I am experimenting with some computer vision techniques, specifically feature detection. I am trying to identify features by conducting auto-correlation between an image and a feature-kernel.
However, the resulting correlation-matrix doesn't make sense to me... Can anyone help me understand how to interpret or visualize this matrix, so that it's apparent where the feature is located?
Feature Kernel:
Original Image:
Code:
import cv2
import pprint
import numpy
import scipy.ndimage
from matplotlib import pyplot as plt
import skimage.feature
# load the image
img = cv2.imread('./lenna.jpg')[:,:,0]
f_kernel = cv2.imread('./lenna_feature.jpg')[:,:,0]
def matched_filter(img, f_kernel, detect_thres):
result = scipy.ndimage.correlate(img, f_kernel)
print("Feature Match Template")
plt.imshow(skimage.feature.match_template(img, f_kernel))
plt.show()
return result
plt.imshow(matched_filter(img,f_kernel,1))
print("Correlation Matrix")
plt.show()
Result:
So, in the first result-image, there's an obvious maximum-point at (150,200). I am interpreting this as the most likely location of the feature.
However, in the second result-image, correlation matrix result, there is no obvious pattern. I was expecting that there would be some, obvious high-correlation point.
Help?
skimage.feature.match_template computes the normalized cross correlation. That is, for each location over the image, the image patch and the template are normalized (subtract mean and divide by standard deviation) and then multiplied together and averaged. This computes the correlation coefficient of the image patch and the template. The correlation coefficient is a value between 1 and -1. A correlation coefficient of 1 indicates that the image patch is a linear modification of the template (i.e. constant1 * template + constant2).
scipy.ndimage.correlate computes the correlation (same as convolution, but without mirroring the kernel). That is, here we do not first normalize the image patch. Places where the image has higher values will automatically also have a higher correlation, even if not at all similar to the template.
I can't reproduce your result. For me, the second image has a shape that looks a bit like Lena.
Anyway, you don't want to use the correlation, you want to use the correlation coefficient to do template matching. Pure correlation is not normalized, so it's rather an averaging filter than a template matching.
Edit: Added the corellation image
Related
Is there an efficient implementation in Python to evaluate the PDF of a multivariate normal distribution when there are missing values in x? I guess the idea would just be that you'd effectively reduce the dimensionality to whatever number of available data points you had for a particular vector for which you are trying to evaluate the probability. But I can't figure out if the scipy implementation has a way to ignore masked values.
e.g.,
from scipy.stats import multivariate_normal as mvnorm
import numpy as np
means = [0.0,0.0,0.0]
cov = np.array([[1.0,0.2,0.2],[0.2,1.0,0.2],[0.2,0.2,1.0]])
d = mvnorm(means,cov)
x = [0.5,-0.2,np.nan]
d.pdf(x)
yields output:
nan
(as expected)
Is there a way to efficiently evaluate the PDF for only values that are present (in this case, making effectively 3D case into a bivariate case?) using this implementation?
This question is a bit of a tricky in terms of math and code. Let me elaborate.
First, the code. scipy.stats does not offer nan-handling as you desire. Speedy code likely requires implementing the multivariate normal distribution PDF by hand and applying it to NumPy arrays directly. Leveraging vectorization is the only way to efficiently offer this functionality for large-scale datasets. On the other hand, the nan-tolerant function nanTol_pdf() below provides the desired functionality while staying true to the multivariate normal distribution as implemented in SciPy. You might find it sufficient for your use case.
def nanTol_pdf(d, x):
'''
Function returns function value of multivariate probability density conditioned on
non-NAN indices of the input vector x
'''
assert isinstance(d, stats._multivariate.multivariate_normal_frozen) and (isinstance(x,list) or isinstance(x,np.ndarray))
# check presence of nan entries
if any(np.isnan(x)):
# indices
subIndex = np.argwhere(~np.isnan(x)).reshape(-1)
# lower-dimensional multiv. Gaussian distribution
lowDim_mean = d.mean[subIndex]
lowDim_cov = cov[np.ix_(subIndex, subIndex)]
lowDim_d = mvnorm(lowDim_mean, lowDim_cov)
return (lowDim_d.pdf(x[subIndex]))
else:
return d.pdf(x)
Regardless, the fact we can do it shouldn't stop us to think if we should.
