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Dilation[image,ker]

gives the morphological dilation of image with respect to the structuring element ker.

Dilation[image,r]

gives the dilation with respect to a range-r square.

Dilation[data,…]

applies dilation to an array of data.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Data  
Parameters  
Options  
Padding  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
Related Guides
History
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Dilation

Dilation[image,ker]

gives the morphological dilation of image with respect to the structuring element ker.

Dilation[image,r]

gives the dilation with respect to a range-r square.

Dilation[data,…]

applies dilation to an array of data.

Details and Options

  • Dilation is also known as Minkowski addition.
  • Dilation works with arbitrary 2D and 3D images, operating separately on each channel, as well as data arrays of any rank.
  • The structuring element ker is a matrix containing s and s.
  • Dilation[image,r] is equivalent to Dilation[image,BoxMatrix[r]].
  • The structuring element is automatically padded with zeros to have odd dimensions. »
  • Dilation takes a Padding option that specifies the values to assume for pixels outside the image.
  • By default, Padding0 is used for images, corresponding to pixel value 0 for all channels.

Examples

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Basic Examples  (3)

Dilation of a binary image:

Wolfram Language code: Dilation[[image], DiskMatrix[4]]

Dilation of a grayscale image:

Wolfram Language code: Dilation[[image], DiskMatrix[4]]

Dilation of a 3D shape:

Wolfram Language code: Dilation[[image], 1]

Scope  (13)

Data  (7)

Dilation of a 2D binary array:

Wolfram Language code: Dilation[(⁠| | | | | | | | - | - | - | - | - | - | | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 1 | 1 | 0 | | 0 | 1 | 1 | 1 | 1 | 0 | | 0 | 0 | 1 | 1 | 0 | 0 | | 0 | 0 | 0 | 1 | 1 | 0 | | 0 | 0 | 0 | 0 | 0 | 0 |⁠), (⁠| | | | | - | - | - | | 0 | 0 | 0 | | 0 | 1 | 1 | | 0 | 0 | 1 |⁠)]//MatrixForm

Dilation of a binary image:

Wolfram Language code: Dilation[[image], (⁠| | | | | - | - | - | | 1 | 0 | 0 | | 0 | 1 | 0 | | 0 | 0 | 1 |⁠)]

Dilation of a numeric array:

Wolfram Language code: Dilation[(⁠| | | | | | | - | - | - | - | - | | 2 | 3 | 2 | 5 | 5 | | 1 | 4 | 1 | 2 | 5 | | 3 | 2 | 1 | 4 | 5 | | 2 | 1 | 2 | 3 | 1 | | 1 | 1 | 2 | 4 | 2 |⁠), 1]//MatrixForm

Dilation of a numeric vector:

Wolfram Language code: data = QuantityMagnitude@Values[FinancialData["AT&T", {{2012, 7, 1}, {2013, 1, 1}, "Day"}]];
Wolfram Language code: ListLinePlot[{data, Dilation[data, 5]}]

Dilation of a grayscale image:

Wolfram Language code: Dilation[[image], 3]

Dilation of a color image:

Wolfram Language code: Dilation[[image], 2]

Dilation on a symbolic array of data:

Wolfram Language code: Dilation[{a, b, c}, 1]

Parameters  (6)

Dilate horizontally:

Wolfram Language code: Dilation[[image], {{1, 1, 1}}]

Dilate vertically:

Wolfram Language code: Dilation[[image], {{1}, {1}, {1}}]

Dilate with radius , equivalent to BoxMatrix[r]:

Wolfram Language code: Dilation[[image], 2]

Dilate with a diagonal structuring element:

Wolfram Language code: Dilation[[image], IdentityMatrix[9]]

Structuring elements with even dimensions are right-padded with zeros:

Wolfram Language code: Dilation[ArrayPad[{{1}}, 2], (⁠| | | | - | - | | 1 | 1 | | 1 | 1 |⁠)] === Dilation[ArrayPad[{{1}}, 2], (⁠| | | | | - | - | - | | 1 | 1 | 0 | | 1 | 1 | 0 | | 0 | 0 | 0 |⁠)]

Dilate a 3D volume using a 3D kernel:

Wolfram Language code: Dilation[[image], DiamondMatrix[{2, 2, 2}]]

Options  (2)

Padding  (2)

By default, the smallest possible number is used for padding when applying dilation to arrays:

Wolfram Language code: Dilation[{1, 2, 3, 4, 5}, 1]

Specify a custom padding:

Wolfram Language code: Dilation[{1, 2, 3, 4, 5}, 1, Padding -> 100]

By default, Padding->0 is used for images:

Wolfram Language code: i = [image];
Wolfram Language code: Dilation[i, 20]

Specify a custom padding:

Wolfram Language code: Dilation[[image], 20, Padding -> Red]

Applications  (2)

Dilation increases the amount of white space in the image, therefore removing smaller, dark features:

Wolfram Language code: Dilation[[image], 1]

Compute external morphological gradient as a difference between dilated and original image:

Wolfram Language code: i = [image]; ImageSubtract[Dilation[i, 1], i]

Properties & Relations  (2)

Binary dilation is extensive if the center of the structuring element is 1:

Wolfram Language code: f = (⁠| | | | | | | | - | - | - | - | - | - | | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 1 | 1 | 0 | | 0 | 1 | 1 | 1 | 1 | 0 | | 0 | 0 | 1 | 1 | 0 | 0 | | 0 | 0 | 0 | 1 | 1 | 0 | | 0 | 0 | 0 | 0 | 0 | 0 |⁠);ker = (⁠| | | | | - | - | - | | 0 | 0 | 0 | | 0 | 1 | 1 | | 0 | 0 | 1 |⁠);
Wolfram Language code: (d = Dilation[f, ker])//MatrixForm

Extensivity means that all elements of f are included in the Dilation[f,ker]:

Wolfram Language code: BitAnd[f, d] === f

Dilation with a box structuring element is the same as MaxFilter:

Wolfram Language code: MaxFilter[[image], 1] == Dilation[[image], 1]

Possible Issues  (1)

Image dilation with a kernel of all zeros will result in a zero image:

Wolfram Language code: Dilation[[image], {{0}}]//ImageData//MatrixForm

Array dilation with an all-zero kernel will result in an array of −∞:

Wolfram Language code: Dilation[{1, 2, 3, 4, 5}, {0}]
Wolfram Research (2008), Dilation, Wolfram Language function, https://reference.wolfram.com/language/ref/Dilation.html (updated 2012).

Text

Wolfram Research (2008), Dilation, Wolfram Language function, https://reference.wolfram.com/language/ref/Dilation.html (updated 2012).

CMS

Wolfram Language. 2008. "Dilation." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012. https://reference.wolfram.com/language/ref/Dilation.html.

APA

Wolfram Language. (2008). Dilation. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Dilation.html

BibTeX

@misc{reference.wolfram_2026_dilation, author="Wolfram Research", title="{Dilation}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/Dilation.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_dilation, organization={Wolfram Research}, title={Dilation}, year={2012}, url={https://reference.wolfram.com/language/ref/Dilation.html}, note=[Accessed: 12-June-2026]}