[Phys-l] Convolution (was Sun's image ...)

[Brian Whatcott <betwys1@sbcglobal.net>]

Anthony Lapinski wrote:
> Not sure my (high school) students have ever heard of "convolution,"  and
> I myself am unfamiliar with the term.
>
> Is there another ("basic") explanation to use when trying to explain this
> optical phenomenon to students?
Let me take a shot at an introductory idea of the convolution concept.

Take  two polynomial equations, say
x^3 +2x^2 + 3x + 4   and  x^3 + 4x^2 + 9x + 16

Just as we can multiply two scalars 3  X 4 = 12,
we can multiply these two polynomial functions in a similar way.
Multiplying these functions in this way is called convolution.
There is a quite compact way of expressing such equations:
a(x)   = [1 2 3 4]    Notice these are the coefficients of the first 
polynomial
b(x)  = [3 4 9 16]   These are the coefficients of the second one.

The convolution of function a and function b expressed in this same 
compact way is
c(x)  = [1 6 20 50 75 84 64 ]
It represents 'dealing out' the first function to places defined by the 
second equation,  or in the familar notation,
c(x) = x^6 + 6x^5 + 20x^4 + 50x^3 + 75x^2 + 84x + 64

Notice the first and last terms are easy to check:
[first] x^3 X x^3 = x^6     an x^6  coefficient of 1
[last]  4 X 16 = 64            an x^0 coefficient of 64


This is a sort of math calculator capability that has been elaborated in 
computer packages for matrix and array and vector algebra, of which 
perhaps the best known is MatLab ("Matrix Lab")
where the same operations are entered like this
a = [1 2 3 4]; b = [1 4 9 16];
c = conv(a,b)                         conv is the convolution operator
Which gives the answer as
c =
1  6  20  50  75  84  64

The process that compares to division of scalars is deconvolution
which calculates the result of dividing a numerator polynomial  by a 
denominator polynomial.

A further elaboration of the idea of convolution of arrays is the 
convolution of two matrices, each of which can describe two-dimensional 
spatial features  of an image.
It was this idea of convolving two dimensional images that cropped up 
earlier in this thread. In that case, a plane wave description is 
convolved with an aperture  description...

Brian W