Detecting Edges from Non-uniform Fourier Data Using Fourier Frames

Gelb, Anne; Song, Guohui

doi:10.1007/s10915-016-0320-8

Cited by 12 publications

(41 citation statements)

References 28 publications

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“…This paper adapts and implements the SBL algorithm from [29] for the non-uniform Fourier edge detection problem using the SSR formulation from [17]. We demonstrate improved accuracy compared with the results of [17] that used 1 regularization in both noise-free and noisy scenarios. The organization of the paper is as follows.…”

Section: Introductionmentioning

confidence: 95%

“…The concentration factor method cannot be extended directly to non-uniform Fourier coefficients because {e iπλ k x } M k=−M is generally not an orthogonal basis. However, it can be adapted by using the finite Fourier frame approximation, [17,26]. We will employ the Fourier frame framework for our new edge detection method.…”

Section: Fourier Frame Approximationmentioning

confidence: 99%

“…We will employ the Fourier frame framework for our new edge detection method. The technique in [17] is briefly described below.…”

Section: Fourier Frame Approximationmentioning

confidence: 99%

“…Groundwork for that reformulation was laid in [16,23,27,31]. The technique in [17] used Fourier frames to construct a forward model for edge detection from given non-uniform Fourier data. An edge map of a piecewise smooth function was then reconstructed using an 1 -norm regularization optimization procedure which is known to encourage sparsity, [11,28].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we set up the problem of edge detection from non-uniform Fourier data. In Section 3, we describe the reformulation of the problem as SSR and reproduce the edge detection results from [17]. Section 4 introduces the probabilistic approach, adapts SBL for the edge detection problem, and explains the resulting algorithm.…”

Section: Introductionmentioning

confidence: 99%

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

Churchill

Gelb

2019

J Sci Comput

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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].

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Section: Introductionmentioning

confidence: 95%

Section: Fourier Frame Approximationmentioning

confidence: 99%

“…We will employ the Fourier frame framework for our new edge detection method. The technique in [17] is briefly described below.…”

Section: Fourier Frame Approximationmentioning

confidence: 99%