Generalized Hough Transform

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Generalized Hough Transform
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Correlation as a base of 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

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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

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Best match in Correlation

• For each translation, compute the correlation
  between the target and the translated query

Best match
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Accumulator Space for Correlation

• Accumulator space like in Hough
• For each translation, compute the correlation
  between the target and the translated query

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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.

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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.

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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.

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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

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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.

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Generalized Hough Transform

• Correlation as Voting

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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.

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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.