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## Copyright (C) 2012 Pantxo Diribarne
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{T} =} cp2tform (@var{rw_pt}, @var{ap_pt}, @var{transtype})
## @deftypefnx {Function File} {@var{T} =} cp2tform (@var{rw_pt}, @var{ap_pt}, @var{transtype}, @var{opt})
## Returns a transformation structure @var{T} (see "help maketform"
## for the form of the structure) that can be further used to
## transform coordinates from one space (here denoted "RW" for "real
## world") to another (here denoted "AP" for "apparent"). The transform
## is infered from two n-by-2 arrays, @var{rw_pt} and @var{ap_pt}, wich
## contain the coordinates of n control points in the two 2D spaces.
## Transform coefficients are stored
## in @var{T}.tdata. Interpretation of transform coefficients depends on the
## requested transform type @var{transtype}:
##
## @table @asis
## @item "affine"
## Return both forward (RW->AP) and inverse (AP->RW) transform
## coefficients @var{T}.tdata.T and @var{T}.tdata.Tinv. Transform
## coefficients are 3x2 matrices which can
## be used as follows:
##
## @example
## @group
## @var{rw_pt} = [@var{ap_pt} ones(rows (ap_pt,1))] * Tinv
## @var{ap_pt} = [@var{rw_pt} ones(rows (rw_pt,1))] * T
## @end group
## @end example
## This transformation is well suited when parallel lines in one space
## are still parallel in the other space (e.g. shear, translation, @dots{}).
##
## @item "nonreflective similarity"
## Same as "affine" except that the transform matrices T and Tinv have
## the form
## @example
## @group
## Tcoefs = [a -b;
## b a;
## c d]
## @end group
## @end example
## This transformation may represent rotation, scaling and
## translation. Reflection is not included.
##
## @item "similarity"
## Same as "nonreflective similarity" except that the transform matrices T and Tinv may also have
## the form
## @example
## @group
## Tcoefs = [a b;
## b -a;
## c d]
## @end group
## @end example
## This transformation may represent reflection, rotation, scaling and
## translation. Generates a warning if the nonreflective similarity is
## better suited.
##
## @item "projective"
## Return both forward (RW->AP) and inverse (AP->RW) transform
## coefficients @var{T}.tdata.T and @var{T}.tdata.Tinv. Transform
## coefficients are 3x3 matrices which can
## be used as follows:
##
## @example
## @group
## [u v w] = [@var{ap_pt} ones(rows (ap_pt,1))] * Tinv
## @var{rw_pt} = [u./w, v./w];
## [x y z] = [@var{rw_pt} ones(rows (rw_pt,1))] * T
## @var{ap_pt} = [x./z y./z];
## @end group
## @end example
## This transformation is well suited when parallel lines in one space
## all converge toward a vanishing point in the other space.
##
## @item "polynomial"
## Here the @var{opt} input argument is the order of the polynomial
## fit. @var{opt} must be 2, 3 or 4 and input control points number must
## be respectively at least 6, 10 and 15. Only the inverse transform
## (AP->RW) is included in the structure @var{T}.
## Denoting x and y the apparent coordinates vector and xrw, yrw the
## the real world coordinates. Inverse transform coefficients are
## stored in a (6,10 or 15)x2 matrix which can be used as follows:
##
## @example
## @group
## Second order:
## [xrw yrw] = [1 x y x*y x^2 y^2] * Tinv
## @end group
## @group
## Third order:
## [xrw yrw] = [1 x y x*y x^2 y^2 y*x^2 x*y^2 x^3 y^3] * Tinv
## @end group
## @group
## Fourth order:
## [xrw yrw] = [1 x y x*y x^2 y^2 y*x^2 x*y^2 x^3 y^3 x^3*y x^2*y^2 x*y^3 x^4 y^4] * Tinv
## @end group
## @end example
## This transform is well suited when lines in one space become curves
## in the other space.
## @end table
## @seealso{tformfwd, tforminv, maketform}
## @end deftypefn
## Author: Pantxo Diribarne <pantxo@dibona>
## Created: 2012-09-05
function trans = cp2tform (crw, cap, ttype, opt)
