我正在使用 openCV 和 c++ 实现图像分析算法，但我发现 openCV 没有任何正式的巴特沃斯带通滤波器功能。在我的项目中，我必须将像素时间序列传递给 Butterworth 5 阶滤波器，该函数将返回过滤后的时间序列像素。Butterworth（像素系列，顺序，频率），如果您有任何想法可以帮助我了解如何开始，请告诉我。谢谢
编辑： 得到帮助后，最后我想出了以下代码。它可以计算分子系数和分母系数，但问题是某些数字与matlab结果不一样。这是我的代码：
#include <iostream>
#include <stdio.h>
#include <vector>
#include <math.h>
using namespace std;
#define N 10 //The number of images which construct a time series for each pixel
#define PI 3.14159
double *ComputeLP( int FilterOrder )
{
double *NumCoeffs;
int m;
int i;
NumCoeffs = (double *)calloc( FilterOrder+1, sizeof(double) );
if( NumCoeffs == NULL ) return( NULL );
NumCoeffs[0] = 1;
NumCoeffs[1] = FilterOrder;
m = FilterOrder/2;
for( i=2; i <= m; ++i)
{
NumCoeffs[i] =(double) (FilterOrder-i+1)*NumCoeffs[i-1]/i;
NumCoeffs[FilterOrder-i]= NumCoeffs[i];
}
NumCoeffs[FilterOrder-1] = FilterOrder;
NumCoeffs[FilterOrder] = 1;
return NumCoeffs;
}
double *ComputeHP( int FilterOrder )
{
double *NumCoeffs;
int i;
NumCoeffs = ComputeLP(FilterOrder);
if(NumCoeffs == NULL ) return( NULL );
for( i = 0; i <= FilterOrder; ++i)
if( i % 2 ) NumCoeffs[i] = -NumCoeffs[i];
return NumCoeffs;
}
double *TrinomialMultiply( int FilterOrder, double *b, double *c )
{
int i, j;
double *RetVal;
RetVal = (double *)calloc( 4 * FilterOrder, sizeof(double) );
if( RetVal == NULL ) return( NULL );
RetVal[2] = c[0];
RetVal[3] = c[1];
RetVal[0] = b[0];
RetVal[1] = b[1];
for( i = 1; i < FilterOrder; ++i )
{
RetVal[2*(2*i+1)] += c[2*i] * RetVal[2*(2*i-1)] - c[2*i+1] * RetVal[2*(2*i-1)+1];
RetVal[2*(2*i+1)+1] += c[2*i] * RetVal[2*(2*i-1)+1] + c[2*i+1] * RetVal[2*(2*i-1)];
for( j = 2*i; j > 1; --j )
{
RetVal[2*j] += b[2*i] * RetVal[2*(j-1)] - b[2*i+1] * RetVal[2*(j-1)+1] +
c[2*i] * RetVal[2*(j-2)] - c[2*i+1] * RetVal[2*(j-2)+1];
RetVal[2*j+1] += b[2*i] * RetVal[2*(j-1)+1] + b[2*i+1] * RetVal[2*(j-1)] +
c[2*i] * RetVal[2*(j-2)+1] + c[2*i+1] * RetVal[2*(j-2)];
}
RetVal[2] += b[2*i] * RetVal[0] - b[2*i+1] * RetVal[1] + c[2*i];
RetVal[3] += b[2*i] * RetVal[1] + b[2*i+1] * RetVal[0] + c[2*i+1];
RetVal[0] += b[2*i];
RetVal[1] += b[2*i+1];
}
return RetVal;
}
double *ComputeNumCoeffs(int FilterOrder)
{
double *TCoeffs;
double *NumCoeffs;
int i;
NumCoeffs = (double *)calloc( 2*FilterOrder+1, sizeof(double) );
if( NumCoeffs == NULL ) return( NULL );
TCoeffs = ComputeHP(FilterOrder);
if( TCoeffs == NULL ) return( NULL );
for( i = 0; i < FilterOrder; ++i)
{
NumCoeffs[2*i] = TCoeffs[i];
NumCoeffs[2*i+1] = 0.0;
}
NumCoeffs[2*FilterOrder] = TCoeffs[FilterOrder];
free(TCoeffs);
return NumCoeffs;
}
double *ComputeDenCoeffs( int FilterOrder, double Lcutoff, double Ucutoff )
{
int k; // loop variables
double theta; // PI * (Ucutoff - Lcutoff) / 2.0
double cp; // cosine of phi
double st; // sine of theta
double ct; // cosine of theta
double s2t; // sine of 2*theta
double c2t; // cosine 0f 2*theta
double *RCoeffs; // z^-2 coefficients
double *TCoeffs; // z^-1 coefficients
double *DenomCoeffs; // dk coefficients
double PoleAngle; // pole angle
double SinPoleAngle; // sine of pole angle
