Generalized Hough Transform
Generalized Hough Transform
Correlation
In order to match a part of a model to a whole, we
can use correlation to find the optimal aligning
transformation:
f ,T (g )   f (x )g T (x ) dx
Generalized Hough Transform
Correlation
For each translation, compute the correlation
between the target and the translated query:
f ,T (g )   f (x )g T (x ) dx
Generalized Hough Transform
Correlation
For each translation, compute the correlation
between the target and the translated query:
f ,T (g )   f (x )g T (x ) dx
Generalized Hough Transform
Correlation
For each translation, compute the correlation
between the target and the translated query:
f ,T (g )   f (x )g T (x ) dx
Generalized Hough Transform
Correlation
For each translation, compute the correlation
between the target and the translated query:
f ,T (g )   f (x )g T (x ) dx
Generalized Hough Transform
Correlation
For each translation, compute the correlation
between the target and the translated query:
f ,T (g )   f (x )g T (x ) dx
Generalized Hough Transform
Correlation
For each translation, compute the correlation
between the target and the translated query:
f ,T (g )   f (x )g T (x ) dx
Generalized Hough Transform
Correlation
For each translation, compute the correlation
between the target and the translated query:
f ,T (g )   f (x )g T (x ) dx
Generalized Hough Transform
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.
Generalized Hough Transform
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.
Generalized Hough Transform
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.
Generalized Hough Transform
Correlation as Voting
The value of the correlation at the point (x0,y0) is:
Corr (f , g , x 0 , y 0 )   f (x , y )g (x  x 0 , y  y 0 )dx dy
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
Generalized Hough Transform
Correlation as Voting
Corr (f , g , x 0 , y 0 )   f (x , y )g (x  x 0 , y  y 0 )dx dy
In the second approach, we treat the pixels as bins
and for every possible translation (x,y), we:
–
–
–
–
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.
Generalized Hough Transform
Correlation as Voting
Generalized Hough Transform
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.
– Fast Voting: O(n2) – for each of O(n) points on the
boundary, cast O(n) votes.
Generalized Hough Transform
Key Idea
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.