MAP GENERALIZATION BY SKELETON RETRACTION
Crust and Anti-Crust: A One-Step Boundary and Skeleton Extraction Algorithm
ADVANTAGES OF THE VORONOI SPATIAL MODEL
C++ implementation of Quad Arc data structure
Fortune's algorithm in Java Script by Raymond Hill
GitHub for Raymond Hill's Javascript
Fortune’s algorithm and implementation
Simplification of objects for adaptive matching and visualization
scipy spacial library "The Delaunay triangulation is a subdivision of a set of points into a non-overlapping set of triangles, such that no point is inside the circumcircle of any triangle. In practice, such triangulations tend to avoid triangles with small angles."
from scipy.spatial import Delaunay
points = np.array([[0, 0], [0, 1.1], [1, 0], [1, 1]])
tri = Delaunay(points)
"scipy.spatial can compute triangulations, Voronoi diagrams, and convex hulls of a set of points, by leveraging the Qhull library."
*note: using QHull "http://qhull.org/html/qvoronoi.htm - Qhull computes the Voronoi diagram via the Delaunay triangulation."*
"To simplify geometry to suit the displayed resolution, various line simplification algorithms exist. While Douglas–Peucker is the most well-known, Visvalingam’s algorithm may be more effective and has a remarkably intuitive explanation: it progressively removes points with the least-perceptible change."
In [126]:
from scipy.spatial import Delaunay, Voronoi, voronoi_plot_2d
import numpy as np
points = np.array([[0, 0], [0, 1.1], [1, 0], [1, 1], [0.5, 0.25], [0.74, 1.3], [0.4, 0.13], \
[0.04, 1.3], [0.24, 0.213], [0.54, 0.83]])
tri = Delaunay(points)
In [127]:
%matplotlib inline
import matplotlib.pyplot as plt
plt.triplot(points[:,0], points[:,1], tri.simplices.copy())
for j, p in enumerate(points):
plt.text(p[0]-0.03, p[1]+0.03, j, ha='right') # label the points
for j, s in enumerate(tri.simplices):
p = points[s].mean(axis=0)
plt.text(p[0], p[1], '#%d' % j, ha='center') # label triangles
plt.xlim(-0.5, 1.5); plt.ylim(-0.5, 1.5)
plt.show()
In [128]:
vor = Voronoi(points)
voronoi_plot_2d(vor)
plt.show()
In [6]:
from IPython.display import IFrame
IFrame("https://en.wikipedia.org/wiki/Fortune's_algorithm#Pseudocode", width=850, height=800)
Out[6]:
In [7]:
from IPython.display import IFrame
IFrame('http://www.raymondhill.net/voronoi/rhill-voronoi.html', width=850, height=800)
Out[7]:
GRASS GIS "Geographic Resources Analysis Support System (commonly termed GRASS GIS) is a geographic information system (GIS) software suite used for geospatial data management and analysis, image processing, producing graphics and maps, spatial and temporal modeling, and visualizing." note: Written in C, C++, Python, Tcl
import sys
from grass.script import core as grass
def main():
drape_map = options['drapemap']
relief_map = options['reliefmap']
brighten = options['brighten']
ret = grass.run_command("d.his", h_map = drape_map, i_map = relief_map, brighten = brighten)
sys.exit(ret)
if __name__ == "__main__":
options, flags = grass.parser()
main()
Capaware "Capaware is a 3D general purpose virtual worlds viewer." home page
NHD "The National Hydrography Dataset (NHD) and Watershed Boundary Dataset (WBD) are used to portray surface water on The National Map."
