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

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Abstract

In recent investigations, the problem of detecting edges given non-uniform Fourier data was reformulated as a sparse signal recovery problem with an \(\ell _1\)-regularized least squares cost function. This result can also be derived by employing a Bayesian formulation. Specifically, reconstruction of an edge map using \(\ell _1\) 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.

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Notes

  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 \(\ell _0\) semi-norm should be used to regularize this problem, the resulting optimization problem is NP-hard. Hence the \(\ell _1\) 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].

  3. A complete derivation explaining how approximating periodic piecewise smooth functions as sums of scaled and shifted ramp functions can be effectively used to design concentration factors from uniform Fourier coefficients can be found in [36].

  4. This is not generally the case given the definition of \({\mathbf {y}}\) in (38). However by construction of (14), if \(\lambda _k\) is close to k, that is, our samples are nearly uniform in the Fourier domain, \({\mathbf B}\) will be diagonally dominant. Hence adopting the probabilistic approach seems reasonable. We also note that it is discussed in [25] that this assumption may not hold. Future investigations will consider the covariance structure in \({\mathbf y}\).

  5. All computations were performed on a MacBook Air with a 1.7 GHz Intel Core i5 processor and 4 GB of memory.

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  1. Department of Mathematics, Dartmouth College, Hanover, USA

    Victor Churchill & Anne Gelb

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  1. Victor Churchill
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  2. Anne Gelb
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Correspondence to Victor Churchill.

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This work is supported in part by the Grants NSF-DMS 1502640, NSF-DMS 1732434, and AFOSR FA9550-18-1-0316.

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Churchill, V., Gelb, A. Detecting Edges from Non-uniform Fourier Data via Sparse Bayesian Learning. J Sci Comput 80, 762–783 (2019). https://doi.org/10.1007/s10915-019-00955-w

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