Multi-objective optimization problem - Euclidean space

Question

I am looking for some clues for an optimization problem.

My problem consists on arriving to a image by optimizing multiple layers with the pixel position probability.

This is an overview of the problem: Please see the overview here

The first image on the left is a zoom of the greyscale image with only two grey values, followed by the probability of each cell taking one value of grey, and the same probability for the value black. Finally, I am looking for a way to arrive to the start image by using only the two layers.

In the end, I will be working with a complete image like this one:

Please see the complete image here

I understand that this will not give me a great result, and that I need to add constraints to my problem, however I am completely lost on the subject. I have found a lot of documentation for optimization, but not a good tutorial or examples for optimization problems defined in an Euclidean space. Is there documentation that I could read, a basic tutorial or code examples?

Answer 1

I believe you should clarify your question a bit more. But the way most optimization problems work is that you need to define a function (cost function) that you need to optimize. Now this function might have multiple parameters or objectives that depend on the problem you are solving. Once you have identified the function, you should use an optimization algorithm to find desired values of the parameters based on the objectives. Some of the methods you can use are Evolutionary Algorithms such as Differential Evolution, Genetic Algorithms or Particle Swarm. You could also use other gradient based methods about which you can probably find extensive literature via Google. I used the following references during my research on shape optimization in aerodynamics:

• Differential evolution: a practical approach to global optimization by Kenneth V. Price, Rainer M. Storn, Jouni A. Lampinen,
• Differential evolution: in search of solutions by Vitaliy Feoktistov,
• Global Optimization Algorithms Theory and Application by Thomas Weise.

Hope this helps.