OpenStreetMap "OpenStreetMap (OSM) is a collaborative project to create a free editable map of the world." home page
Table 1: Quad-Edge Code
{This table gives an object-oriented version of Guibas and Stolfi’s}
{QuakEdgo data structure. its basic functions and example usage.}
TQuad = class
N:TQuad; {next edgeantklockwise}
R:TQuad; {next 1/4 of edge}
V : TPotnt; {vertex}
Index : Interger; {'name' for debugging}
end;
class function TQuad.MakeEdge(Orig, Dest: TPoint) : TQuad
VU
Q0, Q1, Q2, Q3 : TQuad;
begin
(create four new 1/4 edges)
Q0 := TQuad.Create; Q1 := TQuad.Create;
Q2 := TQuad.Create; Q3 := TQuad.Create;
{link the four parts}
Q0.R := Q1; Q1.R := Q2; Q2.R := Q3;
Q3.R := Q0;
(link 0 a 2 to themeelves, 1 a 3 to each other}
Q0.N := Q0; Q1.N := 03; Q2.N := Q2;
Q3.N := Q1;
(met pointers to vertices)
Q0.SetVertex(Origin); Q2.SetVertex(Dest); Result := Q0;
end;
procedure TQuad.Splice(A,B : TQuad); (A B: input Quad-Edges)
var
Alpha, Beta, An. Bn. Aln, Ben : TQuad;
begin
{get neighbouring edges: Alpha a Beta in Guibas a Stolft}
Alpha := A.N.R; Beta := B.N.R;
An :=A.N; Bn := B.N; Aln := Alpha.N; Ben := Beta.N;
(reconnect the four points)
AN := Bn; B.N:= An; Alpha.N := Ben; Beta.N := Aln;
end;
function TQuad.Sym : TQuad; (other end)
begin
Sym := Self.R.R;
end;
function TQuad.Oprev : TOuad; (next edge clockwlse)
Oprev := Self.R.N.R;
end:
function TQuad.Onext : T&ad {next edge anti-cloclwise}
begin
Onext := Self.N;
end;
function TQuad.Lnext : TQuad; (next edge clockwise, other end)
begin
Lnext := Se1f.R.R.R.N.R;
end;
function TQuad.Rprev : TQuad; {next edgo antklockwlae, other end}
begin
Rprev := Self.R.R.N;
end;
functhn TQuad.Vertex : TPoint; {read vertex}
begin
Result := Self.V;
end;
procedure TQuad.SetVsrtox(Ptin Tpoint); (set vertex)
begin
V := Ptin;
end;
procedure TQuad.Delete; (disconnect and free an edge)
begin
Splice(Self. Self.Oprev);
splice(self. sym, sew. Lnext);
Self. Free;
end;
function TQuad.Swap : Boolean; (swap a diagonal in a triangulation)
var
a,b:TQuad;
begin
Result := False
a := Self.Oprev; {get adjacent edges}
b := Self.Lnext;
if (a.Sym.Vertex <> b.Sym.Vertex) then begin
Result := True;
Splke(Self, a); {disconnect diagonal}
Splice(Self.Sym, b);
Splke(Self. aLnext); {reconnect diagonal}
Splice(Setf.Sym, b.Lnext);
SeW.SetVertex(a.SymVertex);
(redefine vertices)
Self.Sym.SetVertex(b.Sym.Vettex);
end;
end;
chris gold website - voronoi.com
possible existing implementation
quad-edge this is really good!
In [8]:
import scipy
class TQuad:
def __init__(self, N, R, V, Index):
self.N = N
self.R = R
self.V = V
self.Index = Index
def MakeEdge(self, Orig, Dest):
result = TQuad()
return
The issue with using a simple averaging of points is that the features are skewed in the direction you average. One idea is to average the points 'N' points at a time to simplify a feature set (usually for resolution or storage see discussion here).
