# Documentation

## Translation Invariant 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.

### 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

$\frac{N||X|{|}_{2}^{2}}{{\sigma }^{2}}$

where N is the length of the signal, ||X||22 is the squared ℓ2 norm, and σ2 is the variance of the noise.

```rng default; [X,XN] = wnoise('bumps',10,sqrt(6)); subplot(211) plot(X); title('Original Signal'); subplot(212) 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 Daubechies' least-asymmetric wavelet with 4 vanishing moments (`sym4`) and denoise the signal down to level 4 using soft thresholding and the universal threshold estimated from the level-1 detail coefficients.

```ydenoise = zeros(length(XN),15); for nn = -7:7 yshift = circshift(XN,[0 nn]); [yd,cyd] = wden(yshift,'sqtwolog' ,'s','sln',4,'sym4'); ydenoise(:,nn+8) = circshift(yd,[0, -nn]); end ydenoise = mean(ydenoise,2);```

Denoise the signal using the orthogonal nonredundant discrete wavelet transform (DWT) with the same parameters. Compare the orthogonal DWT with cycle spinning.

```xd = wden(XN,'sqtwolog','s','sln',4,'sym4'); subplot(211) 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'); subplot(212) 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'); absDiffDWT = norm(X-xd,2) absDiffCycleSpin = norm(X-ydenoise',2)```
```absDiffDWT = 18.0428 absDiffCycleSpin = 15.4778```

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

### 2-D Cycle Spinning

This example shows how to denoise an image using cycle spinning with 82=64 shifts.

Load the sine image and add zero-mean white Gaussian noise with a variance of 5.

```load sinsin; rng default; Xnoisy = X+sqrt(5)*randn(size(X)); subplot(211) imagesc(X); colormap(jet); title('Original Image'); subplot(212) imagesc(Xnoisy); title('Noisy Image');```

Determine the universal threshold from the level-1 detail coefficients. Use the B-spline biorthogonal wavelet with 3 vanishing moments in the reconstruction wavelet and 5 vanishing moments in the decomposition wavelet.

```wname = 'bior3.5'; [C,S] = wavedec2(Xnoisy,1,wname); Cdet = C(4097:end); THR = thselect(Cdet,'sqtwolog');```

Create a grid of 8 shifts in both the X and Y directions. This results in a total of 64 shifts.

```N = 8; [deltaX, deltaY] = ndgrid(0:N-1,0:N-1);```

Allocate a matrix of zeros the size of the image for the cycle spinning result. Specify soft thresholding and set the level to 3.

```Xspin = zeros(size(X)); sorh = 's'; level = 3;```

Use cycle spinning denoising and display the result.

```for nn =1:N^2 Xshift = circshift(Xnoisy, [deltaX(nn) deltaY(nn)]); [coefs,sizes] = wavedec2(Xshift,level,wname); [XDEN,cfsDEN,dimCFS] = wdencmp('gbl',coefs,sizes, ... wname,level,THR,sorh,1); XDEN = circshift(XDEN, -[deltaX(nn) deltaY(nn)]); Xspin = Xspin*(nn-1)/nn+XDEN/nn; end subplot(211) imagesc(X); colormap(jet); title('Original Image'); subplot(212) imagesc(Xspin); title('Cycle Spinning');```

Denoise the image using the identical parameters with the nonredundant DWT. Compare the peak signal-to-noise (PSNR), mean square error, and energy ratios obtained with cycle spinning and the nonredundant DWT.

```[coefs,sizes] = wavedec2(Xnoisy,level,wname); [XDEN,cfsDEN,dimCFS] = wdencmp('gbl',coefs,sizes,wname,3,THR,'s',1); [PSNRcs,MSEcs,~,L2RATcs] = measerr(X,Xspin) [PSNR,MSE,~,L2RAT] = measerr(X,XDEN)```

The error measures show that cycle spinning has improved the image approximation. The PSNR, mean square error, and energy ratio are all better in the image denoised with cycle spinning.