样条曲线(Spline)本质是分段多项式实函数,在实数范围内有: S:[a,b]→R ,在区间 [a,b] 上包含 k 个子区间 [ti−1,ti] ,且有:
对应每一段区间 i 的存在多项式: Pi:[ti−1,ti]→R ,且满足于:
其中, Pi(t) 多项式中最高次项的幂,视为样条的阶数或次数(Order of spline),根据子区间 [ti−1,ti] 的区间长度是否一致分为均匀(Uniform)样条和非均匀(Non-uniform)样条。
满足了公式 (2) 的多项式有很多,为了保证曲线在 S 区间内具有据够的平滑度,一条 n 次样条,同时应具备处处连续且可微的性质:
其中 i=1,…,k−1;j=0,…,n−1 。
按照上述的定义,给定节点:
三次样条曲线满足三个条件:
则三次样条的方程可以写为:
其中, ai,bi,ci,di 分别代表 n 个未知系数。
其中 i=0,1,…,k−1 。
其中 i=0,1,…,k−2 。
令区间长度 hi=ti+1−ti ,则有:
由公式 (6) ,可得: ai=zi ;
由公式 (7) ,可得: ai+bihi+cih2i+dih3i=zi+1 ;
由公式 (8) ,可得:
S′i(ti+1)=bi+2cihi+3dih2i ;
S′i+1(ti+1)=bi+1 ;
⇒bi+2cihi+3dih2i−bi+1=0 ;
由公式 (9) ,可得:
S′′i(ti+1)=2ci+6dihi ;
S′′i+1(ti+1)=2ci+1 ;
⇒2ci+6dihi=2ci+1 ;
设 mi=S′′i(xi)=2ci ,则:
A. mi+6dihi−mi+1=0⇒
di=mi+1−mi6hi ;
B.将 ci,di 代入 zi+bihi+cih2i+dih3i=zi+1⇒
bi=zi+1−zihi−hi2mi−hi6(mi+1−mi) ;
C.将 bi,ci,di 代入 bi+2cihi+3dih2i=bi+1⇒
在上述分析中,曲线段的两个端点 t0 和 tk 是不适用的,有一些常用的端点限制条件,这里只讲解自然边界。
在自然边界下,首尾两端的二阶导函数满足 S′′=0 ,即 m0=0 和 mn=0 ,求解方程组可写为:
其系数矩阵为三对角线矩阵,在该篇博客内会有其讲解。
// CubicSplineInterpolation.h
/* Cubic spline interpolation class. - Editor: Yahui Liu. - Data: 2015-08-16 - Email: [email protected] - Address: Computer Vision and Remote Sensing(CVRS), Lab. */
#ifndef CUBIC_SPLINE_INTERPOLATION_H
#pragma once
#define CUBIC_SPLINE_INTERPOLATION_H
#include <iostream>
#include <vector>
#include <math.h>
#include <cv.h>
#include <highgui.h>
using namespace std;
using namespace cv;
/* Cubic spline interpolation coefficients */
class CubicSplineCoeffs
{
public:
CubicSplineCoeffs( const int &count )
{
a = std::vector<double>(count);
b = std::vector<double>(count);
c = std::vector<double>(count);
d = std::vector<double>(count);
}
~CubicSplineCoeffs()
{
std::vector<double>().swap(a);
std::vector<double>().swap(b);
std::vector<double>().swap(c);
std::vector<double>().swap(d);
}
public:
std::vector<double> a, b, c, d;
};
enum CubicSplineMode
{
CUBIC_NATURAL, // Natural
CUBIC_CLAMPED, // TODO: Clamped
CUBIC_NOT_A_KNOT // TODO: Not a knot
};
enum SplineFilterMode
{
CUBIC_WITHOUT_FILTER, // without filter
CUBIC_MEDIAN_FILTER // median filter
};
/* Cubic spline interpolation */
class CubicSplineInterpolation
{
public:
CubicSplineInterpolation() {}
~CubicSplineInterpolation() {}
public:
/* Calculate cubic spline coefficients. - node list x (input_x); - node list y (input_y); - output coefficients (cubicCoeffs); - ends mode (splineMode). */
void calCubicSplineCoeffs( std::vector<double> &input_x,
std::vector<double> &input_y, CubicSplineCoeffs *&cubicCoeffs,
CubicSplineMode splineMode = CUBIC_NATURAL,
SplineFilterMode filterMode = CUBIC_MEDIAN_FILTER );
/* Cubic spline interpolation for a list. - input coefficients (cubicCoeffs); - input node list x (input_x); - output node list x (output_x); - output node list y (output_y); - interpolation step (interStep). */
void cubicSplineInterpolation( CubicSplineCoeffs *&cubicCoeffs,
std::vector<double> &input_x, std::vector<double> &output_x,
std::vector<double> &output_y, const double interStep = 0.5 );
