Admissible Concentration Factors for Edge Detection from Non-uniform Fourier Data
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Admissible Concentration Factors for Edge Detection from Non-uniform Fourier Data Guohui Song1
· Gabe Tucker1 · Congzhi Xia2
Received: 19 December 2019 / Revised: 5 July 2020 / Accepted: 3 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Edge detection from Fourier data has been emerging in many applications. The concentration factor method has been widely used in detecting edges from Fourier data. We present a theoretic analysis of the concentration factor method for non-uniform Fourier data in this paper. Specifically, we propose admissible conditions for the concentration factors such that the edge detector converges to a smoothed approximation of the jump function. Moreover, we also introduce some specific choices of admissible concentration factors and present estimates of convergence rates correspondingly.
1 Introduction Detecting edges (discontinuities) of a piece-wise smooth functions plays an important role in many applications, such as detection of the bright band in radar data [11], brain tumor detection [12], lane detection in autonomous driving [18]. When the collected data are within the physical domain, many efficient numerical methods of edge detection have been developed, such as Canny edge detector [1] and Sobel operator [16]. On the other hand side, in applications such as magnetic resonance imaging (MRI) and synthetic aperture radar (SAR), the data are frequently collected in the frequency (Fourier) domain. It will be much more challenging to detect edges from Fourier (spectral) data, since the Fourier data are essentially global information of the unknown function and the edges are local features. The concentration factor method has been introduced in [8] to detect edges from uniform Fourier data. In particular, a characterization of admissible concentration factors is
G. Song: Supported in part by National Science Foundation under grants DMS-1521661 and DMS-1939203.
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Guohui Song [email protected] Gabe Tucker [email protected] Congzhi Xia [email protected]
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Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA
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Division of Science and Technology, Quincy University, Quincy, IL 62301, USA 0123456789().: V,-vol
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Page 2 of 16
Journal of Scientific Computing
(2020) 85:3
presented in [8] and convergence analysis are also included there. Moreover, various refined concentration factor methods have been discussed in [5,9,10,15]. There have also been effort in developing concentration factor methods [3,6,7,14,17] for non-uniform Fourier data due to its emerging popularity in applications such as MRI and SAR. In particular, [6] obtain certain analytic forms of the concentration factors through utilization the Fourier frames, while [14,17] try to compute the concentration factors by solving discrete optimization problems. Moreover, [7] incorporates the Fourier frames approximation into an optimization model and solve the concentration factors according to a priori information on the discontinuous pattern of
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