Joint Sparse Recovery Based on Variances

Generalized sparse Bayesian learning and application to image reconstruction

“…x + ν (1) ν (2) =ν , where F (1) and F (2) are the forward models describing how x is mapped to y (1) and y (2) , respectively. Employing the usual likelihood function (2.1) would correspond to assuming that all the components of stacked noise vector ν are i. i. d., which is not true for this example.…”

“…Requiring such a commuting property is often unrealistic in applications, however. 1 Our Contribution. To address these issues, we present a generalized approach to SBL for "almost" general forward and regularization operators, F and R. By "almost" general, we mean that the only restriction on F and R is that their common kernel should be trivial, kernel(F ) ∩ kernel(R) = {0}, a standard assumption in regularized inverse problems [38].…”

“…This is in contrast to the hierarchical Bayesian model which allows for spatially varying regularization. In this regard we note that there are weighted ℓ 1 -regularization methods [14,17,1] that incorporate spatially varying regularization parameters. While such techniques can improve the resolution near the non-zero values in sparse signals, as well as near the internal edges in images, they are still point estimates and thus do not provide additional uncertainty information.…”

“…Recall the piecewise constant signal discussed in subsection 4.2, and assume we want to reconstruct the values of this signal at n = 40 equidistant grid points, denoted by x. We are given two sets of data: y (1) corresponds to direct observations taken at 36 randomly selected locations with added i. i. d. zero-mean normal noise ν (1) with variance σ 2 1 = 5 • 10 −1 , and y (2) corresponds to blurred observations at 24 randomly selected locations with added i. i. d. zero-mean normal noise ν (2) with variance σ 2 2 = 10 −2 . The blurring is again modeled using (4.5) with a Gaussian convolution kernel and convolution parameter γ = 3 • 10 −2 .…”

“…The separate reconstructions by the SBL-based BCD algorithm can be found in Figures 8a and 8b. Both reconstructions are of poor quality, which is due to the high noise variance in the case of y (1) and to the missing information in the case of y (2) . In fact, the reconstruction illustrated in Figure 8b is of reasonable quality except for the region around t = 0.2, where a void of observations causes the reconstruction to miss the jumps at t = 0.15 and t = 0.25.…”