Morphological opening with asymetrical structuring element isn't idempotent

jmarcio · May 9, 2023, 9:28am

Hello,
I noticed that the opening operator isn’t idempotent with asymmetrical structuring elements.
For example, try the following code

import skimage
from skimage.data import camera
from skimage.morphology import opening, disk
import numpy as np
from matplotlib import pyplot as plt

print('Skimage version:', skimage.__version__)

im = camera()

# symmetric SE
se = disk(1) # 4-connexity cross

imOp = opening(im,se)
imOp2 = opening(imOp,se)

print('Difference after opening twice (sym. SE):', np.abs(imOp-imOp2).max())

# non-symmetric SE
se = np.zeros((3,3))
se[0, 2] = 1
se[1, 1] = 1
se[2, 2] = 1

imOp = opening(im,se)
imOp2 = opening(imOp,se)

print('Difference after opening twice (non-sym. SE):', np.abs(imOp-imOp2).max())

plt.figure()
plt.imshow(im, cmap = 'gray')
plt.figure()
plt.imshow(np.abs(imOp2-imOp), cmap = 'gray')

plt.show()

Best regards,
José-Marcio

jmarcio · May 9, 2023, 2:38pm

Sorry,
A colleague of mine already posted this problem at github scikit-image forum, with more informative information.

github.com/scikit-image/scikit-image

<Inaccurate definition of morphological opening/closing (skimage.morphology)>

opened 09:21PM - 08 May 23 UTC

samyblusseau

Bug

### Description: As implemented, the opening and closing and of the skimage.m…orphology package are not always idempotent, whereas they should be. For any structuring element "footprint" and input image im, `imOp = opening(im, footprint)` `imOp2 = opening(imOp, footprint)` should produce two identical images imOp and imOp2. But with skimage.morphology this is not the case whenever footprint is not symmetrical with respect to the central pixel of the footprint (e.g. when you set to zero one non-central pixel of the disk(1) footprint, like: `footprint = disk(1); footprint[0, 1] = 0`). In order to define an opening based on an erosion and a dilation, these erosion and dilation should form an adjunction. In skimage.morphology, erosion(., footprint) and dilation(., footprint) are not adjoint when footprint is not symmetrical. But [erosion(., footprint), dilation(., footprint_reflected)] form an adjunction, where footprint_reflected is the symmetric of footprint with respect to the central pixel. So to get an idempotent opening, a correct implementation would be: `eroded = erosion(image, footprint)` `footprint_reflected = reflect(footprint)` `out = dilation(eroded, footprint_reflected, out=out, shift_x=True, shift_y=True) ` References: see for example https://onlinelibrary.wiley.com/doi/book/10.1002/9781118600788 or https://link.springer.com/book/10.1007/978-3-662-05088-0 or even simply https://en.wikipedia.org/wiki/Opening_(morphology) with the corresponding definitions of dilation https://en.wikipedia.org/wiki/Dilation_(morphology) and erosion https://en.wikipedia.org/wiki/Erosion_(morphology), which do not match those implemented in skimage.morphology. ### Way to reproduce: ``` import skimage from skimage.data import camera from skimage.morphology import opening, disk import numpy as np from matplotlib import pyplot as plt im = camera() # non-symmetric structuring element se = disk(1) se[0, 1] = 0 imOp = opening(im,se) imOp2 = opening(imOp,se) print('Difference after opening twice (non-sym. SE):', np.abs(imOp-imOp2).max()) plt.figure() plt.imshow(im, cmap = 'gray') plt.figure() plt.imshow(np.abs(imOp2-imOp), cmap = 'gray') plt.show() ``` ### Version information: ```Shell 3.8.10 (default, Jun 4 2021, 15:09:15) [GCC 7.5.0] Linux-5.4.0-144-generic-x86_64-with-glibc2.17 scikit-image version: 0.19.3 numpy version: 1.20.2 ```

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stefanv · May 9, 2023, 7:58pm

Thanks, we can handle this on the GitHub issue. It seems to be related to Fix bugs and clean up grayscale morphology operators by crisluengo · Pull Request #6695 · scikit-image/scikit-image · GitHub.