Image Processing with Wavelets

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This numerical tour overviews the use of wavelets for image approximation and denoising.

In [2]:

Wavelet Approximation

Image approximation is obtained by thresholding wavelets coefficients.

First we load an image $f \in \mathbb{R}^N$ of $N = n_0 \times n_0$ pixels.

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name = 'cortex';
n0 = 512;
f = load_image(name,n0);
f = rescale( sum(f,3) );

Display it.

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An orthogonal wavelet basis $ \mathcal{B} = \{ \psi_{j,n}^k \}_{j,n} $ of $\mathbb{R}^N$ is composed of multiscale atoms parameterized by their scale $2^j$, position $2^j n \in [0,1]^2$ and orentation $ k \in \{H,V,D\}$.

A forward wavelet transform computes the set of inner products $$ \Psi f = \{ \langle f,\psi_{j,n}^k\rangle \} \in \mathbb{R}^N $$ where the wavelet atoms are defined as $$ \psi_{j,n}^k(x) = \psi^k\left( \frac{x - 2^j n}{2^j} \right). $$

Set the minimum scale for the transform.

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Jmin = 0;

A short-cut for the wavelet transform $\Psi$:

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Psi = @(f)perform_wavelet_transf(f,Jmin,+1);

A short-cut for the inverse wavelet transform $\Psi^{-1} = \Psi^*$:

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PsiS = @(fw)perform_wavelet_transf(fw,Jmin,-1);

Perform the wavelet transform to compute $\Psi f$.

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fW = Psi(f);

Display the transformed coefficients.

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To perform non-linear image approximation, one remove the small amplitude coefficients. This is performed using a hard thresholding $$ H_T(f,\mathcal{B}) = \Psi^{-1} \circ H_T \circ \Psi (f) = \sum_{|\langle f,\psi_{j,n}^k\rangle| > T} \langle f,\psi_{j,n}^k\rangle \psi_{j,n}^k. $$

$T$ should be adapted to ensure a given number $M$ of non-zero coefficients, and then $f_M = H_T(f,\mathcal{B})$ is the best $M$ terms approximation of $f$ in $\mathcal{B}$.

Select a threshold.

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T = .5;

Shortcut for the thresholding operator $H_T$.

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Thresh = @(fW,T)fW .* (abs(fW)>T);

Perform hard thresholding of the coefficients.

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fWT = Thresh(fW,T);

Exercise 1

Compute the ratio $M/N$ of non-zero coefficients.

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Ratio of non-zero coefficients : M/N = 0.00481.
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%% Insert your code here.

Display the thresholded coefficients.

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title('Original coefficients');

Perform reconstruction using the inverse wavelet transform $\Psi^*$.

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f1 = PsiS(fWT);

Display approximation.

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imageplot(f, 'Image', 1,2,1);
imageplot(clamp(f1), strcat(['Approximation, SNR=' num2str(snr(f,f1),3) 'dB']), 1,2,2);

Number of coefficients for the approximation.

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M = n0^2/16;

Exercise 2

Compute a threshold $T$ to keep only $M$ coefficients.

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In [20]:
%% Insert your code here.

Perform hard thresholding.

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fWT = Thresh(fW,T);

Check the number of non-zero coefficients in |fWT|.

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disp(strcat(['      M=' num2str(M)]));
disp(strcat(['|fWT|_0=' num2str(sum(fWT(:)~=0))]));

Exercise 3

Compute an approximation with an decreasing number of coefficients.

In [23]:
In [24]:
%% Insert your code here.

Orthognal Wavelet Denoising

Image denoising is obtained by thresholding noisy wavelets coefficielts.

Here we consider a simple setting where we intentionnaly add some noise $w$ to a clean image $f$ to obtain $ y = f + w $.

A denoiser computes an estimate $\tilde f$ of $f$ from the observations $y$ alone. In the mathematical model, since $y$ is a random variable depending on $w$, so is $\tilde f$. A mathematical evaluation of the efficiency of the denoiser is the average risk $E_w( \|f-\tilde f\|^2 )$.

Here we consider a single realization of the noise, so we replace the risk by the oracle error $ \|f-\tilde f\|^2$. This allows us to bench the efficiency of the denoising methods by comparing the result to $f$. But you have to keep in mind that for real application, one does not have access to $f$.

We consider a Gaussian white noise $w$ of variance $\sigma^2$.

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sigma = .1;

Generate a noisy image.

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y = f + randn(n0,n0)*sigma;


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imageplot(f, 'Clean image', 1,2,1);
imageplot(clamp(y), ['Noisy image, SNR=' num2str(snr(f,y),3) 'dB'], 1,2,2);

A denoising is obtained by thresholding the wavelet coefficients $$ \tilde f = H_T(f,\mathcal{B}). $$

The asymptotically optimal threshold of Donoho and Johnstone is $T = \sqrt{2 \log(N)} \sigma$. In practice, one observes that much better result are obtained using $T \approx 3 \sigma$.

Compute the noisy wavelet coefficients.

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fW = Psi(y);

Compute the threshold value using the $3\sigma$ heuristic.

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T = 3*sigma;

Perform hard thresholding.

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fWT = Thresh(fW,T);

Display the thresholded coefficients.

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title('Original coefficients');

Perform reconstruction.

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f1 = PsiS(fWT);

Display denoising.

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imageplot(clamp(y), 'Noisy image', 1,2,1);
imageplot(clamp(f1), strcat(['Denoising, SNR=' num2str(snr(f,f1),3) 'dB']), 1,2,2);