| """max_tree.py - max_tree representation of images. |
| |
| This module provides operators based on the max-tree representation of images. |
| A grayscale image can be seen as a pile of nested sets, each of which is the |
| result of a threshold operation. These sets can be efficiently represented by |
| max-trees, where the inclusion relation between connected components at |
| different levels are represented by parent-child relationships. |
| |
| These representations allow efficient implementations of many algorithms, such |
| as attribute operators. Unlike morphological openings and closings, these |
| operators do not require a fixed footprint, but rather act with a flexible |
| footprint that meets a certain criterion. |
| |
| This implementation provides functions for: |
| 1. max-tree generation |
| 2. area openings / closings |
| 3. diameter openings / closings |
| 4. local maxima |
| |
| References: |
| .. [1] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive |
| Connected Operators for Image and Sequence Processing. |
| IEEE Transactions on Image Processing, 7(4), 555-570. |
| :DOI:`10.1109/83.663500` |
| .. [2] Berger, C., Geraud, T., Levillain, R., Widynski, N., Baillard, A., |
| Bertin, E. (2007). Effective Component Tree Computation with |
| Application to Pattern Recognition in Astronomical Imaging. |
| In International Conference on Image Processing (ICIP) (pp. 41-44). |
| :DOI:`10.1109/ICIP.2007.4379949` |
| .. [3] Najman, L., & Couprie, M. (2006). Building the component tree in |
| quasi-linear time. IEEE Transactions on Image Processing, 15(11), |
| 3531-3539. |
| :DOI:`10.1109/TIP.2006.877518` |
| .. [4] Carlinet, E., & Geraud, T. (2014). A Comparative Review of |
| Component Tree Computation Algorithms. IEEE Transactions on Image |
| Processing, 23(9), 3885-3895. |
| :DOI:`10.1109/TIP.2014.2336551` |
| """ |
|
|
| import numpy as np |
|
|
| from ._util import _validate_connectivity, _offsets_to_raveled_neighbors |
| from ..util import invert |
|
|
| from . import _max_tree |
|
|
| unsigned_int_types = [np.uint8, np.uint16, np.uint32, np.uint64] |
| signed_int_types = [np.int8, np.int16, np.int32, np.int64] |
| signed_float_types = [np.float16, np.float32, np.float64] |
|
|
|
|
| |
| def max_tree(image, connectivity=1): |
| """Build the max tree from an image. |
| |
| Component trees represent the hierarchical structure of the connected |
| components resulting from sequential thresholding operations applied to an |
| image. A connected component at one level is parent of a component at a |
| higher level if the latter is included in the first. A max-tree is an |
| efficient representation of a component tree. A connected component at |
| one level is represented by one reference pixel at this level, which is |
| parent to all other pixels at that level and to the reference pixel at the |
| level above. The max-tree is the basis for many morphological operators, |
| namely connected operators. |
| |
| Parameters |
| ---------- |
| image : ndarray |
| The input image for which the max-tree is to be calculated. |
| This image can be of any type. |
| connectivity : unsigned int, optional |
| The neighborhood connectivity. The integer represents the maximum |
| number of orthogonal steps to reach a neighbor. In 2D, it is 1 for |
| a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1. |
| |
| Returns |
| ------- |
| parent : ndarray, int64 |
| Array of same shape as image. The value of each pixel is the index of |
| its parent in the ravelled array. |
| tree_traverser : 1D array, int64 |
| The ordered pixel indices (referring to the ravelled array). The pixels |
