# What are the most popular wavelet or tight frame regularizers for image reconstruction problems?

A common approach to image reconstruction is to solve the convex optimization problem \begin{equation} \text{minimize} \quad \frac12 \| Ax - b \|^2 + \gamma \| Dx \|_1 \end{equation} where $b$ is a vector of measurements (perhaps a blurry, noisy image), $A$ is a linear transformation that maps the reconstructed image to the vector of measurements (often $A$ is a convolution operator), and $D$ is a wavelet or tight frame analysis operator. The optimization variable is $x$, an image stored as a rectangular array of numbers.

What are the most popular choices for $D$ (and what are the most popular and easiest to use software implementations of $D$)? I've experimented with both curvelet and shearlet tight frames.

Is there a good choice of $D$ that is included in Matlab's wavelet toolbox?

• What did you end up doing?
– Royi
Jul 14 at 8:45
• @Royi Although I’ve experimented with curvelets and shearlets and common types of wavelets (like Daubechies, I believe), I still don’t have a clear answer in my mind as to which of these options is best or most popular, or if some other option that I’m not aware of has been found to be superior. I’m most interested in MR image reconstruction, if anyone else would like to chime in. Jul 14 at 9:08 Those dual-tree wavelets originate from Image Analysis Using a Dual-Tree $M$-Band Wavelet Transform, 2006, and the improvement over curvelets was shown, for instance (for standard denoising), in A Nonlinear Stein Based Estimator for Multichannel Image Denoising, 2008.
[EDIT] The 2D code is now available, embedded in a denoising toolbox: $M$-band 2D dual-tree (Hilbert) wavelet multicomponent image denoising, so you might need a little work. I hope I can work on a cleaner implementation. Recently related discussions in How to implement a $j$-level $M$-band wavelet transform of an image? and Code for a wavelet based hilbert transform?