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## Copyright (C) 2004-2011 David Legland <david.legland@grignon.inra.fr>
## Copyright (C) 2004-2011 INRA - CEPIA Nantes - MIAJ (Jouy-en-Josas)
## Copyright (C) 2012 Adapted to Octave by Juan Pablo Carbajal <carbajal@ifi.uzh.ch>
## All rights reserved.
##
## Redistribution and use in source and binary forms, with or without
## modification, are permitted provided that the following conditions are met:
##
## 1 Redistributions of source code must retain the above copyright notice,
## this list of conditions and the following disclaimer.
## 2 Redistributions in binary form must reproduce the above copyright
## notice, this list of conditions and the following disclaimer in the
## documentation and/or other materials provided with the distribution.
##
## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS''
## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR
## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{xc}, @var{yc}] =} polynomialCurveFit (@var{t}, @var{xt}, @var{yt}, @var{order})
## @deftypefnx {Function File} {[@var{xc}, @var{yc}] =} polynomialCurveFit (@var{t}, @var{points}, @var{order})
## @deftypefnx {Function File} {[@var{xc}, @var{yc}] =} polynomialCurveFit (@dots{}, @var{ti}, @var{condi})
## Fit a polynomial curve to a series of points
##
## @var{t} is a Nx1 vector
## @var{xt} and @var{yt} are coordinate for each parameter value (column vectors)
## @var{order} is the degree of the polynomial used for interpolation
## @var{xc} and @var{yc} are polynomial coefficients, given in @var{order}+1 row vectors,
## starting from degree 0 and up to degree @var{order}.
##
## @var{points} specifies coordinate of points in a Nx2 array.
##
## Impose some specific conditions using @var{ti} and @var{condi}.
## @var{ti} is a value of the parametrization
## variable. @var{condi} is a cell array, with 2 columns, and as many rows as
## the derivatives specified for the given @var{ti}. Format for @var{condi} is:
## @var{condi} = @@{X_I, Y_I; X_I', Y_I'; X_I", Y_I"; ...@@};
## with X_I and Y_I being the imposed coordinate at position @var{ti}, X_I' and
## Y_I' being the imposed first derivatives, X_I" and Y_I" the imposed
## second derivatives, and so on...
## To specify a derivative without specifying derivative with lower
## degree, value of lower derivative can be let empty, using '[]'
##
##
## Requires the optimization Toolbox.
##
## Run @command{demo polynomialCurveFit} to see exaples of use.
##
## @seealso{polynomialCurves2d}
## @end deftypefn
function varargout = polynomialCurveFit(t, varargin)
## extract input arguments
# extract curve coordinate
var = varargin{1};
if min(size(var))==1
# curve given as separate arguments
xt = varargin{1};
yt = varargin{2};
varargin(1:2)=[];
else
# curve coordinate bundled in a matrix
if size(var, 1)<size(var, 2)
var = var';
end
xt = var(:,1);
yt = var(:,2);
varargin(1)=[];
end
# order of the polynom
var = varargin{1};
if length(var)>1
Dx = var(1);
Dy = var(2);
else
Dx = var;
Dy = var;
end
varargin(1)=[];
## Initialize local conditions
# For a solution vector 'x', the following relation must hold:
# Aeq * x == beq,
# with:
# Aeq Matrix M*N
# beq column vector with length M
# The coefficients of the Aeq matrix are initialized as follow:
# First point and last point are considered successively. For each point,
# k-th condition is the value of the (k-1)th derivative. This value is
# computed using relation of the form:
# value = sum_i ( fact(i) * t_j^pow(i) )
# with:
# i indice of the (i-1) derivative.
# fact row vector containing coefficient of each power of t, initialized
# with a row vector equals to [1 1 ... 1], and updated for each
# derivative by multiplying by corresponding power minus 1.
# pow row vector of the powers of each monome. It is represented by a
# row vector containing an increasing series of power, eventually
# completed with zeros for lower degrees (for the k-th derivative,
# the coefficients with power lower than k are not relevant).
# Example for degree 5 polynom:
# iter deriv pow fact
# 1 0 [0 1 2 3 4 5] [1 1 1 1 1 1]
# 2 1 [0 0 1 2 3 4] [0 1 2 3 4 5]
# 3 2 [0 0 0 1 2 3] [0 0 1 2 3 4]
# 4 3 [0 0 0 0 1 2] [0 0 0 1 2 3]
# ...
# The process is repeated for coordinate x and for coordinate y.
# Initialize empty matrices
Aeqx = zeros(0, Dx+1);
beqx = zeros(0, 1);
Aeqy = zeros(0, Dy+1);
beqy = zeros(0, 1);
# Process local conditions
while ~isempty(varargin)
if length(varargin)==1
warning('MatGeom:PolynomialCurveFit:ArgumentNumber', ...
