I'm trying to create a Voronoi diagram given a set of scatterplot points. However, several "extra unintended lines" appear to get calculated in the process. Some of these "extra" lines appear to be the infinite edges getting incorrectly calculated. But others are appearing randomly in the middle of the plot as well. How can I only create an extra edge when it's needed/required to connect a polygon to the edge of the plot (e.g. plot boundaries)?
My graph outer boundaries are:
boundaries = np.array([[0, -2], [0, 69], [105, 69], [105, -2], [0, -2]])
Here's the section dealing with the voronoi diagram creation:
def voronoi_polygons(voronoi, diameter):
centroid = voronoi.points.mean(axis=0)
ridge_direction = defaultdict(list)
for (p, q), rv in zip(voronoi.ridge_points, voronoi.ridge_vertices):
u, v = sorted(rv)
if u == -1:
t = voronoi.points[q] - voronoi.points[p] # tangent
n = np.array([-t[1], t[0]]) / np.linalg.norm(t) # normal
midpoint = voronoi.points[[p, q]].mean(axis=0)
direction = np.sign(np.dot(midpoint - centroid, n)) * n
ridge_direction[p, v].append(direction)
ridge_direction[q, v].append(direction)
for i, r in enumerate(voronoi.point_region):
region = voronoi.regions[r]
if -1 not in region:
# Finite region.
yield Polygon(voronoi.vertices[region])
continue
# Infinite region.
inf = region.index(-1) # Index of vertex at infinity.
j = region[(inf - 1) % len(region)] # Index of previous vertex.
k = region[(inf + 1) % len(region)] # Index of next vertex.
if j == k:
# Region has one Voronoi vertex with two ridges.
dir_j, dir_k = ridge_direction[i, j]
else:
# Region has two Voronoi vertices, each with one ridge.
dir_j, = ridge_direction[i, j]
dir_k, = ridge_direction[i, k]
# Length of ridges needed for the extra edge to lie at least
# 'diameter' away from all Voronoi vertices.
length = 2 * diameter / np.linalg.norm(dir_j + dir_k)
# Polygon consists of finite part plus an extra edge.
finite_part = voronoi.vertices[region[inf + 1:] + region[:inf]]
extra_edge = [voronoi.vertices[j] + dir_j * length,
voronoi.vertices[k] + dir_k * length]
combined_finite_edge = np.concatenate((finite_part, extra_edge))
poly = Polygon(combined_finite_edge)
yield poly
Here are the points being used:
['52.629' '24.28099822998047']
['68.425' '46.077999114990234']
['60.409' '36.7140007019043']
['72.442' '28.762001037597656']
['52.993' '43.51799964904785']
['59.924' '16.972000122070312']
['61.101' '55.74899959564209']
['68.9' '13.248001098632812']
['61.323' '29.0260009765625']
['45.283' '36.97500038146973']
['52.425' '19.132999420166016']
['37.739' '28.042999267578125']
['48.972' '2.3539962768554688']
['33.865' '30.240001678466797']
['52.34' '64.94799995422363']
['52.394' '45.391000747680664']
['52.458' '34.79800033569336']
['31.353' '43.14500045776367']
['38.194' '39.24399948120117']
['98.745' '32.15999984741211']
['6.197' '32.606998443603516']
Most likely this is due to the errors associated with floating point arithmetic while computing the voronoi traingulation from your data (esp. the second column).
Assuming that, there is no single solution for such kinds of problems. I urge you to go through this page* of the Qhull manual and try iterating through those parameters in qhull_options before generating the voronoi object that you are inputting in the function. An example would be qhull_options='Qbb Qc Qz QJ'.
Other than that I doubt there is anything that could be modified in the function to avoid such a problem.
*This will take some time though. Just be patient.
