Geometry

Last modified: June 09, 2010

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delaunay_from_points

object delaunay_from_points ([object points], IntVector labels)

Returns:object
Category:Geometry
Defined in:geometry.py
Author:Oliver Christen (based on code by Olivier Devillers)

Computes the Delaunay triangulation directly from a list of points and point labels. The result is a list which contains tuples of adjacent labels, where in each tuple the smaller label is given first.

The arguments points and labels specify the points and nonnegative labels, such that labels[i] is the label of points[i]. Note that the labels need not necessarily all be different, which can be useful for the computation of a neighborship graph from a set of connected components.

The computation of the Delaunay triangulation is based on the Delaunay tree which is a randomized geometric data structure. It is described in O. Devillers, S. Meiser, M. Teillaud: Fully dynamic Delaunay triangulation in logarithmic expected time per operation. Computational Geometry: Theory and Applications 2, pp. 55-80, 1992.

This can be useful for building a neighborhood graph as shown in the following example:

from gamera import graph
from gamera.plugins.geometry import delaunay_from_points

points = [(10,10),(20,30),(32,22),(85,14),(40,70),(80,85)]
labels = range(len(points))
neighbors = delaunay_from_points(points, labels)

g = graph.Graph(graph.UNDIRECTED)
for pair in neighbors:
    g.add_edge(pair[0], pair[1])

labeled_region_neighbors

object labeled_region_neighbors (bool eight_connectivity = True)

Operates on:Image [OneBit]
Returns:object
Category:Geometry
Defined in:geometry.py
Author:Christoph Dalitz

For an image containing labeled regions, a list of all label pairs belonging to touching regions is returned. When eight_connectivity is True (default), 8-connectivity is used for neighborship, otherwise 4-connectivity is used.

This can be useful for building a Delaunay graph from a Voronoi tesselation as in the following example:

#
# build Delaunay graph of neighboring (i.e. adjacent) Cc's
#

# create map label->Cc for faster lookup later
ccs = image.cc_analysis()
label_to_cc = {}
for c in ccs:
   label_to_cc[c.label] = c

# compute area Voronoi tesselation and neighborship list
voronoi = image.voronoi_from_labeled_image()
labelpairs = voronoi.labeled_region_neighbors()

# build map of all neighbors for each label for fast lookup
neighbors = {}
for label in label_to_cc.keys():
   neighbors[label] = []
for n in labelpairs:
   neighbors[n[0]].append(n[1])
   neighbors[n[1]].append(n[0])

# now, all neighbors to a given cc can be looked up with
neighbors_of_cc = [label_to_cc[label] for label in neighbors[cc.label]]

voronoi_from_labeled_image

Image [OneBit|GreyScale] voronoi_from_labeled_image ()

Operates on:Image [OneBit|GreyScale]
Returns:Image [OneBit|GreyScale]
Category:Geometry
Defined in:geometry.py
Author:Christoph Dalitz, based on code by Ullrich Köthe

Computes the area Voronoi tesselation from an image containing labeled Cc's. In the returned onebit image, every pixel is labeled with the label value of the closest Cc in the input image.

To prepare the input image, you can use cc_analysis. When the Cc's only consist of single points, the area Voronoi tesselation is identical to the ordinary Voronoi tesselation.

The implementation applies a watershed algorithm to the distance transform of the input image, a method known as seeded region growing (U. Köthe: Primary Image Segmentation. Proceedings 17th DAGM-Symposium, pp. 554-561, Springer, 1995).

The example shown below is the image voronoi_cells as created with the the following code:

# create an area Voronoi tesselation and
# mark the cells and their edges in color
ccs = image.cc_analysis()  # labels the image
voronoi = image.voronoi_from_labeled_image()
voronoi_cells = voronoi.color_ccs()
voronoi_cells.highlight(image, RGBPixel(0,0,0))
voronoi_edges = voronoi.labeled_region_edges()
voronoi_cells.highlight(voronoi_edges, RGBPixel(255,255,255))

Example 1: voronoi_from_labeled_image

voronoi_from_labeled_image_plugin_00_00 voronoi_from_labeled_image_plugin_00_01

voronoi_from_points

voronoi_from_points ([object points], IntVector labels)

Operates on:Image [OneBit|GreyScale]
Category:Geometry
Defined in:geometry.py
Author:Christoph Dalitz

Computes the Voronoi tesselation from a list of points and point labels. The result is directly written to the input image. Each white pixel is labeled with the label value of the closest point. Non white pixel in the input image are not overwritten.

The arguments points and labels specify the points and labels, such that labels[i] is the label of points[i]. Note that the labels need not necessarily all be different, which can be useful as an approximation of an area Voronoi tesselation.

The algorithm is very simple: it stores the points in a kd-tree and then looks up the closest point for each image pixel. This has a runtime of O(N log(n)), where N is the number of image pixels and n is the number of points. For not too many points, this should be faster than the morphological region growing approach of voronoi_from_labeled_image.

The example shown below is the image voronoi_edges as created with the the following code:

# create a Voronoi tesselation and mark
# the cell edges in a second image
points = [(10,10),(20,30),(32,22),(85,14),(40,70),(80,85)]
voronoi = Image((0,0),(90,90))
voronoi.voronoi_from_points(points,[i+2 for i in range(len(points))])
voronoi_edges = voronoi.labeled_region_edges()
for p in points:
   voronoi_edges.set(p,1)

Example 1: voronoi_from_points

voronoi_from_points_plugin_00_00