To make things easier to keep track of, we will let h(x,y) stand for the blurry image of a star which was originally centered on the source screen (ie at (0,0)), after our crappy camera imaged it.  Since this star is rather point-like, we will treat it as an impulse and say

 

In exact analogy with our wave example, lets say now we take our camera and image the night sky.  The source plane (the cosmos) contains a multitude of stars, each of a different brightness.  What will our image look like?  Well, just knowing how a unit impulse at the origin is imaged, we can figure it out!  Viewing each star as a stretched and translated unit impulse; our image will be a collection of stretched and translated blurs.  We can decompose the night sky into a sum of these modified impulses, and then distribute our imaging function over the sum, blurring each star individually and adding the results.  

 

 

Symbolically say

 

 

So far we have come up with two seemingly different problems; both which have a simple solution if we first understand how an "impulse" is propagated through them.  Now lets take a picture of a portion of the sky with even more stars in it: for example, this small globular cluster which orbits our home galaxy:

 

 

We can apply the exact same reasoning as above here: each star is like an impulse, so to see how each star will look on our final picture, we simply take that impulse, propagate it through the camera until it becomes our response function h, and then add all of them back together.  The resulting blurry photo looks like this:

 

 

It‘s time to make sure the mathematical formalism of this process really makes sense:  at the location of each impulse (star), we apply our "blur" function which models our camera‘s lack of focus.  This blur function spreads the light out from this original point source to a more extended region.  If our star is located on the source plane at point (xi, yi), we need to make sure that our final image of the star is centered on that location as well.  We defined our function h(x,y) to be the blur caused by a unit impulse at the origin, so we need to slide this impulse to occur at our new point (xi, yi) instead.  This is what the term 

 

says.  Its just a translated version of our blurred out impulse, originating from (xi, yi) instead of (0,0).  Now if the stars original brightness was not simply 1 but some other amount, we must scale the brightness of our blur accordingly.  Thus the image of a star of brightness mi located at (xi, yi) will be

 

 

So far, this has just been a more careful repetition of what we have done above.  What we are going to do now seems to be a simple change of symbols, but will actually give us some profound insight into convolution itself.  The number mi represents the brightness of our ith star, which is located at the point (xi, yi) on the source plane.  What this is really saying then is that the brightness of the source plane at location (xi, yi) is given by mi.  That is, we can define a function on the source plane which gives us the brightness at each point.  Lets call this function m(u,v).  Given a point (u,v) of our source plane (the night sky), m tells us what the brightness is there.  Of course, in our star examples so far most points of the plane have a brightness of zero (black sky).  But as we saw in the globular cluster, if we try to image a part of the sky with lots and lots of stars, some of them will seem to touch, and form extended areas of brightness.

 

In fact, in the limit lets say we find a point of the night sky where there are stars everywhere.  There are no dark points, every line of sight ends in a star.  Our brightness function m would be nonzero everywhere, and every single point of the sky would have an impulse (star) located at it.  Can we use what we have learned so far to figure out what our photograph of this area would look like?  Sure we can!  The only difference now is that we have a continuum of stars, instead of discrete points.  But, at each location (u,v) in the night sky, we have an impulse with brightness m(u,v).  We can represent this star by this brightness multiplied by an impulse function shifted to be located at (u,v):

 

(FINAL WAVE PICTURE)

 

Again, we recognize this as a convolution of the wave caused by a single impulse and the pressure field of the extended object:

 

 

Consider for a moment a metal rod which we have heated in some way.  If we let it sit out, it will obviously cool down, but how do we express this quantitatively?  Via a partial differential equation known aptly as the "heat equation", given the initial temperature of our rod (as a function of position), we can solve for its temperature at all future times.  

For a general initial temperature distribution this may be a hard thing to do.  So let‘s see if we can stick to the reasoning that‘s proved fruitful so far, and consider the effect one "impulse" of heat has on our rod.  The mathematical model for this is

 

 

 

 

 

Or, in symbols:

 

 

 

 

By being point-like and centered at the origin, we mean that our impulse is pretty much zero at all other points.  This fits well with our intuitive examples of impulses, the small pebbles and starlight.  How could we go about constructing some function like this? For a starting place, look at the normal curve: