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I'm trying to implement some algorithms in Python from different papers and some require a peak/through detection where local minima and maxima are alternating. That should always be the case of course in an analog signal.
However when working with discrete Signals this is not always the case as you can see with this simple example:
from scipy.signal import argrelmax,argrelmin,argrelextrema
import numpy as np
import matplotlib.pyplot as plt
fig = plt.figure("local extrema")
ax = fig.subplots()
x = range(0,10)
y = np.array([0,1,0,0,1,2,1,0,0,0])
local_minima, = list(argrelmin(np.array(y)))
local_maxima, = list(argrelmax(np.array(y)))
ax.plot(local_maxima, y[local_maxima],"x",color="orange")
ax.plot(local_minima, y[local_minima],"x",color="orange")
ax.plot(x,y)
plt.show()
I know that I can adjust the argrelmax and argrelmin functions but no matter what I do, its always somehow flawed.
Here only two maxima are found. So there are no alternating extrema.
Is there a way to automatically detect extremas and assure they are alternating, or do I have to implement some kind of workaround? (Ideas for that are welcome too)
I found a solution I previously disregarded unfortunately. The Problem with argrelmax,argrelmin and argrelextrema was that they had difficulties detecting flat peaks/throughs because sometimes they found two maxima using np.greater_equal as comparator (beginning and end of flat peak), which lead to non alternating maxima/minima.
After Flows answer I revisited find_peaks from scipy.signal, which I previously disregarded, because in the first few sentences it said: "finds all local maxima by simple comparison of neighboring values" so I thought it had the same flaws as the others.
But in reality it detects flat peaks with only one index at the middle (rounded down) of the peak, no matter how wide apparently.
Thus this solved my Problem for this simple case:
from scipy.signal import argrelmax,argrelmin, argrelextrema, find_peaks
fig = plt.figure("local extrema")
ax = fig.subplots()
x = range(0,10)
y = np.array([0,1,0,0,1,2,1,0,0,0])
peaks,_ = find_peaks(y)
throughs,_ = find_peaks(np.negative(y))
local_maxima = list(peaks)
local_minima = list(throughs)
ax.plot(x,y)
ax.plot(local_maxima, y[local_maxima],"x",color="orange")
ax.plot(local_minima, y[local_minima],"x",color="orange")
plt.show()
As you can see now the minima is detected.
I will have to see if it works on my real world data though, but I think it should. Thx!
Maybe this will be duplicate question but I couldn't find any solution for this.
Normally what I coded should show me a curved line in python. But with this code I cant see it. Is there a problem with my code or pycharm ? This code only shows me an empty graphic with the correct axes.
And I did adding "ro" in plt.plot(at[i], st, "ro"). This showed me the spots on the graph but what I want to see the complete line.
at = [0,1,2,3,4,5,6]
for i in range(len(at)):
st = at[i]**2
plt.plot(at[i], st)
plt.show()
This is how you would normally do this:
import numpy as np
import matplotlib.pyplot as plt
at = np.array([0,1,2,3,4,5,6])
at2 = at ** 2
plt.plot(at,at2)
plt.show()
you can use something like plt.plot(at,at2, c='red', marker='o') to see the spots.
for detailed explanation please read the documentation.
Maybe rather calculate the to be plotted values entirely before plotting.
at = [0,1,2,3,4,5,6]
y = [xi**2 for xi in at]
plt.plot(at, y)
Or do it alternatively with a function
from math import pow
at = [0,1,2,3,4,5,6]
def parabolic(x):
return [pow(xi,2) for xi in x]
plt.plot(at, parabolic(at))
both return the following plot:
the other answers give fixes for your question, but don't tell you why your code is not working.
the reason for not "seeing anything" is that plt.plot(at[i], st) was trying to draw lines between the points you give it. but because you were only ever giving it single values it didn't have anything to draw lines between. as a result, nothing appeared on the plot
when you changed to call plt.plot(at[i], st, 'ro') you're telling it to draw single circles at points and these don't go between points so would appear
the other answers showed you how to pass multiple values to plot and hence matplotlib could draw lines between these values.
one of your comments says "its not parabolic still" and this is because matplotlib isn't a symbolic plotting library. you just give it numeric values and it draws these onto the output device. sympy is a library for doing symbolic computation and supports plotting, e.g:
from sympy import symbols, plot
x = symbols('x')
plot(x**2, (x, 0, 6))
does the right thing for me. the current release (1.4) doesn't handle discontinuities, but this will be fixed in the next release
I am trying to follow the tutorial from scikit-image regarding Template Matching (check it here).