Figured out what was wrong: after each polygon I needed to add a null x and y value or else it would attempt to 'stitch' one polygon to another, drawing an additional unintended line in order to do so. So the data should really look more like this:
GameTime,Half,ObjectType,JerseyNumber,X,Y,PlayerIDEvent,PlayerIDTracking,MatchIDEvent,Position,teamId,i_order,v_vor_x,v_vor_y
0.0,1,1,22,None,None,578478,794888,2257663,3,35179.0,0,22.79645297,6.20866756
0.0,1,1,22,None,None,578478,794888,2257663,3,35179.0,1,17.63464264,3.41230187
0.0,1,1,22,None,None,578478,794888,2257663,3,35179.0,2,20.27639318,34.29191902
0.0,1,1,22,None,None,578478,794888,2257663,3,35179.0,3,32.15600546,36.60432421
0.0,1,1,22,None,None,578478,794888,2257663,3,35179.0,4,38.34639812,33.62806739
0.0,1,1,22,None,None,578478,794888,2257663,3,35179.0,5,22.79645297,6.20866756
0.0,1,1,22,None,None,578478,794888,2257663,3,35179.0,5,nan,nan
0.0,1,1,22,33.865,30.240001678466797,578478,794888,2257663,3,35179.0,,,
0.0,1,0,92,None,None,369351,561593,2257663,1,32446.0,0,46.91696938,29.44801535
0.0,1,0,92,None,None,369351,561593,2257663,1,32446.0,1,55.37574848,29.5855499
0.0,1,0,92,None,None,369351,561593,2257663,1,32446.0,2,58.85876401,23.20381766
0.0,1,0,92,None,None,369351,561593,2257663,1,32446.0,3,57.17455086,21.5228301
0.0,1,0,92,None,None,369351,561593,2257663,1,32446.0,4,44.14237744,22.03925667
0.0,1,0,92,None,None,369351,561593,2257663,1,32446.0,5,45.85962774,28.83613332
0.0,1,0,92,None,None,369351,561593,2257663,1,32446.0,5,nan,nan
0.0,1,0,92,52.629,24.28099822998047,369351,561593,2257663,1,32446.0,,,
0.0,1,0,27,None,None,704169,704169,2257663,2,32446.0,0,65.56965667,33.4292025
0.0,1,0,27,None,None,704169,704169,2257663,2,32446.0,1,57.23303682,32.43809027
0.0,1,0,27,None,None,704169,704169,2257663,2,32446.0,2,55.65704152,38.97814049
0.0,1,0,27,None,None,704169,704169,2257663,2,32446.0,3,60.75304149,44.53251169
0.0,1,0,27,None,None,704169,704169,2257663,2,32446.0,4,65.14170295,40.77562188
0.0,1,0,27,None,None,704169,704169,2257663,2,32446.0,5,65.56965667,33.4292025
0.0,1,0,27,None,None,704169,704169,2257663,2,32446.0,5,nan,nan
I have the following working code to project a single point to every segment in an array.
But I want every point in an array of points to be projected to every segment.
import numpy as np
#find closest segment to single point
#line segment
l1 = np.array([[2,3,0],[7,5,0]])
l2 = np.array([[5,1,0],[8,6,0]])
#point that gets projected
p = np.array([[6,5,0]]) #only single point
#set to origin
line = l2-l1
pv = p-l1
#length of line squared
len_sq = np.sum(line**2, axis = 1) #len_sq = numpy.einsum("ij,ij->i", line, line)
#dot product of 3D vectors with einsum
dot = np.einsum('ij,ij->i',line,pv) #np.sum(line*pv,axis=1)
#percentage of line the pv vector travels in
param = np.array([dot/len_sq])
#param<0 projected point=l1, param>1 pp=l2
clamped_param = np.clip(param,0,1)
#add line fraction to l1 to get projected point
pp = l1+(clamped_param.T*line)
For Example, make
p = np.array([[6,5,0],[3,2,0]]) #multiple points
and return np.array() of 4 projected points.
Maybe you can try something like the following. If project is a function that can do the operation for a single point, then by using apply along axis, you can get it to work on all points in an array of points. The output is yielded as separate generators for each point, which have to be converted back to a single array using a stacking operation.
l1 = np.array([[2,3,0],[7,5,0]])
l2 = np.array([[5,1,0],[8,6,0]])
line = l2-l1
len_sq = np.sum(line**2, axis = 1)
def project(p):
pv = p-l1
dot = np.einsum('ij,ij->i',line,pv)
param = np.array([dot/len_sq])
clamped_param = np.clip(param,0,1)
yield l1+(clamped_param.T*line)
pts = np.array([[6,5,0],
[3,2,0]])
gen = np.apply_along_axis(project, 1, pts)
out = np.hstack([list(G) for G in gen])[0]
I wish to generate this in python:
http://classes.yale.edu/fractals/RandFrac/Market/TradingTime/Example1/Example1.html
but I'm incredibly stuck and new to this concept. Does anybody know of a library or gist for this?
