Stein Unbiased Risk Estimator¶
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This tour uses the Stein Unbiased Risk Estimator (SURE) to optimize the value of parameters in denoising algorithms.
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
Denoising and SURE¶
We consider a simple generative model of noisy images $F = f_0+W$ where $f_0 \in \RR^N$ is a deterministic image of $N$ pixels, and $W$ is a Gaussian white noise distributed according to $\Nn(0,\si^2 \text{Id}_N)$, where $\si^2$ is the variance of noise.
The goal of denoising is to define an estimator $h(F)$ of $f_0$ that depends only on $F$, where $h : \RR^N \rightarrow \RR^N$ is a potentially non-linear mapping.
Note that while $f_0$ is a deterministic image, both $F$ and $h(F)$ are random variables (hence the capital letters).
The goal of denoising is to reduce as much as possible the denoising error given some prior knowledge on the (unknown) image $f_0$. A mathematical way to measure this error is to bound the quadratic risk $\EE_W(\norm{h(F) - f_0}^2)$, where the expectation is computed with respect to the distribution of the noise $W$
For real life applications, one does not have access to the underlying image $f_0$. In this tour, we however assume that $f_0$ is known, and $f = f_0 + w\in \RR^N$ is generated using a single realization of the noise $w$ that is drawn from $W$. We define the estimated deterministic image as $h(f)$ which is a realization of the random vector $h(F)$.
Number $N = n \times n$ of pixels.
n = 128*2
N = n**2
First we load an image $f \in \RR^N$ where $N=n \times n$ is the number of pixels.
f0 = load_image("nt_toolbox/data/hibiscus.bmp",n)
Display it.
plt.figure(figsize = (5,5))
imageplot(f0)
Standard deviation $\si$ of the noise.
sigma = .08
Then we add Gaussian noise $w$ to obtain $f=f_0+w$.
from numpy import random
f = f0 + sigma*random.standard_normal((n,n))
Display the noisy image. Note the use of the clamp function to force the result to be in $[0,1]$ to avoid a loss of contrast of the display.
plt.figure(figsize = (5,5))
imageplot(clamp(f), "Noisy, SNR = %.1f dB" %snr(f0, f))
The Stein Unbiased Risk Estimator (SURE) associated to the mapping $h$ is defined as
$$ \text{SURE}(f) = -N\si^2 + \norm{h(f)-f}^2 + 2\si^2 \text{df}(f) $$where df stands for degree of freedom, and is defined as
$$ \text{df}(f) = \text{div} h(f) = \sum_i \pd{h}{f_i}(f). $$It has been introduced in:
Stein, Charles M. (November 1981). "Estimation of the Mean of a Multivariate Normal Distribution". The Annals of Statistics 9 (6): 1135-1151.
And it has been applied to wavelet-based non-linear denoising in:
Donoho, David L.; Iain M. Johnstone (December 1995). "Adapting to Unknown Smoothness via Wavelet Shrinkage". Journal of the American Statistical Association (Journal of the American Statistical Association, Vol. 90, No. 432) 90 (432): 1200-1244.
If the mapping $f \mapsto h(f)$ is differentiable outside a set of zero measure (or more generally weakly differentiable), then SURE defines an unbiased estimate of the quadratic risk :
$$ \EE_W(\text{SURE}(F)) = \EE_W( \norm{f_0-h(F)}^2 ). $$This is especially useful, since the evaluation of SURE does not necessitate the knowledge of the clean signal $f_0$ (but note however that it requires the knowledge of the noise level $\si$).
In practice, one replaces $\text{SURE}(F)$ from its empirical evaluation $\text{SURE}(f)$ on a single realization $f$. One can then minimize $\text{SURE}(f)$ with respect to a parameter $\la$ that parameterizes the denoiser $h=h_\la$.
Linear Denoising SURE¶
We consider a translation-invariant linear denoising operator, which is thus a convolution
$$ h(f) = f \star g $$where $g \in \RR^N$ is a low pass kernel, and $\star$ denotes the periodic 2-D convolution.
Since we use periodic boundary condition, we compute the convolution as a multiplication over the Fourier domain.
$$ \forall \om, \quad \hat h(f)(\om) = \hat f(\om) \hat g(\om) $$where $\hat g(\om)$ is the frequency $\om$ of the discrete 2-D Fourier transform of $g$ (computed using the pylab function fft2 from the pylab package).
convol = lambda f,g: np.real(pyl.ifft2(pyl.fft2(f)*np.tile(pyl.fft2(g), (1,1))))
We define a parameteric kernel $g_\la$ parameterized by its bandwidth $\la>0$. We use here a Gaussian kernel
$$ g_\la(a) = \frac{1}{Z_\la} e^{ -\frac{\norm{a}}{2 \la^2} } $$where $Z_\la$ ensures that $\sum_a g_\la(a) = 1$.
normalize = lambda f: f/np.sum(f)
x = np.hstack((np.arange(0,n//2+1),np.arange(-n//2 + 1,0)))
[Y, X] = np.meshgrid(x, x)
g = lambda lambd: normalize(np.exp(-(X**2 + Y**2)/(2*lambd**2)))
Define our denoising operator $h=h_\la$ (we make explicit the dependency on $\la$): $$ h_\la(f) = g_\la \star f. $$
h = lambda f, lambd: convol(f, g(lambd))
Example of denoising result.
lambd = 1.5
plt.figure(figsize = (5,5))
imageplot(clamp(h(f, lambd)))
For linear operator, the dregree of freedom is equal to the trace of the operator, and thus in our case it is equal to the sum of the Fourier transform $$ \text{df}_\la(f) = \text{tr}(h_\la) = \sum_{\om} \hat g_\la(\om) $$ Note that we have made explicit the dependency of df with respect to $\la$. Note also that df$(f)$ actually not actually depend on $f$.
df = lambda lambd: np.real(np.sum(pyl.fft2(g(lambd))))
We can now define the SURE=SURE$_\la$ operator, as a function of $f, h(f), \lambda$.
from numpy import linalg
SURE = lambda f,hf,lambd: -N*sigma**2 + linalg.norm(hf-f,"fro")**2 + 2 * sigma**2 * df(lambd)
Exercise 1
For a given $\lambda$, display the histogram of the repartition of the quadratic error $\norm{y-h(y)}^2$ and of $\text{SURE}(y)$. Compute these repartition using Monte-Carlo simulation (you need to generate lots of different realization of the noise $W$). Display in particular the location of the mean of these quantities.
run -i nt_solutions/denoisingadv_9_sure/exo1
## Insert your code here.
In practice, the SURE is used to set up the value of $\la$ from a single realization $f=f_0+w$, by minimizing $\text{SURE}_\la(f)$.
Exercise 2
Compute, for a single realization $f=f_0+w$, the evolution of
$$ E(\la) = \text{SURE}_\la(f) \qandq E_0(\lambda) = \norm{f-h_\la(f)}^2 $$as a function of $\lambda$.
run -i nt_solutions/denoisingadv_9_sure/exo2
## Insert your code here.
Exercise 3
Display the best denoising result $h_{\la^*}(f)$ where $$\la^* = \uargmin{\la} \text{SURE}_\la(f) $$
run -i nt_solutions/denoisingadv_9_sure/exo3