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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")
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 $h=h_\la$.
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 = (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
could not open file /Users/gpeyre/Dropbox/github/numerical-tours/julia/Exos/denoisingadv_9_sure/exo1.jl in include_from_node1(::String) at ./loading.jl:488 in include_from_node1(::String) at /Applications/Julia-0.5.app/Contents/Resources/julia/lib/julia/sys.dylib:? in include_string(::String, ::String) at ./loading.jl:441 in include_string(::String, ::String) at /Applications/Julia-0.5.app/Contents/Resources/julia/lib/julia/sys.dylib:?
## 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("NtSolutions/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("NtSolutions/denoisingadv_9_sure/exo3.jl")
PyObject <matplotlib.text.Text object at 0x33072dd50>
## Insert your code here.
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("NtSolutions/denoisingadv_9_sure/exo4.jl")
WARNING: hist(...) and hist!(...) are deprecated. Use fit(Histogram,...) in StatsBase.jl instead.
in depwarn(::String, ::Symbol) at ./deprecated.jl:64
in #hist!#994(::Bool, ::Function, ::Array{Int64,1}, ::Array{Any,1}, ::LinSpace{Float64}) at ./deprecated.jl:629
in hist(::Array{Any,1}, ::LinSpace{Float64}) at ./deprecated.jl:644
in (::##73#75)(::Array{Any,1}) at /Users/gpeyre/Dropbox/github/numerical-tours/julia/NtSolutions/denoisingadv_9_sure/exo4.jl:28
in include_from_node1(::String) at ./loading.jl:488
in include_from_node1(::String) at /Applications/Julia-0.5.app/Contents/Resources/julia/lib/julia/sys.dylib:?
in include_string(::String, ::String) at ./loading.jl:441
in include_string(::String, ::String) at /Applications/Julia-0.5.app/Contents/Resources/julia/lib/julia/sys.dylib:?
in include_string(::Module, ::String, ::String) at /Users/gpeyre/.julia/v0.5/Compat/src/Compat.jl:577
in execute_request(::ZMQ.Socket, ::IJulia.Msg) at /Users/gpeyre/.julia/v0.5/IJulia/src/execute_request.jl:154
in invokelatest(::Function, ::ZMQ.Socket, ::Vararg{Any,N}) at /Users/gpeyre/.julia/v0.5/Compat/src/Compat.jl:587
in eventloop(::ZMQ.Socket) at /Users/gpeyre/.julia/v0.5/IJulia/src/eventloop.jl:8
in (::IJulia.##18#24)() at ./task.jl:360
while loading /Users/gpeyre/Dropbox/github/numerical-tours/julia/NtSolutions/denoisingadv_9_sure/exo4.jl, in expression starting on line 34
WARNING: hist(...) and hist!(...) are deprecated. Use fit(Histogram,...) in StatsBase.jl instead.
in depwarn(::String, ::Symbol) at ./deprecated.jl:64
in #hist!#994(::Bool, ::Function, ::Array{Int64,1}, ::Array{Any,1}, ::LinSpace{Float64}) at ./deprecated.jl:629
in hist(::Array{Any,1}, ::LinSpace{Float64}) at ./deprecated.jl:644
in (::##73#75)(::Array{Any,1}) at /Users/gpeyre/Dropbox/github/numerical-tours/julia/NtSolutions/denoisingadv_9_sure/exo4.jl:28
in include_from_node1(::String) at ./loading.jl:488
in include_from_node1(::String) at /Applications/Julia-0.5.app/Contents/Resources/julia/lib/julia/sys.dylib:?
in include_string(::String, ::String) at ./loading.jl:441
in include_string(::String, ::String) at /Applications/Julia-0.5.app/Contents/Resources/julia/lib/julia/sys.dylib:?
in include_string(::Module, ::String, ::String) at /Users/gpeyre/.julia/v0.5/Compat/src/Compat.jl:577
in execute_request(::ZMQ.Socket, ::IJulia.Msg) at /Users/gpeyre/.julia/v0.5/IJulia/src/execute_request.jl:154
in invokelatest(::Function, ::ZMQ.Socket, ::Vararg{Any,N}) at /Users/gpeyre/.julia/v0.5/Compat/src/Compat.jl:587
in eventloop(::ZMQ.Socket) at /Users/gpeyre/.julia/v0.5/IJulia/src/eventloop.jl:8
in (::IJulia.##18#24)() at ./task.jl:360
while loading /Users/gpeyre/Dropbox/github/numerical-tours/julia/NtSolutions/denoisingadv_9_sure/exo4.jl, in expression starting on line 40
## 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("NtSolutions/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("NtSolutions/denoisingadv_9_sure/exo6.jl")
SNR = 19
.6 dB
PyObject <matplotlib.text.Text object at 0x331d24050>
## Insert your code here.
To improve the result of soft thresholding, it is possible to threshold blocks of coefficients.
