Hide code cell content 
import mmf_setup;mmf_setup.nbinit()
import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt
from IPython.display import clear_output, display

This cell adds /home/docs/checkouts/readthedocs.org/user_builds/iscimath-583-learning-from-signals/checkouts/latest/src to your path, and contains some definitions for equations and some CSS for styling the notebook. If things look a bit strange, please try the following:

  • Choose "Trust Notebook" from the "File" menu.
  • Re-execute this cell.
  • Reload the notebook.

Denoising#

Here we discuss the following denoising model:

\[\begin{gather*} E[u] = \int\frac{1}{p}\abs{\vect{\nabla}u}^p + \frac{λ}{q}\int\abs{u - d}^q,\\ E'[u] = -\vect{\nabla}\cdot\Bigl(\vect{\nabla}u \abs{\vect{\nabla}u}^{p-2}\Bigr) + λ(u-d)\abs{u-d}^{q-2}. \end{gather*}\]

One Dimension#

To better understand what happens, we start by working in 1D.

from math_583 import denoise

Nx = 16
x = np.arange(Nx)
u1 = x % 2
u2 = np.where(abs(x-Nx/2) < Nx/4, 1.0, 0)

us = [u1, u2]
ims = [denoise.Image(data=u) for u in us]
ds = [denoise.Denoise(im, sigma=0) for im in ims]

fig, axs = denoise.subplots(len(ims))
for ax, im in zip(axs, ims):
    im.show(im.get_data(), ax=ax)
display(fig)

/home/docs/checkouts/readthedocs.org/user_builds/iscimath-583-learning-from-signals/conda/latest/lib/python3.9/site-packages/scipy/__init__.py:146: UserWarning: A NumPy version >=1.16.5 and <1.23.0 is required for this version of SciPy (detected version 1.23.5
  warnings.warn(f"A NumPy version >={np_minversion} and <{np_maxversion}"
../_images/f49601a81116d750cfad669e6824011e5c240446b358970c19162dd8fe0d3ab5.png 
mode = "periodic"
pqs = [(2,2), (2,1), (1,2), (1,1)]
lams = [1.0, 0.1, 0.01]
fig, axs = denoise.subplots(len(ims), len(pqs), aspect=2)
args = dict(eps_p=1e-6, eps_q=1e-6, sigma=0, mode=mode)
for row, (p, q) in enumerate(pqs):
    for col, im in enumerate(ims):
        ax = axs[row, col]
        for lam in lams:
            d = denoise.Denoise(im, lam=lam, p=p, q=q, **args)
            u = d.minimize()
            im.show(u, label=f"{lam=:g}", ax=ax)
        ax.legend()
    axs[row, 0].set(title=f"{p=}, {q=}, {mode=}")
clear_output()
display(fig)
../_images/19f8f7195584941009fc89bf022d2fc8241243632f23ee0e7e71f70842587f4e.png