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Digital Image Processing (2000)

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6.3 Binary Morphological Operators

The fundamental operators of binary morphology are erosion and dilation. We present both the set-theoretic and matrix formulations of these basic morphological operators.
Dilation is the morphological transformation that combines two sets by using vector addition of set elements. Following common practice, we use the definition that is equivalent to Minkowski addition [Ser82, Har87].
Binary dilation of set by set (called the structuring element), denoted , is the union of all translations of set by elements of set .
This loads the package.
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Here are set descriptions of two example images.
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This gives the dilation of set by the structuring element .
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Here we view the two sets using their matrix formulations. The binary arrays corresponding to sets and follow.
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Here, in matrix form, is the description of dilation as a logical OR operation (union) of the translates of set .
    (
0000
1110
0010
0000
)1(
0000
0111
0001
0000
)1(
0000
0000
0111
0001
)1 (
0000
1111
0111
0001
)1
where
stands for the union or logical OR operation. This viewpoint leads in turn to an alternative definition of dilation as a scanning and processing operation between two matrices: an image and a small structuring element. To simplify the notation, we assume all arrays have square dimensions. With A as the N×N binary-valued image, B as the 2 M+1×2 M+1 structuring element, and C=AB as the dilation result, the matrix formulation is as follows:
with 0≤n1,n2≤ N-1, and where
and
are the logical OR (union) and logical AND (intersection) operators, respectively. For the special case of a 3×3 (M=1) structuring element of all ones, we can write Equation (6.3.1) explicitly as
Interestingly, Equations (6.3.1) and (6.3.2) resemble the familiar convolution sum formulation with OR replacing addition and AND replacing multiplication. Thus the structuring element plays an analogous role in morphological signal processing to the FIR filter in LSI signal processing.
Erosion is a morphological dual to dilation [Gon92, Har92, Pra91]. The set-theoretic definition of erosion follows.
The erosion of set by set , denoted , is the set intersection of all negative translates of set by elements of set .
Equivalently, erosion may be defined by the following correlation-like formulation. The erosion of image A with structuring element B, denoted C=AB, has the following matrix formulation:
with 0≤n1,n2≤ N-1. Operators
and
are the logical OR (union) and logical AND (intersection), respectively, and is the point-wise complement of structuring element B. In effect, erosion results in a value of 1 at position (n1,n2) in C, if the spatial arrangement of ones in B fully matches the arrangement of ones in A (i.e., is a subset). In particular, for the special case of a 3×3 (M=1) structuring element of all ones, we can write Equation (6.3.3) explicitly as

Binary morphological functions.

The matrix formulations in Equations (6.3.1) and (6.3.3) assume an odd-sized structuring element with the origin centered in the array. Therefore, we define a 3×3 array B corresponding to set ={{0,0},{0,1},{1,1}} as follows.
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Here is the binary dilation of image A by structuring element B, namely AB.
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Note that this result is same as previously obtained by set-theoretic calculations. This is the binary erosion of array A by structuring element B, namely A B.
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Dilation and erosion are duals of each other. The duality relationships are defined as follows:
where the symbol represents a binary complement of the elements of a binary array A and denotes reflection of B. Here we verify duality of binary erosion and dilation.
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Here we demonstrate the operations of binary erosion and dilation using one of the example images. We threshold the data file to obtain a binary image, and we use a small, odd-sized structuring element.
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This shows the two images.
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This defines the structuring element.
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Here we show the results of dilation and erosion.
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Note how dilation grows the white regions in the image, hence the use of the word dilation for this operator. The operation of erosion, on the other hand, decreases or shrinks the area occupied by white pixels in the example image.
Dilation and erosion are often applied to an image in concatenation. Dilation followed by erosion is called a close operation.
The closing of an image with a compact structuring element without holes, smoothes the contours of objects and eliminates small holes in objects.
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The Open operator, which is the morphological dual of close, consists of erosion followed by dilation.
Opening an image smoothes the contours of objects, eliminates small objects, and breaks narrow strokes.
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Open and close operations satisfy the following duality relationship:
where the symbol represents a binary complement of the elements of a binary array A and denotes the reflection of B. Here we verify the duality relation.
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In the following evaluation, we demonstrate the effect of close and open operations on the binary example image using structuring element S.
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BinaryHitMiss implements the morphological hit-or-miss transformation. This transformation is useful in detecting specific spatial arrangements of image pixels and is used to implement thinning and thickening operators. Given a 2M+1×2M+1 structuring element B and an image array A, the hit-or-miss transform may be defined as
for all i,j such that -M≤i,j≤M and 0≤r,c≤N-1. The following demonstrates the result of computing the hit-or-miss transformation of the example binary image using a wide structuring element.
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