I have two vectors of matching lengths. They are readings from two different sensors (one is from a smartphone and the other is from a wiimote) of the same hand movement. I am trying to find the time offset between them to synchronise the readings for further processing. The readings I get are of the format (Time(ms) Value) for accelerations in the X,Y and Z direction.
For the synchronization, I plotted the cross-correlation function xcorr2() between the two sets. I am getting the same graph (a weird triangle peak and a straight line at the bottom) for Accelerations along the x, y and z directions (which I guess is good) but I don't know how to interpret it. What do the axes in the graph represent?
Can anyone explain to me what xcorr2() means in a qualitative sense. From the correlation function, how do I determine the offset (i.e. how many seconds is sensor1 behind sensor2)?
I concur with the comment made above by Predictor. To align the time series against each-other, I would pick xcorr() without the 2. Consider correlating only the acceleration magnitudes. For example:
a_mag_wii = sqrt(a_x_wii.^2 + a_y_wii.^2 + a_z_wii.^2);
a_mag_phone = sqrt(a_x_phone.^2 + a_y_phone.^2 + a_z_phone.^2);
res = xcorr(a_mag_wiimote, a_mag_smartphone);
Related
I have a huge set of data of a timelapse of 2D laser scans of waves running up and down stairs (see fig.1fig.2fig.3).
There is a lot of noise in the scans, since the water splashes a lot.
Now I want to smoothen the scans.
I have 2 questions:
How do I apply a moving median filter (as recommended by another study dealing with a similar problem)? I can only find instructions for single e.g. (x,y) or (t,y) plots but not for x and y values that vary over time. Maybe an average filter would do it as well, but I do not have a clue on that either.
The scanner is at a fixed point (222m) so all the data spikes point towards that point at the ceiling. Is it possible or necessary to include this into the smoothing process?
This is the part of the code (I hope it's enough to get it):
% Plot data as real time profile
x1=data.x;y1=data.y;
t=data.t;
% add moving median filter here?
h1=plot(x1(1,:),y1(1,:));
axis([210 235 3 9])
ht=title('Scanner data');
for i=1:1:length(t);
set(h1,'XData',x1(i,:),'YData',y1(i,:));set(ht,'String',sprintf('t = %5.2f
s',data.t(i)));pause(.01);end
The data.x values are stored in a (mxn) matrix in which the change in time is arranged vertically and the x values i.e. "laser points" of the scanner are horizontally arranged. The data.y is stored in the same way. The data.t values are stored in a (mx1) matrix.
I hope I explained everything clearly and that somebody can help me. I am already pretty desperate about it... If there is anything missing or confusing, please let me know.
If you're trying to apply a median filter in the x-y plane, then consider using medfilt2 from the Image Processing Toolbox. Note that this function only accepts 2-D inputs, so you'll have to loop over the third dimension.
Also note that medfilt2 assumes that the x and y data are uniformly spaced, so if your x and y data don't fall onto a uniformly spaced grid you may have to manually loop over indices, extract the corresponding patches, and compute the median.
If you can/want to apply an averaging filter instead of a median filter, and if you have uniformly spaced data, then you can use convn to compute a k x k moving average by doing:
y = convn(x, ones(k,k)/(k*k), 'same');
Note that you'll get some bias on the boundaries because you're technically trying to compute an average of k^2 pixels when you have less than that number of values available.
Alternatively, you can use nested calls to movmean since the averaging operation is separable:
y = movmean(movmean(x, k, 2), k, 1);
If your grid is separable, but not uniform, you can still use movmean, just use the SamplePoints name-value pair:
y = movmean(movmean(x, k, 2, 'SamplePoints', yv), k, 1, 'SamplePoints', xv);
You can also control the endpoint handling in movmean with the Endpoints name-value pair.
I have two sensors seperated by some distance which receive a signal from a source. The signal in its pure form is a sine wave at a frequency of 17kHz. I want to estimate the TDOA between the two sensors. I am using crosscorrelation and below is my code
x1; % signal as recieved by sensor1
x2; % signal as recieved by sensor2
len = length(x1);
nfft = 2^nextpow2(2*len-1);
X1 = fft(x1);
X2 = fft(x2);
X = X1.*conj(X2);
m = ifft(X);
r = [m(end-len+1) m(1:len)];
[a,i] = max(r);
td = i - length(r)/2;
I am filtering my signals x1 and x2 by removing all frequencies below 17kHz.
