rotation matrix in openCV

Question

I am creating a rotation matrix in python as follows:

import numpy as np
def make_rot(angle):
    cost = np.cos(np.deg2rad(angle))
    sint = np.sin(np.deg2rad(angle))
    rot = np.array([[cost, -sint, 0],
                   [sint, cost, 0],
                   [0, 0, 1]])
    return rot

This is as defined in the wikipedia page here: http://en.wikipedia.org/wiki/Rotation_matrix

I run it with the angle parameter as 45 degrees and I get something like:

[[ 0.70710678 -0.70710678  0.        ]
 [ 0.70710678  0.70710678  0.        ]
 [ 0.          0.          1.        ]]

Now, I use the OpenCV getRotationMatrix2D API as follows:

import cv2
M = cv2.getRotationMatrix2D((0, 0), 45, 1)

The matrix I get is the inverse of the matrix (the transpose as it is a rotation matrix). The result is as follows:

[[ 0.70710678  0.70710678  0.        ]
 [-0.70710678  0.70710678  0.        ]]

As you can see it is the inverse. I have not found anything in the OpenCV documentation which says anything about this behaviour.

Now, I can use this matrix in OpenCV and skimage to transform an image as follows:

# openCV
M = cv2.getRotationMatrix2D((0, 0), 45, 1)
dst = cv2.warpAffine(image2, M, (coumns, rows))

# skimage
from skimage import transform as tf
tform = tf.AffineTransform(matrix=make_rot(45))
dst = tf.warp(image_2, tform)

The surprising thing is that the result from using my matrix and OpenCV matrix is the same.

My question is why is OpenCV working with the inverse of the transformation? I am wondering if this is something they are not mentioning in the documentation or if I am using this wrong somehow.

Answer 1

I think the problem is that in a traditional right-handed coordinate system the rotation matrix looks exactly like the one you are using.

Whereas in images (and OpenCV works with images) the origin is in the top-left corner, the x-axis goes to the right (as usual), but y-axis goes down rather than up, which means that the coordinate system is left-handed and the rotation matrix is different.

Let's build the rotation matrix in this coordinate system. If we denote:

A - linear transformation (rotation in our case)

then we get

A(i) = (cos(angle), -sin(angle))

where i - is the base vector corresponding to x-axis

A(j) = (sin(angle), cos(angle))

where j- is the base vector for y-axis.

Consequently, rotation matrix looks exactly like in OpenCV:

A(i) is its first column,

A(j) is the second one.