Editing Fractal flame (section)

== Algorithm ==

The algorithm consists of two steps: creating a [[histogram]] and then rendering the histogram.

===Creating the histogram===
[[File:Apophysis-100303-104.jpg|thumb|right|A fractal flame.]]
First, one iterates a set of functions, starting from a randomly chosen point ''P = (P.x,P.y,P.c)'', where the third coordinate indicates the current color of the point.

:Set of flame functions: <math>\begin{cases}
F_1(x,y), \quad p_1 \\
F_2(x,y), \quad p_2 \\
\dots \\
F_n(x,y), \quad p_n
\end{cases}</math>

In each iteration, choose one of the functions above where the probability that ''F<sub>j</sub>'' is chosen is ''p<sub>j</sub>''. Then one computes the next iteration of ''P'' by applying ''F<sub>j</sub>'' on ''(P.x,P.y)''.

Each individual function has the following form:

:<math>F_j(x,y) = \sum_{V_k \in Variations} w_k \cdot V_k(a_j x + b_j y +c_j,d_j x + e_j y +f_j)</math>

where the parameter ''w<sub>k</sub>'' is called the weight of the '''''variation''''' ''V<sub>k</sub>''. Draves suggests
<ref name="dravespdf">{{cite web|url=https://flam3.com/flame_draves.pdf|title=The Fractal Flame Algorithm}}&nbsp;{{small|(22.5&nbsp;MB)}}</ref> that all <math>w_k</math>:s are non-negative and sum to one, but implementations such as Apophysis do not impose that restriction.

The functions ''V<sub>k</sub>'' are a set of predefined functions. 
A few examples<ref name="dravespdf" /> are

* V<sub>0</sub>(''x'',''y'') = (''x'',''y'') (Linear)
* V<sub>1</sub>(''x'',''y'') = (sin ''x'',sin ''y'') (Sinusoidal)
* V<sub>2</sub>(''x'',''y'') = (''x'',''y'')/(''x''<sup>2</sup>+''y''<sup>2</sup>) (Spherical)

The color ''P.c'' of the point is blended with the color associated with the latest applied function ''F<sub>j</sub>'':

: P.c := (P.c + (F<sub>j</sub>)<sub>color</sub>)  / 2

After each iteration, one updates the histogram at the point corresponding to ''(P.x,P.y)''. This is done as follows:

<syntaxhighlight lang="cpp">
histogram[x][y][FREQUENCY] := histogram[x][y][FREQUENCY]+1
histogram[x][y][COLOR] := (histogram[x][y][COLOR] + P.c)/2
</syntaxhighlight>

The colors in the image will therefore reflect what functions were used to get to that part of the image.

===Rendering an image===

To increase the quality of the image, one can use [[supersampling]] to decrease the noise. This involves creating a histogram larger than the image so each pixel has multiple data points to pull from. For example, create a histogram with 300&times;300 cells in order to draw a 100&times;100 px image; each pixel would use a 3&times;3 group of histogram buckets to calculate its value.

For each pixel ''(x,y)'' in the final image, do the following computations:

<syntaxhighlight lang="cpp">
frequency_avg[x][y]  := average_of_histogram_cells_frequency(x,y);
color_avg[x][y] := average_of_histogram_cells_color(x,y);

alpha[x][y] := log(frequency_avg[x][y]) / log(frequency_max);  
//frequency_max is the maximal number of iterations that hit a cell in the histogram.

final_pixel_color[x][y] := color_avg[x][y] * alpha[x][y]^(1/gamma); //gamma is a value greater than 1.
</syntaxhighlight>

The algorithm above uses [[gamma correction]] to make the colors appear brighter. This is implemented in for example the Apophysis software.

To increase the quality even more, one can use gamma correction on each individual color channel, but this is a very heavy computation, since the ''log'' function is slow.

A simplified algorithm would be to let the brightness be linearly dependent on the frequency:

<syntaxhighlight lang="cpp">
final_pixel_color[x][y] := color_avg[x][y] * frequency_avg[x][y]/frequency_max;
</syntaxhighlight>
but this would make some parts of the fractal lose detail, which is undesirable.<ref name="dravespdf" />