## May 30, 2014

### Sparse template-based adaptive filtering

 Significance index related to Student's t-test
The phenomenon arises in several real-life signal processing contexts: acoustic echo-cancellation (AEC) in sound and speech,  non-destructive testing where transmitted waves may rebound at material interfaces (e.g. ultrasounds), or pattern matching in images. Here in seismic reflection or seismology. Weak signals (of interest) are buried under both strong random and structured noise. Provided appropriate templates are obtained, we propose a structured-pattern filtering algorithm (called Ricochet) through constrained adaptive filtering in a  transformed domain. Its generic methodology impose sparsity: in different wavelet frames (Haar, Daubechies, Symmlets) coefficients, using the L-1 or Manhattan norm, as well as on adaptive filter coefficients using concentration measures (for sparser filters in the time domain): L-1, the Frobenius norm squared, and the mixed L-1,2 norms). Regularity properties are constrained as well, for instance slow variation on the adaptive filter coefficients (uniform, Chebychev or L-infinity norm). Quantitative results are given with a significance index, reminiscent of the Student t-test.

Abstract: Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured noises''. As their amplitude may be greater than signals of interest
 Lost in multiples: a creeping primary (flat, bootom-right)
(primaries), additional prior information is especially important in performing efficient signal separation. We address here the problem of multiple reflections, caused by  wave-field bouncing between layers. Since only approximate models of these phenomena are available, we propose a flexible framework for time-varying adaptive filtering of seismic signals, using sparse representations,  based on inaccurate templates. We recast the joint estimation of adaptive filters and primaries in a new convex variational formulation. This approach allows us to incorporate plausible knowledge about noise statistics, data sparsity and slow filter variation in parsimony-promoting wavelet frames.  The designed primal-dual algorithm solves a  constrained  minimization problem that alleviates standard regularization issues in finding hyper-parameters. The approach demonstrates  significantly good performance in low signal-to-noise ratio conditions, both for simulated and real field seismic data.

All the metrics here are convex. Wait a bit for something completely different with non-convex penalties, namely smoothed versions of the ratio of the L1 norm over the L2 norm: Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed $\ell_1/ell_2$ Regularization, covered in Nuit Blanche, and with arxiv page and pdf.