Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

Tadmor, Eitan; Zou, Jing

doi:10.1007/s00041-008-9038-9

Cited by 12 publications

(16 citation statements)

References 19 publications

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“…with this finite subfamily is exactly (25). Therefore it follows that S N is invertible (on span{e iλnx } N n=−N ) and that S N → S uniformly as N → ∞ (see [6], for example).…”

Section: Fourier Framesmentioning

confidence: 99%

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Detection of Edges from Nonuniform Fourier Data

Gelb

Hines

2011

J Fourier Anal Appl

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Abstract. Edge detection is important in a variety of applications. While there are many algorithms available for detecting edges from pixelated images or equispaced Fourier data, much less attention has been given to determining edges from nonuniform Fourier data. There are applications in sensing (e.g. MRI) where the data is given in this way, however. This paper introduces a method for determining the locations of jump discontinuities, or edges, in a one-dimensional periodic piecewise-smooth function from nonuniform Fourier coefficients. The technique employs the use of Fourier frames. Numerical examples are provided.

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“…with this finite subfamily is exactly (25). Therefore it follows that S N is invertible (on span{e iλnx } N n=−N ) and that S N → S uniformly as N → ∞ (see [6], for example).…”

Section: Fourier Framesmentioning

confidence: 99%

“…Finally, we note that there are some advantages in combining the Fourier-based concentration factor edge detection with minimization techniques, [24,25,26]. We will not discuss these ideas here, but may incorporate them into future investigations.…”

Section: Introductionmentioning

confidence: 99%

Detection of Edges from Nonuniform Fourier Data

Gelb

Hines

2011

J Fourier Anal Appl

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“…In [26], the authors describe three extensions of the CF method to combat the effects of partial and/or noisy Fourier data. One extension "fills" in missing Fourier coefficients and suppresses oscillations by minimizing the TV semi-norm.…”

Section: Introductionmentioning

confidence: 99%

“…The jump function approximation is found in a single l 1 minimization step. As with the compressive sensing (CS) edge detection method in [26], we combine ideas from CS with the CF method [9,10]. However we do not suppress the oscillations introduced by the CFs by minimization of the TV semi-norm, but rather remove them by a regularized deconvolution step.…”

Section: Introductionmentioning

confidence: 99%

Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data

et al. 2011

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We present a new method for estimating the edges in a piecewise smooth function from blurred and noisy Fourier data. The proposed method is constructed by combining the so called concentration factor edge detection method, which uses a finite number of Fourier coefficients to approximate the jump function of a piecewise smooth function, with compressed sensing ideas. Due to the global nature of the concentration factor method, Gibbs oscillations feature prominently near the jump discontinuities. This can cause the misidentification of edges when simple thresholding techniques are used. In fact, the true jump function is sparse, i.e. zero almost everywhere with non-zero values only at the edge locations. Hence we adopt an idea from compressed sensing and propose a method that uses a regularized deconvolution to remove the artifacts. Our new method is fast, in the sense that it only needs the solution of a single l 1 minimization. Numerical examples demonstrate the accuracy and robustness of the method in the presence of noise and blur.

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“…As the first algorithm that iteratively uses edges to help recover images, there are no similar algorithms to compare with. We comment that a related paper [30] compares three methods for edge detection from incomplete Fourier measurements, but none of them produces images. Therefore, we compared three EdgeCS algorithms -isotropic, anisotropic, and complex -with RecPF [39], which is based on isotropic TV and does not exploit edge detection.…”

Section: Parameters and Performancementioning

confidence: 99%

Edge Guided Reconstruction for Compressive Imaging

Guo¹,

Yin²

2012

SIAM J. Imaging Sci.

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Abstract. We propose EdgeCS -an edge guided compressive sensing reconstruction approach -to recover images of higher qualities from fewer measurements than the current state-of-the-art methods. Edges are important images features that are used in various ways in image recovery, analysis, and understanding. In compressive sensing, the sparsity of image edges has been widely utilized to recover images. However, edge detectors have not been used on compressive sensing measurements to improve the edge recovery and thus the image recovery. This motivates us to propose EdgeCS, which alternatively performs edge detection and image reconstruction in a mutually beneficial way. The edge detector of EdgeCS is designed to faithfully return partial edges from intermediate image reconstructions even though these reconstructions may still have noise and artifacts. For complex-valued images, it incorporates joint sparsity between the real and imaginary components.EdgeCS has been implemented with both isotropic and anisotropic discretizations of total variation and tested on incomplete k-space (Fourier) samples. It applies to other types of measurements as well. Experimental results on large-scale real/complexvalued phantom and magnetic resonance (MR) images show that EdgeCS is fast and returns high-quality images. For example, it exactly recovers the 256-by-256 Shepp-Logan phantom from merely 7 radial lines (3.03% k-space), which is impossible for most existing algorithms. It is able to accurately reconstruct a 512-by-512 MR image with 0.05 white noise from 20 .87% radial samples. On complex-valued MR images, it obtains recoveries with faithful phases,, which are important in many medical applications. Each of these tests took around 30 seconds on a standard PC. Finally, the algorithm is GPU friendly.

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Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

Cited by 12 publications

References 19 publications

Detection of Edges from Nonuniform Fourier Data

Detection of Edges from Nonuniform Fourier Data

Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data

Edge Guided Reconstruction for Compressive Imaging

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