We define a partition $ \{1,\ldots,N\} = \cup_k b_k $ of the set of wavelet coefficient indexes. The block thresholding is defined as
$$ h_\la(f) = \sum_k \sum_{m \in b_k} a_\la( e_k ) \dotp{f}{\psi_m} \psi_m \qwhereq e_k = \sum_{m \in b_k} \abs{\dotp{f}{\psi_m}}^2, $$where we use the James-Stein attenuation threshold $$ a_\la(e) = \max\pa{ 0, 1 - \frac{\la^2}{e^2} }. $$
The block size $q$.
q = 4;
A function to extract blocks.
include("NtSolutions/src/ndgrid.jl")
(X, Y, dX, dY) = ndgrid(1:q:n-q+1, 1:q:n-q+1, 0:q-1, 0:q-1)
I = X + dX + (Y + dY - 1).*n
I
for i in 1:Base.div(n, q)
for j in Base.div(n, q)
I[i,j, :, :] = transpose(I[i,j, :, :])
end
end
blocks = fw -> fw[I];
could not open file /Users/gpeyre/Dropbox/github/numerical-tours/julia/NtSolutions/src/ndgrid.jl in include_from_node1(::String) at ./loading.jl:488 in include_from_node1(::String) at /Applications/Julia-0.5.app/Contents/Resources/julia/lib/julia/sys.dylib:? in include_string(::String, ::String) at ./loading.jl:441 in include_string(::String, ::String) at /Applications/Julia-0.5.app/Contents/Resources/julia/lib/julia/sys.dylib:?
A function to reconstruct an image from blocks.
function assign(M, I, H)
M_temp = M
M_temp[I] = H
return reshape(M_temp, n,n)
end
unblock = H -> assign(zeros(n,n), I, H)
WARNING: Method definition assign(Any, Any, Any) in module Main at In[36]:2 overwritten at In[80]:2.
(::#77) (generic function with 1 method)
Compute the average energy of each block, and duplicate.
function energy(H)
H_tmp = copy(H)
for i in 1:Base.div(n, q)
for j in 1:Base.div(n, q)
H_tmp[i, j, :, :] = mean(H_tmp[i, j, :, :].^2).*ones(q, q)
end
end
return H_tmp
end
WARNING: Method definition energy(Any) in module Main at In[37]:2 overwritten at In[81]:2.
energy (generic function with 1 method)
Threshold the blocks. We use here a Stein block thresholding. All values within a block are atenuated by the same factor.
S = (H,lambda) -> max(1 - lambda^2 ./ energy(H), 0) .* H
(::#79) (generic function with 1 method)
Block thresholding estimator $h_\lambda(f)$.
h = (f, lambd) -> Ws(unblock(S(blocks(W(f)), lambd)))
(::#81) (generic function with 1 method)
Example of block denoising.
lambd = 1.1*sigma
figure(figsize = (5, 5))
imageplot(clamP(h(f, lambd)))
Since the block-thresholding operates in a block diagonal manner over the wavelet coefficients, it degree of freedom is a sum of the divergence of each block James-Stein operator $$ \text{df}(f) = \sum_{ e_k > \la^2 } \text{tr}( \partial \phi (a_k) ) $$ where $ a_k = (\dotp{f}{\psi_m})_{m \in b_k} $ is the set of coefficients inside a block, that satisfies $\norm{a_k}=e_k$, and where $$ \phi(a) = \pa{ 1 - \frac{\la^2}{\norm{a}^2} } a. $$ One can compute explicitely the derivative of $\phi$ $$ \partial \phi(a) = \pa{ 1 - \frac{\la^2}{\norm{a}^2} } \text{Id} + 2 \frac{\la^2}{\norm{a}^2} \Pi_a $$ where $\Pi_a$ is the orthogonal projector on $a$.
This gives the folowing formula for the degree of freedom $$ \text{df}(f) = \norm{\Ww(h_\la(f))}_0
$$ One can note that the degree of freedom differs from the one of the soft thresholding (it is not in general an integer).