9igizSrSSKrSSKJrJr SSKJr SSKJ r \R\R\R\R/r\R\R \R"\R$/r\R(\R*\R,/rSSjrSS jrSS jrSS jrSS jrSS jrg)a[max_tree.py - max_tree representation of images. This module provides operators based on the max-tree representation of images. A grayscale image can be seen as a pile of nested sets, each of which is the result of a threshold operation. These sets can be efficiently represented by max-trees, where the inclusion relation between connected components at different levels are represented by parent-child relationships. These representations allow efficient implementations of many algorithms, such as attribute operators. Unlike morphological openings and closings, these operators do not require a fixed footprint, but rather act with a flexible footprint that meets a certain criterion. This implementation provides functions for: 1. max-tree generation 2. area openings / closings 3. diameter openings / closings 4. local maxima References: .. [1] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive Connected Operators for Image and Sequence Processing. IEEE Transactions on Image Processing, 7(4), 555-570. :DOI:`10.1109/83.663500` .. [2] Berger, C., Geraud, T., Levillain, R., Widynski, N., Baillard, A., Bertin, E. (2007). Effective Component Tree Computation with Application to Pattern Recognition in Astronomical Imaging. In International Conference on Image Processing (ICIP) (pp. 41-44). :DOI:`10.1109/ICIP.2007.4379949` .. [3] Najman, L., & Couprie, M. (2006). Building the component tree in quasi-linear time. IEEE Transactions on Image Processing, 15(11), 3531-3539. :DOI:`10.1109/TIP.2006.877518` .. [4] Carlinet, E., & Geraud, T. (2014). A Comparative Review of Component Tree Computation Algorithms. IEEE Transactions on Image Processing, 23(9), 3885-3895. :DOI:`10.1109/TIP.2014.2336551` N)_validate_connectivity_offsets_to_raveled_neighbors)invert) _max_treec [R"UR5n[[ UR55H7nS[R "X#S5S'S[R "X#S5S'M9 [ URUSS9upE[R"UR[RS9n[URXE5R[R5n[R"UR5SS9R[R5n[R"UR5UR5R[R 5UUR[R5[R""UR[RS9UR5U5 Xh4$)a! 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) rN)offsetdtypestable)kind)nponesshaperangelenmoveaxisrndimzerosint64rastypeint32argsortravelruint8array) image connectivitymaskk neighborsr parentflat_neighborhoodtree_traversers [/var/www/html/land-doc-ocr/venv/lib/python3.13/site-packages/skimage/morphology/max_tree.pymax_treer(5sI@ 775;; D 3u{{# $%& DQ"&' DQ#%/uzzUR5nUbUc [X5up4[R"UR 5UR 5U5n[R "UR 5UR 5UR 5UUU5 U$)a 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. )copyr(r _compute_arear_direct_filter)rarea_thresholdr r$r&outputareas r' area_openingr1sBZZ\F ~/!)%!>  " "5;;=&,,.. QD       Mr)cUR5nUbUc [X5up4[R"UR 5[ R "UR[ RS9UR 5U5n[R"UR 5UR 5UR 5UUU5 U$)a 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. r ) r+r(r_compute_extensionrrrrrr-)rdiameter_thresholdr r$r&r/diams r'diameter_openingr6 s^ZZ\F ~/!)%!>  ' '  BHH-   D      Mr)cj[U5nUR5nUbUc [XR5up4[R"UR 5UR 5U5n[R "UR 5UR 5UR 5UUU5 [U5nU$)aPerform 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) )rr+r(rr,rr-)rr.r r$r& image_invr/r0s r' area_closingr9osVu I ^^ F ~/!))!B  " "9??#4fllnn UD     F^F Mr)c[U5nUR5nUbUc [XR5up4[R"UR 5[ R"UR[ RS9UR 5U5n[R"UR 5UR 5UR 5UUU5 [U5nU$)a 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) r ) rr+r(rr3rrrrrr-)rr4r r$r&r8r/r5s r'diameter_closingr;stu I ^^ F ~/!))!B  ' ' 1   D    F^F Mr)c [R"UR[RS9nUbUc [ X5up#[ R "UR5UR5UR5U5 U$)aC 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. r )rrruint64r(r_max_tree_local_maximar)rr r$r&r/s r'max_tree_local_maximar?dsd^WWU[[ 2F ~/!)%!> $$ v||~v||~~ Mr))r)@rNN)rNN)rNN)__doc__numpyr_utilrrutilrrruint16uint32r=unsigned_int_typesint8int16rrsigned_int_typesfloat16float32float64signed_float_typesr(r1r6r9r;r?r)r'rRs%NHhh 299bii@GGRXXrxx:jj"**bjj9^"DKOphNRcNKOFNRpfXr)