Using just this example, I would like to find all matching coins (maxima) in the image, not only this one which gave the highest score. I was thinking about using:
maxima = argrelextrema(result, np.greater)
but the problem is that it finds also very small local maxima, which are just a noise. Is there any way to screen numpy array and find the strongest maxima? Thanks!
To find all the coins the documentation suggests "...you should use a proper peak-finding function." The easiest of these is probably peak_local_max (as suggested in the comments) which is also from skimage, and has a manual page here. Using some reasonable numbers in the *args gets the peaks out of the response image.
The second comment about the peaks being displaced is also discussed in the documentation
"Note that the peaks in the output of match_template correspond to the origin (i.e. top-left corner) of the template."
One can manually correct for this (by translating the peaks by the side lengths of the template), or you can set the pad_input bool to True (source), which as a by-product means that the peaks in the response function line up with the center of the template at the point of maximal overlap.
Combining these two bits into a script we get something like:
import numpy as np
import matplotlib.pyplot as plt
from skimage import data
from skimage.feature import match_template
from skimage.feature import peak_local_max # new import!
image = data.coins()
coin = image[170:220, 75:130]
result = match_template(image, coin,pad_input=True) #added the pad_input bool
peaks = peak_local_max(result,min_distance=10,threshold_rel=0.5) # find our peaks
# produce a plot equivalent to the one in the docs
plt.imshow(result)
# highlight matched regions (plural)
plt.plot(peaks[:,1], peaks[:,0], 'o', markeredgecolor='r', markerfacecolor='none', markersize=10)
I have been digging and found some solution but unfortunately I am not sure if I know what exactly is done in the script. I have slightly modified script found here:
neighborhood_size = 20 #how many pixels
threshold = 0.01 #threshold of maxima?
data_max = filters.maximum_filter(result, neighborhood_size)
maxima = (result == data_max)
data_min = filters.minimum_filter(result, neighborhood_size)
diff = ((data_max - data_min) > threshold)
maxima[diff == 0] = 0
x_image,y_image = [], []
temp_size = coin.shape[0]
labeled, num_objects = ndimage.label(maxima)
slices = ndimage.find_objects(labeled)
x, y = [], []
for dy,dx in slices:
x_center = (dx.start + dx.stop - 1)/2
x.append(x_center)
y_center = (dy.start + dy.stop - 1)/2
y.append(y_center)
fig, (raw,found) = plt.subplots(1,2)
raw.imshow(image,cmap=plt.cm.gray)
raw.set_axis_off()
found.imshow(result)
found.autoscale(False)
found.set_axis_off()
plt.plot(x,y, 'ro')
plt.show()
and does this:
I also realized, that coordinates of found peaks are shifted in comparison to raw image. I think the difference comes from the template size. I will update when I will find out more.
EDIT: with slight code modification I was able also to find places on the input image:
x_image_center = (dx.start + dx.stop - 1 + temp_size) / 2
x_image.append(x_image_center)
y_image_center = (dy.start + dy.stop - 1 + temp_size) / 2
y_image.append(y_image_center)
I am trying to plot contour lines of pressure level. I am using a netCDF file which contain the higher resolution data (ranges from 3 km to 27 km). Due to higher resolution data set, I get lot of pressure values which are not required to be plotted (rather I don't mind omitting certain contour line of insignificant values). I have written some plotting script based on the examples given in this link http://matplotlib.org/basemap/users/examples.html.
