Image Approximation with Genetically Selected Cosines October 2nd, 2009
Patrick Stein

input imageI started with the target image here on the right. I then used a genetic algorithm to try to find an approximation of this image as a sum of cosine waves.

Each cosine wave has the following form

f_i(x,y) = A_i \cos \left( 2 \pi \left( x \cos \theta_i + y \cos \theta_i \right) \omega_i + 2 \pi \phi_i \right)
The genetic algorithm can tweak the amplitude A_i, the orientation in the plane \theta_i, the frequency \omega_i, and the phase \phi_i of each wave.

out-0442 The image at the right here, is from the 443rd generation of a population of 200 where each gene represents 100 cosine waves. For the video below, I used a population of 100 genes, each representing the sum of 50 cosine waves.

Image Approximation with Genetically Selected Cosines from Patrick Stein on Vimeo.

Nitty Gritty Details

The genes are represented internally as an array of fixed-point numbers. Each number is five-bits integer, ten-bits fractional, and a sign bit. I scale the amplitudes down by a factor of 50 and scale the frequency and offset up by 50% when I convert them back to floating point from the fixed point representation just to get them into a more useful ballpark. During each generation, the fittest two quartiles of the population are cross-bred (in randomly selected pairs with two-point crossover) to replace the third quartile of the population, and the fourth quartile is replaced by mutations of randomly selected members of the top quartile. I also run through and remove any duplicate genes and replace them with randomly generated ones so that I don’t breed myself into a total corner.

I toyed around with two different fitness functions: RMS and entropy. So, I would calculate the image represented by the gene and subtract that from the original image. Then, I would calculate the RMS (\frac{\sum_{i=1}^{n} v_i^2}{n}) or the sum of the entropies of the possible values (\sum_{v=0}^{255} - p_v \log p_v, where p_v is the proportion of the pixels in the resulting image that had value v).

I bothered with entropy because I was hoping that if I could encode a gene in just a hundred or two hundred bytes that I might be able to compress the remainder (losslessly) and save bytes over just compressing the original image. The big problem with the entropy calculation though is that I get the same answer when every pixel value is off by 100 as I get when every pixel value is right on target. So, for a time, I was multiplying the RMS and the entropy for the fitness function. For the video and images above, I just used the RMS since it reached very close to the same solution as the multiplying method, and I saved myself fifty histograms per generation.

Source Code

Here is the Common Lisp source code for the genetic algorithm. It depends on the CL-PNG library for image input and output. As it appears here, it will create an output image from the fittest member of each generation, an image of the fittest member of the current generation, an image of the original with the fittest member of the current generation subtracted, a Lisp file containing the fittest gene of the current generation, and the current state of the random number generator when the run ends.

For the video above, I quickly put together some code to demonstrate the different parameters that the genetic algorithm can tweak and to show incrementally adding in more of the cosine waves. Here is some general utility code that is display-independent. And, here is the OpenGL display code which depends on the CL-OpenGL libraries. I included the source code here because I had trouble finding an example on the net of using Lisp-created textures with CL-OpenGL. I render the cosine waves into a buffer, copy that buffer into an OpenGL texture, and draw a quad with that texture. Note: it looks a little sluggish when run, but that is because I put a big (SLEEP …) in there to keep things from flying by too quickly.