| are ordered such that every pixel is preceded by its parent (except for |
| the root which has no parent). |
| |
| References |
| ---------- |
| .. [1] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive |
| Connected Operators for Image and Sequence Processing. |
| IEEE Transactions on Image Processing, 7(4), 555-570. |
| :DOI:`10.1109/83.663500` |
| .. [2] Berger, C., Geraud, T., Levillain, R., Widynski, N., Baillard, A., |
| Bertin, E. (2007). Effective Component Tree Computation with |
| Application to Pattern Recognition in Astronomical Imaging. |
| In International Conference on Image Processing (ICIP) (pp. 41-44). |
| :DOI:`10.1109/ICIP.2007.4379949` |
| .. [3] Najman, L., & Couprie, M. (2006). Building the component tree in |
| quasi-linear time. IEEE Transactions on Image Processing, 15(11), |
| 3531-3539. |
| :DOI:`10.1109/TIP.2006.877518` |
| .. [4] Carlinet, E., & Geraud, T. (2014). A Comparative Review of |
| Component Tree Computation Algorithms. IEEE Transactions on Image |
| Processing, 23(9), 3885-3895. |
| :DOI:`10.1109/TIP.2014.2336551` |
| |
| Examples |
| -------- |
| We create a small sample image (Figure 1 from [4]) and build the max-tree. |
| |
| >>> image = np.array([[15, 13, 16], [12, 12, 10], [16, 12, 14]]) |
| >>> P, S = max_tree(image, connectivity=2) |
| """ |
| |
| |
| |
| |
| mask = np.ones(image.shape) |
| for k in range(len(image.shape)): |
| np.moveaxis(mask, k, 0)[0] = 0 |
| np.moveaxis(mask, k, 0)[-1] = 0 |
|
|
| neighbors, offset = _validate_connectivity(image.ndim, connectivity, offset=None) |
|
|
| |
| parent = np.zeros(image.shape, dtype=np.int64) |
|
|
| |
| |
| flat_neighborhood = _offsets_to_raveled_neighbors( |
| image.shape, neighbors, offset |
| ).astype(np.int32) |
|
|
| |
| tree_traverser = np.argsort(image.ravel(), kind="stable").astype(np.int64) |
|
|
| |
| _max_tree._max_tree( |
| image.ravel(), |
| mask.ravel().astype(np.uint8), |
| flat_neighborhood, |
| offset.astype(np.int32), |
| np.array(image.shape, dtype=np.int32), |
| parent.ravel(), |
| tree_traverser, |
| ) |
|
|
| return parent, tree_traverser |
|
|
|
|
| def area_opening( |
| image, area_threshold=64, connectivity=1, parent=None, tree_traverser=None |
| ): |
| """Perform an area opening of the image. |
| |
| Area opening removes all bright structures of an image with |
| a surface smaller than area_threshold. |
| The output image is thus the largest image smaller than the input |
| for which all local maxima have at least a surface of |
| area_threshold pixels. |
| |
| Area openings are similar to morphological openings, but |
| they do not use a fixed footprint, but rather a deformable |
| one, with surface = area_threshold. Consequently, the area_opening |
| with area_threshold=1 is the identity. |
| |
| In the binary case, area openings are equivalent to |
| remove_small_objects; this operator is thus extended to gray-level images. |
| |
| Technically, this operator is based on the max-tree representation of |
| the image. |
| |
| Parameters |
| ---------- |
| image : ndarray |
| The input image for which the area_opening is to be calculated. |
| This image can be of any type. |
| area_threshold : unsigned int |
| The size parameter (number of pixels). The default value is arbitrarily |
| chosen to be 64. |
| connectivity : unsigned int, optional |
| The neighborhood connectivity. The integer represents the maximum |
| number of orthogonal steps to reach a neighbor. In 2D, it is 1 for |
| a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1. |
| parent : ndarray, int64, optional |
| Parent image representing the max tree of the image. The |
| value of each pixel is the index of its parent in the ravelled array. |
