Morphsnakes segmentation

Morphological snakes algorithm for 3D image segmentation. Source: https://github.com/pmneila/morphsnakes

This code was written by P. Márquez Neila and is included in this repository for ease of installation.

LICENSE

Copyright (c) 2013-2015, P. M. Neila All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

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Morphological Snakes

Morphological Snakes [1]_ are a family of methods for image segmentation. Their behavior is similar to that of active contours (for example, Geodesic Active Contours [2]_ or Active Contours without Edges [3]_). However, Morphological Snakes use morphological operators (such as dilation or erosion) over a binary array instead of solving PDEs over a floating point array, which is the standard approach for active contours. This makes Morphological Snakes faster and numerically more stable than their traditional counterpart.

There are two Morphological Snakes methods available in this implementation: Morphological Geodesic Active Contours (MorphGAC, implemented in the function [morphological_geodesic_active_contour](https://nikolas-claussen.github.io/blender-tissue-cartography/Python library/morphsnakes.html#morphological_geodesic_active_contour)) and Morphological Active Contours without Edges (MorphACWE, implemented in the function [morphological_chan_vese](https://nikolas-claussen.github.io/blender-tissue-cartography/Python library/morphsnakes.html#morphological_chan_vese)).

MorphGAC is suitable for images with visible contours, even when these contours might be noisy, cluttered, or partially unclear. It requires, however, that the image is preprocessed to highlight the contours. This can be done using the function [inverse_gaussian_gradient](https://nikolas-claussen.github.io/blender-tissue-cartography/Python library/morphsnakes.html#inverse_gaussian_gradient), although the user might want to define their own version. The quality of the MorphGAC segmentation depends greatly on this preprocessing step.

On the contrary, MorphACWE works well when the pixel values of the inside and the outside regions of the object to segment have different averages. Unlike MorphGAC, MorphACWE does not require that the contours of the object are well defined, and it works over the original image without any preceding processing. This makes MorphACWE easier to use and tune than MorphGAC.

References

[1] A Morphological Approach to Curvature-based Evolution of Curves and Surfaces, Pablo Márquez-Neila, Luis Baumela and Luis Álvarez. In IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 2014, DOI 10.1109/TPAMI.2013.106

[2] Geodesic Active Contours, Vicent Caselles, Ron Kimmel and Guillermo Sapiro. In International Journal of Computer Vision (IJCV), 1997, DOI:10.1023/A:1007979827043

[3] Active Contours without Edges, Tony Chan and Luminita Vese. In IEEE Transactions on Image Processing, 2001, DOI:10.1109/83.902291

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morphological_geodesic_active_contour

 morphological_geodesic_active_contour (gimage, iterations,
                                        init_level_set='circle',
                                        smoothing=1, threshold='auto',
                                        balloon=0, iter_callback=<function
                                        <lambda>>)

*Morphological Geodesic Active Contours (MorphGAC).

Geodesic active contours implemented with morphological operators. It can be used to segment objects with visible but noisy, cluttered, broken borders.*

Type Default Details
gimage (M, N) or (L, M, N) array Preprocessed image or volume to be segmented. This is very rarely the
original image. Instead, this is usually a preprocessed version of the
original image that enhances and highlights the borders (or other
structures) of the object to segment.
morphological_geodesic_active_contour will try to stop the contour
evolution in areas where gimage is small. See
morphsnakes.inverse_gaussian_gradient as an example function to
perform this preprocessing. Note that the quality of
morphological_geodesic_active_contour might greatly depend on this
preprocessing.
iterations uint Number of iterations to run.
init_level_set str circle Initial level set. If an array is given, it will be binarized and used
as the initial level set. If a string is given, it defines the method
to generate a reasonable initial level set with the shape of the
image. Accepted values are ‘checkerboard’ and ‘circle’. See the
documentation of checkerboard_level_set and circle_level_set
respectively for details about how these level sets are created.
smoothing int 1 Number of times the smoothing operator is applied per iteration.
Reasonable values are around 1-4. Larger values lead to smoother
segmentations.
threshold str auto Areas of the image with a value smaller than this threshold will be
considered borders. The evolution of the contour will stop in this
areas.
balloon int 0 Balloon force to guide the contour in non-informative areas of the
image, i.e., areas where the gradient of the image is too small to push
the contour towards a border. A negative value will shrink the contour,
while a positive value will expand the contour in these areas. Setting
this to zero will disable the balloon force.
iter_callback function If given, this function is called once per iteration with the current
level set as the only argument. This is useful for debugging or for
plotting intermediate results during the evolution.
Returns (M, N) or (L, M, N) array Final segmentation (i.e., the final level set)

