noise - Bag of Tricks for Denoising Signals While Maintaining Sharp Transitions

I know this is signal dependent, but when facing a new noisy signal what is your bag of tricks for trying to denoise a signal while maintaining sharp transitions (e.g. so any sort of simple averaging, i.e. convolving with a gaussian, is out). I often find myself facing this question and don't feel like I know what I ought to be trying (besides splines, but they can seriously knock down the right kind of sharp transition as well).

P.S. As a side note, if you know some good methods using wavelets, let me know what it is. Seems like they have a lot of potential in this area, but while there are some papers in the 90s with enough citations to suggest the paper's method turned out well, I can't find anything about what methods ended up winning out as top candidates in the intervening years. Surely some methods turned out to be generally "first things to try" since then.

Answer

L1 norm minimization (compressed sensing) can do a relative better job than conventional Fourier denoising in terms of preserving edges.

The procedure is to minimize an objective function

$$ |x-y|^2 + b|f(y)| $$

where $x$ is the noisy signal, $y$ is the denoised signal, $b$ is the regularziation parameter, and $|f(y)|$ is some L1 norm penalty. Denoising is accomplished by finding the solution $y$ to this optimization problem, and $b$ depends on the noise level.

To preserve edges, depending on the signal $y$, you can choose different penalties such that $f(y)$ is sparse (the spirit of compressed sensing):

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  if $y$ is piece-wise, $f(y)$ can be total variation (TV) penalty;

  
  


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  if $y$ is curve-like (e.g. Sinogram), $f(y)$ can be the expansion coefficients of $y$ with respect to curvelets. (This is for 2D/3D signals, not 1D);

  
  


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  if $y$ has isotropic singularities (edges), $f(y)$ can be the expansion coefficients of $y$ with respect to wavelets.

  
  
  

When $f(y)$ is the the expansion coefficients with respect to some basis functions (like curvelet/wavelet above), solving the optimization problem is equivalent to thresholding the expansion coefficients.

Note this approach can also be applied to deconvolution in which the objective function becomes to $|x-Hy|+ b|f(y)|$, where $H$ is the convolution operator.