John Fremlin's blog: Rotating an image with OpenCV and Python

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OpenCV is the most widely used open-source vision library. It lets you detect faces in photographs or video feeds with very little code.

There are a few tutorials on the Internet explaining how to use an affine transform to rotate an image with OpenCV -- they don't at all handle the issue that rotating a rectangle inside its own bounds will generally cut off the corners, so the shape of the destination image needs to be changed. That's a bit sad, as doing it properly is very little code:

def rotate_about_center(src, angle, scale=1.):
    w = src.shape[1]
    h = src.shape[0]
    rangle = np.deg2rad(angle)  # angle in radians
    # now calculate new image width and height
    nw = (abs(np.sin(rangle)*h) + abs(np.cos(rangle)*w))*scale
    nh = (abs(np.cos(rangle)*h) + abs(np.sin(rangle)*w))*scale
    # ask OpenCV for the rotation matrix
    rot_mat = cv2.getRotationMatrix2D((nw*0.5, nh*0.5), angle, scale)
    # calculate the move from the old center to the new center combined
    # with the rotation
    rot_move = np.dot(rot_mat, np.array([(nw-w)*0.5, (nh-h)*0.5,0]))
    # the move only affects the translation, so update the translation
    # part of the transform
    rot_mat[0,2] += rot_move[0]
    rot_mat[1,2] += rot_move[1]
    return cv2.warpAffine(src, rot_mat, (int(math.ceil(nw)), int(math.ceil(nh))), flags=cv2.INTER_LANCZOS4)

The affine transformation of the rotation has to be combined with the affine transformation of translation, from the center of the original image to the center of the destination image. An affine transformation in 2D is a 2x2 matrix A and a translation 2x1 vector a - it takes a source point p = (x,y) to a destination: Ap + a. Combining two transforms Ap + a and Bp + b, doing A first then B, one gets B(Ap + a) + b - another affine transform with matrix BA and vector Ba + b.

In this case, we are combining a rotation with a translation; A translation as an affine transform has the identity 2x2 matrix I and a movement vector m, so is represented by Ip + m, and we want to first translate to the new center, then rotate about that, so we take the rotation Rp + r after applying Ip + m, which gives Rp + Rm + r, which explains why we have to only add two coefficients.

PS. Sadly, numpy interprets the multiplication operator * not as matrix multiplication if it considers the inputs to be vectors of vectors rather than matrices, so we have to explicitly write np.dot.

PPS. We use the Lanczos interpolation which is generally good for scaling up but not for scaling down very small; that should be adapted given the application.

PPPS. The interaction with Python is much improved with the cv2 module, but there are inescapably some rough edges as numpy has a different co-ordinate ordering than OpenCV. Also, for some reason OpenCV persists in using units like degrees instead of radians, and so on. In numpy, the co-ordinates in an image array are accessed in [y,x] order, as in vertical increasing downwards first, followed by horizontal increasing rightwards second. In OpenCV, sizes are given as (width, height), the opposite order.