Edit:
From what I can understand is that you need to split the fractal in 2 every time. So you have to calculate the y-axis point from the line between the two middle points. Then the two sections need to be formed according to the fractal?
Not 100% sure what you are asking, but as I understood from your comments, you want to generate a realistically looking stock market curve using the recursion described in the link.
As far as I understood the description in the linked page and some of the parent pages, it works like this:
You are given a start and an end point and a number of turning points in the form (t1, v1), (t2, v2), etc., for example start=(0,0), end=(1,1), turns = [(1/4, 1/2), (3/4, 1/4)], where ti and vi are fractions between 0 and 1.
You determine the actual turning points scaled to that interval between start and end and calculate the differences between those points, i.e. how far to go from pi to reach pi+1.
You shuffle those segments to introduce some randomness; when put together, they still cover exactly the same distance, i.e. they connect the original start and end point.
Repeat by recursively calling the function for the different segments between the new points.
Here's some Python code I just put together:
from __future__ import division
from random import shuffle
def make_graph(depth, graph, start, end, turns):
# add points to graph
graph.add(start)
graph.add(end)
if depth > 0:
# unpack input values
fromtime, fromvalue = start
totime, tovalue = end
# calcualte differences between points
diffs = []
last_time, last_val = fromtime, fromvalue
for t, v in turns:
new_time = fromtime + (totime - fromtime) * t
new_val = fromvalue + (tovalue - fromvalue) * v
diffs.append((new_time - last_time, new_val - last_val))
last_time, last_val = new_time, new_val
# add 'brownian motion' by reordering the segments
shuffle(diffs)
# calculate actual intermediate points and recurse
last = start
for segment in diffs:
p = last[0] + segment[0], last[1] + segment[1]
make_graph(depth - 1, graph, last, p, turns)
last = p
make_graph(depth - 1, graph, last, end, turns)
from matplotlib import pyplot
depth = 8
graph = set()
make_graph(depth, graph, (0, 0), (1, 1), [(1/9, 2/3), (5/9, 1/3)])
pyplot.plot(*zip(*sorted(graph)))
pyplot.show()
And here some example output:
I had a similar interest and developed a python3 library to do just what you want.
pip install fractalmarkets
See https://github.com/hyperstripe50/fractal-market-analysis/blob/master/README.md
Using #tobias_k solution and pandas, we can translate and scale the normalized fractal to a time-based one.
import arrow
import pandas as pd
import time
depth = 5
# the "geometry" of fractal
turns = [
(1 / 9, 0.60),
(5 / 9, 0.30),
(8 / 9, 0.70),
]
# select start / end time
t0 = arrow.now().floor("hours")
t1 = t0.shift(days=5)
start = (pd.to_datetime(t0._datetime), 1000)
end = (pd.to_datetime(t1._datetime), 2000)
# create a non-dimensionalized [0,0]x[1,1] Fractal
_start, _end = (0, 0), (1, 1)
graph = set()
make_graph(depth, graph, _start, _end, turns)
# just check graph length
assert len(graph) == (len(turns) + 1) ** depth + 1
# create a pandas dataframe from the normalized Fractal
df = pd.DataFrame(graph)
df.sort_values(0, inplace=True)
df.reset_index(drop=True, inplace=True)
# translate to real coordinates
X = pd.DataFrame(
data=[(start[0].timestamp(), start[1]), (end[0].timestamp(), end[1])]
).T
delta = X[1] - X[0]
Y = df.mul(delta) + X[0]
Y[0] = [*map(lambda x: pd.to_datetime(x, unit="s"), Y[0])]
# now resample and interpolate data according to *grid* size
grid ="min"
Z = Y.set_index(0)
A = Z.resample(grid).mean().interpolate()
# plot both graph to check errors
import matplotlib.pyplot as plt
ax = Z.plot()
A.plot(ax=ax)
plt.show()
showing both graphs:
and zooming to see interpolation and snap-to-grid differences:
I'm am working on a project at the moment where I am trying to create a Hilbert curve using the Python Imaging Library. I have created a function which will generate new coordinates for the curve through each iteration and place them into various lists which then I want to be able to move, rotate and scale. I was wondering if anyone could give me some tips or a way to do this as I am completely clueless. Still working on the a lot of the code.