Second, the math. Mathematically speaking, it is unclear what you attempt to achieve. In your example, SciPy returns nan as you query it with an ill-defined input vector x. Output not-defined, i.e. returning not a number (nan) seems to be the most appropriate answer. Jointly truncating the distribution d and input vector x circumvents numerical problems but opens up statistical questions. In particular, since the probability density function values cannot be understood as (conditional) probabilities. Moreover, the output alone conceals if truncation was applied. Remember that nanTol_pdf() will happily provide a non-negative real number as an output as long as at least one entry in the vector is a real number. Your use case will decide if this is reasonable.
Finally, I would suggest at least considering missing data imputation techniques before moving forward. Let me know if this helps.
I cannot get scipy.optimize.curve_fit to properly fit my data which is visually apparent. I know approximately what the parameter values should be and if I evaluate the function with the given parameters the calculated and experimental data appear to agree well:
However, when I use scipy.optimize.curve_fit the output parameters with the smallest error is a much worse fit (by visual inspection). If I use the "known" parameters as my initial guess and bound the parameters to a relatively narrow window as shown in the example of output from fit function:
I obtain error values ~10^2 times larger but the visual appearance of the fit seems better. The only way I can get a decent looking fit for the data is to bound all the parameters with ~ 0.3 units of the "known" parameter. I plan on using this code to fit more complex data that I will not know the parameters before hand, so I cannot just use the calculated plot.
The relevant code is included below:
import matplotlib.pyplot as plt
import numpy as np
import scipy
from scipy.optimize import curve_fit
d_1= 2.72 #Anstroms
sig_cp_1= 0.44
sig_int_1= 1.03
d_1, sig_cp_1,sig_int_1=float(d_1),float(sig_cp_1),float(sig_int_1)
Ref=[]
Qz_F=[]
Ref_F=[]
g=open("Exp_Fresnal.csv",'rb')#"Test_Fresnal.csv", 'rb')
reader=csv.reader(g)
for row in reader:
Qz_F.append(row[0])
Ref.append(row[1])
Ref_F.append(row[2])
Ref=map(lambda a:float(a),Ref)
Ref_F=map(lambda a:float(a),Ref_F)
Qz_F=map(lambda a:float(a),Qz_F)
Ref_F_arr=np.array((Ref_F))
Qz_arr=np.array((Qz_F))
x=np.array((Qz_arr,Ref_F))
def func(x,d,sig_int,sig_cp):
return (x[1])*(abs(x[0]*d*(np.exp((-sig_int**2)*(x[0]**2)/2)/(1-np.exp(complex(0,1)*x[0]*d)*np.exp((-sig_cp**2)*(x[0]**2)/2)))))**2
DC_ref=func(x,d_1,sig_int_1,sig_cp_1)
Y=np.array((Ref))
popt, pcov=curve_fit(func,x,Y,)#p0=[2.72,1.0,0.44])
perr=np.sqrt(np.diag(pcov))
print "par=",popt;print"Var=",perr
Fit=func(x,*popt)
Fit=func(x,*popt)
Ref=np.transpose(np.array([Ref]))
Qz_F=np.transpose(Qz_F)
plt.plot(Qz_F, Ref, 'bs',label='Experimental')
plt.plot(Qz_F, Fit, 'r--',label='Fit w/ DCM model')
plt.axis([0,3,10**(-10),100])
plt.yscale('log')
plt.title('Reflectivity',fontweight='bold',fontsize=15)
plt.ylabel('Reflectivity',fontsize=15)
plt.xlabel('qz /A^-1',fontsize=15)
plt.legend(loc='upper right',numpoints=1)
plt.show()
The arrays are imported from a file (which I cannot include) and there are no outlier points that would cause the fit to become this distorted. Any help is appreciated.
