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

Abstract

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.

Multilevel Wavelet Decomposition Based Harris Corner Detection Algorithm for Remote-sensing Image

“…And then the mapping relation is established according to the feature points to achieve the goal of image registration. According to the models of the feature points, feature-based image registration algorithms can be divided into three categories: edge point based [3] , corner based [4] and interest operator based [5] .…”

Recovering Piecewise Smooth Functions from Nonuniform Fourier Measurements

“…In [9,18] spectral reprojection techniques were used for this task, and a frame-theoretic approach was introduced in [10]. Recovering the Fourier transform to high accuracy was studied in [16], and in [8,15] the problem of high-order edge detection was addressed. A more detailed discussion is beyond the scope of this paper.…”

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

“…Most recently, the fact that the edges of many signals and images are sparse was utilized to reformulate edge detection from non-uniform Fourier data as a sparse signal recovery (SSR) problem, [17]. 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.…”

“…In both of these applications, data are collected as non-uniform Fourier samples. Detecting edges specifically from non-uniform 1 Fourier data has been explored in [16,17,23]. Most recently, the fact that the edges of many signals and images are sparse was utilized to reformulate edge detection from non-uniform Fourier data as a sparse signal recovery (SSR) problem, [17].…”