vmd_cvm_python

Algorithm for signal denoising according to the VMD_CVM method, which uses the Variational Mode Decomposition (VMD) to identify the relevant modes and the Cramer Von Mises statistics to filter out the noise.

References

• VMD_CVM: Khuram Naveed, Muhammad Tahir Akhtar, Muhammad Faisal Siddiqui, Naveed ur Rehman, "A statistical approach to signal denoising based on data-driven multiscale representation", Digital Signal Processing, Vol. 108, pp. 102896, 2021.
• VMD: K. Dragomiretskiy and D. Zosso, "Variational Mode Decomposition," in IEEE Transactions on Signal Processing, vol. 62, no. 3, pp. 531-544, Feb.1, 2014, doi: 10.1109/TSP.2013.2288675.

Setup

$ git clone https://github.com/albertoCCz/vmd_cvm_python.git
$ cd .\vmd_cvm_python\
$ python -m venv env
$ .\env\Scripts\activate
$ python -m pip install -r requirements.txt

(Linux users must change \ for /, and .\env\Scripts\activate for source env/bin/activate).

Compile for your specific platform and Python version with:

python .\setup.py build_ext --build-lib .\vmd_cvm_cython\

(Already tested with Python versions 3.8.2 and 3.9.6)

Run examples

• Example 1:

  $ python hello_world-vmd_cvm.py -cython

  You should see the hello_world.png image. You can also run it with the -python flag, but it will be slower.
• Example 2:

  $ python VMD_CVM_test.py
  
  
  Input index of signal to denoise {0, 1, 2, ...}:
  50

  You should see something similar to the Signal_denoising.png image.

Usage

Minimal usage example

First we import the function Prop_VMD_CVM from the implementation module we want to use (the Python or Cython version):

import numpy as np
from vmd_cvm_python.Prop_VMD_CVM_python import Prop_VMD_CVM

Then, we set the necessary parameters to run the function

# Input parameters
WIN_LEN = 256       # window length
NIMF    = 10        # number of modes
NP      = 500       # number of consecutive signal points

# Select Pfa using a decaying function e^(-k+1)
opt_Pfa = np.exp(-np.arange(0, NIMF-1))          # probability of false activation

And finally, we use them to remove the noise from a noisy signal:

# Denoise using VMD_CVM method
_, _, denoised = Prop_VMD_CVM(noisy, WIN_LEN, NIMF, opt_Pfa, NP)

Complete usage example

First we make the necessary imports:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

from vmd_cvm_python.Prop_VMD_CVM_python import Prop_VMD_CVM

In this example we are going to use the Python version so, in the import, we set the package and the function with the "_python" suffix. The Prop_VMD_CVM function executes all the algorithm at once, so this is all we need to import from this library in order to use the denoising method.

Then, we fix the parameters for the function:

# Input parameters
WIN_LEN = 256       # window length
NIMF    = 10        # number of modes
NP      = 500       # number of consecutive signal points

# Select Pfa using a decaying function e^(-k+1)
opt_Pfa = np.exp(-np.arange(0, NIMF-1))        # probability of false activation

It's very likely that you'll have to play around with these parameters when denoising your own signals. I recommend reading the References for an in deepth explanation of each of them.

Then we generete a signal, a sine in this example, and we add some noise to it:

# Generate some signal
x      = np.linspace(-2*np.pi, 2*np.pi, 1000)
signal = np.sin(x)

# Add noise
seed  = 123456
rng   = np.random.default_rng(seed)
noise = rng.normal(loc=0.0, scale=1.0, size=x.shape[0])

noisy = signal + 1 * noise

Now we are ready to apply the VMD_CVM algorithm and try to denoise the signal. We do so like this:

# Denoise using VMD_CVM method
_, _, denoised = Prop_VMD_CVM(noisy, WIN_LEN, NIMF, opt_Pfa, NP)

where we just passed the signal to denoise and the previously chosen parameters. The third returned element is the actual "denoised" signal.

We can apply some simple post-processing, like a rolling window that takes the mean, to, sometimes, improve the result.

# Post-processing
denoised = pd.Series(denoised).rolling(window=100, min_periods=1, center=True).mean()

Let's plot all the three signals together to see the result.

# Plot noisy, original and denoised signals
plt.subplots(1, figsize=(10,8))

plt.plot(x, noisy,    linewidth=0.8, label='noisy', alpha=0.5)
plt.plot(x, signal,   linewidth=0.8, label='original')
plt.plot(x, denoised, linewidth=0.8, label='denoised')

plt.title('Signal denoising by VMD_CVM method')
plt.legend()
plt.grid()
plt.tight_layout()
plt.savefig('plots/hello_world.png')
plt.show()

Not bad!

The full script is here:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

from vmd_cvm_python.Prop_VMD_CVM_python import Prop_VMD_CVM

# Input parameters
WIN_LEN = 256       # window length
NIMF    = 10        # number of modes
NP      = 500       # number of consecutive signal points

# Select Pfa using a decaying function e^(-k+1)
opt_Pfa = np.exp(-np.arange(0, NIMF-1))          # probability of false activation

# Generate some signal
x      = np.linspace(-2*np.pi, 2*np.pi, 1000)
signal = np.sin(x)

# Add noise
seed  = 123456
rng   = np.random.default_rng(seed)
noise = rng.normal(loc=0.0, scale=1.0, size=x.shape[0])

noisy = signal + 1 * noise

# Denoise using VMD_CVM method
_, _, denoised = Prop_VMD_CVM(noisy, WIN_LEN, NIMF, opt_Pfa, NP)

# Post-processing
denoised = pd.Series(denoised).rolling(window=100, min_periods=1, center=True).mean()

# Plot noisy, original and denoised signals
plt.subplots(1, figsize=(10,8))

plt.plot(x, noisy,    linewidth=0.8, label='noisy', alpha=0.5)
plt.plot(x, signal,   linewidth=0.8, label='original')
plt.plot(x, denoised, linewidth=0.8, label='denoised')

plt.title('Signal denoising by VMD_CVM method')
plt.legend()
plt.grid()
plt.tight_layout()
plt.savefig('plots/hello_world.png')
plt.show()