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Convolution (Section 3.4)

Convolution (Section 3.4). CS474/674 – Prof. Bebis. g. Output Image. f. Correlation - Review. K x K. Convolution - Review. Same as correlation except that the mask is flipped both horizontally and vertically.

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Convolution (Section 3.4)

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  1. Convolution (Section 3.4) CS474/674 – Prof. Bebis

  2. g Output Image f Correlation - Review K x K

  3. Convolution - Review • Same as correlation except that the mask is flipped both horizontally and vertically. • Note that if w(x,y) is symmetric, that is w(x,y)=w(-x,-y), then convolution is equivalent to correlation!

  4. 1D Continuous Convolution - Definition • Convolution is defined as follows: • Convolution is commutative

  5. Example • Suppose we want to compute the convolution of the following two functions:

  6. Example (cont’d)

  7. Example (cont’d) Step 3:

  8. Example (cont’d)

  9. Example (cont’d)

  10. Example (cont’d)

  11. Example (cont’d)

  12. Example (cont’d)

  13. Example (cont’d)

  14. Important Observations • The extent of f(x) * g(x) is equal to the extent of f(x) plus the extent of g(x) • For every x, the limits of the integral are determined as follows: • Lower limit: MAX (left limit of f(x), left limit of g(x-a)) • Upper limit: MIN (right limit of f(x), right limit of g(x-a))

  15. Example (cont’d)

  16. Example

  17. Convolution with an impulse (i.e., delta function)

  18. Convolution with an “train” of impulses =

  19. Convolution Theorem • Convolution in the time domain is equivalent to multiplication in the frequency domain. • Multiplication in the time domain is equivalent to convolution in the frequency domain. f(x) F(u) g(x) G(u)

  20. Efficient computation of (f * g) • 1. Compute and • 2. Multiply them: • 3. Compute the inverse FT:

  21. Discrete Convolution • Replace integral with summation • Integration variable becomes an index. • Displacements take place in discrete increments

  22. Discrete Convolution (cont’d) 3 samples 5 samples g - 1)

  23. Convolution Theorem in Discrete Case • Input sequences: • Length of output sequence: • Extended input sequences (i.e., pad with zeroes)

  24. Convolution Theorem in Discrete Case (cont’d) • When dealing with discrete sequences, the convolution theorem holds true for the extended sequences only, i.e.,

  25. Why? continuous case discrete case Using DFT, it will be a periodic function with period M (since DFT is periodic)

  26. Why? (cont’d) If M<A+B-1, the periods will overlap If M>=A+B-1, the periods will not overlap

  27. 2D Convolution • Definition • 2D convolution theorem

  28. Discrete 2D convolution • Suppose f(x,y) and g(x,y) are images of size A x B and C x D • The size of f(x,y) * g(x,y) would be N x M where N=A+C-1 and M=B+D-1 • Extended images (i.e., pad with zeroes):

  29. Discrete 2D convolution (cont’d) • The convolution theorem holds true for the extended images.

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