Morphological Dilation and Erosion

Morphologyis a broad set of image processing operations that process images based on shapes. Morphological operations apply a structuring element to an input image, creating an output image of the same size. In a morphological operation, the value of each pixel in the output image is based on a comparison of the corresponding pixel in the input image with its neighbors. By choosing the size and shape of the neighborhood, you can construct a morphological operation that is sensitive to specific shapes in the input image.

The most basic morphological operations are dilation and erosion. Dilation adds pixels to the boundaries of objects in an image, while erosion removes pixels on object boundaries. The number of pixels added or removed from the objects in an image depends on the size and shape of thestructuring elementused to process the image. In the morphological dilation and erosion operations, the state of any given pixel in the output image is determined by applying a rule to the corresponding pixel and its neighbors in the input image. The rule used to process the pixels defines the operation as a dilation or an erosion. This table lists the rules for both dilation and erosion.

Rules for Dilation and Erosion

Operation	Rule
Dilation	The value of the output pixel is themaximumvalue of all the pixels in the input pixel's neighborhood. In a binary image, if any of the pixels is set to the value`1`, the output pixel is set to 1.
Erosion	The value of the output pixel is theminimumvalue of all the pixels in the input pixel's neighborhood. In a binary image, if any of the pixels is set to`0`, the output pixel is set to 0.

下图说明了膨胀binary image. Note how the structuring element defines the neighborhood of the pixel of interest, which is circled. (SeeStructuring Elementsfor more information.) The dilation function applies the appropriate rule to the pixels in the neighborhood and assigns a value to the corresponding pixel in the output image. In the figure, the morphological dilation function sets the value of the output pixel to1because one of the elements in the neighborhood defined by the structuring element is on.

Morphological Dilation of a Binary Image

The following figure illustrates this processing for a grayscale image. The figure shows the processing of a particular pixel in the input image. Note how the function applies the rule to the input pixel's neighborhood and uses the highest value of all the pixels in the neighborhood as the value of the corresponding pixel in the output image.

Morphological Dilation of a Grayscale Image

Processing Pixels at Image Borders (Padding Behavior)

Morphological functions position the origin of the structuring element, its center element, over the pixel of interest in the input image. For pixels at the edge of an image, parts of the neighborhood defined by the structuring element can extend past the border of the image.

To process border pixels, the morphological functions assign a value to these undefined pixels, as if the functions had padded the image with additional rows and columns. The value of these padding pixels varies for dilation and erosion operations. The following table describes the padding rules for dilation and erosion for both binary and grayscale images.

Rules for Padding Images

Operation	Rule
Dilation	Pixels beyond the image border are assigned the minimum value afforded by the data type. For binary images, these pixels are assumed to be set to 0. For grayscale images, the minimum value for`uint8`images is`0`.
Erosion	Pixels beyond the image border are assigned themaximumvalue afforded by the data type. For binary images, these pixels are assumed to be set to`1`. For grayscale images, the maximum value for`uint8`images is`255`.

Operation

Rule

Dilation

Pixels beyond the image border are assigned the minimum value afforded by the data type.

For binary images, these pixels are assumed to be set to 0. For grayscale images, the minimum value foruint8images is0.

Erosion

Pixels beyond the image border are assigned themaximumvalue afforded by the data type.

For binary images, these pixels are assumed to be set to1. For grayscale images, the maximum value foruint8images is255.

Note

By using the minimum value for dilation operations and the maximum value for erosion operations, the toolbox avoidsborder effects, where regions near the borders of the output image do not appear to be homogeneous with the rest of the image. For example, if erosion padded with a minimum value, eroding an image would result in a black border around the edge of the output image.

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