In [18]:
orgpoints = np.array([[0, 0],[0.1, 0.1],[0.11,0.07],[0.2,0.3],[0.07,0.4],[0.06,0.2], [0.03,0.3], [0.06,0.6],[0.12,0.5],[0.15,0.51],[0.3,0.4],[0.5,0.3],[0.45,0.2],[0.35,0.25],[0.25,0.12], [0.4,0.1], [0.7,0.11] ])
#print(orgpoints[:,0])
plt.scatter(orgpoints[:,0],orgpoints[:,1])
plt.plot(orgpoints[:,0],orgpoints[:,1])
plt.show()
In [123]:
#Get the original points into lists which will act like point stacks
xs = orgpoints[:,0].tolist()
ys = orgpoints[:,1].tolist()
#mean lists will hold the mean averaged points
meanxsLR = []
meanysLR = []
#lets look at the Xs to see we got everything
print(xs)
#pr is the point reduction - '3' is reducing every 3 points to 1 or 1/3 of the points
pr = 3
while xs:
temp = []
for i in range(pr):
if xs:
temp.append(xs.pop())
xmean = sum(temp)/len(temp)
meanxsLR.append(xmean)
#print("temp %s, xmeam %s" % (temp, xmean)) #debug output
while ys:
temp = []
for i in range(pr):
if ys:
temp.append(ys.pop())
ymean = sum(temp)/len(temp)
meanysLR.append(ymean)
#print("temp %s, xmeam %s" % (temp, xmean)) #debug output
#lets plot the average on top of the original to see the changes
plt.scatter(orgpoints[:,0],orgpoints[:,1])
plt.plot(orgpoints[:,0],orgpoints[:,1])
plt.scatter(meanxsLR,meanysLR)
plt.plot(meanxsLR,meanysLR)
plt.show()
In [238]:
def lr_pointreduction(orgs, rf, showorgs=True):
#Get the original points into lists which will act like point stacks
xs = orgs[:,0].tolist()
ys = orgs[:,1].tolist()
#mean lists will hold the mean averaged points
meanxsLR = []
meanysLR = []
#rf is the point reduction - '3' is reducing every 3 points to 1 or 1/3 of the points
while xs:
temp = []
for i in range(rf):
if xs:
temp.append(xs.pop())
xmean = sum(temp)/len(temp)
meanxsLR.append(xmean)
while ys:
temp = []
for i in range(rf):
if ys:
temp.append(ys.pop())
ymean = sum(temp)/len(temp)
meanysLR.append(ymean)
#lets plot the average on top of the original to see the changes
ax = plt.axes([0, 0, 3, 3])
if showorgs:
plt.scatter(orgs[:,0],orgs[:,1])
plt.plot(orgs[:,0],orgs[:,1])
plt.scatter(meanxsLR,meanysLR)
plt.plot(meanxsLR,meanysLR)
plt.show()
In [122]:
#Now lets try averaging in the opposite direction
xs = orgpoints[:,0].tolist()
ys = orgpoints[:,1].tolist()
#This time lets reverse the lists to see if the direction we choose to average in makes a difference
xs.reverse()
ys.reverse()
#mean lists will hold the mean averaged points
meanxsRL = []
meanysRL = []
#lets look at the Xs to see we got everything
print(xs)
#pr is the point reduction - '3' is reducing every 3 points to 1 or 1/3 of the points
pr = 3
while xs:
temp = []
for i in range(pr):
if xs:
temp.append(xs.pop())
xmean = sum(temp)/len(temp)
meanxsRL.append(xmean)
#print("temp %s, xmeam %s" % (temp, xmean)) #debug output
while ys:
temp = []
for i in range(pr):
if ys:
temp.append(ys.pop())
ymean = sum(temp)/len(temp)
meanysRL.append(ymean)
#print("temp %s, xmeam %s" % (temp, xmean)) #debug output
#lets plot the average on top of the original to see the changes
plt.scatter(orgpoints[:,0],orgpoints[:,1])
plt.plot(orgpoints[:,0],orgpoints[:,1])
plt.scatter(meanxsRL,meanysRL)
plt.plot(meanxsRL,meanysRL)
plt.show()
In [124]:
#original in blue
plt.scatter(orgpoints[:,0],orgpoints[:,1])
plt.plot(orgpoints[:,0],orgpoints[:,1])
#Left to Right in Green
plt.scatter(meanxsLR,meanysLR)
plt.plot(meanxsLR,meanysLR)
#Right to Left in Red
plt.scatter(meanxsRL,meanysRL)
plt.plot(meanxsRL,meanysRL)
plt.show()
As you can see above the "averaged" graph doesn't retain any of the features of the original (the points are inset, or "shrunk-in", from the original data set). So how do we keep the original features, but reduce the number of points? This is where the skeletal relaxation method discussed in articles above comes into effect.
