Calculating variance image python

Question

Is there an easy way to calculate a running variance filter on an image using Python/NumPy/Scipy? By running variance image I mean the result of calculating sum((I - mean(I))^2)/nPixels for each sub-window I in the image.

Since the images are quite large (12000x12000 pixels), I want to avoid the overhead of converting the arrays between formats just to be able to use a different library and then convert back.

I guess I could do this manually by finding the mean using something like

kernel = np.ones((winSize, winSize))/winSize**2
image_mean = scipy.ndimage.convolve(image, kernel)
diff = (image - image_mean)**2
# Calculate sum over winSize*winSize sub-images
# Subsample result

but it would be much nicer to have something like the stdfilt-function from Matlab.

Can anyone point me in the direction of a library that has this functionality AND supports numpy arrays, or hint at/provide a way to do this in NumPy/SciPy?

Answer 1

Simpler solution and also faster: use SciPy's ndimage.uniform_filter

import numpy as np
from scipy import ndimage 
rows, cols = 500, 500
win_rows, win_cols = 5, 5

img = np.random.rand(rows, cols)
win_mean = ndimage.uniform_filter(img, (win_rows, win_cols))
win_sqr_mean = ndimage.uniform_filter(img**2, (win_rows, win_cols))
win_var = win_sqr_mean - win_mean**2

The "stride trick" is beautiful trick, but 4 slower and not that readable. the generic_filter is 20 times slower than the strides...

Answer 2

You can use numpy.lib.stride_tricks.as_strided to get a windowed view of your image:

import numpy as np
from numpy.lib.stride_tricks import as_strided

rows, cols = 500, 500
win_rows, win_cols = 5, 5

img = np.random.rand(rows, cols)

win_img = as_strided(img, shape=(rows-win_rows+1, cols-win_cols+1,
                                 win_rows, win_cols),
                     strides=img.strides*2)

And now win_img[i, j] is the (win_rows, win_cols) array with the top left corner at position [i, j] :

>>> img[100:105, 100:105]
array([[ 0.34150754,  0.17888323,  0.67222354,  0.9020784 ,  0.48826682],
       [ 0.68451774,  0.14887515,  0.44892615,  0.33352743,  0.22090103],
       [ 0.41114758,  0.82608407,  0.77190533,  0.42830363,  0.57300759],
       [ 0.68435626,  0.94874394,  0.55238567,  0.40367885,  0.42955156],
       [ 0.59359203,  0.62237553,  0.58428725,  0.58608119,  0.29157555]])
>>> win_img[100,100]
array([[ 0.34150754,  0.17888323,  0.67222354,  0.9020784 ,  0.48826682],
       [ 0.68451774,  0.14887515,  0.44892615,  0.33352743,  0.22090103],
       [ 0.41114758,  0.82608407,  0.77190533,  0.42830363,  0.57300759],
       [ 0.68435626,  0.94874394,  0.55238567,  0.40367885,  0.42955156],
       [ 0.59359203,  0.62237553,  0.58428725,  0.58608119,  0.29157555]])

You have to be careful, though, with not converting your windowed view of the image, into a windowed copy of it: in my example that would require 25 times more storage. I believe numpy 1.7 lets you select more than one axis, so you could then simply do:

>>> np.var(win_img, axis=(-1, -2))

I am stuck with numpy 1.6.2, so I cannot test that. The other option, which may fail with not-so-large windows, would be to do, if I remember my math correctly:

>>> win_mean = np.sum(np.sum(win_img, axis=-1), axis=-1)/win_rows/win_cols
>>> win_sqr_mean = np.sum(np.sum(win_img**2, axis=-1), axis=-1)/win_rows/win_cols
>>> win_var = win_sqr_mean - win_mean**2

And now win_var is an array of shape

>>> win_var.shape
(496, 496)

and win_var[i, j] holds the variance of the (5, 5) window with top left corner at [i, j] .

Answer 3

After a bit of optimization we came up with this function for a generic 3D image:

def variance_filter( img, VAR_FILTER_SIZE ):
  from numpy.lib.stride_tricks import as_strided

  WIN_SIZE=(2*VAR_FILTER_SIZE)+1
  if ~ VAR_FILTER_SIZE%2==1:
      print 'Warning, VAR_FILTER_SIZE must be ODD Integer number  '
  # hack -- this could probably be an input to the function but Alessandro is lazy
  WIN_DIMS = [ WIN_SIZE, WIN_SIZE, WIN_SIZE ]


  # Check that there is a 3D image input.
  if len( img.shape ) != 3:
      print "\t variance_filter: Are you sure that you passed me a 3D image?"
      return -1
  else:
      DIMS = img.shape

  # Set up a windowed view on the data... this will have a border removed compared to the img_in
  img_strided = as_strided(img, shape=(DIMS[0]-WIN_DIMS[0]+1, DIMS[1]-WIN_DIMS[1]+1, DIMS[2]-WIN_DIMS[2]+1, WIN_DIMS[0], WIN_DIMS[1], WIN_DIMS[2] ), strides=img.strides*2)

  # Calculate variance, vectorially
  win_mean = numpy.sum(numpy.sum(numpy.sum(img_strided, axis=-1), axis=-1), axis=-1) / (WIN_DIMS[0]*WIN_DIMS[1]*WIN_DIMS[2])

  # As per http://en.wikipedia.org/wiki/Variance, we are removing the mean from every window,
  #   then squaring the result.
  # Casting to 64 bit float inside, because the numbers (at least for our images) get pretty big
  win_var = numpy.sum(numpy.sum(numpy.sum((( img_strided.T.astype('<f8') - win_mean.T.astype('<f8') )**2).T, axis=-1), axis=-1), axis=-1) / (WIN_DIMS[0]*WIN_DIMS[1]*WIN_DIMS[2])

  # Prepare an output image of the right size, in order to replace the border removed with the windowed view call
  out_img = numpy.zeros( DIMS, dtype='<f8' )
  # copy borders out...
  out_img[ WIN_DIMS[0]/2:DIMS[0]-WIN_DIMS[0]+1+WIN_DIMS[0]/2, WIN_DIMS[1]/2:DIMS[1]-WIN_DIMS[1]+1+WIN_DIMS[1]/2, WIN_DIMS[2]/2:DIMS[2]-WIN_DIMS[2]+1+WIN_DIMS[2]/2, ] = win_var

  # output
  return out_img.astype('>f4')

Answer 4

You can use scipy.ndimage.generic_filter . I can't test with matlab, but perhaps this gives you what you're looking for:

import numpy as np
import scipy.ndimage as ndimage     
subs = 10  # this is the size of the (square) sub-windows
img = np.random.rand(500, 500)
img_std = ndimage.filters.generic_filter(img, np.std, size=subs)

You can make the sub-windows of arbitrary sizes using the footprint keyword. See this question for an example.