| tree_traverser : 1D array, int64, optional |
| The ordered pixel indices (referring to the ravelled array). The pixels |
| are ordered such that every pixel is preceded by its parent (except for |
| the root which has no parent). |
| |
| Returns |
| ------- |
| output : ndarray |
| Output image of the same shape and type as the input image. |
| |
| See Also |
| -------- |
| skimage.morphology.area_closing |
| skimage.morphology.diameter_opening |
| skimage.morphology.diameter_closing |
| skimage.morphology.max_tree |
| skimage.morphology.remove_small_objects |
| skimage.morphology.remove_small_holes |
| |
| References |
| ---------- |
| .. [1] Vincent L., Proc. "Grayscale area openings and closings, |
| their efficient implementation and applications", |
| EURASIP Workshop on Mathematical Morphology and its |
| Applications to Signal Processing, Barcelona, Spain, pp.22-27, |
| May 1993. |
| .. [2] Soille, P., "Morphological Image Analysis: Principles and |
| Applications" (Chapter 6), 2nd edition (2003), ISBN 3540429883. |
| :DOI:`10.1007/978-3-662-05088-0` |
| .. [3] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive |
| Connected Operators for Image and Sequence Processing. |
| IEEE Transactions on Image Processing, 7(4), 555-570. |
| :DOI:`10.1109/83.663500` |
| .. [4] Najman, L., & Couprie, M. (2006). Building the component tree in |
| quasi-linear time. IEEE Transactions on Image Processing, 15(11), |
| 3531-3539. |
| :DOI:`10.1109/TIP.2006.877518` |
| .. [5] Carlinet, E., & Geraud, T. (2014). A Comparative Review of |
| Component Tree Computation Algorithms. IEEE Transactions on Image |
| Processing, 23(9), 3885-3895. |
| :DOI:`10.1109/TIP.2014.2336551` |
| |
| Examples |
| -------- |
| We create an image (quadratic function with a maximum in the center and |
| 4 additional local maxima. |
| |
| >>> w = 12 |
| >>> x, y = np.mgrid[0:w,0:w] |
| >>> f = 20 - 0.2*((x - w/2)**2 + (y-w/2)**2) |
| >>> f[2:3,1:5] = 40; f[2:4,9:11] = 60; f[9:11,2:4] = 80 |
| >>> f[9:10,9:11] = 100; f[10,10] = 100 |
| >>> f = f.astype(int) |
| |
| We can calculate the area opening: |
| |
| >>> open = area_opening(f, 8, connectivity=1) |
| |
| The peaks with a surface smaller than 8 are removed. |
| """ |
| output = image.copy() |
|
|
| if parent is None or tree_traverser is None: |
| parent, tree_traverser = max_tree(image, connectivity) |
|
|
| area = _max_tree._compute_area(image.ravel(), parent.ravel(), tree_traverser) |
|
|
| _max_tree._direct_filter( |
| image.ravel(), |
| output.ravel(), |
| parent.ravel(), |
| tree_traverser, |
| area, |
| area_threshold, |
| ) |
| return output |
|
|
|
|
| def diameter_opening( |
| image, diameter_threshold=8, connectivity=1, parent=None, tree_traverser=None |
| ): |
| """Perform a diameter opening of the image. |
| |
| Diameter opening removes all bright structures of an image with |
| maximal extension smaller than diameter_threshold. The maximal |
| extension is defined as the maximal extension of the bounding box. |
| The operator is also called Bounding Box Opening. In practice, |
| the result is similar to a morphological opening, but long and thin |
| structures are not removed. |
| |
| Technically, this operator is based on the max-tree representation of |
| the image. |
| |
| Parameters |
| ---------- |
| image : ndarray |
| The input image for which the area_opening is to be calculated. |
| This image can be of any type. |
| diameter_threshold : unsigned int |
| The maximal extension parameter (number of pixels). The default value |
| is 8. |
| connectivity : unsigned int, optional |
| The neighborhood connectivity. The integer represents the maximum |