Below is the Voronoi of our graph
In [186]:
ax = plt.axes([0, 0, 3, 3], frame_on=True, xticks=[], yticks=[])
vor = Voronoi(orgpoints)
voronoi_plot_2d(vor,ax=ax)
plt.plot(orgpoints[:,0],orgpoints[:,1])
plt.show()
In [140]:
tri = Delaunay(orgpoints)
plt.plot(orgpoints[:,0],orgpoints[:,1])
plt.triplot(orgpoints[:,0], orgpoints[:,1], tri.simplices.copy())
plt.show()
In [145]:
vor = Voronoi(orgpoints)
voronoi_plot_2d(vor)
tri = Delaunay(orgpoints)
plt.plot(orgpoints[:,0],orgpoints[:,1])
plt.triplot(orgpoints[:,0], orgpoints[:,1], tri.simplices.copy())
plt.show()
In [146]:
#morepoints = np.array([[13.889598,-28.28427],[13.889597,-28.28427],[25.001275,-31.8198],[25.001276,-31.81981],[58.08377,53.28555],[58.08377,53.28554],[-32.82995,31.31473],[-32.82996,31.31473],[-35.1028,10.6066],[-35.102804,10.6066],[-17.930208,29.04189],[-17.930208,29.04189],[7.323606,32.82996],[7.323606,32.82995],[33.840108,15.90989],[33.84011,15.90989],[22.22335,-21.46573],[22.22336,-21.46573],[0,0],[-6.69226,-9.97525],[-13.76333,-19.95051],[-7.07107,-9.97525],[-14.52094,-19.95051],[-16.03617,-19.95051],[-1.51523,0],[-10.85914,-6.56599],[-19.82424,-13.13198],[-8.9651,-6.566],[-17.5514,-13.13199],[-17.5514,-13.13199],[9.84899,-14.89975],[9.84899,-14.89975],[31.31472,3.03046],[31.31473,3.03046],[36.61803,7.32361],[36.61803,7.3236],[16.16244,-10.85914],[16.16244,-10.85914],[-11.36422,-15.9099],[-11.36421,-15.90991],[-34.85026,6.566],[-34.85026,6.56599],[-7.32361,-6.56599],[-7.32361,-6.566],[7.57614,-11.11167],[7.57615,-11.11168],[0,0],[7.95495,-4.16688],[16.98319,-7.95495],[9.02824,-3.78807],[19.12976,-7.19734],[23.42291,-5.68211],[4.29315,1.51523],[23.10724,8.33376],[40.84805,14.77348],[17.7408,6.43973],[34.40832,12.50064],[34.40832,12.50064],[18.68782,10.85914],[18.68782,10.85914],[30.55711,-24.24366],[30.55712,-24.24367],[7.57614,-44.69925],[7.57615,-44.69925],[-22.72843,-26.76904],[-22.72843,-26.76904],[-51.51778,-8.58629],[-51.51778,-8.5863],[-24.4962,14.14214],[-24.4962,14.14213],[1.76776,21.2132],[1.76777,21.2132],[23.73859,15.65737],[23.73858,15.65736],[31.8198,-7.57614],[31.81981,-7.57614],[-6.31346,-23.23351],[-6.31345,-23.23351],[23.23351,12.87945],[23.23351,12.87944],[9.34391,32.07234],[9.34391,32.07234],[-24.74874,16.66752],[-24.74873,16.66752],[-40.15357,-12.62691],[-40.15356,-12.62691],[-21.46574,-18.18274],[-21.46574,-18.18275],[-17.1726,-39.39595],[-17.17259,-39.39595],[7.82868,-18.94036],[7.82869,-18.94036],[0,0],[22.09708,-2.14657],[44.50984,-4.41941],[22.41276,-2.27285],[45.1412,-4.67196],[46.40389,-5.17704],[1.26269,-0.50508],[23.86485,-2.27284],[46.15134,-3.91434],[22.28649,-1.64149],[44.25731,-3.15672],[44.25731,-3.15672],[49.24493,26.5165],[49.24494,26.51651],[26.51651,44.44671],[26.5165,44.44671],[-20.20305,41.16372],[-20.20305,41.16371],[-57.32616,15.65737],[-57.32616,15.65736],[-3