numpy - distance matrix of curves in python

Question

I have a set of curves defined as 2D arrays (number of points, number of coordinates). I am calculating a distance matrix for them using Hausdorff distance. My current code is as follows. Unfortunately it is too slow with 500-600 curves each having 50-100 3D points. Is there any faster way for that?

def distanceBetweenCurves(C1, C2):
    D = scipy.spatial.distance.cdist(C1, C2, 'euclidean')

    #none symmetric Hausdorff distances
    H1 = np.max(np.min(D, axis=1))
    H2 = np.max(np.min(D, axis=0))

    return (H1 + H2) / 2.

def distanceMatrixOfCurves(Curves):
    numC = len(Curves)

    D = np.zeros((numC, numC))
    for i in range(0, numC-1):
        for j in range(i+1, numC):
            D[i, j] = D[j, i] = distanceBetweenCurves(Curves[i], Curves[j])

    return D

Answer 1

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This is kind of a hard problem. A possible way would be to implement the euclidian distance on your own, completely abandon scipy and make use of pypy's JIT compiler. But most likely this will not make you gane much.

Personally, I would recommend you to write the routine in C.

The problem is less the implementation but the way you approach this problem. You chose a brute force approach by calculating the euclidian distance for each distinct pair of points in each possible pair of the metric space subsets. This is computationally demanding:

• Assume you have 500 curves and each of them has 75 points. With the brute force approach you end up calculating the euclidean distance 500 * 499 * 75 * 75 = 1 403 437 500 times. It is not further surprising that this approach takes forever to run.

I'm not an expert with this but I know that the Hausdorff distance is extensively used in image processing. I would suggest you to browse the literature for speed optimized algorithms. A starting point might be this, or this paper. Also, often mentioned in combination with the Hausdorff distance is the Voroni diagram.

I hope these links might help you with this problem.