| number of orthogonal steps to reach a neighbor. In 2D, it is 1 for |
| a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1. |
| parent : ndarray, int64, optional |
| Parent image representing the max tree of the image. The |
| value of each pixel is the index of its parent in the ravelled array. |
| tree_traverser : 1D array, int64, optional |
| The ordered pixel indices (referring to the ravelled array). The pixels |
| are ordered such that every pixel is preceded by its parent (except for |
| the root which has no parent). |
| |
| Returns |
| ------- |
| output : ndarray |
| Output image of the same shape and type as the input image. |
| |
| See Also |
| -------- |
| skimage.morphology.area_opening |
| skimage.morphology.area_closing |
| skimage.morphology.diameter_closing |
| skimage.morphology.max_tree |
| |
| References |
| ---------- |
| .. [1] Walter, T., & Klein, J.-C. (2002). Automatic Detection of |
| Microaneurysms in Color Fundus Images of the Human Retina by Means |
| of the Bounding Box Closing. In A. Colosimo, P. Sirabella, |
| A. Giuliani (Eds.), Medical Data Analysis. Lecture Notes in Computer |
| Science, vol 2526, pp. 210-220. Springer Berlin Heidelberg. |
| :DOI:`10.1007/3-540-36104-9_23` |
| .. [2] Carlinet, E., & Geraud, T. (2014). A Comparative Review of |
| Component Tree Computation Algorithms. IEEE Transactions on Image |
| Processing, 23(9), 3885-3895. |
| :DOI:`10.1109/TIP.2014.2336551` |
| |
| Examples |
| -------- |
| We create an image (quadratic function with a maximum in the center and |
| 4 additional local maxima. |
| |
| >>> w = 12 |
| >>> x, y = np.mgrid[0:w,0:w] |
| >>> f = 20 - 0.2*((x - w/2)**2 + (y-w/2)**2) |
| >>> f[2:3,1:5] = 40; f[2:4,9:11] = 60; f[9:11,2:4] = 80 |
| >>> f[9:10,9:11] = 100; f[10,10] = 100 |
| >>> f = f.astype(int) |
| |
| We can calculate the diameter opening: |
| |
| >>> open = diameter_opening(f, 3, connectivity=1) |
| |
| The peaks with a maximal extension of 2 or less are removed. |
| The remaining peaks have all a maximal extension of at least 3. |
| """ |
| output = image.copy() |
|
|
| if parent is None or tree_traverser is None: |
| parent, tree_traverser = max_tree(image, connectivity) |
|
|
| diam = _max_tree._compute_extension( |
| image.ravel(), |
| np.array(image.shape, dtype=np.int32), |
| parent.ravel(), |
| tree_traverser, |
| ) |
|
|
| _max_tree._direct_filter( |
| image.ravel(), |
| output.ravel(), |
| parent.ravel(), |
| tree_traverser, |
| diam, |
| diameter_threshold, |
| ) |
| return output |
|
|
|
|
| def area_closing( |
| image, area_threshold=64, connectivity=1, parent=None, tree_traverser=None |
| ): |
| """Perform an area closing of the image. |
| |
| Area closing removes all dark structures of an image with |
| a surface smaller than area_threshold. |
| The output image is larger than or equal to the input image |
| for every pixel and all local minima have at least a surface of |
| area_threshold pixels. |
| |
| Area closings are similar to morphological closings, but |
| they do not use a fixed footprint, but rather a deformable |
| one, with surface = area_threshold. |
| |
| In the binary case, area closings are equivalent to |
| remove_small_holes; this operator is thus extended to gray-level images. |
| |
| Technically, this operator is based on the max-tree representation of |
| the image. |
| |
| Parameters |
| ---------- |
| image : ndarray |
| The input image for which the area_closing is to be calculated. |
| This image can be of any type. |
| area_threshold : unsigned int |
| The size parameter (number of pixels). The default value is arbitrarily |
| chosen to be 64. |
