## Translation Invariant Wavelet Denoising with Cycle Spinning

Cycle spinning compensates for the lack of shift invariance in the critically-sampled wavelet transform by averaging over denoised cyclically-shifted versions of the signal or image. The appropriate inverse circulant shift operator is applied to the denoised signal/image and the results are averaged together to obtain the final denoised signal/image.

There are *N* unique cyclically-shifted versions of a signal of
length, *N*. For an *M*-by-*N* image,
there are *MN* versions. This makes using all possible shifted versions
computationally prohibitive. However, in practice, good results can be obtained by using
a small subset of the possible circular shifts.

The following example shows how you use `wdenoise`

and `circshift`

to denoise a 1-D signal using
cycle spinning. For denoising grayscale and RGB images, `wdenoise2`

supports cycle spinning.

### 1-D Cycle Spinning

This example shows how to denoise a 1-D signal using cycle spinning and the shift-variant orthogonal nonredundant wavelet transform. The example compares the results of the two denoising methods.

Create a noisy 1-D *bumps* signal with a signal-to-noise ratio of 6. The signal-to-noise ratio is defined as $$\frac{N||X||\genfrac{}{}{0ex}{}{2}{2}}{\sigma}$$ where $$N$$ is the length of the signal, $$||X||\genfrac{}{}{0ex}{}{2}{2}$$ is the squared L2 norm, and $${\sigma}^{2}$$ is the variance of the noise.

rng default [X,XN] = wnoise('bumps',10,sqrt(6)); subplot(2,1,1) plot(X) title('Original Signal') subplot(2,1,2) plot(XN) title('Noisy Signal')

Denoise the signal using cycle spinning with 15 shifts, 7 to the left and 7 to the right, including the zero-shifted signal. Use `wdenoise`

with default settings. By default, `wdenoise`

uses Daubechies' least-asymmetric wavelet with four vanishing moments, `sym4`

. Denoising is down to the minimum of `floor(log2(N))`

and `wmaxlev(N,'sym4')`

where `N`

is the number of samples in the data.

ydenoise = zeros(length(XN),15); for nn = -7:7 yshift = circshift(XN,[0 nn]); [yd,cyd] = wdenoise(yshift); ydenoise(:,nn+8) = circshift(yd,[0, -nn]); end ydenoise = mean(ydenoise,2);

Denoise the signal using `wdenoise`

. Compare with the cycle spinning results.

xd = wdenoise(XN); subplot(2,1,1) plot(ydenoise,'b','linewidth',2) hold on plot(X,'r') axis([1 1024 -10 10]) legend('Denoised Signal','Original Signal','Location','SouthEast') ylabel('Amplitude') title('Cycle Spinning Denoising') hold off subplot(2,1,2) plot(xd,'b','linewidth',2) hold on plot(X,'r') axis([1 1024 -10 10]) legend('Denoised Signal','Original Signal','Location','SouthEast') xlabel('Sample') ylabel('Amplitude') title('Standard Orthogonal Denoising') hold off

absDiffDWT = norm(X-xd,2)

absDiffDWT = 12.4248

absDiffCycleSpin = norm(X-ydenoise',2)

absDiffCycleSpin = 10.6124

Cycle spinning with only 15 shifts has reduced the approximation error.