Structure from sound - minimal problem in 2D
TOA 2D - minimal inverse problem 3 'sounds' and 3 'microphones
Here is a new solver for the problem of determining the 2D positions of 3 'microphones' and 3 'sounds sources' given the 9 distances between each sound source and each microphone. The code is available for download at github:
https://github.com/kalleastrom/toa2D
The problem was solved in Henrik Stewenius Thesis from 2005:
https://lup.lub.lu.se/search/publication/24960
Henrik has a solver using his technique. It is available at
http://www.vis.uky.edu/~stewe/code/audio_abs_flat/
Here we present a new solver based on technique presented in
https://lup.lub.lu.se/search/publication/3910305
The input to the solver is a 3x3 matrix D, where the element D(i,j) is the distance from microphone i to sound source j. There can be several solutions. The output are two cell arrays, Rc and Sc, with all the real solutions to the problem, i.e. Rc{k} and Sc{k} are both 2x3 matrices for a solutions k, where Rc{k} represents the receiver positions in the plane and Sc{k} represents the sender positions in the plane.
Contents
Generate data, receiver positions R and transmitter positions S
R = [[0;0] randn(2,2)]; R(1,3)=0 S = randn(2,3)
R =
0 0.3923 0
0 0.0017 0.3302
S =
-1.9455 0.4918 -0.3845
0.1514 0.0178 -0.0059
Calculate 9 distances from 3 transmitters to 3 receivers
d=toa_calc_d_from_xy(R,S)
d =
1.9514 0.4921 0.3846
2.3426 0.1008 0.7769
1.9537 0.5826 0.5107
Run solver on distances, to obtain all real minimal solutions
[Rc,Sc]=toa_2D_33(d); Rc Sc
Rc =
Columns 1 through 4
[2x3 double] [2x3 double] [2x3 double] [2x3 double]
Column 5
[2x3 double]
Sc =
Columns 1 through 4
[2x3 double] [2x3 double] [2x3 double] [2x3 double]
Column 5
[2x3 double]
Plot results (plot maximum of four of the real solutions)
close all; for k = 1:min(length(Rc),4); figure(1); subplot(2,2,k); hold off; plot(Rc{k}(1,:),Rc{k}(2,:),'o'); hold on; plot(Sc{k}(1,:),Sc{k}(2,:),'*'); for i = 1:3, for j = 1:3, plot([Rc{k}(1,i) Sc{k}(1,j)],[Rc{k}(2,i) Sc{k}(2,j)],'-'); xx = (Rc{k}(1,i)+Sc{k}(1,j))/2; yy = (Rc{k}(2,i)+Sc{k}(2,j))/2; text(xx,yy,num2str(d(i,j),2)); endendend;
