I have 3D accelerometer signals which are obviously noisy. I am trying to use Kalman filter to remove bias and drift from these signals. As an extra input I have rotation matrix.
I can also use a moving average filter and it will be fine but how can I use Kalaman filter to smooth these signals using rotation matrix?
Best Regards
Chintan
Please have a look the documentation how to use scalar Kamlan Filter:
http://www.swarthmore.edu/NatSci/echeeve1/Ref/Kalman/ScalarKalman.html
If you can run the code in Matlab and see how the scalar Kalman Filter works:
http://www.swarthmore.edu/NatSci/echeeve1/Ref/Kalman/scalarkalman.m
Once you are familiar with the basic and use of scalar kalman filter then you can follow the practical implementation guide:
http://robotsforroboticists.com/kalman-filtering/
Rotation is included and the linear model is replaced by a general one. Next two filters are combined and finally the federated Kalman Filter is included:
http://campar.in.tum.de/Chair/KalmanFilter
Related
I found a Matlab implementation of the LKT algorithm here and it is based on the brightness constancy equation.
The algorithm calculates the Image gradients in x and y direction by convolving the image with appropriate 2x2 horizontal and vertical edge gradient operators.
The brightness constancy equation in the classic literature has on its right hand side the difference between two successive frames.
However, in the implementation referred to by the aforementioned link, the right hand side is the difference of convolution.
It_m = conv2(im1,[1,1;1,1]) + conv2(im2,[-1,-1;-1,-1]);
Why couldn't It_m be simply calculated as:
it_m = im1 - im2;
As you mentioned, in theory only pixel by pixel difference is stated for optical flow computation.
However, in practice, all natural (not synthetic) images contain some degree of noise. On the other hand, differentiating is some kind of high pass filter and would stress (high pass) noise ratio to the signal.
Therefore, to avoid artifact caused by noise, usually an image smoothing (or low pass filtering) is carried out prior to any image differentiating (we have such process in edge detection too). The code does exactly this, i.e. apply and moving average filter on the image to reduce noise effect.
It_m = conv2(im1,[1,1;1,1]) + conv2(im2,[-1,-1;-1,-1]);
(Comments converted to an answer.)
In theory, there is nothing wrong with taking a pixel-wise difference:
Im_t = im1-im2;
to compute the time derivative. Using a spatial smoother when computing the time derivative mitigates the effect of noise.
Moreover, looking at the way that code computes spatial (x and y) derivatives:
Ix_m = conv2(im1,[-1 1; -1 1], 'valid');
computing the time derivate with a similar kernel and the valid option ensures the matrices It_x, It_y and Im_t have compatible sizes.
The temporal partial derivative(along t), is connected to the spatial partial derivatives (along x and y).
Think of the video sequence you are analyzing as a volume, spatio-temporal volume. At any given point (x,y,t), if you want to estimate partial derivatives, i.e. estimate the 3D gradient at that point, then you will benefit from having 3 filters that have the same kernel support.
For more theory on why this should be so, look up the topic steerable filters, or better yet look up the fundamental concept of what partial derivative is supposed to be, and how it connects to directional derivatives.
Often, the 2D gradient is estimated first, and then people tend to think of the temporal derivative secondly as independent of the x and y component. This can, and very often do, lead to numerical errors in the final optical flow calculations. The common way to deal with those errors is to do a forward and backward flow estimation, and combine the results in the end.
One way to think of the gradient that you are estimating is that it has a support region that is 3D. The smallest size of such a region should be 2x2x2.
if you do 2D gradients in the first and second image both using only 2x2 filters, then the corresponding FIR filter for the 3D volume is collected by averaging the results of the two filters.
The fact that you should have the same filter support region in 2D is clear to most: thats why the Sobel and Scharr operators look the way they do.
You can see the sort of results you get from having sanely designed differential operators for optical flow in this Matlab toolbox that I made, in part to show this particular point.
I am currently looking into some image processing project and just wondering how to obtain the low, middle and high frequency components of an image? For example, as this picture showed (I got it from googling without detailed description how to obtained this picture, but presumably using some filtering).
Also, I came across this post of using discrete cosine transform (DCT), and it can help us to get the low and high frequency components of an image. Just wondering how to use DCT to get the middle frequency component?
Link of DCT
I also have very basic knowledge about filtering. I think there are also Gaussian high/low pass filters available to use. And also wavelet based filtering. Just wondering what are the differences between Gaussian, Wavelet and DCT based filtering? Which one should I use?
Typical steps would be:
use a Fourier Transform to bring the image into frequency domain
apply filtering by zero-ing out areas of the fft image
reverse the fourier transform to bring image back to spatial domain
This is a really good example of high/low/mid pass filters in frequency domain: http://paulbourke.net/miscellaneous/imagefilter/
You will want to use MatLab's built in fft our fast fourier transform function. Fourier transforms are an extremely powerful method to filter frequencies. http://www.mathworks.com/help/matlab/ref/fft.html has some great examples on how to use the fft. Once you find the frequencies that make up the image you can take out the undesired frequencies to fit and then reverse fourier transform to obtain the new image.
I am new to matlab and signal processing methods, but i am trying to use its filter properties over a set of data I have. I have a collection of amplitude values obtained at different timestamps. When this is plotted, I get a waveform with several peaks that I can identify. I then perform calculations to derive the time between each consecutive peak and I want to eliminate the rates that are around the range of 48-52peaks per second.