.53553,16.16244],[-3.53553,16.16244],[10.10152,17.67767],[10.10153,17.67767],[38.89087,11.11168],[38.89087,11.11168],[4.29315,-10.35406],[4.29315,-10.35407],[-21.21321,-4.79822],[-21.2132,-4.79823],[-4.29315,-17.17259],[-4.29314,-17.1726],[23.99112,-9.59644],[23.99112,-9.59645],[23.23351,3.283],[23.23351,3.28299],[29.04188,-26.01143],[29.04189,-26.01143],[7.3236,-46.71956],[7.32361,-46.71955],[-33.0825,-33.33503],[-33.08249,-33.33504],[10.85914,-20.45558],[10.85914,-20.45559],[23.73858,11.86929],[23.73859,11.86929],[15.65736,20.70812],[15.65737,20.70813]])
morepoints = np.array([[13.889598,-28.28427],[13.889597,-28.28427],[25.001275,-31.8198],[25.001276,-31.81981],[58.08377,53.28555],[58.08377,53.28554],[7.323606,32.82996],[7.323606,32.82995],[33.840108,15.90989],[33.84011,15.90989],[22.22335,-21.46573],[22.22336,-21.46573],[10.85914,-20.45558],[10.85914,-20.45559],[23.73858,11.86929],[23.73859,11.86929],[15.65736,20.70812],[15.65737,20.70813]])
plt.scatter(morepoints[:,0],morepoints[:,1])
plt.plot(morepoints[:,0],morepoints[:,1])
plt.show()
In [162]:
from PIL import Image
from pylab import *
im = np.array(Image.open('green_leaves_PNG3678.png').convert('L'))
xpixels = im.shape[1]
ypixels = im.shape[0]
dpi = 72
scalefactor = 1
xinch = xpixels * scalefactor / dpi
yinch = ypixels * scalefactor / dpi
fig = plt.figure(figsize=(xinch,yinch))
ax = plt.axes([0, 0, 1, 1], frame_on=False, xticks=[], yticks=[])
#contour(im, levels=[240], colors='black', origin='image', linewidths=[3])
contour(im, levels=[245], colors='black', origin='image')
plt.savefig('same_size.png', dpi=dpi, transparent=True)
fig.clf()
plt.close(fig)
In [163]:
from PIL import Image
im_orig = np.array(Image.open('green_leaves_PNG3678.png'))
im_cnt = np.array(Image.open('same_size.png').convert('L'))
print(im_orig.shape)
print(im_cnt.shape)
#print('Original dimensions: %s ' % (im_orig.shape))
#print('Contour dimensions: %s' % (im_orig.shape))
xpixels = im_orig.shape[1]
ypixels = im_orig.shape[0]
dpi = 72
scalefactor = 1
xinch = xpixels * scalefactor / dpi
yinch = ypixels * scalefactor / dpi
fig = plt.figure(figsize=(xinch,yinch))
ax = plt.axes([0, 0, 1, 1], frame_on=False, xticks=[], yticks=[])
ax.imshow(im_orig)
ax.imshow(np.ma.masked_equal(im_cnt, 255), cmap=plt.cm.Greys_r)
ax.imshow(im_orig)
plt.savefig('overlay2.png', dpi=dpi, transparent=True)
In [181]:
maplepoints = [[ 770.5 , 1195.51606426], [ 770.59756098, 1195.5 ], [ 770.5 , 1195.40243902], [ 770.48393574, 1195.5 ], [ 770.5 , 1195.51606426],[ 772.5 , 1195.53921569], [ 772.53921569, 1195.5 ], [ 772.5 , 1195.46078431], [ 772.28723404, 1195.5 ], [ 772.5 , 1195.53921569],[ 774.5 , 1191.53030303], [ 774.51209677, 1191.5 ], [ 774.5 , 1191.43617021], [ 774.46103896, 1191.5 ], [ 774.5 , 1191.53030303],[ 769.5 , 1189.56622517], [ 769.62048193, 1189.5 ], [ 