if (nargin < 3)
print_usage ();
endif
if (! all (size (crw) == size (cap)) || columns (crw) != 2)
error ("cp2tform: expect the 2 first input arguments to be (m x 2) matrices")
elseif (! ischar (ttype))
error ("cp2tform: expect a string as third input argument")
endif
ttype = lower (ttype);
switch ttype
case {'nonreflective similarity', 'similarity', 'affine', 'projective'}
trans = gettrans (ttype, cap, crw);
case 'polynomial'
if (nargin < 4)
error ("cp2tform: expect a fourth input argument for 'polynomial'")
elseif (! isscalar (opt))
error ("cp2tform: expect a scalar as fourth argument")
endif
trans = gettrans (ttype, cap, crw, round (opt));
otherwise
error ("cp2tform: expect 'nonreflective similarity', 'similarity', 'affine' or 'polynomial' as third input argument")
endswitch
endfunction
function trans = gettrans (ttype, cap, crw, ord = 0)
switch ttype
case "nonreflective similarity"
x = cap(:,1);
y = cap(:,2);
u = crw(:,1);
v = crw(:,2);
tmp0 = zeros(size(u));
tmp1 = ones(size(u));
A = [u v tmp1 tmp0 ; v -u tmp0 tmp1];
B = [x; y];
tmat = A\B;
tmat = [tmat(1) -tmat(2);
tmat(2) tmat(1);
tmat(3) tmat(4)];
trans = maketform ("affine", tmat);
case "similarity"
x = cap(:,1);
y = cap(:,2);
u = crw(:,1);
v = crw(:,2);
tmp0 = zeros(size(u));
tmp1 = ones(size(u));
#without reflection
A = [u v tmp1 tmp0 ; v -u tmp0 tmp1];
B = [x; y];
tmat1 = A\B;
resid = norm (A*tmat1 - B);
#with reflection
A = [u v tmp1 tmp0 ; -v u tmp0 tmp1];
tmat2 = A\B;
if (norm (A*tmat2 - B) < resid)
tmat = [tmat2(1) tmat2(2);
tmat2(2) -tmat2(1);
tmat2(3) tmat2(4)];
else
tmat = [tmat1(1) -tmat1(2);
tmat1(2) tmat1(1);
tmat1(3) tmat1(4)];
warning ("cp2tform: reflection not included.")
endif
trans = maketform ("affine", tmat);
case "affine"
tmat = [crw ones(rows(crw), 1)]\cap;
trans = maketform ("affine", tmat);
case "projective"
x = cap(:,1);
y = cap(:,2);
u = crw(:,1);
v = crw(:,2);
tmp0 = zeros(size(u));
tmp1 = ones(size(u));
A = [-u -v -tmp1 tmp0 tmp0 tmp0 x.*u x.*v x;
tmp0 tmp0 tmp0 -u -v -tmp1 y.*u y.*v y];
B = - A(:,end);
A(:,end) = [];
tmat = A\B;
tmat(9) = 1;
tmat = reshape (tmat, 3, 3);
trans = maketform ("projective", tmat);
case "polynomial"
x = cap(:,1);
y = cap(:,2);
u = crw(:,1);
v = crw(:,2);
tmp1 = ones(size(x));
ndims_in = 2;
ndims_out = 2;
forward_fcn = [];
inverse_fcn = @inv_polynomial;
A = [tmp1, x, y, x.*y, x.^2, y.^2];
B = [u v];
switch ord
case 2
case 3
A = [A, y.*x.^2 x.*y.^2 x.^3 y.^3];
case 4
A = [A, y.*x.^2 x.*y.^2 x.^3 y.^3];
A = [A, x.^3.*y x.^2.*y.^2 x.*y.^3 x.^4 y.^4];
otherwise
error ("cp2tform: supported polynomial orders are 2, 3 and 4.")
endswitch
tmat = A\B;
trans = maketform ("custom", ndims_in, ndims_out, ...
forward_fcn, inverse_fcn, tmat);
otherwise
error ("cp2tform: invalid TRANSTYPE %s.", ttype);
endswitch
endfunction
function out = inv_polynomial (x, pst)
out = [];
for ii = 1:2
p = pst.tdata(:,ii);
if (rows (p) == 6)
## 2nd order
out(:,ii) = p(1) + p(2)*x(:,1) + p(3)*x(:,2) + p(4)*x(:,1).*x(:,2) + ...
p(5)*x(:,1).^2 + p(6)*x(:,2).^2;
elseif (rows (p) == 10)
## 3rd order
out(:,ii) = p(1) + p(2)*x(:,1) + p(3)*x(:,2) + p(4)*x(:,1).*x(:,2) + ...
