%matlab notes for 5-15-2012
clear all

%joint gaussian distribution, zero mean, standard deviation 1
p = 0 %p is correlation
[X,Y] = meshgrid(-5:.01:5,-5:.01:5);
size(X)
size(Y)
imagesc(exp(-(X.^2 - 2*p*X.*Y + Y.^2)/2/(1-p^2))/(2*pi)/(sqrt(1-p^2)));
p = .5;
imagesc(exp(-(X.^2 - 2*p*X.*Y + Y.^2)/2/(1-p^2))/(2*pi)/(sqrt(1-p^2))); axis xy;
p = -.5;
imagesc(exp(-(X.^2 - 2*p*X.*Y + Y.^2)/2/(1-p^2))/(2*pi)/(sqrt(1-p^2))); axis xy;
p = .9
imagesc(exp(-(X.^2 - 2*p*X.*Y + Y.^2)/2/(1-p^2))/(2*pi)/(sqrt(1-p^2))); axis xy;

mesh(exp(-(X.^2 - 2*p*X.*Y + Y.^2)/2/(1-p^2))/(2*pi)/(sqrt(1-p^2)));

%how to use randn to generate zero mean Gaussian random numbers with
%a given covariance matrix C
C = [1 .5;.5 1/3]
det(C)
[V,D] = eigs(C)
V*D*V'
S = sqrt(D)*V'
N1 = randn(1,1000);
N2 = randn(1,1000);
X = S'*[N1;N2];
cov(X(1,:),X(2,:))
C
scatter(X(1,:),X(2,:))

%use SVD for image compression
f_in = double(imread('elvis.bmp')); %load image
f = f_in(:,:,1);
imagesc(f)
colormap gray
[mr,mc] = size(f);
m = mr*mc;
g = zeros(mr,mc);
mr
mc
help svd
[U,S,V] = svd(f);
imagesc(S)
sv = diag(S); %singular values
sv(1)
sv(2)
sv(3)
svlen = length(find(sv>0))
mr
r = .1
s = round(r*svlen);
s
Uc = U(:,1:s);
svc = sv(1:s);
Vc = V(:,1:s);
size(Uc)
g = Uc*(svc(:,ones(1,mc)).*Vc');
imagesc(g)
%now throw away even more information
s = .01
r = .01
s = round(r*svlen); %number of singular values to keep
%compress
Uc = U(:,1:s);
svc = sv(1:s);
Vc = V(:,1:s);
g = Uc*(svc(:,ones(1,mc)).*Vc');
imagesc(g)