Edit
I included additional code and the input data to go along with the code but you will have to re-save it as a MS-Dos .CSV
#WarrenWeckesser has a really good point, but further note that the y axis is logarithmic. That apparently huge error at the right end is something like 1e-5 in magnitude, while the points on the top left have reflectivity values of around 0.1. The square error coming from the tail is simply insignificant compared to the huge terms on the left.
I'm sure curve_fit works great. If you want a better visual fit, I suggest trying a fit to log(y) with the log() of your model (either that, or weight your points during fitting); then the result might be more stable visually (and from a physical point of view). Since you're probably trying to give an overall broad-spectrum description of your system, this might be closer to what you expect (but this will inevitably lead to a less precise fit where the reflectivity is high).
I am working with IFFT and have a set of real and imaginary values with their respective frequencies (x-axis). The frequencies are not equidistant, I can't use a discrete IFFT, and I am unable to fit my data correctly, because the values are so jumpy at the beginning. So my plan is to "stretch out" my frequency data points on a lg-scale, fit them (with polyfit) and then return - somehow - to normal scale.
f = data[0:27,0] #x-values
re = daten[0:27,5] #y-values
lgf = p.log10(f)
polylog_re = p.poly1d(p.polyfit(lgf, re, 6))
The fit works definitely better (http://imgur.com/btmC3P0), but is it possible to then transform my polynom back into the normal x-scaling? Right now I'm using those logarithmic fits for my IFFT and take the log10 of my transformed values for plotting etc., but that probably defies all mathematical logic and results in errors.
Your fit is perfectly valid but not a regular polynomial fit. By using log_10(x), you use another model function. Something like y(x)=sum(a_i * 10^(x_i^i). If this is okay for you, you are done. When you wan't to do some more maths, I would suggest using the natural logarithm instead the one to base 10.
I am trying to use simple FFT to make Fourier transform of some function, but apparently the numpy and scipy FFT doesn't work so well even for 1024 points.
For example, suppose I want to make FFT of sin(50x)+cos(80x). Then, at k=50 point should be purely imaginary and k=80 should be purely real. Generally there is some error, but working with a number of points as large as 1024 generally gives quite satisfactory output. But here there is quite a bit of error in the output. The result doesn't improve much with increasing number of points.
Can someone explain the reason of this?
I have tried the following code in Python:
from __future__ import division
import numpy as np
import matplotlib.pyplot as plt
import scipy.fftpack
pi = np.pi
#no. of points
N = 1024
#real axis
x = np.linspace(0,2*pi,N)
#real fn
f_x = np.sin(50*x)+np.cos(80*x)
f_k = (2/N)*scipy.fftpack.fft(f_x)
print f_k[50]
print f_k[80]
and it gives the following output:
(0.155273271152-0.983403030451j)
(0.960302223459+0.242617561413j)
which should have been, 0-1j and 1+0j. With 1024 points I was expecting a more accurate result.
I have also tried transforming by using explicit FT formula, and numpy instead of scipy. Both give the same accuracy.
For ideal, infinite-length signals it would be 0-1j and 1+0j. However, this is a finite-length, digital signal. Due to windowing and the limitations in representing floating-point numbers on a computer, it is never going to perfectly match the ideal case.
Your input length looks like it is off by one.
An FFT only give exact results for sinusoids that exactly integer periodic in the FFT length. For the infinitely many other frequencies, you can improve the results by parabolic or Sinc interpolation between FFT result bins.
I attach a zip archive with all the files needed to illustrate and reproduce the problem.
(I don't have permissions to upload images yet...)
I have an image (test2.png in the zip archive ) with curved lines.
I try to warp it so the lines are straight.
I thought of using scikit-image transform, and in particular transform.PolynomialTransform because the transformation involves high order distortions.
So first I measure the precise position of each line at regular intervals in x to define the input interest points (in the file source_test2.csv).
Then I compute the corresponding desired positions, located along a straight line (in the file destination_test2.csv).
The figure correspondence.png shows how it looks like.
Next, I simply call transform.PolynomialTransform() using a polynomial of order 3.
It finds a solution, but when I apply it using transform.warp(), the result is crazy, as illustrated in the file Crazy_Warped.png
Anybody can tell what I am doing wrong?
