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Several important optical
effects can be described in terms of convolutions.
Let us examine the concepts using 1D continuous functions.
The convolution of two functions f(x) and g(x), written f(x)*g(x), is defined
by the integral
For example, let us take two top hat functions of the type
described earlier. Let be the top hat function shown in
and let be as shown in Fig. 13,
Fig. 13 Another top hat:
- is the reflection of this function in the vertical
- is the latter shifted to the right by a distance x.
- Thus for a given value of x, integrated over all
is the area of overlap of these two top hats, as has unit
- An example is shown for x in the range in
Fig. 14 Convolving two top hats
If we now consider x moving from to , we can see
- for or , there is no overlap;
- as x goes from -1 to 0 the area
of overlap steadily increases from 0 to 1/2;
- as x increases from 0 to 1,
the overlap area remains at 1/2;
- and finally as x increases from 1 to 2, the
overlap area steadily decreases again from 1/2 to 0.
- Thus the convolution of
f(x) and g(x), f(x)*g(x), in this case has the form shown in
Fig. 15 Convolution of two top hats
Mathematically the convolution is expressed by:
David Marshall 1994-1997