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.
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:
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.
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Ludwik Kowalski, a retired physics teacher and an amateur journalist.
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