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.
using PyPlot
using NtToolBox
using Autoreload
arequire("NtToolBox")
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("NtToolBox/src/data/hibiscus.png", n);
Display it.
figure(figsize = (5, 5))
imageplot(f0)
Standard deviation $\si$ of the noise.
sigma = .08;
Then we add Gaussian noise $w$ to obtain $f=f_0+w$.
using Distributions
f = f0 .+ sigma.*rand(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.
figure(figsize = (5, 5))
imageplot(clamP(f), @sprintf("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
$$ h(f) = g \star h $$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 = (f, g) -> real(plan_ifft((plan_fft(f)*f).*(plan_fft(g)*g))
*((plan_fft(f)*f).*(plan_fft(g)*g)))
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 $\sum_a g_\la(a) = 1$.
include("NtToolBox/src/ndgrid.jl")
normalize = f -> f./sum(f)
x = [collect(0 : Base.div(n, 2)); collect(-Base.div(n, 2) + 1 : -1)]
(Y, X) = meshgrid(x, x)
g = lambd -> normalize(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 = (f, lambd) -> convol(f, g(lambd));
Example of denoising result.
lambd = 1.5
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 = lambd -> real(sum(plan_fft(g(lambd))*g(lambd)));
We can now define the SURE=SURE$_\la$ operator, as a function of $f, h(f), \lambda$.
SURE = (f ,hf, lambd) -> -N*sigma^2 + vecnorm(hf - f)^2 + 2*sigma^2*df(lambd); # vecnorm is for Frobenius norm
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.
include("Exos/denoisingadv_9_sure/exo1.jl") #It takes time to run
# ntrials = 100
# nlaunch = 20
# E0 = []
# E = []
# for i in 1:nlaunch
# f = repeat(f0, inner = [1, 1, ntrials]) + sigma.*rand(Normal(), n, n, ntrials)
# hf = h(f, lambd)
# #quadratic error
# e = sum((hf - repeat(f0, inner = [1, 1, ntrials])).^2, (1, 2))
# E0 = [E0; e[:]]
# #sure error
# e = -N*sigma^2 + sum((hf - f).^2, (1, 2)) + 2*sigma^2*df(lambd)
# E = [E; e[:]]
# end
# v_true = mean(E0)
# v_sure = mean(E)
# a = v_true - 8*stdm(E0, mean(E0))
# b = v_true + 8*stdm(E0, mean(E0))
# t = linspace(a, b, 31)
# mybar = e -> hist(e[collect((i > a) & (i < b) for i in E0)], t)
# figure(figsize = (10, 7))
# subplot(2,1,1)
# s = mybar(E0)[2]
# s = [s; 0]
# bar(t[1 : end], s, width = (b-a)/31, color = "darkblue", edgecolor = "white")
# axvline(v_true, color = "red", linewidth = 3)
# subplot(2,1,2)
# s = mybar(E)[2]
# s = [s; 0]
# bar(t[1 : end], s, width = (b-a)/31, color = "darkblue",edgecolor = "white")
# axvline(v_sure, color = "red", linewidth = 3)
# show()
## 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 $\lambda$.
include("Exos/denoisingadv_9_sure/exo2.jl")
## Insert your code here.
Exercise 3
Display the best denoising result $h_{\la^*}(f)$ where $$\la^* = \uargmin{\la} \text{SURE}_\la(f) $$
include("Exos/denoisingadv_9_sure/exo3.jl")
## Insert your code here.
Soft Thresholding SURE¶
In order to enhance the denoising results for piecewise regular signal and image, it is possible to use non-linear thresholding in an orthogonal wavelet basis $ \Bb = \{ \psi_m \}_{m} $ where $\psi_m \in \RR^N$ is a wavelet element.
Re-generate a noisy image.
f = f0 + sigma.*rand(Normal(), n, n);
The soft-thresholding estimator thus reads $$ h_\la(f) = \sum_m s_\la( \dotp{f}{\psi_m} ) \psi_m \qwhereq s_\la(\al) = \max\pa{0, 1-\frac{\la}{\abs{\al}}} \al. $$ It can be conveniently written as $$ h_\la = \Ww^* \circ S_\la \circ \Ww $$ where $\Ww$ and $\Ww^*$ are forward and inverse wavelet transform $$ \Ww(f) = ( \dotp{f}{\psi_m} )_m \qandq \Ww^*(x) = \sum_m x_m \psi_m, $$ and $ S_\la $ is the diagonal soft thresholding operator $$ S_\la(x) = ( s_\la(x_m) )_m. $$
Define the wavelet transform and its inverse.
h_daub = compute_wavelet_filter("Daubechies", 4)
W = f1 -> NtToolBox.perform_wavortho_transf(f1,0,+1,h_daub)
Ws = x -> NtToolBox.perform_wavortho_transf(x,0,-1,h_daub);
Display the wavelet transform $\Ww(f_0)$ of the original image.
figure(figsize = (10,10))
plot_wavelet(W(f0), 1)
show()
Define the soft thresholding operator.
S = (x, lambd) -> max(0, 1 - lambd./max(1e-9, abs(x)) ) .* x;
Define the denoising operator.
h = (f1, lambd) -> Ws(S(W(f1), lambd));
Example of denoising result.
lambd = 3*sigma/2
figure(figsize = (5, 5))
imageplot(clamP(h(f,lambd)))
Since $Ww$ is an orthogonal transform, one has $$ \text{df}(f) = \text{div}( S_\la )( \Ww(f) ) = \sum_m s_\la'( \dotp{f}{\psi_m} ) = \norm{\Ww(h(f))}_0 $$ where $ s_\la' $ is the derivative of the 1-D function $s_\la$, and $\norm{\cdot}_0$ is the $\ell^0$ pseudo-norm $$ \norm{x}_0 = \abs{ \enscond{m}{x_m \neq 0} }. $$
To summarize, the degree of freedom is equal to the number of non-zero coefficients in the wavelet coefficients of $h(f)$.
df = (hf, lambd) -> sum(abs(W(hf)) .> 1e-8);
We can now define the SURE operator, as a function of $f, h(f), \lambda$.
SURE = (f, hf, lambd) -> -N*sigma^2 + vecnorm(hf - f)^2 + 2*sigma^2*df(hf, lambd);
Exercise 4
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. Hint: you can do the computation directly over the wavelet domain, i.e. consider that the noise is added to the wavelet transform.
include("Exos/denoisingadv_9_sure/exo4.jl")
## Insert your code here.
Exercise 5
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 $\lambda$.
include("Exos/denoisingadv_9_sure/exo5.jl")
## Insert your code here.
Exercise 6
Display the best denoising result $h_{\la^*}(f)$ where $$\la^* = \uargmin{\la} \text{SURE}_\la(f) $$
include("Exos/denoisingadv_9_sure/exo6.jl")