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morphological_chan_vese

 morphological_chan_vese (image, iterations,
                          init_level_set='checkerboard', smoothing=1,
                          lambda1=1, lambda2=1, iter_callback=<function
                          <lambda>>)

*Morphological Active Contours without Edges (MorphACWE)

Active contours without edges implemented with morphological operators. It can be used to segment objects in images and volumes without well defined borders. It is required that the inside of the object looks different on average than the outside (i.e., the inner area of the object should be darker or lighter than the outer area on average).*

Type Default Details
image (M, N) or (L, M, N) array Grayscale image or volume to be segmented.
iterations uint Number of iterations to run
init_level_set str checkerboard Initial level set. If an array is given, it will be binarized and used
as the initial level set. If a string is given, it defines the method
to generate a reasonable initial level set with the shape of the
image. Accepted values are ‘checkerboard’ and ‘circle’. See the
documentation of checkerboard_level_set and circle_level_set
respectively for details about how these level sets are created.
smoothing int 1 Number of times the smoothing operator is applied per iteration.
Reasonable values are around 1-4. Larger values lead to smoother
segmentations.
lambda1 int 1 Weight parameter for the outer region. If lambda1 is larger than
lambda2, the outer region will contain a larger range of values than
the inner region.
lambda2 int 1 Weight parameter for the inner region. If lambda2 is larger than
lambda1, the inner region will contain a larger range of values than
the outer region.
iter_callback function If given, this function is called once per iteration with the current
level set as the only argument. This is useful for debugging or for
plotting intermediate results during the evolution.
Returns (M, N) or (L, M, N) array Final segmentation (i.e., the final level set)

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inverse_gaussian_gradient

 inverse_gaussian_gradient (image, alpha=100.0, sigma=5.0)

*Inverse of gradient magnitude.

Compute the magnitude of the gradients in the image and then inverts the result in the range [0, 1]. Flat areas are assigned values close to 1, while areas close to borders are assigned values close to 0.

This function or a similar one defined by the user should be applied over the image as a preprocessing step before calling morphological_geodesic_active_contour.*

Type Default Details
image (M, N) or (L, M, N) array Grayscale image or volume.
alpha float 100.0 Controls the steepness of the inversion. A larger value will make the
transition between the flat areas and border areas steeper in the
resulting array.
sigma float 5.0 Standard deviation of the Gaussian filter applied over the image.
Returns (M, N) or (L, M, N) array Preprocessed image (or volume) suitable for
morphological_geodesic_active_contour.

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checkerboard_level_set

 checkerboard_level_set (image_shape, square_size=5)

Create a checkerboard level set with binary values.

Type Default Details
image_shape tuple of positive integers Shape of the image.
square_size int 5 Size of the squares of the checkerboard. It defaults to 5.
Returns array with shape image_shape Binary level set of the checkerboard.

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ellipsoid_level_set

 ellipsoid_level_set (image_shape, center=None, semi_axis=None)

Create a ellipsoid level set with binary values.

Type Default Details
image_shape tuple of positive integers Shape of the image
center NoneType None Coordinates of the center of the ellipsoid.
If not given, it defaults to the center of the image.
semi_axis NoneType None Lengths of the semi-axis of the ellispoid.
If not given, it defaults to the half of the image dimensions.
Returns array with shape image_shape Binary level set of the ellipsoid with the given center
and semi_axis.

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circle_level_set

 circle_level_set (image_shape, center=None, radius=None)

Create a circle level set with binary values.

Type Default Details
image_shape tuple of positive integers Shape of the image
center NoneType None Coordinates of the center of the circle given in (row, column). If not
given, it defaults to the center of the image.
radius NoneType None Radius of the circle. If not given, it is set to the 75% of the
smallest image dimension.
Returns array with shape image_shape Binary level set of the circle with the given radius and center.

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inf_sup

 inf_sup (u)

IS operator.

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sup_inf

 sup_inf (u)

SI operator.