Information and Communication Technology Seminar, Vol. 1 No. 1, August 2005
ISSN 1858-1633 2005 ICTS 6
Fig. 8. Granulometry and size density function
The r-times magnification of B, denoted rB, is usually defined in the context of mathematical
morphology as follows:
where {0} denotes a single dot at the origin. Let us consider a differentiation of the size
distribution function. In the case of discrete sizes, it is equivalent to the area differences of the image pairs
corresponding to adjacent sizes in X
B
, X
2B
, X
3B
, .... For example, the area difference between X
2B
and X
3B
corresponds to the part included inX
2B
but excluded fromX
3B
, that is, the part whose size is exactly 2. The sequence of the areas corresponding to each size
exactly, derived as the above, is called pattern spectrum [13], and the sequence of the areas relative
to the area of the original object is called size density function [14]. An example of granulometry and size
density function is illustrated in Fig. 8.
Size distribution function and size density function have similar properties to probability distribution
function and probability density function, respectively, so that such names are given to these functions.
Similarly to probability distributions, the average and the variance of size of objects in an image can be
considered. Higher moments of a size distribution can be also defined, which are called granulometric
moments, and image objects can be characterized
using these moments [14–16]
.
4.2. Application to texture analysis
As described in Sec. 2., morphological opening is a regeneration of an image by arranging the structuring
element, and removes smaller white regions in binary case or smaller regions composed of brighter pixels
than its neighborhood in gray scale case than the structuring element. Thus opening is effective for
eliminating noisy pixels that are brighter than its neighborhood.
Since opening generates the resultant image by an arrangement of the structuring element, the hape and
pixel value distribution of the structuring element directly appear in the resultant image. It causes
artifacts if the shape and pixel value distribution are not related to the original image.
This artifact can be suppressed by using a structuring element resembling the shape and pixel
value distribution contained in the original image. Such structuring element cannot be defined generally,
but can be estimated for texture images, since texture is composed an arrangement of small objects
appearing repetitively in the texture.
We explain in this subsection a method of developing the optimal artifact-free opening for noise
removal in texture images [17]. This method estimates the structuring element which resembles small objects
appearing repetitively in the target texture. This is achieved based on Primitive, Grain, and Point
Configuration PGPC texture model, which we have proposed to describe a texture, and an optimization
method with respect to the size distribution function.
The optimal opening of suppressing the artifacts is achieved by using the estimated structuring element.
In the case of noise removal, the primitive cannot be estimated by the target image itself, since the original
uncorrupted image corresponding to the target image is unknown. This problem is similar to that of the
image processing by learning, which estimates the optimal filter parameters by giving an example of
corrupted image and its ideal output to a learning mechanism [18–20]. In the case of texture image,
however, if a sample uncorrupted part of the texture similar to the target corrupted image is available, the
primitive can be estimated from this sample, since the sample and the target image are different realization
but have common textural characteristics.
4.2.1. PGPC texture model and estimation of the optimal structuring element: The PGPC
texture model regards a texture as an image composed by a regular or irregular arrangement of objects that
are much smaller than the size of image and resemble each other. The objects arranged in a texture are called
grains, and the grains are regarded to be derived from one or a few typical objects called primitives.
