1. Generalized Hough Transform

2. Correlation as a base of Generalized Hough Transform
In order to match a part of a model to a whole, we can use correlation to find the optimal aligning transformation:

3. Calculating Correlation
For each translation, compute the correlation between the target and the translated query:

4. Calculating Correlation
For each translation, compute the correlation between the target and the translated query:

5. Calculating Correlation
For each translation, compute the correlation between the target and the translated query:

6. Calculating Correlation
For each translation, compute the correlation between the target and the translated query:

7. Calculating Correlation
For each translation, compute the correlation between the target and the translated query:

8. Best match in Correlation
For each translation, compute the correlation between the target and the translated query:
Best match

9. Accumulator Space for Correlation
Accumulator space like in Hough
For each translation, compute the correlation between the target and the translated query:

10. Complexity of calculating Correlation
Complexity for binary n * n grids with O(n) non-zero points:
Brute Force: O(n4) – for each of O(n2) translations, compute the O(n2) dot product.

11. Complexity of calculating Correlation
Complexity for binary nxn grids with O(n) non-zero points:
Brute Force: O(n4) – for each of O(n2) translations, compute the O(n2) dot product.
Fast Integration: O(n3) – for each of O(n2) translations, compute the O(n) dot product.

12. Complexity of calculating Correlation
Complexity for binary nxn grids with O(n) non-zero points:
Brute Force: O(n4) – for each of O(n2) translations, compute the O(n2) dot product.
Fast Integration: O(n3) – for each of O(n2) translations, compute the O(n) dot product.
Fourier: O(n2 logn) – compute the FFT, multiply frequency components, compute the IFFT.

13. Correlation as Voting
The value of the correlation at the point (x0,y0) is:
This can be understood in two ways:
g(x-x0, y-y0) is the function g(x, y) translated by (x0,y0)
g(x-x0, y-y0) is the function g(x0, y0) translated by (x,y) and flipped about the origin

14. Correlation as Voting
In the second approach, we treat the pixels as bins and for every possible translation (x,y), we do the following:
Translate g by (x,y)
Flip g
Scale by the value f(x,y)
Update the values of all the bins by the values of the transformed g.

15. Generalized Hough Transform
Correlation as Voting

16. Complexity of Correlation
Complexity for binary n * n grids with O(n) non-zero points:
Brute Force: O(n4) – for each of O(n2) translations, compute the O(n2) dot product.
Fast Integration: O(n3) – for each of O(n2) translations, compute the O(n) dot product.
Fourier: O(n2 logn) – compute the FFT, multiply frequency components, compute the IFFT.
Fast Voting: O(n2) – for each of O(n) points on the boundary, cast O(n) votes.

17. The main idea of a Generalized Hough Transform
When we compute the correlation by voting, we spend most of the time casting bad votes.
Use extra shape information (e.g. gradients) to cast fewer votes:
O(n) complexity: For each of O(n) points on the boundary, cast O(1) votes.