/* Cubic spline interpolation for a value. - input coefficients (cubicCoeffs); - input a value(x); - output interpolation value(y); */
void cubicSplineInterpolation2( CubicSplineCoeffs *&cubicCoeffs,
std::vector<double> &input_x, double &x, double &y );
/* calculate tridiagonal matrices with Thomas Algorithm(TDMA) : example: | b1 c1 0 0 0 0 | |x1 | |d1 | | a2 b2 c2 0 0 0 | |x2 | |d2 | | 0 a3 b3 c3 0 0 | |x3 | = |d3 | | ... ... | |...| |...| | 0 0 0 0 an bn | |xn | |dn | Ci = ci/bi , i=1; ci / (bi - Ci-1 * ai) , i = 2, 3, ... n-1; Di = di/bi , i=1; ( di - Di-1 * ai )/(bi - Ci-1 * ai) , i = 2, 3, ..., n-1 xi = Di - Ci*xi+1 , i = n-1, n-2, 1; */
bool caltridiagonalMatrices( cv::Mat_<double> &input_a,
cv::Mat_<double> &input_b, cv::Mat_<double> &input_c,
cv::Mat_<double> &input_d, cv::Mat_<double> &output_x );
/* Calculate the curve index interpolation belongs to */
int calInterpolationIndex( double &pt, std::vector<double> &input_x );
/* median filtering */
void cubicMedianFilter( std::vector<double> &input, const int filterSize = 5 );
double cubicSort( std::vector<double> &input );
// double cubicNearestValue( std::vector );
};
#endif // CUBIC_SPLINE_INTERPOLATION_H
// CubicSplineInterpolation.cpp
#include "CubicSplineInterpolation.h"
void CubicSplineInterpolation::calCubicSplineCoeffs(
std::vector<double> &input_x,
std::vector<double> &input_y,
CubicSplineCoeffs *&cubicCoeffs,
CubicSplineMode splineMode /* = CUBIC_NATURAL */,
SplineFilterMode filterMode /*= CUBIC_MEDIAN_FILTER*/ )
{
int sizeOfx = input_x.size();
int sizeOfy = input_y.size();
if ( sizeOfx != sizeOfy )
{
std::cout << "Data input error!" << std::endl <<
"Location: CubicSplineInterpolation.cpp" <<
" -> calCubicSplineCoeffs()" << std::endl;
return;
}
/* hi*mi + 2*(hi + hi+1)*mi+1 + hi+1*mi+2 = 6{ (yi+2 - yi+1)/hi+1 - (yi+1 - yi)/hi } so, ignore the both ends: | - - - 0 ... 0 | |m0 | | h0 2(h0+h1) h1 0 ... 0 | |m1 | | 0 h1 2(h1+h2) h2 0 ... | |m2 | | ... ... 0 | |...| | 0 ... 0 h(n-2) 2(h(n-2)+h(n-1)) h(n-1) | | | | 0 ... ... - | |mn | */
std::vector<double> copy_y = input_y;
if ( filterMode == CUBIC_MEDIAN_FILTER )
{
cubicMedianFilter(copy_y, 5);
}
const int count = sizeOfx;
const int count1 = sizeOfx - 1;
const int count2 = sizeOfx - 2;
const int count3 = sizeOfx - 3;
cubicCoeffs = new CubicSplineCoeffs( count1 );
std::vector<double> step_h( count1, 0.0 );
// for m matrix
cv::Mat_<double> m_a(1, count2, 0.0);
cv::Mat_<double> m_b(1, count2, 0.0);
cv::Mat_<double> m_c(1, count2, 0.0);
cv::Mat_<double> m_d(1, count2, 0.0);
cv::Mat_<double> m_part(1, count2, 0.0);
cv::Mat_<double> m_all(1, count, 0.0);
// initial step hi
for ( int idx=0; idx < count1; idx ++ )
{
step_h[idx] = input_x[idx+1] - input_x[idx];
}
// initial coefficients
for ( int idx=0; idx < count3; idx ++ )
{
m_a(idx) = step_h[idx];
m_b(idx) = 2 * (step_h[idx] + step_h[idx+1]);
m_c(idx) = step_h[idx+1];
}
// initial d
for ( int idx =0; idx < count3; idx ++ )
{
m_d(idx) = 6 * (
(copy_y[idx+2] - copy_y[idx+1]) / step_h[idx+1] -
(copy_y[idx+1] - copy_y[idx]) / step_h[idx] );
}
//cv::Mat_<double> matOfm( count2, )
