Robust Nonlinear Fitting by RANSAC (Outlier Tolerent Nonlinear Regression)

Robust Nonlinear Optimization MATLAB Code Download [File->Download]

Please have a look in RANSAC_4_Nonlinear.m This code is not heavily tested. Therefore, please be warned! 

All credit goes to the RANSAC toolbox creator! I just tested it for a simple nonlinear function. 

Contents

• Clearing and loading fresh data
• setting RANSAC options
• Running RANSAC
• GOG Parameter estimation from Inliers using levenberg-marquardt algorithm
• Results Visualization

%
% Objective: RANSAC for Robust Nonlinear Model Estimation
% Requirements: Optimization Toolbox (I am sorry, but its possible to modify the code to remove this dependency!)
% Important Acknowledgement: This RANSAC code is based on the RANSAC Toolbox [1][2]
% Thanks to the author for sharing but offcourse I do not take any
% responsibility neither for my or anyone else code! :-)
% --------------------------------------------------
% ACKNOWLEGEMENT:
%---------------------------------------------------
% [1] http://www.mathworks.com/matlabcentral/linkexchange/links/1814-ransac-toolbox
% [2] http://vision.ece.ucsb.edu/~zuliani/Research/RANSAC/RANSAC.shtml

%---------------------------------------------------

%---------------------------------------------------
% Input:
%---------------------------------------------------
% This simple script will try to fit the following nonlinear model
% Y=offset+gain*(X.^gamma); The equation is formally defined in the
% function titled "mygogcurve"
% Here, X is the independent variable and saved in the first row of matrix "xdataydata"
% And Y is the dependent variable and saved in the second row of matrix
% titled "xdataydata". Our Objective is to find the 3 parameters (offset,gain and gamma respectively)

% User defined functions

% estimate_gog_line: This function is used by main RANSAC.m in two ways.
% First, its used to compute k, where k means minimum number of data
% required to estimate the model. Here, I understand that we need at least 3
% points to estimate my nonlinear model. But to capture the nonlinearity, I have forcefully demands at
% least 5% data to start the model. See the function code to see the detail.

% residual_error_gog_line: this function returns squared error(E) and degrees of freedom(d for dof)
% Since, we are searching for 3 parameters dof will be k-3. Squared error
% is computed by fit=Theta(1)*(X(1,:).^Theta(2))+Theta(3); E=(fit-(X(2,:))).^2;

%---------------------------------------------------
% Output:
%---------------------------------------------------
% Estimated parameters in matrix "parameter_hat" (offset,gain and gamma respectively)
% --------------------------------------------------
% SEE ALSO:
%---------------------------------------------------

% Nonlinear model part is written by:
% SHEIKH FARIDUL Hasan
% hasan.sheikh.faridul@univ-st-etienne.fr
% Hasan.Sheikh-Faridul@technicolor.com
% https://sites.google.com/site/infofaridulhasan/

Clearing and loading fresh data

close all
clear all
% clc
tic

load RGB_im2_im2

xdata=RGB_ref(:,1)'; % You may test xdata=RGB_ref(:,2)'; with ydata=RGB_test(:,2)'; OR xdata=RGB_ref(:,3)'; with ydata=RGB_test(:,3)';
ydata=RGB_test(:,1)';
xdataydata=[xdata; ydata];

setting RANSAC options

options.epsilon = 1e-6;
options.P_inlier = 0.95;
options.sigma = 1;
options.est_fun = @estimate_gog_line;
options.man_fun = @residual_error_gog_line;
options.mode = 'MSAC';
options.Ps = [];
options.notify_iters = [];
options.min_iters = 50;
options.fix_seed = false;
options.reestimate = true;
options.stabilize = false;
options.max_iters=100;

Running RANSAC

[results, options] = RANSAC(xdataydata, options);

Starting RANSAC
Minimal sample set dimension = 16
Squared noise threshold = 22.708231, (assuming Gaussian noise, for sigma = 1.000000)
Iteration =     1/       37. Inliers =    313/   336 (rank is r = -3.65524372)
Iteration =     2/       16. Inliers =    325/   336 (rank is r = -1.64444872)
Iteration =    10/       16. Inliers =    325/   336 (rank is r = -1.56384809)
Iteration =    24/       16. Inliers =    325/   336 (rank is r = -1.56014290)
Iteration =    28/       16. Inliers =    325/   336 (rank is r = -1.54867447)
Iteration =    45/       16. Inliers =    325/   336 (rank is r = -1.54532717)
Restimating the parameter vector... Done
Final number of inliers = 325/336
Converged in 51 iterations (0.898319 seconds)

GOG Parameter estimation from Inliers using levenberg-marquardt algorithm

ind = results.CS;

options_myfit=optimset('Algorithm',{'levenberg-marquardt',.005});
[parameter_hat,resnorm,residual,exitflag,output,lambda,jacobian] = lsqnonlin(@mygogcurve,[0;1;1],[],[],options_myfit,xdataydata(1, ind),xdataydata(2, ind));

toc

Local minimum found.

Optimization completed because the size of the gradient is less than
the default value of the function tolerance.



Elapsed time is 1.054063 seconds.

Results Visualization

figure;
hold on
plot(xdataydata(1, ind),xdataydata(2, ind), '.g')
plot(xdataydata(1, ~ind), xdataydata(2, ~ind), '.r')
plot([1:255],parameter_hat(1)+parameter_hat(2)*([1:255].^parameter_hat(3)),'b')
legend('Inliers', 'Outliers','LM-fit from inliers')

xlabel('x-reference color')
ylabel('y-test color')
title('RANSAC outlier detection and GOG Param by LM estimation')

Published with MATLAB® 7.13

Conclusion: Will be completed later