#! usr/bin/python
import Image, ImageDraw
import math
# Set the starting shape
img = Image.new('RGB', (1000, 1000))
draw = ImageDraw.Draw(img)
curve_X = [0, 0, 1, 1]
curve_Y = [0, 1, 1, 0]
combinedCurve = zip(curve_X, curve_Y)
draw.line((combinedCurve), fill=(220, 255, 250))
iterations = 5
# Start the loop
for i in range(0, iterations):
# Make 4 copies of the curve
copy1_X = list(curve_X)
copy1_Y = list(curve_Y)
copy2_X = list(curve_X)
copy2_Y = list(curve_Y)
copy3_X = list(curve_X)
copy3_Y = list(curve_Y)
copy4_X = list(curve_X)
copy4_Y = list(curve_Y)
# For copy 1, rotate it by 90 degree clockwise
# Then move it to the bottom left
# For copy 2, move it to the top left
# For copy 3, move it to the top right
# For copy 4, rotate it by 90 degrees anticlockwise
# Then move it to the bottom right
# Finally, combine all the copies into a big list
combinedCurve_X = copy1_X + copy2_X + copy3_X + copy4_X
combinedCurve_Y = copy1_Y + copy2_Y + copy3_Y + copy4_Y
# Make the initial curve equal to the combined one
curve_X = combinedCurve_X[:]
curve_Y = combinedCurve_Y[:]
# Repeat the loop
# Scale it to fit the canvas
curve_X = [x * xSize for x in curve_X]
curve_Y = [y * ySize for y in curve_Y]
# Draw it with something that connects the dots
curveCoordinates = zip(curve_X, curve_Y)
draw.line((curveCoordinates), fill=(255, 255, 255))
img2=img.rotate(180)
img2.show()
Here is a solution working on matrices (which makes sense for this type of calculations, and in the end, 2D coordinates are matrices with 1 column!),
Scaling is pretty easy, just have to multiply each element of the matrix by the scale factor:
scaled = copy.deepcopy(original)
for i in range(len(scaled[0])):
scaled[0][i]=scaled[0][i]*scaleFactor
scaled[1][i]=scaled[1][i]*scaleFactor
Moving is pretty easy to, all you have to do is to add the offset to each element of the matrix, here's a method using matrix multiplication:
import numpy as np
# Matrix multiplication
def mult(matrix1,matrix2):
# Matrix multiplication
if len(matrix1[0]) != len(matrix2):
# Check matrix dimensions
print 'Matrices must be m*n and n*p to multiply!'
else:
# Multiply if correct dimensions
new_matrix = np.zeros(len(matrix1),len(matrix2[0]))
for i in range(len(matrix1)):
for j in range(len(matrix2[0])):
for k in range(len(matrix2)):
new_matrix[i][j] += matrix1[i][k]*matrix2[k][j]
return new_matrix
Then create your translation matrix
import numpy as np
TranMatrix = np.zeros((3,3))
TranMatrix[0][0]=1
TranMatrix[0][2]=Tx
TranMatrix[1][1]=1
TranMatrix[1][2]=Ty
TranMatrix[2][2]=1
translated=mult(TranMatrix, original)
And finally, rotation is a tiny bit trickier (do you know your angle of rotation?):
import numpy as np
RotMatrix = np.zeros((3,3))
RotMatrix[0][0]=cos(Theta)
RotMatrix[0][1]=-1*sin(Theta)
RotMatrix[1][0]=sin(Theta)
RotMatrix[1][1]=cos(Theta)
RotMatrix[2][2]=1
rotated=mult(RotMatrix, original)
Some further reading on what I've done:
http://en.wikipedia.org/wiki/Transformation_matrix#Affine_transformations
http://en.wikipedia.org/wiki/Homogeneous_coordinates
http://www.essentialmath.com/tutorial.htm (concerning all the algebra transformations)
So basically, it should work if you insert those operations inside your code, multiplying your vectors by the rotation / translation matrices
EDIT
I just found this Python library that seems to provide all type of transformations: http://toblerity.org/shapely/index.html
I have a raster file and a WGS84 lat/lon point.
