Wavelet Block Thresholding¶

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This numerical tour presents block thresholding methods, that makes use of the structure of wavelet coefficients of natural images to perform denoising. Theoretical properties of block thresholding were investigated in CaiSilv Cai99 HallKerkPic99

In [44]:

from __future__ import division

import numpy as np
import scipy as scp
import pylab as pyl
import matplotlib.pyplot as plt

from nt_toolbox.general import *
from nt_toolbox.signal import *

import warnings
warnings.filterwarnings('ignore')

%matplotlib inline
%load_ext autoreload
%autoreload 2

The autoreload extension is already loaded. To reload it, use:
  %reload_ext autoreload

Generating a Noisy Image¶

Here we use an additive Gaussian noise.

Size of the image of $N=n \times n$ pixels.

In [45]:

n = 256

First we load an image $f_0 \in \RR^N$.

In [46]:

f0 = rescale(load_image("nt_toolbox/data/boat.bmp", n))

Display it.

In [47]:

plt.figure(figsize = (5,5))
imageplot(f0)

Noise level.

In [48]:

sigma = .08

Generate a noisy image $f=f_0+\epsilon$ where $\epsilon \sim \Nn(0,\si^2\text{Id}_N)$.

In [49]:

from numpy import random

f = f0 + sigma*random.standard_normal((n,n))

Display it.

In [50]:

plt.figure(figsize = (5,5))
imageplot(clamp(f))

Orthogonal Wavelet Thresholding¶

We first consider the traditional wavelet thresholding method.

Parameters for the orthogonal wavelet transform.

In [51]:

Jmin = 4

Shortcuts for the foward and backward wavelet transforms.

In [52]:

from nt_toolbox.perform_wavelet_transf import *

wav  = lambda f  : perform_wavelet_transf(f, Jmin, +1)
iwav = lambda fw : perform_wavelet_transf(fw, Jmin, -1)

Display the original set of noisy coefficients.

In [53]:

plt.figure(figsize=(10,10))
plot_wavelet(wav(f), Jmin)
plt.show()