JPNM Physics

Convolution Math

So from the graphical introduction to convolution, we find that we have shifted the system response function, we have scaled it according to the x-ray intensity, and we have summed all these scaled responses together - to repeat -shift, scale, and add.

If the input function is f(x,y), a 2-D function, and the system response is h(x,y), then the mathematical expression for shifting to point (x',y') is

h(x-x',y-y')

and the scaling by the input function intensity is

f(x,y)h(x-x',y-y') .

To get the output image g(x,y), we sum up over all points (x',y'):

This is the mathematical definition of the process called "convolution." Note one more time that both graphically and mathematically we have shifted the system response function h(x,y), scaled it by the x-ray intensity function f(x,y), and added (integrated) to produce the output response function g(x,y). Note also that the system response function h(x,y) is the system response to a delta function (or very narrow) input.

Time response

Not only is the process of convolution used for spatial distributions, but it is also used for temperal functions. For example, say we have a narrow injection bolus which produces a time activity curve response h(t). The response in time of an injection which fluctuates in time according to the equation f(t) will produce a time activity curve g(t) given by:

At the risk of boring you to tears, you'll note that the system response function h(t) has been shifted to time t', scaled by the fluctuation of the input f(t) at that time, and added to produce the output time activity response g(t).

Simplifying Notation

When we wish to convolve the input function f(t) with the system response function h(t) to obtain output g(t), we can either write out the integral shown above or use the following short-hand notation involving the "star" symbol:

g(t)=f(t) * h(t)

Note that one should be careful to distinguish this symbol for convolution from the multiplication sign!

Next (Fourier Transform, part 1)

Previous (convolution introduction)

JPNM Physics Home Page

Douglas J. Wagenaar, Ph.D., wagenaar@nucmed.bih.harvard.edu