I would like to know what value in the raster corresponds with the point.
My feeling is that I should use GetSpatialRef() on the raster object or one of its bands and then apply a ogr.osr.CoordinateTransformation() to the point to map it to the raster's space.
My hope would then be that I could simply ask the rasters' bands what is at that point.
However, the raster object doesn't seem to have a GetSpatialRef() or a way to access a geo-located point, so I'm somewhat at a loss for how to do this.
Any thoughts?
Say i have a geotiff file test.tif. Then followin code should look up value somewhere near the pixel. I am not that confident for the part looking up cell, and will fix there is error. This page should help, "GDAL Data Model"
Also, you may go to gis.stackexchange.com to find experts, if you haven't.
import gdal, osr
class looker(object):
"""let you look up pixel value"""
def __init__(self, tifname='test.tif'):
"""Give name of tif file (or other raster data?)"""
# open the raster and its spatial reference
self.ds = gdal.Open(tifname)
srRaster = osr.SpatialReference(self.ds.GetProjection())
# get the WGS84 spatial reference
srPoint = osr.SpatialReference()
srPoint.ImportFromEPSG(4326) # WGS84
# coordinate transformation
self.ct = osr.CoordinateTransformation(srPoint, srRaster)
# geotranformation and its inverse
gt = self.ds.GetGeoTransform()
dev = (gt[1]*gt[5] - gt[2]*gt[4])
gtinv = ( gt[0] , gt[5]/dev, -gt[2]/dev,
gt[3], -gt[4]/dev, gt[1]/dev)
self.gt = gt
self.gtinv = gtinv
# band as array
b = self.ds.GetRasterBand(1)
self.arr = b.ReadAsArray()
def lookup(self, lon, lat):
"""look up value at lon, lat"""
# get coordinate of the raster
xgeo,ygeo,zgeo = self.ct.TransformPoint(lon, lat, 0)
# convert it to pixel/line on band
u = xgeo - self.gtinv[0]
v = ygeo - self.gtinv[3]
# FIXME this int() is probably bad idea, there should be
# half cell size thing needed
xpix = int(self.gtinv[1] * u + self.gtinv[2] * v)
ylin = int(self.gtinv[4] * u + self.gtinv[5] * v)
# look the value up
return self.arr[ylin,xpix]
# test
l = looker('test.tif')
lon,lat = -100,30
print l.lookup(lon,lat)
lat,lon =28.816944, -96.993333
print l.lookup(lon,lat)
Yes, the API isn't consistent. The raster (the data source) has a GetProjection() method instead (which returns WKT).
Here is a function that does what you want (drawn from here):
def extract_point_from_raster(point, data_source, band_number=1):
"""Return floating-point value that corresponds to given point."""
# Convert point co-ordinates so that they are in same projection as raster
point_sr = point.GetSpatialReference()
raster_sr = osr.SpatialReference()
raster_sr.ImportFromWkt(data_source.GetProjection())
transform = osr.CoordinateTransformation(point_sr, raster_sr)
point.Transform(transform)
# Convert geographic co-ordinates to pixel co-ordinates
x, y = point.GetX(), point.GetY()
forward_transform = Affine.from_gdal(*data_source.GetGeoTransform())
reverse_transform = ~forward_transform
px, py = reverse_transform * (x, y)
px, py = int(px + 0.5), int(py + 0.5)
# Extract pixel value
band = data_source.GetRasterBand(band_number)
structval = band.ReadRaster(px, py, 1, 1, buf_type=gdal.GDT_Float32)
result = struct.unpack('f', structval)[0]
if result == band.GetNoDataValue():
result = float('nan')
return result
Its documentation is as follows (drawn from here):
spatial.extract_point_from_raster(point, data_source, band_number=1)
data_source is a GDAL raster, and point is an OGR point object. The
function returns the value of the pixel of the specified band of
data_source that is nearest to point.
point and data_source need not be in the same reference system, but
they must both have an appropriate spatial reference defined.
If the point does not fall in the raster, RuntimeError is raised.
project = self.ds.GetProjection()
srPoint = osr.SpatialReference(wkt=project)
done... with that, the vector file has adopted the projection from input raster file