How can I use scipy.ndimage.interpolation.affine_transform to rotate an image about its centre?

Question

I am perplexed by the API to scipy.ndimage.interpolation.affine_transform. And judging by this issue I\'m not the only one. I\'m actually wanting to do more interesting th

Answer 1

Based on the insight from @timday that matrix and offset are defined in the output coordinate system, I would offer the following reading of the issue, which fits with standard notations in linear algebra and allows to understand the scaling of images as well. I use here T.inv=T^-1 as pseudo-python notation to mean the inverse of a matrix and * to mean the dot product.

For each point o in the output image, affine_transform finds the corresponding point i in the input image as i=T.inv*o+s, where matrix=T.inv is the inverse of the 2x2 transformation matrix that one would use to define the forward affine transformation and offset=s is the translation defined in the output coordinates. For a pure rotation T=R=[[cos,-sin],[sin,cos]], and in this special case matrix=T.inv=T.T, which is the reason why @timday had to apply the transposition still (alternatively one could just use the negative angle).

The value for the offset s is found exactly the way described by @timday: if c_in is supposed to be positioned, after the affine transformation, at c_out (e.g. the input centre should be placed at the output centre) then c_in=T.inv*c_out+s or s=c_in-T.inv*c_out (note the conventional mathematical order of the matrix product used here, matrix*vector, which is why @timday, who used the revers order, didn't need a transposition at this point in his code).

If one wants a scaling S first and then a rotation R it holds that T=R*S and therefore T.inv=S.inv*R.inv (note the reversed order). For example, if one wants to make the image double as wide in the columns direction ('x'), then S=diag((1, 2)), hence S.inv=diag((1, 0.5)).

src = scipy.misc.lena()
c_in = 0.5 * array(src.shape)
dest_shape = (512, 1028)
c_out = 0.5 * array(dest_shape)
for i in xrange(0, 7):
    a = i * 15.0 * pi / 180.0
    rot = array([[cos(a), -sin(a)], [sin(a), cos(a)]])
    invRot = rot.T
    invScale = diag((1.0, 0.5))
    invTransform = dot(invScale, invRot)
    offset = c_in - dot(invTransform, c_out)
    dest = scipy.ndimage.interpolation.affine_transform(
        src, invTransform, order=2, offset=offset, output_shape=dest_shape, cval=0.0, output=float32
    )
    subplot(1, 7, i + 1);axis('off');imshow(dest, cmap=cm.gray)
show()

If the image is to be first rotated, then stretched, the order of the dot product needs to be reversed:

invTransform = dot(invRot, invScale)