I am having two problems with the above code:
1. With the sensors and source at the same place, I am getting different values of 'td' at each time. I am not sure what is wrong. Is it because of the noise? If so can anyone please provide a solution? I have read many papers and went through other questions on stackoverflow so please answer with code along with theory instead of just stating the theory.
2. The value of 'td' is sometimes not matching with the delay as calculated using xcorr. What am i doing wrong? Below is my code for td using xcorr
[xc,lags] = xcorr(x1,x2);
[m,i] = max(xc);
td = lags(i);
One problem you might have is the fact that you only use a single frequency. At f = 17 kHz, and an estimated speed-of-sound v = 340 m/s (I assume you use ultra-sound), the wavelength is lambda = v / f = 2 cm. This means that your length measurement has an unambiguity range of 2 cm (sorry, cannot find a good link, google yourself). This means that you already need to know your distance to better than 2 cm, before you can use the result of your measurement to refine the distance.
Think of it in another way: when taking the cross-correlation between two perfect sines, the result should be a 'comb' of peaks with spacing equal to the wavelength. If they overlap perfectly, and you displace one signal by one wavelength, they still overlap perfectly. This means that you first have to know which of these peaks is the right one, otherwise a different peak can be the highest every time purely by random noise. Did you make a plot of the calculated cross-correlation before trying to blindly find the maximum?
This problem is the same as in interferometry, where it is easy to measure small distance variations with a resolution smaller than a wavelength by measuring phase differences, but you have no idea about the absolute distance, since you do not know the absolute phase.
The solution to this is actually easy: let your source generate more frequencies. Even using (band-limited) white-noise should work without problems when calculating cross-correlations, and it removes the ambiguity problem. You should see the white noise as a collection of sines. The cross-correlation of each of them will generate a comb, but with different spacing. When adding all those combs together, they will add up significantly only in a single point, at the delay you are looking for!
White Noise, Maximum Length Sequency or other non-periodic signals should be used as the test signal for time delay measurement using cross correleation. This is because non-periodic signals have only one cross correlation peak and there will be no ambiguity to determine the time delay. It is possible to use the burst type of periodic signals to do the job, but with degraded SNR. If you have to use a continuous periodic signal as the test signal, then you can only measure a time delay within one period of the periodic test signal. This should explain why, in your case, using lower frequency sine wave as the test signal works while using higher frequency sine wave does not. This is demonstrated in these videos: https://youtu.be/L6YJqhbsuFY, https://youtu.be/7u1nSD0RlwY .
Please can you help me understand how to calculate the Mean-Squared Displacement for a single particle moving randomly within a given period of time. I have read a lot of articles on this (including Saxton,1991,Single-Particle Tracking: The Distribution of Diffusion Coefficients), but still confused (not getting the right answer).
Let me start by showing you how I do it and please correct me if I'm wrong:
The way I'm doing it is as follows:
1.Run the program from t=0 to t=100
2.Calculate the displacement, (s(t)-s(t+tau)), at each timestep (ie. at t=1,2,3,...100) and store it in a vector
3.Square the answer to number 2
4.find the mean to the answer of 3
In essence, this is what I'm doing in Matlab
%Initialise the lattice with a square consisting of 16 nonzero lattice sites then proceed %as follows to calculate the MSD:
for t=1:tend
% Allow the particle to move randomly in the lattice. Then do the following
[row,col]=find(lattice>0);
centroid=mean([row col]);
xvec=[xvec centroid(2)];
yvec=[yvec centroid(1)];
k=length(xvec)-1; % Time
dt=1;
diffx = xvec(1:k) - xvec((1+dt):(k+dt));
diffy = yvec(1:k) - yvec((1+dt):(k+dt));
xsquare = diffx.^2;
ysquare = diffy.^2;
MSD=mean(xsquare+ysquare);
end
I'm trying to find the MSD in order to compute the diffusion co-efficient. Note that I'm modelling a collection of lattice sites (16) to represent a single particle (more biologically realistic), instead of just one. I have been brief with the comment within the for loop as it is quite long, but I'm happy to send it to you.
So far, I'm getting very small MSD values (in the range of 0.001-1), whereas I'm supposed to get values in the range of (10-50). The particle moves very large distances so surely my range of 0.001-1 cannot be right!
