[SOLVED] finding matrix through optimisation

finding matrix through optimisation

I am looking for algorithm to solve the following problem :

I have two sets of vectors, and I want to find the matrix that best approximate the transformation from the input vectors to the output vectors.

vectors are 3x1, so matrix is 3x3.

This is the general problem. My particular problem is I have a set of RGB colors, and another set that contains the desired color. I am trying to find an RGB to RGB transformation that would give me colors closer to the desired ones.

There is correspondence between the input and output vectors, so computing an error function that should be minimized is the easy part. But how can I minimize this function ?

Solution

You don't specify a language, but here's how I would approach the problem in Matlab.

v1 is a 3xn matrix, containing your input colors in vertical vectors
v2 is also a 3xn matrix containing your output colors

You want to solve the system

M*v1 = v2
M = v2*inv(v1)

However, v1 is not directly invertible, since it's not a square matrix. Matlab will solve this automatically with the mrdivide operation (M = v2/v1), where M is the best fit solution.

eg: 
>> v1 = rand(3,10);
>> M = rand(3,3);
>> v2 = M * v1;
>> v2/v1 - M

ans =

   1.0e-15 *

    0.4510    0.4441   -0.5551
    0.2220    0.1388   -0.3331
    0.4441    0.2220   -0.4441

>> (v2 + randn(size(v2))*0.1)/v1 - M
ans =

    0.0598   -0.1961    0.0931
   -0.1684    0.0509    0.1465
   -0.0931   -0.0009    0.0213

This gives a more language-agnostic solution on how to solve the problem.