| connectivity : unsigned int, optional |
| The neighborhood connectivity. The integer represents the maximum |
| number of orthogonal steps to reach a neighbor. In 2D, it is 1 for |
| a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1. |
| parent : ndarray, int64, optional |
| Parent image representing the max tree of the inverted image. The |
| value of each pixel is the index of its parent in the ravelled array. |
| See Note for further details. |
| tree_traverser : 1D array, int64, optional |
| The ordered pixel indices (referring to the ravelled array). The pixels |
| are ordered such that every pixel is preceded by its parent (except for |
| the root which has no parent). |
| |
| Returns |
| ------- |
| output : ndarray |
| Output image of the same shape and type as input image. |
| |
| See Also |
| -------- |
| skimage.morphology.area_opening |
| skimage.morphology.diameter_opening |
| skimage.morphology.diameter_closing |
| skimage.morphology.max_tree |
| skimage.morphology.remove_small_objects |
| skimage.morphology.remove_small_holes |
| |
| References |
| ---------- |
| .. [1] Vincent L., Proc. "Grayscale area openings and closings, |
| their efficient implementation and applications", |
| EURASIP Workshop on Mathematical Morphology and its |
| Applications to Signal Processing, Barcelona, Spain, pp.22-27, |
| May 1993. |
| .. [2] Soille, P., "Morphological Image Analysis: Principles and |
| Applications" (Chapter 6), 2nd edition (2003), ISBN 3540429883. |
| :DOI:`10.1007/978-3-662-05088-0` |
| .. [3] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive |
| Connected Operators for Image and Sequence Processing. |
| IEEE Transactions on Image Processing, 7(4), 555-570. |
| :DOI:`10.1109/83.663500` |
| .. [4] Najman, L., & Couprie, M. (2006). Building the component tree in |
| quasi-linear time. IEEE Transactions on Image Processing, 15(11), |
| 3531-3539. |
| :DOI:`10.1109/TIP.2006.877518` |
| .. [5] Carlinet, E., & Geraud, T. (2014). A Comparative Review of |
| Component Tree Computation Algorithms. IEEE Transactions on Image |
| Processing, 23(9), 3885-3895. |
| :DOI:`10.1109/TIP.2014.2336551` |
| |
| Examples |
| -------- |
| We create an image (quadratic function with a minimum in the center and |
| 4 additional local minima. |
| |
| >>> w = 12 |
| >>> x, y = np.mgrid[0:w,0:w] |
| >>> f = 180 + 0.2*((x - w/2)**2 + (y-w/2)**2) |
| >>> f[2:3,1:5] = 160; f[2:4,9:11] = 140; f[9:11,2:4] = 120 |
| >>> f[9:10,9:11] = 100; f[10,10] = 100 |
| >>> f = f.astype(int) |
| |
| We can calculate the area closing: |
| |
| >>> closed = area_closing(f, 8, connectivity=1) |
| |
| All small minima are removed, and the remaining minima have at least |
| a size of 8. |
| |
| Notes |
| ----- |
| If a max-tree representation (parent and tree_traverser) are given to the |
| function, they must be calculated from the inverted image for this |
| function, i.e.: |
| >>> P, S = max_tree(invert(f)) |
| >>> closed = diameter_closing(f, 3, parent=P, tree_traverser=S) |
| """ |
| |
| image_inv = invert(image) |
| output = image_inv.copy() |
|
|
| if parent is None or tree_traverser is None: |
| parent, tree_traverser = max_tree(image_inv, connectivity) |
|
|
| area = _max_tree._compute_area(image_inv.ravel(), parent.ravel(), tree_traverser) |
|
|
| _max_tree._direct_filter( |
| image_inv.ravel(), |
| output.ravel(), |
| parent.ravel(), |
| tree_traverser, |
| area, |
| area_threshold, |
| ) |
|
|
| |
| output = invert(output) |
|
|
| return output |
|
|
|
|
| def diameter_closing( |
| image, diameter_threshold=8, connectivity=1, parent=None, tree_traverser=None |
| ): |
| """Perform a diameter closing of the image. |
| |
