How to obtain the scale and rotation angle from LogPolar transform

Question

I'm trying to use LogPolar transform to obtain the scale and the rotation angle from two images. Below are two 300x300 sample images. The first rectangle is 100x100, and the second rectangle is 150x150, rotated by 45 degree.

The algorithm:

1. Convert both images to LogPolar.
2. Find the translational shift using Phase Correlation.
3. Convert the translational shift to scale and rotation angle (how to do this?).

My code:

#include <opencv2/imgproc/imgproc.hpp>
#include <opencv2/imgproc/imgproc_c.h>
#include <opencv2/highgui/highgui.hpp>
#include <iostream>

int main()
{
    cv::Mat a = cv::imread("rect1.png", 0);
    cv::Mat b = cv::imread("rect2.png", 0);
    if (a.empty() || b.empty())
        return -1;

    cv::imshow("a", a);
    cv::imshow("b", b);

    cv::Mat pa = cv::Mat::zeros(a.size(), CV_8UC1);
    cv::Mat pb = cv::Mat::zeros(b.size(), CV_8UC1);
    IplImage ipl_a = a, ipl_pa = pa;
    IplImage ipl_b = b, ipl_pb = pb;
    cvLogPolar(&ipl_a, &ipl_pa, cvPoint2D32f(a.cols >> 1, a.rows >> 1), 40);
    cvLogPolar(&ipl_b, &ipl_pb, cvPoint2D32f(b.cols >> 1, b.rows >> 1), 40);

    cv::imshow("logpolar a", pa);
    cv::imshow("logpolar b", pb);

    cv::Mat pa_64f, pb_64f;
    pa.convertTo(pa_64f, CV_64F);
    pb.convertTo(pb_64f, CV_64F);

    cv::Point2d pt = cv::phaseCorrelate(pa_64f, pb_64f);

    std::cout << "Shift = " << pt 
              << "Rotation = " << cv::format("%.2f", pt.y*180/(a.cols >> 1)) 
              << std::endl;

    cv::waitKey(0);

    return 0;
}

The log polar images:

For the sample image images above, the translational shift is (16.2986, 36.9105). I have successfully obtain the rotation angle, which is 44.29. But I have difficulty in calculating the scale. How to convert the given translational shift to obtain the scale?

Answer 1

You have two Images f1, f2 with f1(m, n) = f2(m/a , n/a) That is f1 is scaled by factor a

In logarithmic notation that is equivalent to f1(log m, log n) = f2(logm − log a, log n − log a) where log a is the shift in your phasecorrelated image.

Compare B. S. Reddy, B. N. Chatterji: An FFT-Based Technique for Translation, Rotation and Scale-Invariant Image Registration, IEEE Transactions On Image Processing Vol. 5 No. 8, IEEE, 1996

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.185.4387&rep=rep1&type=pdf

Answer 2

here is python version

which tells

ir = abs(ifft2((f0 * f1.conjugate()) / r0))
    i0, i1 = numpy.unravel_index(numpy.argmax(ir), ir.shape)
    angle = 180.0 * i0 / ir.shape[0]
    scale = log_base ** i1