Adaptive Edge Detectors for Piecewise Smooth Data Based on the minmod Limiter

We propose three novel methods for recovering edges in piecewise smooth functions from their possibly incomplete and noisy spectral information. The proposed methods utilize three different approaches: #1. The randomly-based sparse Inverse Fast Fourier Transform (sIFT); #2. The Total Variation-based (TV) compressed sensing; and #3. The modified zero crossing. The different approaches share a common feature: edges are identified through separation of scales. To this end, we advocate here the use of concentration kernels (Tadmor, Acta Numer. 16:305-378, 2007), to convert the global spectral data into an approximate jump function which is localized in the immediate neighborhoods of the edges. Building on these concentration kernels, we show that the sIFT method, the TV-based compressed sensing and the zero crossing yield effective edge detectors, where finitely many jump discontinuities are accurately recovered. One-and two-dimensional numerical results are presented.

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“…To demonstrate the performance of the sFFT-based method, we compare it with the edge detector based on the minmod limiter employed in [14,Sect. 4…”

Section: Numerical Results For Sift Edge Detectionmentioning

confidence: 99%

Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

Tadmor

Zou

2008

J Fourier Anal Appl

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“…Three different concentration factors presented in [16] are displayed in Figure 1: i. (12) σ exp (x) = iπCxe 1…”

Section: Concentration Factor Edge Detectionmentioning

confidence: 99%

“…Unfortunately, this type of behavior means that standard thresholding techniques for detecting edges directly from (7) may fail. This difficulty was addressed in [12], where nonlinear enhancement techniques were introduced to better pinpoint the edge locations. Specifically, a better approximation of [f ](x) can be obtained by combining the results of (7) using several different concentration factors as (14) […”

Section: Concentration Factor Edge Detectionmentioning

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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“…Edge detection is also useful in itself as applications such as target identification and image segmentation rely heavily on accurate information about internal boundary structures. Since edge detection from uniform Fourier data is a well studied problem (see e.g., [13]) one option may be to first interpolate the data to uniform modes. To do this would be computationally less efficient and may even be less accurate, especially if large interpolation errors in the high frequency region are incurred.…”

Section: Introductionmentioning

confidence: 99%

Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm

2014

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Detecting edges in images from a finite sampling of Fourier data is important in a variety of applications. For example, internal edge information can be used to identify tissue boundaries of the brain in a magnetic resonance imaging (MRI) scan, which is an essential part of clinical diagnosis. Likewise, it can also be used to identify targets from synthetic aperture radar (SAR) data. Edge information is also critical in determining regions of smoothness so that high resolution reconstruction algorithms, i.e. those that do not "smear over" the internal boundaries of an image, can be applied. In some applications, such as MRI, the sampling patterns may be designed to oversample the low frequency while more sparsely sampling the high frequency modes. This type of non-uniform sampling creates additional difficulties in processing the image. In particular, there is no fast reconstruction algorithm, since the FFT is not applicable. However, interpolating such highly non-uniform Fourier data to the uniform coefficients (so that the FFT can be employed) may introduce large errors in the high frequency modes, which is especially problematic for edge detection. Convolutional gridding, also referred to as the non-uniform FFT (NFFT), is a forward method that uses a convolution process to obtain uniform Fourier data so that the FFT can be directly applied to recover the underlying image. Carefully chosen parameters ensure that the algorithm retains accuracy in the high frequency coefficients. Similarly, the convolutional gridding edge detection algorithm developed in this paper provides an efficient and robust way to calculate edges. We demonstrate our technique in one and two dimensional examples.

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Adaptive Edge Detectors for Piecewise Smooth Data Based on the minmod Limiter

Cited by 54 publications

References 14 publications

Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

Detection of Edges from Nonuniform Fourier Data

Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm

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