30-11-2012, 04:50 PM
Convolution – Theory and Applications
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Discrete Convolution
Let f(x) and h(x) be the two sequences, with the respective support regions N
and M:
• If f(x) and h(x) are infinite, the convolution g(x) is linear; g(x) takes the
support region of length N+M-1 ;
• If f(x) and h(x) are periodic and having the same period N, the
convolution g(x) is circular. In this case, outside the display region of
f(x), the data is periodically repeated; and outside the support region of
h(x), the data is zero, extended to the end of the period of dimension N.
• Example of f(x) and h(x) :
Circular Convolution
It is possible to implement any linear or invariant displacement filter with the
convolution theory;
The circular convolution theorem indicates that the circular convolution can
be implemented with Fourier Transforms (DFT) and vice versa.
Experimentation to illustrate this theorem (Lab exercise):
1. Apply a convolution in the spatial domain with the Robert mask (2x2) ;
2. Apply a convolution in the Fourier space:
1. Take the DFT of the original image;
2. Take the DFT of the Robert mask;
3. Multiply the two DFT images;
4. Inverse the resulting image of the convolution in the Fourier space ;
3. Compare the two results.
Conclusion :
A consequence of the convolution theorem is that for all filtering masks in the
spatial domain, there exists a corresponding mask in the frequency domain,
and vice versa.
Why this duality?
• Measurements on the execution speed showed that discrete convolution
is faster for small-size masks, and the Fourier method is faster for large
masks.
• In the case of filtering coherent noises, the design of the filter and its
processing is performed in the frequency domain. Fourier method is
more adequate.
• For simple operations such as mean neighborhood, Gaussian or
Laplacian filtering, that use small-size masks, convolution in the spatial
domain is usually more appropriate.
It is important to note that all linear filters and all invariant displacement filters
can be implemented in the two domains