double CosPoleAngle; // cosine of pole angle
double a; // workspace variables
cp = cos(PI * (Ucutoff + Lcutoff) / 2.0);
theta = PI * (Ucutoff - Lcutoff) / 2.0;
st = sin(theta);
ct = cos(theta);
s2t = 2.0*st*ct; // sine of 2*theta
c2t = 2.0*ct*ct - 1.0; // cosine of 2*theta
RCoeffs = (double *)calloc( 2 * FilterOrder, sizeof(double) );
TCoeffs = (double *)calloc( 2 * FilterOrder, sizeof(double) );
for( k = 0; k < FilterOrder; ++k )
{
PoleAngle = PI * (double)(2*k+1)/(double)(2*FilterOrder);
SinPoleAngle = sin(PoleAngle);
CosPoleAngle = cos(PoleAngle);
a = 1.0 + s2t*SinPoleAngle;
RCoeffs[2*k] = c2t/a;
RCoeffs[2*k+1] = s2t*CosPoleAngle/a;
TCoeffs[2*k] = -2.0*cp*(ct+st*SinPoleAngle)/a;
TCoeffs[2*k+1] = -2.0*cp*st*CosPoleAngle/a;
}
DenomCoeffs = TrinomialMultiply(FilterOrder, TCoeffs, RCoeffs );
free(TCoeffs);
free(RCoeffs);
DenomCoeffs[1] = DenomCoeffs[0];
DenomCoeffs[0] = 1.0;
for( k = 3; k <= 2*FilterOrder; ++k )
DenomCoeffs[k] = DenomCoeffs[2*k-2];
return DenomCoeffs;
}
void filter(int ord, double *a, double *b, int np, double *x, double *y)
{
int i,j;
y[0]=b[0] * x[0];
for (i=1;i<ord+1;i++)
{
y[i]=0.0;
for (j=0;j<i+1;j++)
y[i]=y[i]+b[j]*x[i-j];
for (j=0;j<i;j++)
y[i]=y[i]-a[j+1]*y[i-j-1];
}
for (i=ord+1;i<np+1;i++)
{
y[i]=0.0;
for (j=0;j<ord+1;j++)
y[i]=y[i]+b[j]*x[i-j];
for (j=0;j<ord;j++)
y[i]=y[i]-a[j+1]*y[i-j-1];
}
}
int main(int argc, char *argv[])
{
//Frequency bands is a vector of values - Lower Frequency Band and Higher Frequency Band
//First value is lower cutoff and second value is higher cutoff
double FrequencyBands[2] = {0.25,0.375};//these values are as a ratio of f/fs, where fs is sampling rate, and f is cutoff frequency
//and therefore should lie in the range [0 1]
//Filter Order
int FiltOrd = 5;
//Pixel Time Series
/*int PixelTimeSeries[N];
int outputSeries[N];
*/
//Create the variables for the numerator and denominator coefficients
double *DenC = 0;
double *NumC = 0;
//Pass Numerator Coefficients and Denominator Coefficients arrays into function, will return the same
NumC = ComputeNumCoeffs(FiltOrd);
for(int k = 0; k<11; k++)
{
printf("NumC is: %lf\n", NumC[k]);
}
//is A in matlab function and the numbers are correct
DenC = ComputeDenCoeffs(FiltOrd, FrequencyBands[0], FrequencyBands[1]);
for(int k = 0; k<11; k++)
{
printf("DenC is: %lf\n", DenC[k]);
}
double y[5];
double x[5]={1,2,3,4,5};
filter(5, DenC, NumC, 5, x, y);
return 1;
}
我的代码得到了这个结果：
B= 1,0,-5,0,10,0,-10,0,5,0,-1 A= 1.000000000000000, -4.945988709743181, 13.556489496973796, -24.700711850327743, 32.994881546824828, -33.180726698160655, 25.546126213403539, -14.802008410165968, 6.285430089797051, -1.772929809750849, 0.277753012228403
但是如果我想在 MATLAB 中测试相同频段的系数，我会得到以下结果：
>> [B, A]=butter(5, [0.25,0.375])B = 0.0002, 0, -0.0008, 0, 0.0016, 0, -0.0016, 0, 0.0008, 0, -0.0002
A = 1.0000、-4.9460、13.5565、-24.7007、32.9948、-33.1806、25.5461、-14.8020、6.2854、-1.7729、0.2778
我已经测试了这个网站：http://www.exstrom.com/journal/sigproc/ 代码，但结果和我的一样，不是matlab。有人知道为什么吗？或者我怎样才能得到与 matlab 工具箱相同的结果？