Is it possible to apply this to a 2D signal (x,y position)?

[Cram3r95]

Hi, this is an amazing content. I would like to filter my 2D signal to obtain both the velocity and acceleration at a time t, in order to estimate a CTRA model. Which method would you recommend to me? I am using the Argoverse Motion Forecasting dataset.

An example of what I am trying to do is as following:

[espdev]

Hello @Cram3r95,

sgolay2 works only for surfaces on regular Cartesian grids.

You can consider csaps smoothing for your task. See the documentation for multivariate smoothing (2D case): https://csaps.readthedocs.io/en/latest/tutorial.html#multivariate-smoothing

If you need to use sgolay filtering, you can use 1D savgol_filter from scipy and parametrize by parameter t for x and y separately.

[Cram3r95]

Got it. I would like to preserve the velocity and acceleration information, not to obtain equidistant points. If I conduct a separate filtering for each coordinate, and then join then, is this correct?

[espdev]

These are not necessarily equidistant points. It all depends on the t data site grid. Could you provide a sample of your data?

[espdev]

If I conduct a separate filtering for each coordinate, and then join then, is this correct?

Yes, you can use 1-D savgol filter for each coordinate.

[Cram3r95]

Here is my data:

(Pdb) x array([-0.31418146, -0.32084206, -0.43827689, -0.43749745, -0.40649263, -0.24785958, -0.21867927, -0.21718585, -0.20703234, -0.1752735 , -0.04367442, -0.30753632, -0.14336099, -0.01123529, -0.11978056, -0.10602833, -0.08356585, 0.10507356, 0.12562331, 0. ])

(Pdb) y array([14.48325329, 14.00158717, 13.2696119 , 12.78406973, 11.74274226, 11.55458092, 11.06965875, 9.44117946, 8.8930401 , 8.3431558 , 7.56183614, 6.96244131, 6.65170151, 5.79408915, 4.25335575, 3.63905006, 3.0565611 , 2.19078549, 1.61893788, 0. ])

I could also implement a Bayesian filter solution, but first I want to explore a solution as efficient as possible -> I want to filter my signal in order to best estimate my velocity and acceleration in the last observation frame (that is, where X and Y are equal to 0, or the center of the image, as illustrated above).

[espdev]

For example:

import numpy as np
from scipy.signal import savgol_filter
import matplotlib.pyplot as plt

x = ...  # X data vector
y = ...  # Y data vector

xy = np.vstack((x, y))
xy_f = savgol_filter(xy, window_length=5, polyorder=3, axis=1)

x_f = xy_f[0, :]
y_f = xy_f[1, :]

plt.plot(x, y, ':o', x_f, y_f, '.-')
plt.show()

[Cram3r95]

Yes, but doing this you are only smoothing one of the coordinates, right? Not both, since you assume one of them as the time-axis.

[espdev]

No, in this case we smooth both coordinates separately by axis=1.

For your sample we are getting:

[Cram3r95]

This is another tricky example. The observation seems to present a constant velocity, but the observation is quite noisy.

Input data:

(Pdb) x array([-2.90470876, -2.45423747, -2.47054609, -2.15517243, -1.11500226, -1.01270935, -1.00007073, -0.76081561, -0.25430657, -0.47272535, -0.06071316, 0.07222567, -0.31588703, 0.13088221, -0.00530504, -0.0374395 , 0.20050868, 0.02133935, -0.16461405, 0. ]) (Pdb) y array([17.91807666, 17.18600004, 16.85460763, 16.12626906, 13.91504899, 13.24139842, 13.42173635, 12.01677173, 10.80650343, 10.13585118, 8.83081164, 8.03904163, 6.62514378, 5.91366662, 4.5361813 , 3.8069856 , 2.85839244, 1.56333415, 0.80128267, 0. ])

[espdev]

Could you try to use the code with savgol_filter and plot filtered data on your image?

[Cram3r95]

Ah, got it, you stack both variables and then smooth along the time-axis, as a positional encoding, makes sense.

[Cram3r95]

Yes, give me a moment. I assume that with this solution, we are going to get smoothed vectors with the same length (20 observations in this case), in order to not lose the positional information to compute the velocity and acceleration.

[espdev]

For second data sample I get:

But we need to use right scaling of axes so that it makes sense.

[Cram3r95]

Nice, I am integrating this code with the image. Please, try this:

(Pdb) x array([ 0.51638492, 0.24687597, 0.24687597, 0.61462248, 0.1597019 , 0.67346952, 0.55337124, 0.45054865, 0.45638668, 0.34816387, 0.31673042, 0.26071881, 0.26717796, 0.26230824, 0.1894866 , 0.23076022, 0.19717661, -0.06022335, 0.06737089, 0. ]) (Pdb) y array([-22.86125362, -22.1318296 , -22.1318296 , -19.69528495, -19.07734936, -16.82239551, -15.99187992, -15.15671999, -14.21899416, -12.36447778, -11.40892178, -10.2670728 , -9.35220874, -7.18156316, -6.22068484, -5.270861 , -4.20995223, -2.0349487 , -1.16746912, 0. ])

I am wondering if my Deep Learning model should have this noise in the input to prevent overfitting or a smoothed version of the input.

[espdev]

Is smoothing x and y by savgol_filter working right for you? It seems the approach preserves the velocity and acceleration information. And you don't have to set any grids.

I am wondering if my Deep Learning model should have this noise in the input to prevent overfitting or a smoothed version of the input.

I don't know, sorry. :) I think you need to experiment and try here.