Inverse Wavelet Transform [/xpost signalprocessing]

Question

Main Problem: How can the scipy.signal.cwt() function be inversed.

I have seen where Matlab has an inverse continuous wavelet transform function which will return the original form of the data by inputting the wavelet transform, although you can filter out the slices you don't want.

MATALAB inverse cwt funciton

Since scipy doesn't appear to have the same function, I have been trying to figure out how to get the data back in the same form, while removing the noise and background. How do I do this? I tried squaring it to remove negative values, but this gives me values way to large and not quite right.

Here is what I have been trying:

# Compute the wavelet transform
widths = range(1,11) 
cwtmatr = signal.cwt(xy['y'], signal.ricker, widths)

# Maybe we multiple by the original data? and square?
WT_to_original_data = (xy['y'] * cwtmatr)**2

And here is a fully compilable short script to show you the type of data I am trying to get and what I have etc.:

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

# Make some random data with peaks and noise
def make_peaks(x):
    bkg_peaks = np.array(np.zeros(len(x)))
    desired_peaks = np.array(np.zeros(len(x)))
    # Make peaks which contain the data desired
    # (Mid range/frequency peaks)
    for i in range(0,10):
        center = x[-1] * np.random.random() - x[0]
        amp = 60 * np.random.random() + 10
        width = 10 * np.random.random() + 5
        desired_peaks += amp * np.e**(-(x-center)**2/(2*width**2))
    # Also make background peaks (not desired)
    for i in range(0,3):
        center = x[-1] * np.random.random() - x[0]
        amp = 40 * np.random.random() + 10
        width = 100 * np.random.random() + 100
        bkg_peaks += amp * np.e**(-(x-center)**2/(2*width**2))
    return bkg_peaks, desired_peaks

x = np.array(range(0, 1000))
bkg_peaks, desired_peaks = make_peaks(x)
y_noise = np.random.normal(loc=30, scale=10, size=len(x))
y = bkg_peaks + desired_peaks + y_noise
xy = np.array( zip(x,y), dtype=[('x',float), ('y',float)])

# Compute the wavelet transform
# I can't figure out what the width is or does?
widths = range(1,11) 
# Ricker is 2nd derivative of Gaussian
# (*close* to what *most* of the features are in my data)
# (They're actually Lorentzians and Breit-Wigner-Fano lines)
cwtmatr = signal.cwt(xy['y'], signal.ricker, widths)

# Maybe we multiple by the original data? and square?
WT = (xy['y'] * cwtmatr)**2


# plot the data and results
fig = plt.figure()
ax_raw_data = fig.add_subplot(4,3,1)
ax = {}
for i in range(0, 11):
    ax[i] = fig.add_subplot(4,3, i+2)

ax_desired_transformed_data = fig.add_subplot(4,3,12)

ax_raw_data.plot(xy['x'], xy['y'], 'g-')  
for i in range(0,10):
    ax[i].plot(xy['x'], WT[i])

ax_desired_transformed_data.plot(xy['x'], desired_peaks, 'k-')

fig.tight_layout()
plt.show()

This script will output this image:

Where the first plot is the raw data, the middle plots are the wavelet transforms and the last plot is what I want to get out as the processed (background and noise removed) data.

Does anyone have any suggestions? Thank you so much for the help.

Katerinekates

Thanks @MrE I have looked at the documentation, but don't really provide much detail on regarding what's happening. I was hoping there would be a scipy.signal.icwt function that could get the inverse and hide all of the math and technicalities, but it appears you have to perform that yourself. So, the problem is I don't understand the math of how to do the inverse of the cwt() function. They don't really describe enough in the documentation of how to do that. – Katerinekates

Sorry I misread your question, I thought you hadn't seen that function – Rankins

Answer 1

I ended up finding a package which provides an inverse wavelet transform function called mlpy. The function is mlpy.wavelet.uwt. This is the compilable script I ended up with which may interest people if they are trying to do noise or background removal:

import numpy as np
from scipy import signal
import matplotlib.pyplot as plt
import mlpy.wavelet as wave