p(5)*x(:,1).^2 + p(6)*x(:,2).^2 + p(7)*x(:,2).*x(:,1).^2 + ...
p(8)*x(:,1).*x(:,2).^2 + p(9)*x(:,1).^3 + p(10)*x(:,2).^3;
elseif (rows (p) == 15)
## 4th order
out(:,ii) = p(1) + p(2)*x(:,1) + p(3)*x(:,2) + p(4)*x(:,1).*x(:,2) + ...
p(5)*x(:,1).^2 + p(6)*x(:,2).^2 + p(7)*x(:,2).*x(:,1).^2 + ...
p(8)*x(:,1).*x(:,2).^2 + p(9)*x(:,1).^3 + p(10)*x(:,2).^3 + ...
p(11)*x(:,2).*x(:,1).^3 + p(12)*x(:,2).^2.*x(:,1).^2+ ...
p(13)*x(:,1).*x(:,2).^3 + p(14)*x(:,1).^4 + p(15)*x(:,2).^4;
endif
endfor
endfunction
%!function [crw, cap] = coords (npt = 1000, scale = 2, dtheta = pi/3, dx = 2, dy = -6, sig2noise = 1e32)
%! theta = (rand(npt, 1)*2-1)*2*pi;
%! R = rand(npt,1);
%! y = R.*sin(theta);
%! x = R.*cos(theta);
%! crw = [y x];
%!
%! thetap = theta + dtheta;
%! Rap = R * scale;
%!
%! yap = Rap.*sin(thetap);
%! yap = yap + dy;
%! yap = yap + rand (size (yap)) * norm (yap) / sig2noise;
%!
%! xap = Rap.*cos(thetap);
%! xap = xap + dx;
%! xap = xap + rand (size (xap)) * norm (xap) / sig2noise;
%! cap = [yap xap];
%!endfunction
%!test
%! npt = 100000;
%! [crw, cap] = coords (npt);
%! ttype = 'projective';
%! T = cp2tform (crw, cap, ttype);
%! crw2 = tforminv (T, cap);
%! finalerr = norm (crw - crw2)/npt;
%! assert (finalerr < eps, "norm = %3.2e ( > eps)", finalerr)
%!test
%! npt = 100000;
%! [crw, cap] = coords (npt);
%! ttype = 'affine';
%! T = cp2tform (crw, cap, ttype);
%! crw2 = tforminv (T, cap);
%! finalerr = norm (crw - crw2)/npt;
%! assert (finalerr < eps, "norm = %3.2e ( > eps)", finalerr)
%!test
%! npt = 100000;
%! [crw, cap] = coords (npt);
%! ttype = 'nonreflective similarity';
%! T = cp2tform (crw, cap, ttype);
%! crw2 = tforminv (T, cap);
%! finalerr = norm (crw - crw2)/npt;
%! assert (finalerr < eps, "norm = %3.2e ( > eps)", finalerr)
%!test
%! npt = 100000;
%! [crw, cap] = coords (npt);
%! cap(:,2) *= -1; % reflection around y axis
%! ttype = 'similarity';
%! T = cp2tform (crw, cap, ttype);
%! crw2 = tforminv (T, cap);
%! finalerr = norm (crw - crw2)/npt;
%! assert (finalerr < eps, "norm = %3.2e ( > eps)", finalerr)
%!xtest
%! npt = 100000;
%! [crw, cap] = coords (npt);
%! ttype = 'polynomial';
%! ord = 2;
%! T = cp2tform (crw, cap, ttype, ord);
%! crw2 = tforminv (T, cap);
%! finalerr = norm (crw - crw2)/npt;
%! assert (finalerr < eps, "norm = %3.2e ( > eps)", finalerr)
%!xtest
%! npt = 100000;
%! [crw, cap] = coords (npt);
%! ttype = 'polynomial';
%! ord = 3;
%! T = cp2tform (crw, cap, ttype, ord);
%! crw2 = tforminv (T, cap);
%! finalerr = norm (crw - crw2)/npt;
%! assert (finalerr < eps, "norm = %3.2e ( > eps)", finalerr)
%!xtest
%! npt = 100000;
%! [crw, cap] = coords (npt);
%! ttype = 'polynomial';
%! ord = 4;
%! T = cp2tform (crw, cap, ttype, ord);
%! crw2 = tforminv (T, cap);
%! finalerr = norm (crw - crw2)/npt;
%! assert (finalerr < eps, "norm = %3.2e ( > eps)", finalerr)