numpy - python convolution with different dimension

Question

I'm trying to implement convolutional neural network in Python.
However, when I use signal.convolve or np.convolve, it can not do convolution on X, Y(X is 3d, Y is 2d). X are training minibatches. Y are filters. I don't want to do for loop for every training vector like:

for i in xrange(X.shape[2]):
    result = signal.convolve(X[:,:,i], Y, 'valid')
    ....

So, is there any function I can use to do convolution efficiently?

Answer 1

Scipy implements standard N-dimensional convolutions, so that the matrix to be convolved and the kernel are both N-dimensional.

A quick fix would be to add an extra dimension to Y so that Y is 3-Dimensional:

result = signal.convolve(X, Y[..., None], 'valid')

I'm assuming here that the last axis corresponds to the image index as in your example [width, height, image_idx] (or [height, width, image_idx]). If it is the other way around and the images are indexed in the first axis (as it is more common in C-ordering arrays) you should replace Y[..., None] with Y[None, ...].

The line Y[..., None] will add an extra axis to Y, making it 3-dimensional [kernel_width, kernel_height, 1] and thus, converting it to a valid 3-Dimensional convolution kernel.

NOTE: This assumes that all your input mini-batches have the same width x height, which is standard in CNN's.

---

EDIT: Some timings as @Divakar suggested.

The testing framework is setup as follows:

def test(S, N, K):
    """ S: image size, N: num images, K: kernel size"""
    a = np.random.randn(S, S, N)
    b = np.random.randn(K, K)
    valid = [slice(K//2, -K//2+1), slice(K//2, -K//2+1)]

    %timeit signal.convolve(a, b[..., None], 'valid')
    %timeit signal.fftconvolve(a, b[..., None], 'valid')
    %timeit ndimage.convolve(a, b[..., None])[valid]

Find bellow tests for different configurations:

• Varying image size S:

  >>> test(100, 50, 11) # 100x100 images
  1 loop, best of 3: 909 ms per loop
  10 loops, best of 3: 116 ms per loop
  10 loops, best of 3: 54.9 ms per loop
  
  >>> test(1000, 50, 11) # 1000x1000 images
  1 loop, best of 3: 1min 51s per loop
  1 loop, best of 3: 16.5 s per loop
  1 loop, best of 3: 5.66 s per loop
  

• Varying number of images N:

  >>> test(100, 5, 11) # 5 images
  10 loops, best of 3: 90.7 ms per loop
  10 loops, best of 3: 26.7 ms per loop
  100 loops, best of 3: 5.7 ms per loop
  
  >>> test(100, 500, 11) # 500 images
  1 loop, best of 3: 9.75 s per loop
  1 loop, best of 3: 888 ms per loop
  1 loop, best of 3: 727 ms per loop
  

• Varying kernel size K:

  >>> test(100, 50, 5) # 5x5 kernels
  1 loop, best of 3: 217 ms per loop
  10 loops, best of 3: 100 ms per loop
  100 loops, best of 3: 11.4 ms per loop
  
  >>> test(100, 50, 31) # 31x31 kernels
  1 loop, best of 3: 4.39 s per loop
  1 loop, best of 3: 220 ms per loop
  1 loop, best of 3: 560 ms per loop
  

So, in short, ndimage.convolve is always faster, except when the kernel size is very large (as K = 31 in the last test).