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From: jesjones@milton.u.washington.edu (Jesse Jones)
Subject: Transforming Fractal Image
Message-ID: <1992May3.161234.21102@u.washington.edu>
Summary: Discusses eight different ways to transform a fractal image
Keywords: Mandella Transform Convolution Evert Delta etc.
Sender: news@u.washington.edu (USENET News System)
Organization: University of Washington, Seattle
Date: Sun, 3 May 1992 16:12:34 GMT
Lines: 188
Fractal Image Transforms 2.0
by Jesse Jones
ABOUT THIS FILE
This file discusses eight different ways to transform fractal images.
For
example the Sobel transform converts a fractal image into a fascinating
pseudo
3D image. These transforms have all been implemented in a Mac program
called
Mandella.
The transforms all operate on an array containing the dwell values for
the
fractal. The transformed image is then displayed in a new window. Two of
these
transforms result in a full color image. The transforms I'll discuss are:
convolution, Blur, Delta, Bands, Evert, Log, Full Color, and Color Sobel.
Mandella also provides a 3D transform which I've described in a seperate
article.
CONVOLUTION
Convolution is a common image processing technique. In this method an
array
is superimposed on a pixel and its neighbors. Each neighbor and the pixel
itself are multiplied by its entry in the array and then summed together
to get
a new value for the pixel. As an example the 3x3 array
-1
-1
-1
-1
8
-1
-1
-1
-1
will act as an edge detector: pixels that have neighbors with different
count
values will be hilited. Pixels whose neighbors all equal its count value
will
be assigned a zero count value. These arrays are called kernels.
There are a few sticky points when using convolution. Namely how
should
negative count values be treated and what happens at the edges of the
image?
Mandella forces the new count values to be in the range 0..hiCount. To
avoid
the edge problem Mandella resizes the new window so that it is smaller
then
the original window. A 3x3 kernel results in a window that is two pixels
smaller in the x and y directions. A 5x5 kernel results in a new window
six
pixels smaller.
Mandella comes with eight predefined kernels. An editor is built-in so
that
the user can change, delete, or add kernels. The kernels included are:
Edges:
-1
-1
-1
-1
8
-1
-1
-1
-1
GradNW:
-2 -1 0
-1 0 +1
0 +1 +2
GradW:
0 -1
-1 -2
-1 -2
-1 -2
-1 -2
-1 -2
0 -1
Hat:
0
0
-1
0
0
0
-1
-2
-1
0
Laplace:
-1 -1
-1 -1
-1 -1
-1 -1
-1 -1
-1
-2
-3
-3
-3
-2
-1
0
0
0
0
0
0
0
-1
-2
17
-2
-1
-1
-1
25
-1
-1
Sharpen:
-1
-1
1
2
3
3
3
2
1
0
-1
-2
-1
0
-1
-1
-1
-1
-1
1
2
2
2
2
2
1
0
1
1
1
1
1
0
0
0
-1
0
0
-1
-1
-1
-1
-1
-1
Sharpen is pretty neat.
-1
-1
Sobel:
-1
bottom
0
1
9
-1
-1
-1
-2
-1
0
2
0
2
But Sobel is great. And yes, the two in the
right corner is really a two.
BLUR
Blur is related to convolution. A kernel that consists of all ones is
used.
The resulting number is then divided by the size of the kernel. Mandella
uses
a 3x3 kernel so it divides the results by nine.
DELTA
In this method a pixel is colored according to how many of its
neighbors
have count values different from the pixel. Since there are eight
neighbors
plus the center pixel this method only displays 9 colors. But with the
right
color selection it produces excellent results.
BANDS
Here the modulus function is used to restrict the image to a user
defined
number of colors. The algorithm looks like this:
offset := NoColors DIV bands;
index := count MOD bands;
color := FirstColor + index*offset;
EVERT
This transform everts a circle. In other words it maps every point
inside
the circle outside. Every point outside the circle is mapped inside. The
equation w = 1/z (where w and z are complex) will evert the unit circle.
To
evert an arbitrary circle the equation w = R^2/(z - a) + a can be used. R
is
the radius of the circle and a is its location.
For best results the eversion should be done in reverse: if w
represents a
point in the transformed image and z a point in the original image. Then
for
every w find the corresponding pixel in the original image: z = R^2/(w a) +
a. It is also neccesary to test z for large values. As w goes to a z will
go to
infinity.
LOG
In this method we use Ln and the hiCount to smoothly shade the image.
To
spice the image up we also use the pixels distance from the origin. This
adds
an interesting tiling effect. The algorithm looks like this:
index := NoColors*Ln(count)/Ln(hiCount);
dist := (h*h + v*v) and 15;
(* just use lower 4 bits *)
index := index + dist;
FULL COLOR
This method provided an easy way for me to take advantage of the truecolor
abilities of 32-Bit QuickDraw. I simply construct an expanded palette
with 20
times the number of colors by interpolating between the old colors. Then
the
transform maps counts into palette entries and outputs the color at that
entry.
The CountToEntry procedure is identical to the one used when drawing in
256
colors. This winds up looking very nice on Macs with true-color video
boards.
COLOR SOBEL
The Sobel transform is basicly a fancy edge detector. One of the
convolution
kernels is a simplified form of the real thing. The full Sobel transform
uses
two convolution kernels to produce a vector field. I again create a truecolor
image using the direction for the hue, the stength for the value, and the
saturation fixed at 50000. The direction is ArcTan(Y/X) the strength is
X*X +
Y*Y. X and Y are derived from the below convolution kernels:
X =
-1
-2
-1
0
0
0
1
2
1
Y =
1
0
-1
2
0
-2
1
0
-1
REFRENCES
A good introduction to image processing and convolution is "Digital
Image
Processing" by William Green ISBN 0-442-23052-4. Another source is
"Introduction to Image Processing" by Benjamin Dawson, March 1987 BYTE. A
good
text on complex math is "Complex Variables for Mathematics and
Engineering" by
John Mathews ISBN 0-697-06764-5.
CHANGE HISTORY
Version 2.0
1) Changed formatting.
2) Added Full Color, Color Sobel, and Blur transforms.
3) Changed discussion of convolution.
4) Added several new convolution kernels.
5) Added references section.
Version 1.1
1) Added Evert transform.
Version 1.0
1) First version uploaded (to CompuServe).
ADDRESS
Jesse Jones
Usenet: jesjones@milton.u.washington.edu
CServe: 73627,152