Convolutions

Saturday, November 18. 2006

Convolutions

(Q26) Meanwhile, I know that a convolution of one function $f$ with another function $g$ means that $f$ will be “ $g$ -smeared”, and vice versa. I visualized some convolutions using Octave. I was prepared to implement $f*g(k)=\sum_jf(j)g(k-j)$ for two vectors, but then stepped over HGFei’s conv2fei which makes the surprising use of the identity $\widehat{f*g}=\hat f\cdot\hat g$ ; I wouldn’t have come to the idea to do it this way, you only have to multiply two FFT’s and then do the inverse FFT.

First I convolved a square with a small Gaussian—this is nothing less than a Gaussian blur:

$*$

$=$

Note that the blur would already occur even if the second object were solid! Look at the square convolved with itself:

$*$

$=$

Then I watched how these bars convolve with each other:

	$*$		$=$
	$*$		$=$

Notice how the blurness shrinks when the two objects can’t overlap much.

Posted by Stephan Paukner in Master's Thesis at 17:48 | Comments (0) | Trackback (1)

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