'Wrong number of arguments in polynomialCurvefit');
end
# extract parameter t, and cell array of local conditions
ti = varargin{1};
cond = varargin{2};
# factors for coefficients of each polynomial. At the beginning, they
# all equal 1. With successive derivatives, their value increase by the
# corresponding powers.
factX = ones(1, Dx+1);
factY = ones(1, Dy+1);
# start condition initialisations
for i = 1:size(cond, 1)
# degrees of each polynomial
powX = [zeros(1, i) 1:Dx+1-i];
powY = [zeros(1, i) 1:Dy+1-i];
# update conditions for x coordinate
if ~isempty(cond{i,1})
Aeqx = [Aeqx ; factY.*power(ti, powX)]; ##ok<AGROW>
beqx = [beqx; cond{i,1}]; ##ok<AGROW>
end
# update conditions for y coordinate
if ~isempty(cond{i,2})
Aeqy = [Aeqy ; factY.*power(ti, powY)]; ##ok<AGROW>
beqy = [beqy; cond{i,2}]; ##ok<AGROW>
end
# update polynomial degrees for next derivative
factX = factX.*powX;
factY = factY.*powY;
end
varargin(1:2)=[];
end
## Initialisations
# ensure column vectors
t = t(:);
xt = xt(:);
yt = yt(:);
# number of points to fit
L = length(t);
## Compute coefficients of each polynomial
# main matrix for x coordinate, size L*(degX+1)
T = ones(L, Dx+1);
for i = 1:Dx
T(:, i+1) = power(t, i);
end
# compute interpolation
# Octave compatibility - JPi 2013
xc = lsqlin (T, xt, zeros(1, Dx+1), 1, Aeqx, beqx)';
# main matrix for y coordinate, size L*(degY+1)
T = ones(L, Dy+1);
for i = 1:Dy
T(:, i+1) = power(t, i);
end
# compute interpolation
# Octave compatibility - JPi 2013
yc = lsqlin (T, yt, zeros(1, Dy+1), 1, Aeqy, beqy)';
## Format output arguments
if nargout <= 1
varargout{1} = {xc, yc};
else
varargout{1} = xc;
varargout{2} = yc;
end
endfunction
function x = lsqlin (C, d, A, b, Aeq, beq)
H = C'*C;
q = -C'*d;
x0 = zeros (size(C,2),size(d,2));
x = qp (x0, H, q, Aeq, beq, [], [],[], A, b);
endfunction
%!demo
%! # defines a curve (circle arc) with small perturbations
%! N = 50;
%! t = linspace(0, 3*pi/4, N)';
%! xp = cos(t) + 5e-2*randn(size(t));
%! yp = sin(t) + 5e-2*randn(size(t));
%!
%! [xc yc] = polynomialCurveFit(t, xp, yp, 3);
%!
%! figure(1);
%! clf;
%! drawPolynomialCurve(t([1 end]), xc, yc);
%! hold on
%! plot(xp,yp,'.g');
%! hold off
%! axis tight
%! axis equal
%!demo
%! # defines a curve (circle arc) with small perturbations
%! N = 100;
%! t = linspace(0, 3*pi/4, N)';
%! xp = cos(t) + 7e-2*randn(size(t));
%! yp = sin(t) + 7e-2*randn(size(t));
%!
%! # plot the points
%! figure (1); clf; hold on;
%! axis ([-1.2 1.2 -.2 1.2]); axis equal;
%! drawPoint (xp, yp, ".g");
%!
%! # fit without knowledge on bounds
%! [xc0 yc0] = polynomialCurveFit (t, xp, yp, 5);
%! h = drawPolynomialCurve (t([1 end]), xc0, yc0);
%! set(h, "color", "b")
%!
%! # fit by imposing coordinate on first point
%! [xc1 yc1] = polynomialCurveFit (t, xp, yp, 5, 0, {1, 0});
%! h = drawPolynomialCurve (t([1 end]), xc1, yc1);
%! set(h, "color", "r")
%!
%! # fit by imposing coordinate (1,0) and derivative (0,1) on first point
%! [xc2 yc2] = polynomialCurveFit (t, xp, yp, 5, 0, {1, 0;0 1});
%! h = drawPolynomialCurve (t([1 end]), xc2, yc2);
%! set(h, "color", "m")
%!
%! # fit by imposing several conditions on various points
%! [xc3 yc3] = polynomialCurveFit (t, xp, yp, 5, ...
%! 0, {1, 0;0 1}, ... # coord and first derivative of first point
%! 3*pi/4, {-sqrt(2)/2, sqrt(2)/2}, ... # coord of last point
%! pi/2, {[], [];-1, 0}); # derivative of point on the top of arc
%! h = drawPolynomialCurve (t([1 end]), xc3, yc3);
%! set(h, "color", "k")
%! axis tight
%! axis equal