normCompare.m

% Benjamin Shih
% Section 5.1: Sensitivity to Normalization

tic

close all
clear all

p = 100 * [-2 -1 0 1 2; 10 2 1 2 10];
p = [p; ones(1, length(p))];
ptest = [0; 300; 1];
    

pSize = size(p);

% Parameters for the normal distribution.
mu = 0;
sigma = 1;

numTrials = 1000;
noNormed = zeros(3, numTrials);
normed = zeros(3, numTrials);

for i=1:numTrials
    % Introduce Gaussian noise to the data.
    noise = mu + sigma .* randn(pSize);
    p_corrupt = p + noise;

    % Generate the homographies between the original and corrupted data.
    % The homographies produce 3x3 transformation matrices. 
    H2to1 = computeH(p, p_corrupt);
    H2to1norm = computeH_norm(p, p_corrupt);
    
    % Apply the normalized and unnormalized homographies to the test point.
    noNormed(:,i) = H2to1*ptest;
    normed(:,i) = H2to1norm*ptest;

    
end

% Normalize results.
noNormed(1,1:end) = noNormed(1,1:end)./noNormed(3,1:end);
noNormed(2,1:end) = noNormed(2,1:end)./noNormed(3,1:end);
noNormed(3,1:end) = noNormed(3,1:end)./noNormed(3,1:end);

normed(1,1:end) = normed(1,1:end)./normed(3,1:end);
normed(2,1:end) = normed(2,1:end)./normed(3,1:end);
normed(3,1:end) = normed(3,1:end)./normed(3,1:end);



%% Plot Results
% Plot the resulting point sets in a single plot in order to compare the
% normalized and un-normalized results. 

% noNormed = normBS(noNormed);
% normed = normBS(normed);

figure;
hold on;
plot(noNormed(1,1:end), noNormed(2,1:end), 'rx');
plot(normed(1,1:end), normed(2,1:end), 'bo');


%% Covariance Comparison
% Covariance of the transformed test point for both the normalized and
% un-normalized solution. 

% Remove the 3rd row from the point matrices such that we just have the 2xN
% matrices of points.
% Input: 3x1000
% Output 2x1000
normed = normed(1:2,:);
noNormed = noNormed(1:2,:);

% Find the covariance of the test points.
% Input: 2x1000
% Output: 1000x1000
covNormed = cov(normed);
covNoNormed = cov(noNormed);

% Find the standard deviations of the test points.
% Input: 1000x1000
% Output: 1000x1
stdevNormed = sqrt(diag(covNormed));
stdevNoNormed = sqrt(diag(covNoNormed));

% Find the radio of the stdev between the unnormalized and normalized
% points.
ratio = stdevNoNormed ./ stdevNormed;




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