# 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 $$\text{minimize} \quad \frac12 \| Ax - b \|^2 + \gamma \| Dx \|_1$$ 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?

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?