bool isSucceed = caltridiagonalMatrices(m_a, m_b, m_c, m_d, m_part);
if ( !isSucceed )
{
std::cout<<"Calculate tridiagonal matrices failed!"<<std::endl<<
"Location: CubicSplineInterpolation.cpp -> " <<
"caltridiagonalMatrices()"<<std::endl;
return;
}
if ( splineMode == CUBIC_NATURAL )
{
m_all(0) = 0.0;
m_all(count1) = 0.0;
for ( int i=1; i<count1; i++ )
{
m_all(i) = m_part(i-1);
}
for ( int i=0; i<count1; i++ )
{
cubicCoeffs->a[i] = copy_y[i];
cubicCoeffs->b[i] = ( copy_y[i+1] - copy_y[i] ) / step_h[i] -
step_h[i]*( 2*m_all(i) + m_all(i+1) ) / 6;
cubicCoeffs->c[i] = m_all(i) / 2.0;
cubicCoeffs->d[i] = ( m_all(i+1) - m_all(i) ) / ( 6.0 * step_h[i] );
}
}
else
{
std::cout<<"Not define the interpolation mode!"<<std::endl;
}
}
void CubicSplineInterpolation::cubicSplineInterpolation(
CubicSplineCoeffs *&cubicCoeffs,
std::vector<double> &input_x,
std::vector<double> &output_x,
std::vector<double> &output_y,
const double interStep )
{
const int count = input_x.size();
double low = input_x[0];
double high = input_x[count-1];
double interBegin = low;
for ( ; interBegin < high; interBegin += interStep )
{
int index = calInterpolationIndex(interBegin, input_x);
if ( index >= 0 )
{
double dertx = interBegin - input_x[index];
double y = cubicCoeffs->a[index] + cubicCoeffs->b[index] * dertx +
cubicCoeffs->c[index] * dertx * dertx +
cubicCoeffs->d[index] * dertx * dertx * dertx;
output_x.push_back(interBegin);
output_y.push_back(y);
}
}
}
void CubicSplineInterpolation::cubicSplineInterpolation2(
CubicSplineCoeffs *&cubicCoeffs,
std::vector<double> &input_x, double &x, double &y)
{
const int count = input_x.size();
double low = input_x[0];
double high = input_x[count-1];
if ( x<low || x>high )
{
std::cout<<"The interpolation value is out of range!"<<std::endl;
}
else
{
int index = calInterpolationIndex(x, input_x);
if ( index > 0 )
{
double dertx = x - input_x[index];
y = cubicCoeffs->a[index] + cubicCoeffs->b[index] * dertx +
cubicCoeffs->c[index] * dertx * dertx +
cubicCoeffs->d[index] * dertx * dertx * dertx;
}
else
{
std::cout<<"Can't find the interpolation range!"<<std::endl;
}
}
}
bool CubicSplineInterpolation::caltridiagonalMatrices(
cv::Mat_<double> &input_a,
cv::Mat_<double> &input_b,
cv::Mat_<double> &input_c,
cv::Mat_<double> &input_d,
cv::Mat_<double> &output_x )
{
int rows = input_a.rows;
int cols = input_a.cols;
if ( ( rows == 1 && cols > rows ) ||
(cols == 1 && rows > cols ) )
{
const int count = ( rows > cols ? rows : cols ) - 1;
output_x = cv::Mat_<double>::zeros(rows, cols);
cv::Mat_<double> cCopy, dCopy;
input_c.copyTo(cCopy);
input_d.copyTo(dCopy);
if ( input_b(0) != 0 )
{
cCopy(0) /= input_b(0);
dCopy(0) /= input_b(0);
}
else
{
return false;
}
for ( int i=1; i < count; i++ )
{
double temp = input_b(i) - input_a(i) * cCopy(i-1);
if ( temp == 0.0 )
{
return false;
}
cCopy(i) /= temp;
dCopy(i) = ( dCopy(i) - dCopy(i-1)*input_a(i) ) / temp;
}
output_x(count) = dCopy(count);
for ( int i=count-2; i > 0; i-- )
{
output_x(i) = dCopy(i) - cCopy(i)*output_x(i+1);