| Diameter closing removes all dark structures of an image with |
| maximal extension smaller than diameter_threshold. The maximal |
| extension is defined as the maximal extension of the bounding box. |
| The operator is also called Bounding Box Closing. In practice, |
| the result is similar to a morphological closing, but long and thin |
| structures are not removed. |
| |
| Technically, this operator is based on the max-tree representation of |
| the image. |
| |
| Parameters |
| ---------- |
| image : ndarray |
| The input image for which the diameter_closing is to be calculated. |
| This image can be of any type. |
| diameter_threshold : unsigned int |
| The maximal extension parameter (number of pixels). The default value |
| is 8. |
| connectivity : unsigned int, optional |
| The neighborhood connectivity. The integer represents the maximum |
| number of orthogonal steps to reach a neighbor. In 2D, it is 1 for |
| a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1. |
| parent : ndarray, int64, optional |
| Precomputed parent image representing the max tree of the inverted |
| image. This function is fast, if precomputed parent and tree_traverser |
| are provided. See Note for further details. |
| tree_traverser : 1D array, int64, optional |
| Precomputed traverser, where the pixels are ordered such that every |
| pixel is preceded by its parent (except for the root which has no |
| parent). This function is fast, if precomputed parent and |
| tree_traverser are provided. See Note for further details. |
| |
| Returns |
| ------- |
| output : ndarray |
| Output image of the same shape and type as input image. |
| |
| See Also |
| -------- |
| skimage.morphology.area_opening |
| skimage.morphology.area_closing |
| skimage.morphology.diameter_opening |
| skimage.morphology.max_tree |
| |
| References |
| ---------- |
| .. [1] Walter, T., & Klein, J.-C. (2002). Automatic Detection of |
| Microaneurysms in Color Fundus Images of the Human Retina by Means |
| of the Bounding Box Closing. In A. Colosimo, P. Sirabella, |
| A. Giuliani (Eds.), Medical Data Analysis. Lecture Notes in Computer |
| Science, vol 2526, pp. 210-220. Springer Berlin Heidelberg. |
| :DOI:`10.1007/3-540-36104-9_23` |
| .. [2] Carlinet, E., & Geraud, T. (2014). A Comparative Review of |
| Component Tree Computation Algorithms. IEEE Transactions on Image |
| Processing, 23(9), 3885-3895. |
| :DOI:`10.1109/TIP.2014.2336551` |
| |
| Examples |
| -------- |
| We create an image (quadratic function with a minimum in the center and |
| 4 additional local minima. |
| |
| >>> w = 12 |
| >>> x, y = np.mgrid[0:w,0:w] |
| >>> f = 180 + 0.2*((x - w/2)**2 + (y-w/2)**2) |
| >>> f[2:3,1:5] = 160; f[2:4,9:11] = 140; f[9:11,2:4] = 120 |
| >>> f[9:10,9:11] = 100; f[10,10] = 100 |
| >>> f = f.astype(int) |
| |
| We can calculate the diameter closing: |
| |
| >>> closed = diameter_closing(f, 3, connectivity=1) |
| |
| All small minima with a maximal extension of 2 or less are removed. |
| The remaining minima have all a maximal extension of at least 3. |
| |
| Notes |
| ----- |
| If a max-tree representation (parent and tree_traverser) are given to the |
| function, they must be calculated from the inverted image for this |
| function, i.e.: |
| >>> P, S = max_tree(invert(f)) |
| >>> closed = diameter_closing(f, 3, parent=P, tree_traverser=S) |
| """ |
| |
| image_inv = invert(image) |
| output = image_inv.copy() |
|
|
| if parent is None or tree_traverser is None: |
| parent, tree_traverser = max_tree(image_inv, connectivity) |
|
|
| diam = _max_tree._compute_extension( |
| image_inv.ravel(), |
| np.array(image_inv.shape, dtype=np.int32), |
