Shaping filters and the 1#1-norm

We prove that the 1#1-norm solves the problem highlighted in the preceding section. Now our goal is to estimate one shaping filter 313#313 that minimizes the objective function  

 316#316 (133)

To achieve this, the shaping filter is estimated iteratively using a nonlinear conjugate gradient solver (NLCG) as described in (). The objective function we actually minimize is  

 317#317 (134)

with

318#318 (135)

where r_i is the residual for one component of the data space, and 12#12a constant we choose a priori. Equation () is minimized with the standard iteratively re-weighted least-squares approach ((); (); ()) The objective function in equation () amounts to the 1#1 measure when 319#319 is large and amounts to the 312#312 measure when 320#320 with a smooth transition between the two.

In Figure , we display the result of the adaptive subtraction when the 1#1-norm is utilized to estimate the shaping filter [equation () with a small 12#12]. The estimated signal in Figure a is perfect, and so is the estimated noise. It is easy to check that the energy in Figure a (e=2) is less than the energy in Figure a (e=3.2) if we use the 1#1 norm. Figure shows the shaping filter associated with the 1#1-norm. This filter is a spike at lag=0. This simple 1D example demonstrates that the 1#1 should be utilized each time significant amplitude differences exist between multiples and primaries. In the next section, we show another synthetic example where internal multiples are attenuated.

 

1Dl1
Figure 4 (a) The signal estimated with the 1#1-norm. (b) The noise estimated with the 1#1-norm.

 

filterl1 Figure 5 Shaping filter estimated for the 1D problem with the 1#1-norm. This filter is a single spike at lag=0.