What would be the correct way to go about processing this data step by step? Would a bandstop or notch filter be better if I want to eliminate those frequencies and not attenuate it simply? I am completely lost in the parameters required to feed into the filters for this. Please help...
periodogram is OK, but I would suggest using pwelch instead. It makes a more reasonable PSD estimate and the default parameters are well thought out (Hann windows, 50% overlap of segments, etc.)
If what you want is to remove signals in a wide band (e.g. 48-52 Hz) equally, rather than a single and unchanging frequency, than a bandstop filter is ideal. For example:
fs = 2048;
y = rand(fs*8, 1);
[b,a] = ellip(4, 2, 40, [46 54]/(fs/2));
yy = filter(b,a,y);
This will use a 4th order elliptic bandstop filter to filter the random data variable 'y'. filtfilt.m is also a nice function; it applies the filter forwards and backwards so you get twice the filter action and none of the phase lag or dispersion.
I am currently doing something similar to what you are doing.
I am processing a lot of signals from the Inertial Measurement Unit and motor drives. They all are asynchronously obtained, i.e. they all have very different timestamp and also very different acquisition frequency.
First thing I did was to interpolate all signals data in order to have all signals with same timestamp. You can use the matlab function interp to do this.
After this, you will have all signals with same sample frequency and also timestamp, which will be good in further analysis.
Ok, another thing you can do to analyse the frequency of the peaks is to perform the fourier transform. For beginners i advice the use of periodogram function and not the fft function.
Imagine you signal is x and your sample frequency (after interpolation) is Fs.
You can now use the function periodogram available in matlab like this:
[P,f] = periodogram(x,[],[length(t)],Fs);
This will give the power vs frequency of your signal. After that you will be able to plot and take a look at the frequencies of your signal. In other words, you be able to see the frequencies of the signals that make your acquired signal.
Plot the data this way:
plot(f,P); or semilogy(f,P);
The second is the same thing as the first, but with a logarithmic scale.
After this analysis you can use the Filter Desing and Analysis Tool to design you filter. Just type fdatool in matlab and it will open the design window. Choose the filter type, the cut and pass frequencies and click in design. This tool is very intuitive.
After designing you can export the filter to workspace.
Finally you can use the filter you designed in your signal to see if its what you wanted.
Use the functions filter os filtfilt for this.
Search in the web of the matlab help for the functions I wrote to get more details.
There are a lot of examples availables too.
I hope I could help you.
Good luck.
For a homework assignment I have to design a simple bandpass filter in Matlab that filters out everything between 250Hz and 1000 Hz. What I did so far:
- using the 'enframe' function to create half overlapping windows with 512 samples each. On the windows I apply the hann window function.
- On each window I apply an fft. After this I reconstruct the original signal with the function ifft, that all goes well.
But the problem is how I have to interpret the result of the fft function and how to filter out a frequency band.
Unless I'm mistaken, it sounds like you're taking the wrong approach to this.
If your assignment is to manipulate a signal specifically by manipulating its FFT then ignore me. Otherwise.. read on.
The FFT is normally used to analyse a signal in the frequency domain. If you start fiddling with the complex coefficients that an FFT returns then you're getting into a complicated mathematical situation. This is particularly the case since your cut-off frequencies aren't going to lie nicely on FFT bin frequencies. Also, remember that the FFT is not a perfect transform of the signal you're analysing. It will always introduce artefacts of its own due to scalloping error, and convolution with your hann window.
So.. let's leave the FFT for analysis, and build a filter.
If you're doing band-pass design in your class I'm going to assume you understand what they do. There's a number of functions in Matlab to generate the coefficients for different types of filter i.e. butter, kaiser cheby1. Look up their help pages in Matlab for loads more info. The values you plug in to these functions will be dependent on your filter specification, i.e. you want "X"dB rolloff and "Y"dB passband ripple. You'll need some idea of the how these filters work, and knowledge of their transfer functions to understand how their filter order relates to your specification.
Once you have your coefficients, it's just a case of running them through the filter function (again.. check the help page if you're not sure how this works).
The mighty JOS has a great walkthrough of bandpass filter design here.
One other little niggle.. in your question you mentioned that you want your filter to "filter out" everything between 250Hz and 1000Hz. This is a bit ambiguous. If you're designing a bandpass filter you would want to "pass" everything between 250Hz and 1000Hz. If you do in fact want to "filter out" everything in this range you want a band-stop filter instead.
It all depends on the sampling rate you use.
If you sample right according to the Nyquist-Shannon sampling theorem then you can try and interpret the samples of your fft using the definition of the DFT.
For understanding which frequencies correspond with which samples in the dft results, I think it's best to look at the inverse transformation. You multiply coefficient k with
exp(i*2*pi*k/N*n)
which can be interpreted to be a cosine with Euler's Formula. So each coefficient gets multiplied by a sine of a certain frequency.
Good luck ;)
I am new to using matlab and signal processing in general and can't figure out how to implement a low pass filter. I have these digital signal that's a vector and would like to run it through a low pass fiter of 40hz or if at all possible a bandpass filter .53-40hz (12db/oct). I have access to the signal processing toolbox but can't figure out how. Any help is appreciated.
There is more then one option to applying a digital filter in Matlab. The doc is very good, so it is a good starting point to look at it and follow the examples:
Applying 1D digital filter
Using conv for filtering using a FIR filter
FIR filter design
IIR filter design