After plotting the image looks like this
From the image I have encircled the contours which are small and not required to be plotted. Also, I would like to plot all the contour lines smoother as mentioned in the above image. Overall I would like to get the contour image like this:-
Possible solution I think of are
Find out the number of points required for plotting contour and mask/omit those lines if they are small in number.
or
Find the area of the contour (as I want to omit only circled contour) and omit/mask those are smaller.
or
Reduce the resolution (only contour) by increasing the distance to 50 km - 100 km.
I am able to successfully get the points using SO thread Python: find contour lines from matplotlib.pyplot.contour()
But I am not able to implement any of the suggested solution above using those points.
Any solution to implement the above suggested solution is really appreciated.
Edit:-
# Andras Deak
I used print 'diameter is ', diameter line just above del(level.get_paths()[kp]) line to check if the code filters out the required diameter. Here is the filterd messages when I set if diameter < 15000::
diameter is 9099.66295612
diameter is 13264.7838257
diameter is 445.574234531
diameter is 1618.74618114
diameter is 1512.58974168
However the resulting image does not have any effect. All look same as posed image above. I am pretty sure that I have saved the figure (after plotting the wind barbs).
Regarding the solution for reducing the resolution, plt.contour(x[::2,::2],y[::2,::2],mslp[::2,::2]) it works. I have to apply some filter to make the curve smooth.
Full working example code for removing lines:-
Here is the example code for your review
#!/usr/bin/env python
from netCDF4 import Dataset
import matplotlib
matplotlib.use('agg')
import matplotlib.pyplot as plt
import numpy as np
import scipy.ndimage
from mpl_toolkits.basemap import interp
from mpl_toolkits.basemap import Basemap
# Set default map
west_lon = 68
east_lon = 93
south_lat = 7
north_lat = 23
nc = Dataset('ncfile.nc')
# Get this variable for later calucation
temps = nc.variables['T2']
time = 0 # We will take only first interval for this example
# Draw basemap
m = Basemap(projection='merc', llcrnrlat=south_lat, urcrnrlat=north_lat,
llcrnrlon=west_lon, urcrnrlon=east_lon, resolution='l')
m.drawcoastlines()
m.drawcountries(linewidth=1.0)
# This sets the standard grid point structure at full resolution
x, y = m(nc.variables['XLONG'][0], nc.variables['XLAT'][0])
# Set figure margins
width = 10
height = 8
plt.figure(figsize=(width, height))
plt.rc("figure.subplot", left=.001)
plt.rc("figure.subplot", right=.999)
plt.rc("figure.subplot", bottom=.001)
plt.rc("figure.subplot", top=.999)
plt.figure(figsize=(width, height), frameon=False)
# Convert Surface Pressure to Mean Sea Level Pressure
stemps = temps[time] + 6.5 * nc.variables['HGT'][time] / 1000.
mslp = nc.variables['PSFC'][time] * np.exp(9.81 / (287.0 * stemps) * nc.variables['HGT'][time]) * 0.01 + (
6.7 * nc.variables['HGT'][time] / 1000)
# Contour only at 2 hpa interval
level = []
for i in range(mslp.min(), mslp.max(), 1):
if i % 2 == 0:
if i >= 1006 and i <= 1018:
level.append(i)
# Save mslp values to upload to SO thread
# np.savetxt('mslp.txt', mslp, fmt='%.14f', delimiter=',')
P = plt.contour(x, y, mslp, V=2, colors='b', linewidths=2, levels=level)
# Solution suggested by Andras Deak
for level in P.collections:
for kp,path in enumerate(level.get_paths()):
# include test for "smallness" of your choice here:
# I'm using a simple estimation for the diameter based on the
# x and y diameter...
verts = path.vertices # (N,2)-shape array of contour line coordinates
diameter = np.max(verts.max(axis=0) - verts.min(axis=0))
if diameter < 15000: # threshold to be refined for your actual dimensions!
