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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 l.jpg

Convolution (Section 3.4)

CS474/674 – Prof. Bebis


Correlation review l.jpg

g

Output

Image

f

Correlation - Review

K x K


Convolution review l.jpg
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!


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1D Continuous Convolution - Definition

  • Convolution is defined as follows:

  • Convolution is commutative


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Example

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










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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))




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



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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)


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Efficient computation of (f * g)

  • 1. Compute and

  • 2. Multiply them:

  • 3. Compute the inverse FT:


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Discrete Convolution

  • Replace integral with summation

  • Integration variable becomes an index.

  • Displacements take place in discrete increments


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Discrete Convolution (cont’d)

3 samples

5 samples

g - 1)


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Convolution Theorem in Discrete Case

  • Input sequences:

  • Length of output sequence:

  • Extended input sequences (i.e., pad with zeroes)


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Convolution Theorem in Discrete Case (cont’d)

  • When dealing with discrete sequences, the convolution theorem holds true for the extended sequences only, i.e.,


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Why?

continuous case

discrete case

Using DFT, it will be a periodic function

with period M (since DFT is periodic)


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Why? (cont’d)

If M<A+B-1, the periods

will overlap

If M>=A+B-1, the periods

will not overlap


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2D Convolution

  • Definition

  • 2D convolution theorem


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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):


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Discrete 2D convolution (cont’d)

  • The convolution theorem holds true for the extended images.


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