How to remove the bump from the signal using filters?

Question

I want to have only the slope of the signal. How can I do? I have tried to apply some filters but so far I have not been able to hit the nail on the head. I use Matlab.

Answer 1

A bump like this one is likely to be wide-band, especially with the sharp onset. Plus, the line may be hard to deal with in the Fourier domain. Hence, the combination is complicated to remove with a classical linear filter. The problem is very akin to baseline, background or trend removal, answered elsewhere here.

Several options are possible, for instance:

• use a non-linear filter, based on a median, or a minimum/maximum statistics,
• use morphological operators: a rolling ball, lot of straight segments, etc.
• use a knowledge on the data model, like a linear equation: $y=ax+b$, or the fact that the bump is "above",
• combine the above in a variational formulation, using appropriate data fidelity and penalty.

In your example, I suspect that a classical linear fit with robust distance (like a least-absolute distortion) could do the job. I will call all the above filters, in the wide sense that you will remplace a value with respect to some sort of combination of the others.

You can also call the following robust regression, LAD fitting. An example at work:

% Standard and Robust fit of a degree 1 polynomial w/ a bump
nSample = 1000;
% Create a similar composite signal
time = linspace(0,5,nSample)';
polyCoef = [0.2  0];
dataLine = polyval(polyCoef,time);
dataParabola = -8*(time-2).*(time-3);
dataParabola(dataParabola < 0) = 0;
data = dataLine+dataParabola;

% Use Matlab curve fitting toolbox
optsRobust = fitoptions('Method','LinearLeastSquares','Robust','LAR');
[fitObject,gof] = fit(time,data,'poly1',optsRobust);
h1=plot(fitObject,time,data);
grid on