We assume here that the grains are derived from one primitive by homothetic magnification.We also
assume that the primitive is expressed by a structuring element B, and let X be the target texture image. In
this case, X
rB
is regarded as the texture image composed by the arrangement of rB only. It follows
that rB − r + 1B indicates the region included in the
arrangement of rB but not included in that of r+1B. Consequently, X
rB
−X
r+1B
is the region where r-size grains are arranged if X is expressed by employing an
Mathematical Morphology and Its Application – Akira Asano
ISSN 1858-1633 2005 ICTS 7
arrangement of grains which are preferably large magnifications of the primitive. The sequence X
− X
B
, X
B
− X
2B
, . . . , X
rB
− X
r+1B
, . . . , is the decomposition of the target texture to the arrangement of the grains of
each size. Since the sequence can be derived by using any
structuring element, it is necessary to estimate the appropriate primitive that is a really typical
representative of the grains. We employ an idea that the structuring element yielding the simplest grain
arrangement is the best estimate of the primitive, similarly to the principle of minimum description
length MDL. The simple arrangement locates a few number of large magnifications for the expression of a
large part of the texture image, contrarily to the arrangement of a large number of small-size
magnifications. We derive the estimate by finding the structuring element minimizing the integral of 1
− Fr, where Fr is the size distribution function with
respect to size r. The function 1 − Fr is 0 for r = 0
and monotonically increasing, and 1 for the maximum size required to compose the texture by the
magnification of this size. Consequently, if the integral of 1
−Fr is minimized as illustrated in Fig. 9, the sizes of employed magnifications concentrate to
relatively large sizes, and the structuring element in this case expresses the texture using the largest
possible magnifications. We regard this structuring element as the estimate of primitive.
We estimate the gray scale structuring element in two steps: the shape of structuring element is
estimated by the above method in the first step, and the gray scale value at each pixel in the primitive
estimated in the first step is then estimated. However, if the above method is applied to the gray scale
estimation, the estimate often has a small number of high-value pixel and other pixels whose values are
almost zero. This is because the umbra of any object can be composed by arranging the umbra of one-pixel
structuring element, as illustrated in Fig. 10. This is absolutely not a desired estimate. Thus we modify the
above method, and minimize 1
− F1, i. e. the residual area of X
B
instead of the above method. In this case, the composition by this structuring element
and its magnification is the most admissible when the residual area is the minimum, since the residual region
cannot be composed of even the smallest magnification.
The exploration of the structuring element can be performed by the simulated annealing, which iterates a
modification of the structuring element and find the best estimate minimizing the evaluation function
described in the above [21].
Fig. 9. Function 1 − Fr. Size r is actually discrete for digital
images.aFunction and its integral. b Minimization of the integral.
Fig. 10. Any object can be composed by arranging one-pixel structuring element.
4.2.2. Experimental results: Figures 11 and 12 show the example experimental results of noise
removal using the estimated primitives as the structuring elements. All images contains 64×64 8-bit
gray scale pixels. In each example, the gray scale primitive shown in b is estimated for the example
image a. Each small square in b corresponds to one pixel in the primitive, and the shape is expressed by
the arrangement of white squares. The primitive is explored from connected figures of nine pixels within
5 × 5-pixel square. The gray scale value is explored by setting the initial pixel value to 50 and modifying the
value in the range of 0 to 100.
The opening using the primitive b as the structuring element is performed on the corrupted
image c. This image is generated by adding a uniformly distributed random value, which is in the
range between 0 and 255, to 1000 randomly selected pixels of an image extracted from an image which is a
different realization of the same texture as a. Opening eliminates brighter peaks of small extent, so
that this kind of noise is employed for this experiment.
Information and Communication Technology Seminar, Vol. 1 No. 1, August 2005
ISSN 1858-1633 2005 ICTS 8
The result using the estimate primitive is shown in d, that using the flat structuring element whose shape is
the same as b is shown in e, and that using the 3 × 3-pixel square flat structuring element is shown in f.
The “MSE” attached to each resultant image is defined as the sum of pixelwise difference between
each image and the original uncorrupted image of c, which is not shown here, divided by the number of
pixels in the image.
Fig. 11. Experimental results 1.
The results d show high effectiveness of our method in noise removal and detail preservation. The
results using the square structuring element contain artifacts since the square shape appears directly in the
results, and the results using the binary primitives yield regions of unnaturally uniform pixel values. The
comparison of d and e indicates that the optimization of binary structuring elements is
insufficient and the grayscale optimization is necessary.
Fig. 12. Experimental results 2.
In these examples, the assumptions that the grains are derived from one primitive by homothetic
magnification and the primitive is expressed by one structuring element are not exactly satisfied. However,
the results indicate that our method is applicable to these cases practically.
5. CONCLUSIONS