# Make some random data with peaks and noise
############################################################
def gen_data():
    def make_peaks(x):
        bkg_peaks = np.array(np.zeros(len(x)))
        desired_peaks = np.array(np.zeros(len(x)))
        # Make peaks which contain the data desired
        # (Mid range/frequency peaks)
        for i in range(0,10):
            center = x[-1] * np.random.random() - x[0]
            amp = 100 * np.random.random() + 10
            width = 10 * np.random.random() + 5
            desired_peaks += amp * np.e**(-(x-center)**2/(2*width**2))
        # Also make background peaks (not desired)
        for i in range(0,3):
            center = x[-1] * np.random.random() - x[0]
            amp = 80 * np.random.random() + 10
            width = 100 * np.random.random() + 100
            bkg_peaks += amp * np.e**(-(x-center)**2/(2*width**2))
        return bkg_peaks, desired_peaks

    # make x axis
    x = np.array(range(0, 1000))
    bkg_peaks, desired_peaks = make_peaks(x)
    avg_noise_level = 30
    std_dev_noise = 10
    size = len(x)
    scattering_noise_amp = 100
    scat_center = 100
    scat_width = 15
    scat_std_dev_noise = 100
    y_scattering_noise = np.random.normal(scattering_noise_amp, scat_std_dev_noise, size) * np.e**(-(x-scat_center)**2/(2*scat_width**2))
    y_noise = np.random.normal(avg_noise_level, std_dev_noise, size) + y_scattering_noise
    y = bkg_peaks + desired_peaks + y_noise
    xy = np.array( zip(x,y), dtype=[('x',float), ('y',float)])
    return xy
# Random data Generated
#############################################################

xy = gen_data()

# Make 2**n amount of data
new_y, bool_y = wave.pad(xy['y'])
orig_mask = np.where(bool_y==True)

# wavelet transform parameters
levels = 8
wf = 'h'
k = 2

# Remove Noise first
# Wave transform
wt = wave.uwt(new_y, wf, k, levels)
# Matrix of the difference between each wavelet level and the original data
diff_array = np.array([(wave.iuwt(wt[i:i+1], wf, k)-new_y) for i in range(len(wt))])
# Index of the level which is most similar to original data (to obtain smoothed data)
indx = np.argmin(np.sum(diff_array**2, axis=1))
# Use the wavelet levels around this region
noise_wt = wt[indx:indx+1]
# smoothed data in 2^n length
new_y = wave.iuwt(noise_wt, wf, k)

# Background Removal
error = 10000
errdiff = 100
i = -1
iter_y_dict = {0:np.copy(new_y)}
bkg_approx_dict = {0:np.array([])}
while abs(errdiff)>=1*10**-24:
    i += 1
    # Wave transform
    wt = wave.uwt(iter_y_dict[i], wf, k, levels)

    # Assume last slice is lowest frequency (background approximation)
    bkg_wt = wt[-3:-1]
    bkg_approx_dict[i] = wave.iuwt(bkg_wt, wf, k)

    # Get the error
    errdiff = error - sum(iter_y_dict[i] - bkg_approx_dict[i])**2
    error = sum(iter_y_dict[i] - bkg_approx_dict[i])**2

    # Make every peak higher than bkg_wt
    diff = (new_y - bkg_approx_dict[i])
    peak_idxs_to_remove = np.where(diff>0.)[0]
    iter_y_dict[i+1] = np.copy(new_y)
    iter_y_dict[i+1][peak_idxs_to_remove] = np.copy(bkg_approx_dict[i])[peak_idxs_to_remove]

# new data without noise and background
new_y = new_y[orig_mask]
bkg_approx = bkg_approx_dict[len(bkg_approx_dict.keys())-1][orig_mask]
new_data = diff[orig_mask] 

##############################################################
# plot the data and results
fig = plt.figure()

ax_raw_data = fig.add_subplot(121)
ax_WT = fig.add_subplot(122)

ax_raw_data.plot(xy['x'], xy['y'], 'g')
for bkg in bkg_approx_dict.values():
    ax_raw_data.plot(xy['x'], bkg[orig_mask], 'k')

ax_WT.plot(xy['x'], new_data, 'y')


fig.tight_layout()
plt.show()

And here is the output I am getting now: As you can see, there is still a problem with the background removal (it shifts to the right after each iteration), but it is a different question which I will address here.

Katerinekates

You can correctly inverse a part of wavelet coefficients by first aligning them with one of the two align functions (and you'll have to stick to corresponding wavelet). Then you can restore each scale separately--I couldn't make it inverse a range at once. – Shrubby

I'm now replacing mlpy with pywt, so far looks good. It doesn't include inverse transform, though, but I found it here: groups.google.com/d/msg/pywavelets/tLh9ObKJaoc/hksGDLHwGpMJ . – Shrubby