%% Example:  Projections

% First, we'll construct some orthonormal basis vectors that might be
% interesting:
x=linspace(-pi,pi);  %Returns a vector of 100 evenly spaced points between -pi and pi.
v1=sin(x)';
v2=sin(3*x)';
v3=cos(x');

V=[v1 v2 v3];

%  Does V have orthonormal columns?
V'*V

temp=sqrt(diag(V'*V));
V=V./repmat(temp',100,1);
V'*V

% Now we have a three dimensional subspace of R^100.  Create some a new
% vector in R^100, then project it into the three dimensional subspace.

y=exp(-x.^2).*sin(3*x)-cos(4*x);
y=y'; %Force y to be a column vector

% y is a vector in R^100.  Give the three dimensional representation:
yc=V'*y

% Now get the projection of y into the three dimensional space:
yRecon=V*yc;

%% Fun with Clowns!

% Just for fun:
load clown
figure(1)
image(X);  colormap(map);

x=linspace(-pi,pi,200)';
v1=sin(x); v2=sin(3*x); v3=sin(5*x); v4=sin(7*x);
V=[v1 v2 v3 v4];
temp=sqrt(diag(V'*V));
V=V./repmat(temp',200,1);


Coords=V'*X;
figure(2)
plot3(Coords(1,:),Coords(2,:),Coords(3,:),'.')


Recon=V*Coords;
figure(3)
imagesc(Recon)