Mathematical Morphology for Erosion and Dilation

Mathematical Morphology was born in 1964 from the collaborative work of Georges Matheron and Jean Serra, at the École des Mines de Paris, France. Matheron supervised the PhD thesis of Serra, devoted to the quantification of mineral characteristics from thin cross sections, and this work resulted in a novel practical approach, as well as theoretical advancements in integral geometry and topology.

In 1968, the Centre de Morphologie Mathématique was founded by the École des Mines de Paris in Fontainebleau, France, lead by Matheron and Serra. During the rest of the 1960's and most of the 1970's, MM dealt essentially with binary images, treated as sets, and generated a large number of binary operators and techniques: Hit-or-miss transform, dilation, erosion, opening, closing, granulometry, thinning, skeletonization, ultimate erosion, conditional bisector, and others. A random approach was also developed, based on novel image models. Most of the work in that period was developed in Fontainebleau.

From mid-1970's to mid-1980's, MM was generalized to grayscale functions and images as well. Besides extending the main concepts (such as dilation, erosion, etc...) to functions, this generalization yielded new operators, such as morphological gradients,

top-hat transform and the Watershed (MM's main segmentation approach). Z× and that f and b are functions that assign a gray-level value (a real number from Z the set of real numbers, R ) to each distinct pair of coordinates ( , )x y . If the gray levels also are integers, Z replaces R .

2.1.1 Dilation

The dilation method [2, 3] was introduced by Georges Matheron and Jean Serra, it is the basic operation with morphological image process.

N. Desikachari and Robert M. Haralick [14] propose recursive binary dilation and erosion using digital line structuring elements in arbitrary orientations, the Su Chen and Robert M. Haralick [15] propose recursive erosion and dilation transforms.

We have detail description as above.

Gray-scale dilation of f by , denoted b f ⊕ , is defined as b and have to be in the domain of , is analogous to the condition in the binary definition of dilation, where the two sets have to overlap by at least one element. Note

y b

)}

also that the form of Eq. (2.1.1-1) is similar to 2-D convolution, with the max operation replacing the sums of convolution and the addition replacing the products of convolution.

We illustrate the notation and mechanics of Eq. (2.1.1-1) by means of simple 1-D functions. For functions of one variable, Eq. (2.1.1-1) reduces to the expression

mirrored with respect to the origin of the x axis. As in convolution, the function

( )

f s−x moves to the right for positive s and to the left for negative s. The requirements that the value of (s− has to be in the domain of f and that the value x) of x has to be in the domain of b imply that f an b overlap. Eq. (2.1.1-1) could be written so that b undergoes translation instead of

d

f . However, if the domain of b is s aller than the domain of m f (a condition almost always found in practice), the form given in Eq. (2.1.1-1) is simpler in terms of indexing and achieves the same result.

Conceptually, f sliding by b is really no different than b sliding by f .

The general effect of performing dilation on a gray-scale image is twofold: (1) If all the values of the structuring element are positive, the output image

tends to be brighter than the input. (2) Dark details either are reduced or eliminated, depending on how their values and shapes relate to the structuring element used for dilation.

2.1.2 Erosion

The erosion method [2, 3] is the same as dilation, the difference between them is to narrow the images, here we use the basic part. We have detail description as above.

Gray-scale erosion, denoted f ⊖ , is defined as b have to be in the domain of , is analogous to the condition in the binary definition of erosion, where the structuring element has to be complete contained by the set being eroded. Note that the form of Eq. (2.1.2-1) is similar in form to 2-D correlation, with the min operation replacing the sums of correlation and subtraction replacing the products of correlation.

y

b

We illustrate the mechanics of Eq. (2.1.2-1) by eroding a simple 1-D functions. For functions of one variable, the expression for erosion reduces to

the right for negative . The requirements that

s

s (s x)+ have to be in the domain of f and x have to be in the domain of imply that the range of b is completely contained within the range of the displaced

b f .

Finally, unlike the binary definition of erosion, f , rather than the structuring element b, is shifted. Eq. (2.1.2-1) could be written so that is the

function translated, resulting in a more complicated expression in terms of indexing.

Because

b

f sliding past b conceptually is the same as sliding pastb f .

The general effect of performing erosion on a gray-scale image is twofold: (1) If all the elements of the structuring element are positive, the output image tends to be darker than the input image. (2) The effect of bright details in the input image that is smaller in area than the structuring element is reduced, with the degree of the reduction being determined by the gray-level values surrounding the bright detail and by the shape and amplitude values of the structuring element itself.

Gray-scale dilation and erosion are duals with respect to function complementation and reflection. That is,

[ f ⊖ b ]^c( , )s t =(f^c⊕ b s t)( , ) (2.1.2-3) Where ( , )f^c = −f x y and b= − −b( x, y).

2.1.3 Example

Figure 2.1 A binary image requiring careful definition of object and background

connectivity.

(a) Dilation D(A,B) (b) Erosion E(A,B)

Figure 2.2 A binary image containing two object sets A and B. The three pixels in B are

"color-coded" as is their effect in the result. And (a) (b) is Dilation and Erosion respect.

While either set A or B can be thought of as an "image", A is usually considered as the image and B is called a structuring element (SE). The structuring element is to mathematical morphology what the convolution kernel is to linear filter theory.

Dilation, in general, causes objects to dilate or grow in size; erosion causes objects to shrink. The amount and the way that they grow or shrink depend upon the choice of the structuring element. Dilating or eroding without specifying the structural element makes no more sense than trying to lowpass filter an image without specifying the filter. The two most common structuring elements (given a Cartesian grid) are the 4-connected and 8-connected sets, N4 and N8.

Figure 2.3 The standard structuring elements N4 and N₈.