#print 'diameter is ', diameter
del(level.get_paths()[kp]) # no remove() for Path objects:(
#level.remove() # This does not work. produces ValueError: list.remove(x): x not in list
plt.gcf().canvas.draw()
plt.savefig('dummy', bbox_inches='tight')
plt.close()
After the plot is saved I get the same image
You can see that the lines are not removed yet. Here is the link to mslp array which we are trying to play with http://www.mediafire.com/download/7vi0mxqoe0y6pm9/mslp.txt
If you want x and y data which are being used in the above code, I can upload for your review.
Smooth line
You code to remove the smaller circles working perfectly. However the other question I have asked in the original post (smooth line) does not seems to work. I have used your code to slice the array to get minimal values and contoured it. I have used the following code to reduce the array size:-
slice = 15
CS = plt.contour(x[::slice,::slice],y[::slice,::slice],mslp[::slice,::slice], colors='b', linewidths=1, levels=levels)
The result is below.
After searching for few hours I found this SO thread having simmilar issue:-
Regridding regular netcdf data
But none of the solution provided over there works.The questions similar to mine above does not have proper solutions. If this issue is solved then the code is perfect and complete.
General idea
Your question seems to have 2 very different halves: one about omitting small contours, and another one about smoothing the contour lines. The latter is simpler, since I can't really think of anything else other than decreasing the resolution of your contour() call, just like you said.
As for removing a few contour lines, here's a solution which is based on directly removing contour lines individually. You have to loop over the collections of the object returned by contour(), and for each element check each Path, and delete the ones you don't need. Redrawing the figure's canvas will get rid of the unnecessary lines:
# dummy example based on matplotlib.pyplot.clabel example:
import matplotlib
import numpy as np
import matplotlib.cm as cm
import matplotlib.mlab as mlab
import matplotlib.pyplot as plt
delta = 0.025
x = np.arange(-3.0, 3.0, delta)
y = np.arange(-2.0, 2.0, delta)
X, Y = np.meshgrid(x, y)
Z1 = mlab.bivariate_normal(X, Y, 1.0, 1.0, 0.0, 0.0)
Z2 = mlab.bivariate_normal(X, Y, 1.5, 0.5, 1, 1)
# difference of Gaussians
Z = 10.0 * (Z2 - Z1)
plt.figure()
CS = plt.contour(X, Y, Z)
for level in CS.collections:
for kp,path in reversed(list(enumerate(level.get_paths()))):
# go in reversed order due to deletions!
# include test for "smallness" of your choice here:
# I'm using a simple estimation for the diameter based on the
# x and y diameter...
verts = path.vertices # (N,2)-shape array of contour line coordinates
diameter = np.max(verts.max(axis=0) - verts.min(axis=0))
if diameter<1: # threshold to be refined for your actual dimensions!
del(level.get_paths()[kp]) # no remove() for Path objects:(
# this might be necessary on interactive sessions: redraw figure
plt.gcf().canvas.draw()
Here's the original(left) and the removed version(right) for a diameter threshold of 1 (note the little piece of the 0 level at the top):
Note that the top little line is removed while the huge cyan one in the middle doesn't, even though both correspond to the same collections element i.e. the same contour level. If we didn't want to allow this, we could've called CS.collections[k].remove(), which would probably be a much safer way of doing the same thing (but it wouldn't allow us to differentiate between multiple lines corresponding to the same contour level).
To show that fiddling around with the cut-off diameter works as expected, here's the result for a threshold of 2:
All in all it seems quite reasonable.
