Image Denoising with Wavelets¶

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This numerical tour uses wavelets to perform non-linear image denoising.

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Image Denoising¶

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 $\tilde F$ of $f_0$ that depends only on $F$, i.e. $\tilde F = \phi(F)$ where $\phi : \RR^N \rightarrow \RR^N$ is a potentially non-linear mapping.

Note that while $f_0$ is a deterministic image, both $F$ and $\tilde F$ 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 $\EE_w(\norm{\tilde F - f_0}^2)$, where the expectation is computed with respect to the distribution of the noise $W$.

Image loading and adding Gaussian Noise¶

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 $\tilde f = \phi(f)$ which is a realization of the random vector $\tilde F$.

First we load an image $f \in \RR^N$ where $N=n \times n$ is the number of pixels.

In [3]:

n = 256;
name = 'hibiscus';
f0 = rescale( load_image(name,n) );
if using_matlab()
    f0 = rescale( sum(f0,3) );
end

Display it.

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clf;
imageplot(f0);

Standard deviation $\si$ of the noise.

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sigma = .08;

Then we add Gaussian noise $w$ to obtain $f=f_0+w$.

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f = f0 + sigma*randn(size(f0));

Display the noisy image. Note the use of the |clamp| function to saturate the result to $[0,1]$ to avoid a loss of contrast of the display.

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clf;
imageplot(clamp(f), strcat(['Noisy, SNR=' num2str(snr(f0,f),3)]));

Hard Thresholding in Wavelet Bases¶

A simple but efficient non-linear denoising estimator is obtained by thresholding the coefficients of $f$ in a well chosen orthogonal basis $\Bb = \{\psi_m\}_m$ of $\RR^N$.

In the following, we will focuss on a wavelet basis, which is efficient to denoise piecewise regular images.

The hard thresholding operator with threshold $T \geq 0$ applied to some image $f$ is defined as $$ S_T^0(f) = \sum_{\abs{\dotp{f}{\psi_m}}>T} \dotp{f}{\psi_m} \psi_m = \sum_m s_T^0(\dotp{f}{\psi_m}) \psi_m $$ where the hard thresholding operator is $$ s_T^0(\alpha) = \choice{ \alpha \qifq \abs{\al}>T, \\ 0 \quad\text{otherwise}. }$ $$

The denoising estimator is then defined as $$ \tilde f = S_T^0(f). $$

Set the threshold value.

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T = 1;

Display the function $s_T^0(\al)$.

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alpha = linspace(-3,3,1000);
clf;
plot(alpha, alpha.*(abs(alpha)>T));
axis tight;

Parameters for the orthogonal wavelet transform.

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options.ti = 0;
Jmin = 4;

First we compute the wavelet coefficients $a$ of the noisy image $f$.

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a = perform_wavelet_transf(f,Jmin,+1,options);

Display the noisy coefficients.

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clf;
plot_wavelet(a,Jmin);