I tried polynomial of order 2 without luck...
I managed to get a good transformation for a sub-image (the first 400 columns only).
Is transform.PolynomialTransform() completely unstable in a case like mine?
Here is the entire code:
import numpy as np
import matplotlib.pyplot as plt
import asciitable
import matplotlib.pylab as pylab
from skimage import io, transform
# read image
orig=io.imread("test2.png",as_grey=True)
# read tables with reference points and their desired transformed positions
source=asciitable.read("source_test2.csv")
destination=asciitable.read("destination_test2.csv")
# format as numpy.arrays as required by scikit-image
# (need to add 1 because I started to count positions from 0...)
source=np.column_stack((source["x"]+1,source["y"]+1))
destination=np.column_stack((destination["x"]+1,destination["y"]+1))
# Plot
plt.imshow(orig, cmap='gray', interpolation='nearest')
plt.plot(source[:,0],source[:,1],'+r')
plt.plot(destination[:,0],destination[:,1],'+b')
plt.xlim(0,orig.shape[1])
plt.ylim(0,orig.shape[0])
# Compute the transformation
t = transform.PolynomialTransform()
t.estimate(destination,source,3)
# Warping the image
img_warped = transform.warp(orig, t, order=2, mode='constant',cval=float('nan'))
# Show the result
plt.imshow(img_warped, cmap='gray', interpolation='nearest')
plt.plot(source[:,0],source[:,1],'+r')
plt.plot(destination[:,0],destination[:,1],'+b')
plt.xlim(0,img_warped.shape[1])
plt.ylim(0,img_warped.shape[0])
# Save as a file
io.imsave("warped.png",img_warped)
Thanks in advance!
There are a couple of things wrong here, mainly they have to do with coordinate conventions. For example, if we examine the code where you plot the original image, and then put the clicked point on top of it:
plt.imshow(orig, cmap='gray', interpolation='nearest')
plt.plot(source[:,0],source[:,1],'+r')
plt.xlim(0,orig.shape[1])
plt.ylim(0,orig.shape[0])
(I've taken out the destination points to make it cleaner) then we get the following image:
As you can see, the y-axis is flipped, if we invert the y-axis with:
source[:,1] = orig.shape[0] - source[:,1]
before plotting, then we get the following:
So that is the first problem (don't forget to invert the destination points as well), the second has to do with the transform itself:
t.estimate(destination,source,3)
From the documentation we see that the call takes the source points first, then the destination points. So the order of those arguments should be flipped.
Lastly, the clicked points are of the form (x,y), but the image is stored as (y,x), so we have to transpose the image before applying the transform and then transpose back again:
img_warped = transform.warp(orig.transpose(), t, order=2, mode='constant',cval=float('nan'))
img_warped = img_warped.transpose()
When you make these changes, you get the following warped image:
These lines aren't perfectly flat but it makes much more sense.
Thank you very much for the detailed answer! I cannot believe I did not see the axis inversion problem... Thanks for catching it!
But I am afraid your final solution does not solve my problem... The image you get is still crazy. It should be continuous, no have such big holes and weird distortions... (see final solution below)
I found I could get a reasonable solution using RANSAC:
from skimage.measure import ransac
t, inliers = ransac((destination,source), transform.PolynomialTransform, min_samples=20,residual_threshold=1.0, max_trials=1000)
outliers = inliers == False
I then get the following result
Note that I think I was right using (destination,source) in that order! I think it has to do with the fact that transform.warp requires the inverse_map as input for the transformation object, not the forward map. But maybe I am wrong? The good result I am getting suggest it's correct.
I guess that Polynomial transforms are too unstable, and using RANSAC allows to get a reasonable solution.
My problem was then to find a way to change the polynomial order in the RANSAC call...
transform.PolynomialTransform() does not take any parameters, and uses by default a 2nd order polynomial, but from the result I can see I would need a 3rd or 4th order polynomial.
So I opened a new question, and got a solution from Stefan van der Walt. Follow the link to see how to do it.
Thanks again for your help!