}
return true;
}
else
{
return false;
}
}
int CubicSplineInterpolation::calInterpolationIndex(
double &pt, std::vector<double> &input_x )
{
const int count = input_x.size()-1;
int index = -1;
for ( int i=0; i<count; i++ )
{
if ( pt > input_x[i] && pt <= input_x[i+1] )
{
index = i;
return index;
}
}
return index;
}
void CubicSplineInterpolation::cubicMedianFilter(
std::vector<double> &input, const int filterSize /* = 5 */ )
{
const int count = input.size();
for ( int i=filterSize/2; i<count-filterSize/2; i++ )
{
std::vector<double> temp(filterSize, 0.0);
for ( int j=0; j<filterSize; j++ )
{
temp[j] = input[i+j - filterSize/2];
}
input[i] = cubicSort(temp);
std::vector<double>().swap(temp);
}
for ( int i=0; i<filterSize/2; i++ )
{
std::vector<double> temp(filterSize, 0.0);
for ( int j=0; j<filterSize; j++ )
{
temp[j] = input[j];
}
input[i] = cubicSort(temp);
std::vector<double>().swap(temp);
}
for ( int i=count-filterSize/2; i<count; i++ )
{
std::vector<double> temp(filterSize, 0.0);
for ( int j=0; j<filterSize; j++ )
{
temp[j] = input[j];
}
input[i] = cubicSort(temp);
std::vector<double>().swap(temp);
}
}
double CubicSplineInterpolation::cubicSort( std::vector<double> &input )
{
int iCount = input.size();
for ( int j=0; j<iCount-1; j++ )
{
for ( int k=iCount-1; k>j; k-- )
{
if ( input[k-1] > input[k] )
{
double tp = input[k];
input[k] = input[k-1];
input[k-1] = tp;
}
}
}
return input[iCount/2];
}
// main.cpp
#include "CubicSplineInterpolation.h"
void main()
{
double x[22] = {
926.500000,
928.000000,
929.500000,
931.000000,
932.500000,
934.000000,
935.500000,
937.000000,
938.500000,
940.000000,
941.500000,
943.000000,
944.500000,
946.000000,
977.500000,
980.500000,
982.000000,
983.500000,
985.000000,
986.500000,
988.000000,
989.500000};
double y[22] = {
381.732239,
380.670530,
380.297786,
379.853896,
379.272647,
378.368584,
379.319757,
379.256485,
380.233150,
378.183257,
377.543639,
376.948999,
376.253935,
198.896327,
670.369434,
374.273702,
372.498821,
373.149402,
372.139661,
372.510891,
372.772791,
371.360553};
std::vector<double> input_x(22), input_y(22);
for ( int i=0; i<22; i++)
{
input_x[i] = x[i];
input_y[i] = y[i];
}
CubicSplineCoeffs *cubicCoeffs;
CubicSplineInterpolation cubicSpline;
cubicSpline.calCubicSplineCoeffs(input_x, input_y, cubicCoeffs, CUBIC_NATURAL, CUBIC_MEDIAN_FILTER);
std::vector<double> output_x, output_y;
cubicSpline.cubicSplineInterpolation( cubicCoeffs, input_x, output_x, output_y );
double xx(946.0), yy(0.0);
cubicSpline.cubicSplineInterpolation2(cubicCoeffs, input_x, xx, yy);
std::cout<<yy<<std::endl;
std::ofstream outfile( "E:\\test.txt", std::ios::out );
if ( outfile )
{
for ( int i=0; i<output_y.size(); i++ )
{
outfile<<std::fixed<<setprecision(3)<<output_x[i]<<" "<<output_y[i]<<std::endl;
}
}
outfile.close();
}
运行结果:
其中单独点插值的运行结果分别为:
198.896 // yy, CUBIC_WITHOUT_FILTER
376.949 // yy, CUBIC_MEDIAN_FILTER
参考文献:
1.https://en.wikipedia.org/wiki/Spline_(mathematics)
2.http://www.cnblogs.com/xpvincent/archive/2013/01/26/2878092.html