can be performed at a higher level than the more simple "shrinking" and "growing" operations. This is referred to as high level morphological transformation. Another apect of of morphology is the ability to perform logical operations on gray level images. One approach to this has been to map gray level to a third dimension and perform 3-D mathematical morphology on the binary volume. Another approach, which has been taken by the optical community, is to use optical correlation followed by thresholding to perform the gray level operations. In this case the input is the gray level image and the structuring element is a linear filter. We have extended this correlation based research to include complex spatial domain structuring elements in the form of linear filters. In addition to this, several filters are used in a filter bank to attain a more robust performance. We refer to this approach as Complex Linear Morphology (CLM) and its application is high level gray scale morphology.

EXAMPLE OF CLM

Michael E. Lhamon, Laurence G. Hassebrook and Jyoti Chatterjee "Complex Spatial Images for Multi-Parameter Distortion-Invariant Optical Pattern Recognition and High Level Morphological Transformations," SPIE Proceedings, (April 1996).

Figure 1: Target class, space shuttle.

Figure 2: Magnitude of Superimage formed from shifted target images, each weighted by a complex exponential. The output dot pattern is a "T" in this case.

Figure 3: Phase of Superimage in Fig. 2.

Figure 4: Magnitude of the filter impulse response. Superimage is rotated and weighted by complex exponential to form the impulse response of a rotation-invariant filter.

Figure 5: Phase of the filter impulse response.

Figure 6: Clutter or non-target class Hubble telescope.

Figure 7: Input test image with rotated, edge enhanced, target and clutter class images.

Figure 8: Filter System output given Fig. 7 is input scene.

Figure 9: Thresholded result of Fig. 8.