This is an extract from the article which I'm trying to reproduce their figure:
" We began by running some simulations in 1D for a single
cell. We allowed the cell to move for a given number of
Monte Carlo time steps (MCS), worked out the mean square
distance traveled in that time, repeated this process 500
times, and evaluate the mean squared distance for this t.
We then repeated this process ten times to get the mean of
. The reason for this choice of repetitions was to
keep the time required to run the simulations within a reasonable
level yet ensuring that the standard deviation of the
mean was relatively small (<7%)".
You can access the article here "From discrete to a continuous model of biological cell movement, 2004, by Turner et al., Physical Review E".
Any hints are greatly appreciated.
How many dimensions does the particle move along ?
I don't have Matlab right now, but here is how I'd do that over one dimension :
% pos is the vector of positions
delta = pos(2:100) - pos(1:99);
meanSquared = mean(delta .* delta);
First of all, why have a particle cover multiple lattice sites? What counts for MSD, in the end, is the displacement of the centroid, which can be represented as a point. If your particle (or cell) is large, or only takes large steps, you can always just make a wider grid. Also, if you're trying to reproduce a figure from somewhere else, you should really use the same algorithm.
For your Monte Carlo simulation, what do you do? If all you really want is get a displacement, you can generate a bunch of random movement vectors in one go (using rand or randi), and use cumsum to calculate the positions. Also, have you plotted your random walks to make sure the data is sensible?
Then, your code looks a bit funny (see comments). Why don't you just use the code provided in this answer to calculate MSD from the positions?
for t=1:tend
% Allow the particle to move randomly in the lattice. Then do the following
[row,col]=find(lattice>0); %# what do you do this for?
centroid=mean([row col]);
xvec=[xvec centroid(2)];
yvec=[yvec centroid(1)]; %# till here, I have no idea what you want to do
k=length(xvec)-1; % Time %# you should subtract dt here
dt=1; %# dt should depend on t!
diffx = xvec(1:k) - xvec((1+dt):(k+dt));
diffy = yvec(1:k) - yvec((1+dt):(k+dt));
xsquare = diffx.^2;
ysquare = diffy.^2;
MSD=mean(xsquare+ysquare);
end
I am trying to program a 3 body problem using matlab. I was given the formula for the moon's trajectory in its rotational frame in space. It's basically the ydotdot, xdotdot=GM/(x^2+y^2)^3/2 formula. What the formula is, is not that important.
THe problem I am facing is that, I am supposed to code up a program that will numerically solve the moon's trajectory equation. I'm using ODE45 to compare with since my goal is to get the same results as ODE45. My ultimate problem is that, I want to iterate through time in terms of days so tspan= [0 365]. The thins is when I convert Gravitational constant to seconds and then do tspace= [0 365] I get a completely different result then If I were to do [0 365*3600*34] representing the seconds in a year and G= 6.67e-11. It seems that my units are very weird.
I was wondering if anyone can explain why this is happening when I use ODE 45. Why can't I convert seconds to days clearly using ODE45? Is there an extra step I have to do? The only other variables in my problem is radius, distance, and the mass of the 3 bodies.
Thank you so much. I've been working on this for a very very long time. Any help would be much appreciated.
That formula for gravitational acceleration along each axis isn't correct.
Put the earth, with mass M_e, at the origin, with the moon (mass M_m) at (x,y). Then
the earth-moon distance is given by:
R_em = sqrt(x^2 + y^2)
The total earth-moon force is given by:
F_em = G*M_e*M_m/R_em^2
The total acceleration due to Earth's gravity is given by:
a_em = F_em/M_m = G*M_e/R_em^2
and is directed toward the origin. The acceleration along
each axis is then:
xdotdot = -F_em*cos(theta) = -F_em*x/R_em = -G*M_e*x/R_em^(3/2)
ydotdot = -F_em*sin(theta) = -F_em*y/R_em = -G*M_e*y/R_em^(3/2)
Note the x and y factors, which are missing from the formula you stated.
I'm not sure what you mean by "converting the gravitational constant to seconds".
The value you're using for G has units of newton-meter^2/kg^2. So it's already
expressed in the MKS (meter-kilogram-second) system, and the accelerations calculated
using this value will have units of meters/sec^2.
