Question

I am implementing an image analysis algorithm using openCV and c++, but I found out openCV doesnt have any function for Butterworth Bandpass filter officially.
in my project I have to pass a time series of pixels into the Butterworth 5 order filter and the function will return the filtered time series pixels. Butterworth(pixelseries,order, frequency), if you have any idea to help me of how to start please let me know. Thank you

EDIT :
after getting help, finally I come up with the following code. which can calculate the Numerator Coefficients and Denominator Coefficients, but the problem is that some of the numbers is not as same as matlab results. here is my code:

#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;
}

I get this resutls for my code :

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

but if I want to test the coefficinets in same frequency band in MATLAB, I get the following results:

>> [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

I have test this website :http://www.exstrom.com/journal/sigproc/ code, but the result is equal as mine, not matlab. anybody knows why? or how can I get the same result as matlab toolbox?

Answer 1

I finally found it.
I just need to implement the following code from matlab source code to c++ . “the_mandrill” were right, I need to add the normalizing constant into the coefficient:

kern = exp(-j*w*(0:length(b)-1));
b = real(b*(kern*den(:))/(kern*b(:)));

EDIT:
and here is the final edition, which the whole code will return numbers exactly equal to MATLAB :

double *ComputeNumCoeffs(int FilterOrder,double Lcutoff, double Ucutoff, double *DenC)
{
    double *TCoeffs;
    double *NumCoeffs;
    std::complex<double> *NormalizedKernel;
    double Numbers[11]={0,1,2,3,4,5,6,7,8,9,10};
    int i;

    NumCoeffs = (double *)calloc( 2*FilterOrder+1, sizeof(double) );
    if( NumCoeffs == NULL ) return( NULL );

    NormalizedKernel = (std::complex<double> *)calloc( 2*FilterOrder+1, sizeof(std::complex<double>) );
    if( NormalizedKernel == 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];
    double cp[2];
    double Bw, Wn;
    cp[0] = 2*2.0*tan(PI * Lcutoff/ 2.0);
    cp[1] = 2*2.0*tan(PI * Ucutoff / 2.0);

    Bw = cp[1] - cp[0];
    //center frequency
    Wn = sqrt(cp[0]*cp[1]);
    Wn = 2*atan2(Wn,4);
    double kern;
    const std::complex<double> result = std::complex<double>(-1,0);

    for(int k = 0; k<11; k++)
    {
        NormalizedKernel[k] = std::exp(-sqrt(result)*Wn*Numbers[k]);
    }
    double b=0;
    double den=0;
    for(int d = 0; d<11; d++)
    {
        b+=real(NormalizedKernel[d]*NumCoeffs[d]);
        den+=real(NormalizedKernel[d]*DenC[d]);
    }
    for(int c = 0; c<11; c++)
    {
        NumCoeffs[c]=(NumCoeffs[c]*den)/b;
    }

    free(TCoeffs);
    return NumCoeffs;
}