CGII/framework/include/cgv/math/bi_polynomial.h

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#pragma once
#include <cgv/math/polynomial.h>
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namespace cgv {
namespace math {
/**
* bivariate polynomials are stored in matrices in the following order:
* [a,b,c,d a*x1^3*x2^3 + b*x1^2*x2^3 + c*x1*x2^3 + d*x2^3 +
* e,f,g,h e*x1^3*x2^2 + f*x1^2*x2^2 + g*x1*x2^2 + h*x2^2 +
* i,j,k,l i*x1^3*x2 + j*x1^2*x2 + k*x1*x2 + l*x2 +
* m,n,o,p] m*x1^3 + n*x1^2 + o*x1 + p
*/
/// evaluate a bivariate polynomial p at (x1,x2)
/// p is the coefficients matrix of the bivariate polynomial
/// evaluation is done by using the horner scheme
template <typename T>
T bipoly_val(const mat<T>& bp, const T& x1, const T& x2)
{
unsigned n = bp.ncols();
unsigned m = bp.nrows();
assert(n > 0);
assert(m > 0);
vec<T> r(m);
for(unsigned i = 0; i < m;i++)
r(i) = poly_val(bp.row(i),x1);
return poly_val(r,x2);
}
/// evaluate a bivariate polynomial p at 2d position x
/// p is the coefficients matrix of the bivariate polynomial
/// evaluation is done by using the horner scheme
template <typename T>
T bipoly_val(const mat<T>& bp, const vec<T>& x)
{
unsigned n = bp.ncols();
unsigned m = bp.nrows();
assert(n > 0);
assert(m > 0);
assert(v.size() == 2)
vec<T> r(n);
for(unsigned i = 0; i < m;i++)
r(i) = poly_val(bp.row(i),x(0));
return poly_val(r,x(1));
}
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//returns vandermonde matrix for bivariate polynomial fitting
template <typename T>
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cgv::math::mat<T> vander2(unsigned degree1, unsigned degree2,
const vec<T>& X1,const vec<T> X2)
{
assert(X1.size()>0);
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assert(X1.size() == X2.size());
unsigned n = X1.size();
unsigned m = (degree1+1)*(degree2+1);
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cgv::math::mat<T> Vxy(n,m);
cgv::math::mat<T> Vx = vander(degree1,X1);
cgv::math::mat<T> Vy = vander(degree2,X2);
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for(unsigned r = 0; r < n; r++)
{
for(unsigned i = 0; i < degree2+1; i++)
for(unsigned j =0; j < degree1+1; j++)
Vxy(r,j*(degree2+1)+i) = Vy(r,i)*Vx(r,j);
}
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return Vxy;
}
//fit a bivariate polynomial
template <typename T>
mat<T> bipoly_fit(unsigned degree1, unsigned degree2,
const vec<T>& X1,const vec<T> X2,const vec<T> Y)
{
assert(X1.size() == X2.size());
assert(X1.size() == Y.size());
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mat<T> V = cgv::math::vander2(degree1,degree2,X1,X2);
mat<T> A;
vec<T> b,bp;
AtA(V,A);
Atx(V,Y,b);
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svd_solve(A,b,bp);
return reshape(bp,degree2+1,degree1+1);
}
/*
template <typename T>
mat<T> bipoly_fit2(unsigned degree1, unsigned degree2,
const vec<T>& X1,const vec<T> X2,const vec<T> Y)
{
assert(X1.size() == X2.size());
assert(X1.size() == Y.size());
vec<T> C = poly_fit<T>(degree2,X2,Y);
//std::cout << C << "\n";
mat<T> B = dyad(cgv::math::ones<T>(X1.size()),C);
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mat<T> V = cgv::math::vander(degree1,X1);
mat<T> A;
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AtA(V,A);
mat<T> D,bp;
AtB(V,B,D);
svd_solve(A,D,bp);
//std::cout <<":\n"<< A*bp-D;
return bp;
}*/
////analytic derivation of bivariate polynomial dp(x1,x2)/dx1
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template <typename T>
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mat<T> bipoly_der_x1(const mat<T>& bp)
{
assert(bp.ncols() > 0 && bp.nrows() > 0);
if(bp.ncols() == 1)
return zeros<T>(1,1);
mat<T> m(bp.nrows(),bp.ncols()-1);
for(unsigned i = 0; i < bp.nrows(); i++)
m.set_row(i,poly_der(bp.row(i)));
return m;
}
////analytic derivation of bivariate polynomial dp(x1,x2)/dx2
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template <typename T>
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mat<T> bipoly_der_x2(const mat<T>& bp)
{
assert(bp.ncols() > 0 && bp.nrows() > 0);
if(bp.nrows() == 1)
return zeros<T>(1,1);
T n = (T)(bp.nrows()-1);
mat<T> m(bp.nrows()-1,bp.ncols());
for(unsigned i = 0; i < bp.nrows()-1; i++)
m.set_row(i,(n-(T)i)* bp.row(i));
return m;
}
///analytic integration of bivariate polynomial int p(x1,x2) dx1
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template <typename T>
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mat<T> bipoly_int_x1(const mat<T>& bp,const T& k=0)
{
mat<T> m(bp.nrows(),bp.ncols()+1);
for(unsigned i = 0; i < bp.nrows();i++)
m.set_row(i,poly_int(bp.row(i),k));
return m;
}
///analytic integration of bivariate polynomial int p(x1,x2) dx2
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template <typename T>
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mat<T> bipoly_int_x2(const mat<T>& bp,const T& k=0)
{
mat<T> v(1,bp.ncols());
v.zeros();
v(0,bp.ncols()-1)=k;
mat<T> m = vertcat(bp,v);
for(unsigned i = 0; i < m.nrows()-2;i++)
for(unsigned j = 0; j < m.ncols();j++)
m(i,j)=m(i,j)/(m.nrows()-1-i);
return m;
}
}
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}