Detecting Edges from Non-uniform Fourier Data via Sparse Bayesian Learning

Abstract

In recent investigations, the problem of detecting edges given non-uniform Fourier data was reformulated as a sparse signal recovery problem with an 1 -regularized least squares cost function. This result can also be derived by employing a Bayesian formulation. Specifically, reconstruction of an edge map using 1 regularization corresponds to a so-called type-I (maximum a posteriori ) Bayesian estimate. In this paper, we use the Bayesian framework to design an improved algorithm for detecting edges from non-uniform Fourier data. In particular, we employ what is known as type-II Bayesian estimation, specifically a method called sparse Bayesian learning. We also show that our new edge detection method can be used to improve downstream processes that rely on accurate edge information like image reconstruction, especially with regards to compressed sensing techniques. 2 regularization, [10]. The second category is type-II, or evidence maximization Bayesian estimation which employs a hierarchical, flexible parametrized prior that is learned from the given data. For SSR, [20] provides * Department of Mathematics, Dartmouth College, USA 1 Note that while here we only explicitly consider non-uniform Fourier samples, all methods described here apply to uniform Fourier samples as well.2 Although ideally the 0 semi-norm should be used to regularize this problem, the resulting optimization problem is NP-hard. Hence the 1 norm has become a popular convex surrogate that makes the problem computationally tractable and also offers theoretical guarantees for exact reconstruction, [8], as well as a variety of other benefits related to compressed sensing, [14].

Empirical Bayesian Inference using Joint Sparsity

“…7 As noted previously, in the case of sparse signals, SBL with EM yields a comparable posterior, in the sense that it uses an empirically based support informed prior, to that in (2.19b). It is not computationally efficient in higher dimensions, however, [15,34,35]. To further simplify the analysis we choose the non-zero values of the underlying signal to be exactly 1.…”

Generalized sparse Bayesian learning and application to image reconstruction

“…, β k ) is a diagonal inverse covariance matrix. Ideas discussed in [50,52,16,11,15,7,5,13,19,20] suggest that conditionally Gaussian priors of the form (1.4) are particularly suited to promote sparsity of Rx. For example, the model proposed in [50] is designed to recover sparse representations of kernel approximations, coining the term sparse Bayesian learning (SBL).…”

“…Promoting sparse solutions, as done in [50], corresponds to using R = I ∈ R n×n as a regularization operator in (1.4). Further investigations that made use of SBL to promote sparse solutions include [52,55,54,13,19]. In many applications, however, it is some linear transformation Rx that is desired to be sparse.…”

“…In particular, some preliminary results on signal and image reconstruction indicate its potential use for denoising, deblurring, and magnetic resonance imaging. Finally, we relax the usual assumption that ν in (1.1) corresponds to independent and identically distributed (i. i. d.) normal noise, as done in [50,52,16,11,15,7,5,19,20]. In contrast, we allow for more degrees of freedom in the noise by only restricting ν to be independent but not necessarily identically distributed.…”

“…, β k ) is referred to as the inverse prior convariance matrix. See [50,52,16,11,15,7,5,13,19,20] and references therein. The conditionally Gaussian prior (2.7) can be justified by its asymptotic behavior [11].…”