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The convolution method was said to useful to solve problems similar to
one shown below. Referring to this problem, John D. wrote: "It's
trivial. Just code the convolution from the definition and run the
program." I do not know how to do this.
= = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = =
PROBLEM TO SOLVE:
1) Suppose the source of light is a flat square (each side is 10 cm)
in the z=0 plane. That square is subdivided into 20 strips of columns
and 20 strips of rows, forming 400 tiny cells. Each cell can be
represented by a point-like source of light (a pixel) situated in its
center. Each "pixel" emits equal amount of light (100 units)
isotropically.
2) Suppose a sheet of black paper, located in the plane z=10 cm,
absorbs light everywhere, except in three circular holes (apertures).
The radius of each circle is 1 cm.
Centers of circles are located at (x=0, y=0), at (x=5 cm, y=5 cm), and
at (x=5 cm, y=-5 cm)
3) Suppose the resulting light spot is observed on a screen situated
in the plane z=30 cm.
4) What is the distribution of light on the screen?
More specifically, suppose the screen is subdivided into a grid of
adjacent rectangular cells, each of 1cm by 1 cm. Centers of these
cells are identified by coordinates, such as (x=0 cm, y=0 cm), (x=0
cm, y=1 cm), (x=0 cm, y=2 cm), (x=1 cm, y=1 cm), etc. How much light
arrives to a cell whose coordinates are specified. For example,
a) how many units of light arrive to a cell whose center is at (x=7
cm, y=10 cm)?
b) how many units of light arrive to a cell whose center is at (x=-7
cm, y=10 cm)?
= = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = =
5) In an earlier message John D. wrote:
". . . Also: As a starting place, and for many serious real-world
applications, the convolution is a sum not an integral. You can with
a small amount of cleverness do the sums using a spreadsheet; here's
an example that you can play with:
http://www.av8n.com/physics/convolution-intro.xls
Feel free to change the kernel (in the light green area) and the main
input (in the light blue area)." I opened this file but it did not
help me. Answering (a) and (b) is not at all trivial to me.
You can demonstrate the whole idea in less time than it takes to tell about it. Start with a board with three pinholes in it. Let the arrangement of pinholes be non-symmetric enough to be recognizable. This board will be called the _mask_ and will serve as the lens in a slightly peculiar pinhole camera obscura.
Also arrange for a bright scene to serve as the _object_. Form an image on a screen in the usual way.
Case 1: This the intermediate case, where each pinhole produces a separate subimage that does not overlap with the other subimages. Each subimage will be a clear representation of the object.
Case 2: In this case, let the object be very small (but still bright) so that each subimage becomes small and structureless. In this case, the image as a whole is a one-to-one representation of the mask.
Case 3: This is the opposite extreme from case 2. In this case, let the object be large and/or let the spacing between pinholes be small. Then there will be nearly 100% overlap between the subimages, so that in effect there is only one image. It is a single representation of the object, just slightly blurred.
In all three cases, the image as a whole is the convolution of a function representing the object with a function representing the mask. The mask can be considered the kernel of the convolution.
This thread started by calling attention to the two extreme cases (case 2 and case 3). I suggest that the intermediate case (case 1) is easier to understand. There's nothing tricky about it. No calculus, No math of any kind. Just look at the three subimages.