| parent.ravel(), |
| tree_traverser, |
| ) |
|
|
| _max_tree._direct_filter( |
| image_inv.ravel(), |
| output.ravel(), |
| parent.ravel(), |
| tree_traverser, |
| diam, |
| diameter_threshold, |
| ) |
| output = invert(output) |
| return output |
|
|
|
|
| def max_tree_local_maxima(image, connectivity=1, parent=None, tree_traverser=None): |
| """Determine all local maxima of the image. |
| |
| The local maxima are defined as connected sets of pixels with equal |
| gray level strictly greater than the gray levels of all pixels in direct |
| neighborhood of the set. The function labels the local maxima. |
| |
| Technically, the implementation is based on the max-tree representation |
| of an image. The function is very efficient if the max-tree representation |
| has already been computed. Otherwise, it is preferable to use |
| the function local_maxima. |
| |
| Parameters |
| ---------- |
| image : ndarray |
| The input image for which the maxima are to be calculated. |
| connectivity : unsigned int, optional |
| The neighborhood connectivity. The integer represents the maximum |
| number of orthogonal steps to reach a neighbor. In 2D, it is 1 for |
| a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1. |
| parent : ndarray, int64, optional |
| The value of each pixel is the index of its parent in the ravelled |
| array. |
| tree_traverser : 1D array, int64, optional |
| The ordered pixel indices (referring to the ravelled array). The pixels |
| are ordered such that every pixel is preceded by its parent (except for |
| the root which has no parent). |
| |
| Returns |
| ------- |
| local_max : ndarray, uint64 |
| Labeled local maxima of the image. |
| |
| See Also |
| -------- |
| skimage.morphology.local_maxima |
| skimage.morphology.max_tree |
| |
| References |
| ---------- |
| .. [1] Vincent L., Proc. "Grayscale area openings and closings, |
| their efficient implementation and applications", |
| EURASIP Workshop on Mathematical Morphology and its |
| Applications to Signal Processing, Barcelona, Spain, pp.22-27, |
| May 1993. |
| .. [2] Soille, P., "Morphological Image Analysis: Principles and |
| Applications" (Chapter 6), 2nd edition (2003), ISBN 3540429883. |
| :DOI:`10.1007/978-3-662-05088-0` |
| .. [3] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive |
| Connected Operators for Image and Sequence Processing. |
| IEEE Transactions on Image Processing, 7(4), 555-570. |
| :DOI:`10.1109/83.663500` |
| .. [4] Najman, L., & Couprie, M. (2006). Building the component tree in |
| quasi-linear time. IEEE Transactions on Image Processing, 15(11), |
| 3531-3539. |
| :DOI:`10.1109/TIP.2006.877518` |
| .. [5] Carlinet, E., & Geraud, T. (2014). A Comparative Review of |
| Component Tree Computation Algorithms. IEEE Transactions on Image |
| Processing, 23(9), 3885-3895. |
| :DOI:`10.1109/TIP.2014.2336551` |
| |
| Examples |
| -------- |
| We create an image (quadratic function with a maximum in the center and |
| 4 additional constant maxima. |
| |
| >>> w = 10 |
| >>> x, y = np.mgrid[0:w,0:w] |
| >>> f = 20 - 0.2*((x - w/2)**2 + (y-w/2)**2) |
| >>> f[2:4,2:4] = 40; f[2:4,7:9] = 60; f[7:9,2:4] = 80; f[7:9,7:9] = 100 |
| >>> f = f.astype(int) |
| |
| We can calculate all local maxima: |
| |
| >>> maxima = max_tree_local_maxima(f) |
| |
| The resulting image contains the labeled local maxima. |
| """ |
|
|
| output = np.ones(image.shape, dtype=np.uint64) |
|
|
| if parent is None or tree_traverser is None: |
| parent, tree_traverser = max_tree(image, connectivity) |
|
|
| _max_tree._max_tree_local_maxima( |
| image.ravel(), output.ravel(), parent.ravel(), tree_traverser |
| ) |
|
|
| return output |
|
|