With a third body (say, the sun) at (x_s, y_s), you compute a new R_s representing
the moon-sun distance, and compute new acceleration vectors as above, using the
sun's mass M_s (except the acceleration is now in the direction of (x_s, y_s), rather than (0,0)). The accelerations of the moon from the gravity of the earth and the sun just add component-wise, once everything is put into a common coordinate system (here,
geocentric coordinates -- although heliocentric might be a more convenient choice, if you're simulating the sun-earth-moon system). That, plus the initial positions and velocities, should be all you need to compute the positions and velocities at the next time step.
I have a 1D accelerometer signal (one axis only). I would like to create a robust algorithm, which would be able to recognize some shapes in the signal.
At first I apply a moving average filter to the raw signal. On the attached picture the raw signal is coloured red and the averaged signal is black. As seen from the picture, some trends are visible from the averaged (black) signal - the signal contains 10 repetitions of a peak like pattern, where acceleration climbs to a maximum and then drops back down. I have marked the beginnings and endings of those patterns with a cross.
So my goal is to find the marked positions automatically. The problem making the pattern extraction difficult are:
the start of the pattern could have a different y value than the end of the pattern
the pattern could have more than one peak
I do not have any concrete time information (from start to the end of the pattern it takes A time units)
I have tried different approaches, which are pretty much home-brew, so I won't mention them - I don't want you to be biased by my way of thinking. Are there some standard or by the books approaches for doing that kind of pattern extraction? Or maybe does anyone know how to tackle the problem in a robust way?
Any idea will be appreciated.
Keep it simple!
It appears the moving average is a good enough damper device; keep it as-is, maybe only increasing or decreasing its sample count if you notice that it either leaves too much noise or removes too much signal respectively. You then work off the this averaged signal exclusively.
The pattern markers you seek seems relatively easy to detect. Expressed in English, these markers are:
Targets = the points of inflection in the averaged readings curve, when the slope goes markedly negative to positive.
You should therefore be able to detect this situation by comparison of the slope values, calculated along with the moving average, as each new reading value comes available (of course with a short delay, as of course the slope at a given point can only be calculated when the averaged reading for the next [few] point[s] is available)
To avoid false detection, however, you'd need to define a few parameters aimed at filtering the undesirable patterns. These paremeters will define more precisely the meaning of "markedly" in the above target definition.
Tentatively the formula for detecting a point of interest could be as simple as this
(-1 * S(t-1) + St ) > Min_delta_Slope
where
S is the slope (more on this) at time t-1 and t, respectively
Min_delta_Slope is a parameter defining how "sharp" a change in slope we want at a minimum.
Assuming normalized t and Y units, we can set the Min_delta_Slope parameter close to or even past 1. Intuitively a value of 1 (again in normalized units) would indicate that we target points where the curved "arrived" with a downward slope of say 50% and left the point with a upward slope of 50% (or 40% + 60% or .. 10% i.e almost flat and 90% i.e. almost vertical).
To avoid detecting points in the case when this is merely a small dip in the curve, we can take more points into consideration, with a fancier formula such as say
(Pm2 * S(t-2) + Pm1 * S(t-1) + P0 * St + Pp1 S(t+1) ) > Min_delta_Slope
where
Pm2, Pm1, P0 and Pp1 are coefficients giving relative importance to the slope at various point before and after the point of interest. (Pm2 and Pm1 typically negative values unless we use only positive parameter and use negative signs in the formula)
St +/- n is the slope a various times
and Min_delta_Slope is a parameter defining how "sharp" a change in slope we want at a minimum.
Intuitively, this 4 points formula would take into account the shape of the curve at a point two readings prior and two reading past the point of interest (in addition to considering the point right before and after it). Given the proper values for the parameters, the formula would require that the curve be steadily coming "down" for two time slices, then steadily going up for the next two time slices, hence avoiding to mark smaller dips in the curve.
An alternative way to achieve this, may be to compute the slope by using the difference in Y value between the [averaged] reading from two (or more) time slices ago and that of the current [averaged] reading. These two approaches are similar but would produce slightly different result; generally we'd have more say on the desired shape of the curve with the Pm2, Pm1, P0 and P1 parameters.
You might want to look at watershed segmentation, which does a related kind of thing (dividing landscapes into their separate catchment basins). Oddly enough, I'm actually writing a PhD thesis which uses watershed a lot at the moment (seriously :))