Your actual case
Since you've added your actual data, here's the application to your case. Note that you can directly generate the levels in a single line using np, which will almost give you the same result. The exact same can be achieved in 2 lines (generating an arange, then selecting those that fall between p1 and p2). Also, since you're setting levels in the call to contour, I believe the V=2 part of the function call has no effect.
import numpy as np
import matplotlib.pyplot as plt
# insert actual data here...
Z = np.loadtxt('mslp.txt',delimiter=',')
X,Y = np.meshgrid(np.linspace(0,300000,Z.shape[1]),np.linspace(0,200000,Z.shape[0]))
p1,p2 = 1006,1018
# this is almost the same as the original, although it will produce
# [p1, p1+2, ...] instead of `[Z.min()+n, Z.min()+n+2, ...]`
levels = np.arange(np.maximum(Z.min(),p1),np.minimum(Z.max(),p2),2)
#control
plt.figure()
CS = plt.contour(X, Y, Z, colors='b', linewidths=2, levels=levels)
#modified
plt.figure()
CS = plt.contour(X, Y, Z, colors='b', linewidths=2, levels=levels)
for level in CS.collections:
for kp,path in reversed(list(enumerate(level.get_paths()))):
# go in reversed order due to deletions!
# include test for "smallness" of your choice here:
# I'm using a simple estimation for the diameter based on the
# x and y diameter...
verts = path.vertices # (N,2)-shape array of contour line coordinates
diameter = np.max(verts.max(axis=0) - verts.min(axis=0))
if diameter<15000: # threshold to be refined for your actual dimensions!
del(level.get_paths()[kp]) # no remove() for Path objects:(
# this might be necessary on interactive sessions: redraw figure
plt.gcf().canvas.draw()
plt.show()
Results, original(left) vs new(right):
Smoothing by resampling
I've decided to tackle the smoothing problem as well. All I could come up with is downsampling your original data, then upsampling again using griddata (interpolation). The downsampling part could also be done with interpolation, although the small-scale variation in your input data might make this problem ill-posed. So here's the crude version:
import scipy.interpolate as interp #the new one
# assume you have X,Y,Z,levels defined as before
# start resampling stuff
dN = 10 # use every dN'th element of the gridded input data
my_slice = [slice(None,None,dN),slice(None,None,dN)]
# downsampled data
X2,Y2,Z2 = X[my_slice],Y[my_slice],Z[my_slice]
# same as X2 = X[::dN,::dN] etc.
# upsampling with griddata over original mesh
Zsmooth = interp.griddata(np.array([X2.ravel(),Y2.ravel()]).T,Z2.ravel(),(X,Y),method='cubic')
# plot
plt.figure()
CS = plt.contour(X, Y, Zsmooth, colors='b', linewidths=2, levels=levels)
You can freely play around with the grids used for interpolation, in this case I just used the original mesh, as it was at hand. You can also play around with different kinds of interpolation: the default 'linear' one will be faster, but less smooth.
Result after downsampling(left) and upsampling(right):
Of course you should still apply the small-line-removal algorithm after this resampling business, and keep in mind that this heavily distorts your input data (since if it wasn't distorted, then it wouldn't be smooth). Also, note that due to the crude method used in the downsampling step, we introduce some missing values near the top/right edges of the region under consideraton. If this is a problem, you should consider doing the downsampling based on griddata as I've noted earlier.
This is a pretty bad solution, but it's the only one that I've come up with. Use the get_contour_verts function in this solution you linked to, possibly with the matplotlib._cntr module so that nothing gets plotted initially. That gives you a list of contour lines, sections, vertices, etc. Then you have to go through that list and pop the contours you don't want. You could do this by calculating a minimum diameter, for example; if the max distance between points is less than some cutoff, throw it out.
That leaves you with a list of LineCollection objects. Now if you make a Figure and Axes instance, you can use Axes.add_collection to add all of the LineCollections in the list.
I checked this out really quick, but it seemed to work. I'll come back with a minimum working example if I get a chance. Hope it helps!
Edit: Here's an MWE of the basic idea. I wasn't familiar with plt._cntr.Cntr, so I ended up using plt.contour to get the initial contour object. As a result, you end up making two figures; you just have to close the first one. You can replace checkDiameter with whatever function works. I think you could turn the line segments into a Polygon and calculate areas, but you'd have to figure that out on your own. Let me know if you run into problems with this code, but it at least works for me.
