Why sift feature is also invariant to planar homography transform

Question

I have read sift features' paper, and I understand why it is rotation invariant. But I do not understand why does it also invariant to planar homography transform, as my test code shows. In the homography transform between two images, the change does not only include rotation and scale. For example, a rectangle may be transformed to other quadrangle with every corner less or larger than 90 degrees. You can image that the shape of the object is changed, but why the feature of the key point still match?

As to the details of the algorithm, when the key point's surrounding pixel changed without rotating the same degree, the keypoint's 128 dimension feature's value will be different when they subtract the keypoint's gradient angle.

Can someone explain why?

Answer 1

As far as I know, the SIFT descriptor is not invariant to a projective transformation (homography). However, it works well enough when the actual homography is sufficiently close to a similarity transformation.

This paper by Mikolajczyk and Schmid proposes an interest point detector, which is affine-invariant. They also make the descriptor affine-invariant by transforming the image patch from which it is computed.