Synthesizing images using simple equations

January 13th, 2006

Synthesizing images using simple equations

In 1988, Gerard Holzmann of AT&T Bell Labs (back when there was such a thing) published a fun book called Beyond Photography: The Digital Darkroom (http://tinyurl.com/b9rb9). The book shows how some simple math on pixel values and coordinates can transform images in fascinating ways.

In the first part of the book, Holzmann shows how to synthesize interesting images using functions of both Cartesian and polar coordinates. Let's see how to do that in MATLAB.

MATLAB functions featured: meshgrid, cart2pol

Image Processing Toolbox functions featured: imshow

Start with meshgrid
Concentric rings
Using polar coordinates

Start with meshgrid

The MATLAB function meshgrid is extremely useful for computing a function of two Cartesian coordinates, and you can make some interesting images this way.

meshgrid is kind of hard to explain in words. It's easier to just look at what it does.

x = 1:3;
y = 10:14;
[xx, yy] = meshgrid(x, y)

xx =

     1     2     3
     1     2     3
     1     2     3
     1     2     3
     1     2     3


yy =

    10    10    10
    11    11    11
    12    12    12
    13    13    13
    14    14    14

Concentric rings

Let's make an image out of the equation:

x = linspace(-pi, pi, 201);
% If you pass meshgrid only one vector, it uses that vector for both the x
% and the y coordinates.
[xx,yy] = meshgrid(x);
A = 10;
I = sin(A*(xx.^2 + yy.^2));

% Specify the range -1 to 1 when displaying the image.
imshow(I, [-1 1])

Using polar coordinates

If you want to construct an image from a function of polar coordinates, use cart2pol in conjunction with meshgrid.

(The interesting patterns you see in the center of the image below result from ''aliasing,'' but that's a topic for another day.)

[xx,yy] = meshgrid(-125:125);
[theta,R] = cart2pol(xx,yy);
I = sin(50*theta);
imshow(I, [-1 1])

Get the MATLAB code

Published with MATLAB® 7.1

By Steve Eddins

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6 Responses to “Synthesizing images using simple equations”

Ike replied on January 14th, 2006 at 08:38 UTC :

excellent idea , the Image proc. blog.
i think that i order to select from all the suggested subjects maybe specifying a bit on each subject would raise comments.
i for one am new to matlab , and have no clue for what a queue is…..
still basically IMO any subject you choose, will involve usage of common functions , and by that will be useful to me, at least ….

philip batchelor replied on January 16th, 2006 at 08:59 UTC :

Yes, great idea. In my lab (medical image processing, split between computer sciences and medical physics), we have regular informal meetings to discuss topics on image processing-either new stuff, or old stuff one should know but doesn’t.

I expect this blog to provide such topics, and maybe even we’d have some to suggest at some point…

philip batchelor replied on January 16th, 2006 at 09:32 UTC :

I see the remark:
‘meshgrid is kind of hard to explain in words. ‘
Couldn’t one say that it returns the x and y coordinates of a grid on the grid?
[X,Y] = meshgrid(1:5,1:5);
for k = 1:size(X,1)
for l =1:size(Y,2)
XY{k,l} = ['(',num2str(X(k,l)),',',num2str(Y(k,l)),')'];
end
end
kind of illustrates?

Steve replied on February 10th, 2006 at 08:35 UTC :

Peter – see my January 26 post.

Simon replied on August 13th, 2011 at 19:14 UTC :

The book referenced “Beyond Photography: The Digital Darkroom”
can be found as an online version here:
http://spinroot.com/pico/

Steve replied on August 17th, 2011 at 02:14 UTC :

Simon—Thanks.

Steve on Image Processing

January 13th, 2006