769.59638554, 1188.5 ], [ 769.5 , 1188.36885246], [ 769.46837945, 1188.5 ], [ 769.46078431, 1189.5 ], [ 769.5 , 1189.56622517],[ 777.5 , 1187.53921569], [ 777.53921569, 1187.5 ], [ 777.5 , 1187.40566038], [ 777.46078431, 1187.5 ], [ 777.5 , 1187.53921569],[ 766.5 , 1180.53921569], [ 766.63333333, 1180.5 ], [ 766.5 , 1180.27272727], [ 766.16666667, 1180.5 ], [ 766.5 , 1180.53921569],[ 777.5 , 1178.51492537], [ 777.50406504, 1178.5 ], [ 777.5 , 1178.47777778], [ 777.48717949, 1178.5 ], [ 777.5 , 1178.51492537],[ 765.5 , 1177.55681818], [ 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In [192]:
mapoints = np.array(maplepoints)
In [191]:
ax = plt.axes([0, 0, 3, 3])
mapoints = np.array(maplepoints)
vor = Voronoi(maplepoints)
voronoi_plot_2d(vor, ax=ax)
tri = Delaunay(mapoints)
plt.plot(mapoints[:,0],mapoints[:,1])
plt.triplot(mapoints[:,0], mapoints[:,1], tri.simplices.copy())
plt.show()
In [193]:
plt.scatter(mapoints[:,0],mapoints[:,1])
plt.plot(mapoints[:,0],mapoints[:,1])
plt.show()
In [200]:
maple_segments = cs.collections[0].get_segments()
ax = plt.axes([0, 0, 3, 3])
for seg in maple_segments:
#print(seg)
temp = np.array(seg)
plt.scatter(temp[:,0],temp[:,1])
plt.plot(temp[:,0],temp[:,1])
plt.show()
In [211]:
# read image to array
im = array(Image.open('leaf-of-a-maple-contour-vector-442364.jpg').convert('L'))
# create a new figure
figure()
# show contours with origin upper left corner
cs = contour(im, levels=[250], colors='black', origin='image', linewidths=[3])
axis('equal')
show()
#print(cs.collections[0].get_segments())#.get_array())#.get_segments())#print(cs.collections[0].get_paths())
In [223]:
maple_segments = cs.collections[0].get_segments()
ax = plt.axes([0, 0, 3, 3])
combined_paths = np.array([]).reshape(0,2)
count = 0
seg_longest = 0
for seg in maple_segments:
if seg_longest < len(seg):
seg_longest = len(seg)
combined_paths = np.array(seg)
count = count + 1
temp = np.array(seg)
combined_paths = np.r_[combined_paths,temp]
plt.scatter(combined_paths[:,0],combined_paths[:,1])
plt.plot(combined_paths[:,0],combined_paths[:,1])
vor = Voronoi(combined_paths)
voronoi_plot_2d(vor, ax=ax)
tri = Delaunay(combined_paths)
plt.plot(combined_paths[:,0],combined_paths[:,1])
plt.triplot(combined_paths[:,0], combined_paths[:,1], tri.simplices.copy())
plt.show()
In [227]:
ax = plt.axes([0, 0, 3, 3])
vor = Voronoi(combined_paths)
voronoi_plot_2d(vor, ax=ax)
plt.show()
In [226]:
ax = plt.axes([0, 0, 3, 3])
tri = Delaunay(combined_paths)
plt.plot(combined_paths[:,0],combined_paths[:,1])
plt.triplot(combined_paths[:,0], combined_paths[:,1], tri.simplices.copy())
plt.show()
In [228]:
ax = plt.axes([0, 0, 3, 3])
plt.scatter(combined_paths[:,0],combined_paths[:,1])
plt.plot(combined_paths[:,0],combined_paths[:,1])
plt.show()
In [245]:
lr_pointreduction(combined_paths,190,False)
In [ ]: