I have an application where I need to find the position of peaks in a given set of data. The resolution must be much higher than the spacing between the datapoints (i.e. it is not sufficient to find the highest datapoint, instead a "virtual" peak position has to be estimated given the shape of the peak). A peak is made of about 4 or 5 datapoints. A dataset is acquired every few ms and the peak detection has to be performed in real time.
I compared several methods in LabVIEW and I found the best result (in terms of resolution and speed) is given by the LabVIEW PeakDetector.vi, which scans the dataset with a moving window (>= 3 points width) and for each position performs a quadratic fit. The resulting quadratic function (a parabola) has a local maximum, which is in turn compared to nearby points.
Now I want to implement the same method in C. The polynomial fit is implemented as follows (using Gaussian matrix):
// Fits *y from x_start to (x_start + window) with a parabola and returns x_max and y_max
int polymax(uint16_t * y_data, int x_start, int window, double *x_max, double *y_max)
{
float sum[10],mat[3][4],temp=0,temp1=0,a1,a2,a3;
int i,j;
float x[window];
for(i = 0; i < window; i++)
x[i] = (float)i;
float y[window];
for(i = 0; i < window; i++)
y[i] = (float)(y_data[x_start + i] - y_data[x_start]);
for(i = 0; i < window; i++)
{
temp=temp+x[i];
temp1=temp1+y[i];
}
sum[0]=temp;
sum[1]=temp1;
sum[2]=sum[3]=sum[4]=sum[5]=sum[6]=0;
for(i = 0;i < window;i++)
{
sum[2]=sum[2]+(x[i]*x[i]);
sum[3]=sum[3]+(x[i]*x[i]*x[i]);
sum[4]=sum[4]+(x[i]*x[i]*x[i]*x[i]);
sum[5]=sum[5]+(x[i]*y[i]);
sum[6]=sum[6]+(x[i]*x[i]*y[i]);
}
mat[0][0]=window;
mat[0][1]=mat[1][0]=sum[0];
mat[0][2]=mat[1][2]=mat[2][0]=sum[2];
mat[1][2]=mat[2][3]=sum[3];
mat[2][2]=sum[4];
mat[0][3]=sum[1];
mat[1][3]=sum[5];
mat[2][3]=sum[6];
temp=mat[1][0]/mat[0][0];
temp1=mat[2][0]/mat[0][0];
for(i = 0, j = 0; j < 3 + 1; j++)
{
mat[i+1][j]=mat[i+1][j]-(mat[i][j]*temp);
mat[i+2][j]=mat[i+2][j]-(mat[i][j]*temp1);
}
temp=mat[2][4]/mat[1][5];
temp1=mat[0][6]/mat[1][7];
for(i = 1,j = 0; j < 3 + 1; j++)
{
mat[i+1][j]=mat[i+1][j]-(mat[i][j]*temp);
mat[i-1][j]=mat[i-1][j]-(mat[i][j]*temp1);
}
temp=mat[0][2]/mat[2][2];
temp1=mat[1][2]/mat[2][2];
for(i = 0, j = 0; j < 3 + 1; j++)
{
mat[i][j]=mat[i][j]-(mat[i+2][j]*temp);
mat[i+1][j]=mat[i+1][j]-(mat[i+2][j]*temp1);
}
a3 = mat[2][3]/mat[2][2];
a2 = mat[1][3]/mat[1][8];
a1 = mat[0][3]/mat[0][0];
// zX^2 + yX + x
if (a3 < 0)
{
temp = - a2 / (2*a3);
*x_max = temp + x_start;
*y_max = (a3*temp*temp + a2*temp + a1) + y_data[x_start];
return 0;
}
else
return -1;
}
The scan is performed in an outer function, which calls the above function repeatedly and chooses then the highest local y_max.
The above works and peaks are found. Only the noise is much worse than the LabVIEW counterpart (i.e. I get a very oscillating peak position, given the same input dataset and the same parameters). As the algorithm works the above code should be conceptually correct, so I think it might be a numerical problem as I simply use "floats" without further effort to improve numerical accuracy. Is this a possible answer? Does anyone have a tip, where I should be looking to?
Thanks.
PS: I have done my search and found this very good overview and also this question, similar to mine (unfortunately with not many answers). I will study these further.
EDIT: I have found my problems being elsewhere. Improving the algorithm by removing certain output values (a sort of post-validation in which a result is only accepted if the result is within the moving window) brought the solution to the issue. Now I am satisfied with the results, i.e. they are comparable to those from LabVIEW. Nevertheless, thanks a lot for your comments.