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
def checkDiameter(seg, tol=.3):
# Function for screening line segments. NB: Not actually a proper diameter.
diam = (seg[:,0].max() - seg[:,0].min(),
seg[:,1].max() - seg[:,1].min())
return not (diam[0] < tol or diam[1] < tol)
# Create testing data
x = np.linspace(-1,1, 21)
xx, yy = np.meshgrid(x,x)
z = np.exp(-(xx**2 + .5*yy**2))
# Original plot with plt.contour
fig0, ax0 = plt.subplots()
# Make sure this contour object actually has a tiny contour to remove
cntrObj = ax0.contour(xx,yy,z, levels=[.2,.4,.6,.8,.9,.95,.99,.999])
# Primary loop: Copy contours into a new LineCollection
lineNew = list()
for lineOriginal in cntrObj.collections:
# Get properties of the original LineCollection
segments = lineOriginal.get_segments()
propDict = lineOriginal.properties()
propDict = {key: value for (key,value) in propDict.items()
if key in ['linewidth','color','linestyle']} # Whatever parameters you want to carry over
# Filter out the lines with small diameters
segments = [seg for seg in segments if checkDiameter(seg)]
# Create new LineCollection out of the OK segments
if len(segments) > 0:
lineNew.append(mpl.collections.LineCollection(segments, **propDict))
# Make new plot with only these line collections; display results
fig1, ax1 = plt.subplots()
ax1.set_xlim(ax0.get_xlim())
ax1.set_ylim(ax0.get_ylim())
for line in lineNew:
ax1.add_collection(line)
plt.show()
FYI: The bit with propDict is just to automate bringing over some of the line properties from the original plot. You can't use the whole dictionary at once, though. First, it contains the old plot's line segments, but you can just swap those for the new ones. But second, it appears to contain a number of parameters that are in conflict with each other: multiple linewidths, facecolors, etc. The {key for key in propDict if I want key} workaround is my way to bypass that, but I'm sure someone else can do it more cleanly.
I'm trying to fit some data points in order to find the center of a circle. All of the following points are noisy data points around the circumference of the circle:
data = [(2.2176383052987667, 4.218574252410221),
(3.3041214516913033, 5.223500807396272),
(4.280815855023374, 6.461487709813785),
(4.946375258539319, 7.606952538212697),
(5.382428804463699, 9.045717060494576),
(5.752578028217334, 10.613667377465823),
(5.547729017414035, 11.92662513852466),
(5.260208374620305, 13.57722448066025),
(4.642126672822957, 14.88238955729078),
(3.820310290976751, 16.10605425390148),
(2.8099420132544024, 17.225880123445773),
(1.5731539516426183, 18.17052077121059),
(0.31752822350872545, 18.75261434891438),
(-1.2408437559671106, 19.119355580780265),
(-2.680901948575409, 19.15018791257732),
(-4.190406775175328, 19.001321726517297),
(-5.533990404926917, 18.64857428377178),
(-6.903383826792998, 17.730112542165955),
(-8.082883753215347, 16.928080323602334),
(-9.138397388219254, 15.84088004983959),
(-9.92610373064812, 14.380575762984085),
(-10.358670204629814, 13.018017342781242),
(-10.600053524240247, 11.387283417089911),
(-10.463673966507077, 10.107554951600699),
(-10.179820255235496, 8.429558128401448),
(-9.572153386953028, 7.1976672709797676),
(-8.641475289758178, 5.8312286526738175),
(-7.665976739804268, 4.782663065707469),
(-6.493033077746997, 3.8549965442534684),
(-5.092340806635571, 3.384419909199452),
(-3.6530364510489073, 2.992272643733981),
(-2.1522365767310796, 3.020780664301393),
(-0.6855406924835704, 3.0767643753777447),
(0.7848958776292426, 3.6196842530995332),
(2.0614188482646947, 4.32795711960546),
(3.2705467984691508, 5.295836809444288),
(4.359297538484424, 6.378324784240816),
(4.981264502955681, 7.823851404553242)]
I was trying to use some library like Scipy, but I'm having trouble using the available functions.
