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106 Chapter 3 ■ Digital Morphology

There is also a way, similar to that of global erosion, to encode all possible
openings as one grey-level image, and all possible closings can be computed
at the same time. First, as in global erosion, the distance map of the image
is found. Then all pixels that do not have at least one neighbor nearer to the
background and one neighbor more distant are located and marked: these will
be called nodal pixels. Figure 3.14c shows the nodal pixels associated with the
object of Figure 3.14a. If the distance map is thought of as a three-dimensional
surface where the distance from the background is represented as height, every
pixel can be thought of as being the peak of a pyramid having a standardized
slope. Peaks not included in any other pyramid are the nodal pixels. One way
to locate nodal pixels is to scan the distance map, looking at all object pixels;
find the minimum and maximum value of all neighbors of the target pixel, and
compute MAX-MIN. If this value is less than the maximum possible, which is
2 when using 8-distance, the pixel is nodal.
To encode all openings of the object, a digital disk is drawn centered at each
nodal point. The pixel values and the extent of the disk are equal to the value
of the nodal pixel. If a pixel has already been drawn, it will take on the larger
of its current value or the new one being painted. The resulting object has
the same outline as the original binary image, so the object can be re-created
from the nodal pixels alone. In addition, the grey levels of this globally opened
image represent an encoding of all possible openings. As an example, consider
the disk shaped object in Figure 3.15a and the corresponding distance map of
Figure 3.15b.
There are nine nodal points: four have the value 3, and the remainder have
the value 5. Thresholding the encoded image yields an opening having depth
equal to the threshold.
All possible closings can be encoded along with the openings if the distance
map is changed to include the distance of background pixels from an object.
Closings are coded as values less than some arbitrary central value (say, 128)
and openings are coded as values greater than this central value.
As a practical case, consider an example from geology. To a geologist, the
pores that exist in oil-bearing (reservoir) rock are of substantial interest; oil
resides in these pores. Porosity of reservoir rock can be measured by slicing
the rock into thin sections after filling the pores with a colored resin. The
slices show microscopic features of grain and pore space, and one method of
characterizing the shapes of the pores is to examine the differences between
openings of increasing depth. Openings of higher orders are smoother than
those of lower orders, and the difference between the order N opening and the
order N + 1 opening is referred to as the roughness of order N. The histogram of
the pixels in each opened pore image by order of roughness yields a roughness
spectrum, which has been extensively applied to the classification of pore shape.