% converts the 10 qchirp2 parameters to the 8 pchirp parameters using the
% 4 corners of a presumed image as the common points to define the relation
% between the qchirp2 parameters and the pchirp2 parameters

function p2=approximationatcorners(p1)

if nargin==0
  error('qchirp22p: you must have an input argument')
end%if

if length(p1) ~= 10
  error('qchirp22p: input vector, must have length 10')
end%if

if min(size(p1)) ~= 1
  error('qchirp22p: input must be a vector and not a matrix')
end%if

x1=0; y1=0;
x2=0; y2=1;
x3=1; y3=0;
x4=1; y4=1;

x=[x1;x2;x3;x4];
y=[y1;y2;y3;y4];

u=p1(1)*ones(4,1) + p1(2)*x + p1(3)*y +p1(4).*x.*x + p1(5).*x.*y;
v=p1(6)*ones(4,1) + p1(7)*x + p1(8)*y +p1(9).*y.*y + p1(10).*x.*y;

p2=corners2r([u(1) v(1) u(2) v(2) u(3) v(3) u(4) v(4)]);

% now p2 would map the first image into the second but we wish to map
% the second to the first

%p2=reshape([p2(:);1],3,3).';   % .' because want across rows
%p2=reshape(inv(p2),9,1).';     % .' because want across rows
%p2=p2(1:8);

p2=pinverse(p2);  % same as 3 commented lines above, but neater