Blog reader Alex asked for an explanation of nonflat grayscale dilation and erosion.
In the most commonly-used form of dilation, the structuring element defines a neighborhood of a pixel. In the output image, a pixel is computed by taking the maximum of all the input pixels in the neighborhood. For erosion, an output image pixel is computed by taking the minimum of all the input pixels in the neighborhood.
In nonflat dilation and erosion, each neighbor has an associated height. A dilation output pixel is computed by first adding the neighbor heights to the neighbor input pixels, and then taking the maximum.
Here's how Digital Image Processing Using MATLAB describes the operation, including the mathematical equation. (This excerpt is from section 9.6.1, pages 366-367.)
"The gray-scale dilation of f by structuring element b [...] is defined as
where D_b is the domain of b, and f(x,y) is assumed equal to [minus infinity] outside the domain of f. This equation implements a process similar to spatial convolution [...]. Conceptually, we can think of rotation the structuring element about its origin and translating it to all locations in the image, just as the convolution kernel is rotated and then translated about the image. At each translated location, the rotated structuring element values are added to the image pixel values and the maximum is computed."
The 'ball' option of the strel function produces a nonflat structuring element.
se = strel('ball',5,5)
se =
Nonflat STREL object containing 109 neighbors.
Decomposition: 8 STREL objects containing a total of 24 neighbors
Neighborhood:
0 0 1 1 1 1 1 1 1 0 0
0 1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
0 1 1 1 1 1 1 1 1 1 0
0 0 1 1 1 1 1 1 1 0 0
Height:
Columns 1 through 10
-Inf -Inf 0 0.6248 1.2497 1.8745 1.2497 0.6248 0 -Inf
-Inf 0.6248 1.2497 1.8745 2.4994 2.4994 2.4994 1.8745 1.2497 0.6248
0 1.2497 1.8745 2.4994 3.1242 3.1242 3.1242 2.4994 1.8745 1.2497
0.6248 1.8745 2.4994 3.1242 3.7491 3.7491 3.7491 3.1242 2.4994 1.8745
1.2497 2.4994 3.1242 3.7491 4.3739 4.3739 4.3739 3.7491 3.1242 2.4994
1.8745 2.4994 3.1242 3.7491 4.3739 4.9987 4.3739 3.7491 3.1242 2.4994
1.2497 2.4994 3.1242 3.7491 4.3739 4.3739 4.3739 3.7491 3.1242 2.4994
0.6248 1.8745 2.4994 3.1242 3.7491 3.7491 3.7491 3.1242 2.4994 1.8745
0 1.2497 1.8745 2.4994 3.1242 3.1242 3.1242 2.4994 1.8745 1.2497
-Inf 0.6248 1.2497 1.8745 2.4994 2.4994 2.4994 1.8745 1.2497 0.6248
-Inf -Inf 0 0.6248 1.2497 1.8745 1.2497 0.6248 0 -Inf
Column 11
-Inf
-Inf
0
0.6248
1.2497
1.8745
1.2497
0.6248
0
-Inf
-Inf
As far as I know, flat dilation and erosion are far more commonly used. If you know of an image processing application particularly well-suited for applying nonflat dilation and erosion, send me a note - I'd love to hear about it.
Get
the MATLAB code
Published with MATLAB® 7.3
Dear steve,
please can you give me some references available online, about Nonflat grayscale morphological analysis.
thank you in advance.
Amin - the reference I recommend for morphology is the book by P. Soille, Morphological Image Analysis. Nonflat grayscale dilation and erosion is also covered, briefly, in Digital Image Processing Using MATLAB. As for online references, I have no doubt that Web search engines will lead you to helpful information.
Dear Steve,
Could you explain why f(x,y) is assumed equal to [minus infinity] outside the domain of f.
thank you in advance.
DKS - The basic answer is to avoid boundary effects. Specifically, if f(x,y) equals minus infinity outside the image boundaries, then the output of the dilation operation will take on only defined values of f. Perhaps it would have been less surprising if I had said that f(x,y) equals 0 outside the image boundaries. That works fine when image pixels are nonnegative. The key idea is that the assumed value outside the boundary be no greater than any possible image pixel value.
There is a more detailed treatment of this question in Soille, Morphological Image Analysis: Principles and Applications, 2nd ed. See pages 51, 67, and 73. I’ve made a note to look at this issue again and see whether it’s worth writing more about.
Dear Steve,
Could you help me on the following?
What does ( x-x’, y-y’) mean? What happens when x-x’ is negative? in the formula of dilataion
(f + b)(x,y) = max {f(x-x’,y-y’)+b(x’,y ’)| (x’,y’)belongs to Db}
Why in dilation it is x-x’ and y-y’, whilst in erosion it is x+x’ and y+y’?
(f – b)(x,y) = min {f(x+x’,y+y’)+b(x’,y’)| (x’,y’)belongs to Db}
Please let me know at your earliest convenience. It’s urgent
Thank you very much.
DKS
DKS - The notation refers to function f defined on a Cartesian grid, (x,y). (x’,y’) is a point in the domain of the structuring element. For example, if you have a flat, zero-valued structuring element whose domain is -1 < = x' <= 1, -1 <= y' <= 1, then there'll be 9 terms inside the max{} operator: f(x - (-1), y - (-1)), f(x - 0, y - (-1)), ..., f(x - 1, y - 1).
For any position (x-x’, y-y’) outside the defined domain of f, you have to assume a value for f, or you have to omit those values from the max{} operator. I talked about choosing those “pad” values in my blog last week.
The choice of equations for dilation and erosion is somewhat a matter of convention. I have the impression that the equations you quoted are more common in the US, whereas a different convention is more often used in Europe. It doesn’t matter that much, except that the choice affects the definition of morphological opening and closing. With the equations you quoted, opening is erosion followed by dilation with the same structuring element. With the European convention, the structuring element has to be reflected for the erosion step.
Recently,I found that some grayscale erosion or dilation using a scaled hemispheric elements.Would you please tell me what the scaled hemispheric elements,and how to use them,thank you very much
clcclcclc—That’s an interesting name that you have. How do you pronounce it?
I haven’t seen the term scaled hemispheric element before, but I’d guess that it’s the kind of structuring element returned by strel(’ball’, …).
See the strel doc for more information.
Oh,thank you very much.Recently I saw a paper named
“FACE DETECTION IN COLOR IMAGES” by Rein-Lien Hsu§, Mohamed Abdel-Mottaleb*, and Anil K. Jain §,in which mentioned that “In our eye detection algorithm, the greyscale dilation and erosion using hemispheric structuring elements at an estimated scale are applied independently to construct EyeMap in the luma”.
Now I know the algorithm,but I don’t know the structure to be used.
I am a Chinese student,thank you again for your answer