[python] Computing cross-correlation function?

In R, I am using ccf or acf to compute the pair-wise cross-correlation function so that I can find out which shift gives me the maximum value. From the looks of it, R gives me a normalized sequence of values. Is there something similar in Python's scipy or am I supposed to do it using the fft module? Currently, I am doing it as follows:

xcorr = lambda x,y : irfft(rfft(x)*rfft(y[::-1]))
x = numpy.array([0,0,1,1])
y = numpy.array([1,1,0,0])
print xcorr(x,y)

If you are looking for a rapid, normalized cross correlation in either one or two dimensions I would recommend the openCV library (see http://opencv.willowgarage.com/wiki/ http://opencv.org/). The cross-correlation code maintained by this group is the fastest you will find, and it will be normalized (results between -1 and 1).

While this is a C++ library the code is maintained with CMake and has python bindings so that access to the cross correlation functions is convenient. OpenCV also plays nicely with numpy. If I wanted to compute a 2-D cross-correlation starting from numpy arrays I could do it as follows.

import numpy
import cv

#Create a random template and place it in a larger image
templateNp = numpy.random.random( (100,100) )
image = numpy.random.random( (400,400) )
image[:100, :100] = templateNp

#create a numpy array for storing result
resultNp = numpy.zeros( (301, 301) )

#convert from numpy format to openCV format
templateCv = cv.fromarray(numpy.float32(template))
imageCv = cv.fromarray(numpy.float32(image))
resultCv =  cv.fromarray(numpy.float32(resultNp))

#perform cross correlation
cv.MatchTemplate(templateCv, imageCv, resultCv, cv.CV_TM_CCORR_NORMED)

#convert result back to numpy array
resultNp = np.asarray(resultCv)

For just a 1-D cross-correlation create a 2-D array with shape equal to (N, 1 ). Though there is some extra code involved to convert to an openCV format the speed-up over scipy is quite impressive.

I just finished writing my own optimised implementation of normalized cross-correlation for N-dimensional arrays. You can get it from here.

It will calculate cross-correlation either directly, using scipy.ndimage.correlate, or in the frequency domain, using scipy.fftpack.fftn/ifftn depending on whichever will be quickest.

For 1D array, numpy.correlate is faster than scipy.signal.correlate, under different sizes, I see a consistent 5x peformance gain using numpy.correlate. When two arrays are of similar size (the bright line connecting the diagonal), the performance difference is even more outstanding (50x +).

# a simple benchmark
res = []
for x in range(1, 1000):
    list_x = []
    for y in range(1, 1000): 

        # generate different sizes of series to compare
        l1 = np.random.choice(range(1, 100), size=x)
        l2 = np.random.choice(range(1, 100), size=y)

        time_start = datetime.now()
        np.correlate(a=l1, v=l2)
        t_np = datetime.now() - time_start

        time_start = datetime.now()
        scipy.signal.correlate(in1=l1, in2=l2)
        t_scipy = datetime.now() - time_start

        list_x.append(t_scipy / t_np)
    res.append(list_x)
plt.imshow(np.matrix(res))

As default, scipy.signal.correlate calculates a few extra numbers by padding and that might explained the performance difference.

>> l1 = [1,2,3,2,1,2,3]
>> l2 = [1,2,3]
>> print(numpy.correlate(a=l1, v=l2))
>> print(scipy.signal.correlate(in1=l1, in2=l2))

[14 14 10 10 14]
[ 3  8 14 14 10 10 14  8  3]  # the first 3 is [0,0,1]dot[1,2,3]