There is for example:
# == METHOD 2 ==
from scipy import optimize
method_2 = "leastsq"
def calc_R(xc, yc):
""" calculate the distance of each 2D points from the center (xc, yc) """
return sqrt((x-xc)**2 + (y-yc)**2)
def f_2(c):
""" calculate the algebraic distance between the data points and the mean circle centered at c=(xc, yc) """
Ri = calc_R(*c)
return Ri - Ri.mean()
center_estimate = x_m, y_m
center_2, ier = optimize.leastsq(f_2, center_estimate)
xc_2, yc_2 = center_2
Ri_2 = calc_R(*center_2)
R_2 = Ri_2.mean()
residu_2 = sum((Ri_2 - R_2)**2)
But this seems to be using a single xy? Any ideas on how to plug this function to my data example?
Your data points seem fairly clean and I see no outliers, so many circle fitting algorithms will work.
I recommend you to start with the Coope method, which works by magically linearizing the problem:
(X-Xc)² + (Y-Yc)² = R² is rewritten as
2 Xc X + 2 Yc Y + R² - Xc² - Yc² = X² + Y², then
A X + B Y + C = X² + Y², solved by linear least squares.
As a follow up to Bas Swinckels post, I figured I'd post my code implementing the Halir and Flusser method of fitting an ellipse
https://github.com/bdhammel/least-squares-ellipse-fitting
pip install lsq-ellipse
Now make the data into a numpy array using
import numpy as np
data=np.array(data)
Using the above code you can find the center with the following method.
from ellipse import LsqEllipse
import numpy as np
import matplotlib.pyplot as plt
import statistics
from statistics import mean
from matplotlib.patches import Ellipse
lsqe = LsqEllipse()
lsqe.fit(data)
center, width, height, phi = lsqe.as_parameters()
plt.close('all')
fig = plt.figure(figsize=(6,6))
ax = fig.add_subplot(111)
ax.axis('equal')
ax.plot(data[:,0], data[:,1], 'ro', label='test data', zorder=1)
ellipse = Ellipse(xy=center, width=2*width, height=2*height, angle=np.rad2deg(phi),
edgecolor='b', fc='None', lw=2, label='Fit', zorder = 2)
ax.add_patch(ellipse)
print('center of fitted circle =',center, '\n','radius =', mean([width,height]),'+/- stddev=',statistics.stdev([width,height]))
plt.legend()
plt.show()
Here we just take average of height and width of this ellipse as the radius of the circular fit and their standard deviation as the error. This can be modified
I do not have any experience fitting circles, but I have worked with the more general case of fitting ellipses. Doing this in a correct way with noisy data is not trivial. For this problem, the algorithm described in Numerically stable direct least squares fitting of ellipses by Halir and Flusser works pretty well. The paper includes Matlab code, which should be straightforward to translate to Numpy. Maybe you could use this algorithm to fit an ellipse and then take the average of the two axis as the radius or so. Some of the references in the paper also mention fitting circles, you might want to look those up.
I know this is an old question, but in 2019 there's a circle fitting library in python called circle-fit.
pip install circle-fit
you can use one of two algorithms to solve, least_squares_circle or hyper_fit.
import circle_fit as cf
xc,yc,r,_ = cf.least_squares_circle((data)
then you get xc, yc as the coordinate pair for the solution circle center.
Ian Coope's algorithm (paper here) linearizes the problem using a variable substitution. It's the most generally robust and fastest algorithm I've come across for circle fitting so far. I've implemented the algorithm in python in scikit-guess. Using the function skg.nsphere_fit for a one-line solution:
>>> r, c = skg.nsphere_fit(data)
>>> r
8.138962707494084
>>> c
array([-2.45595128, 11.04352796])
Here is a plot of the result:
t = np.linspace(0, 2 * np.pi, 1000, endpoint=True)
plt.scatter(*np.array(data).T, color='r')
plt.plot(r * np.cos(t) + c[0], r * np.sin(t) + c[1])
plt.axis('equal')
plt.show()