Reducing frequency-domain artifacts of binary image due to coarse sampling by repeated interpolation and smoothing of Radon projections

Abstract

We develop a method to calculate 2D spectrum of a binary image with better quality than that obtained via direct 2D-FFT. With FFT, jagged edges of objects due to coarse sampling introduce artifacts into the frequency domain, especially in the high-frequency area. With the proposed method, Radon projections of the binary image along lines at different viewing angles are calculated. Each projection is extended by interpolation, then smoothed and decimated. The interpolation–smoothing–decimation operation is repeated several times to reduce ruggedness and improve quality of the Radon projections considerably. One-dimensional FFT of each refined Radon projection is calculated, resulting in a set of frequency-domain samples distributed on a polar coordinate system. These samples are interpolated onto a Cartesian grid to give the required 2D spectrum of the sampled binary image. Numerical computations on several objects show that the method can provide significant improvement to the spectrum as compared with direct 2D-FFT.

Introduction

The present work is motivated by a recent study on the reversed use of the Fourier diffraction theorem to predict patterns of acoustic scattering [1]. In that work, we consider a 2D sound scattering problem in which an infinitely long cylindrical object is immersed in water and insonified by a plane sound wave. The incident wave propagates towards the cylinder, perpendicular to the axis. The cross-sectional distributions of acoustical properties, i.e., refraction indices, of the object and the water form a binary image. The Fourier diffraction theorem, which is the foundation of acoustic tomography [2], states that the forward- and backward-scattered field distributions correspond to 1D inverse Fourier transform of spectral values taken along a pair of half-circles in the 2D Fourier transform of the image. The values taken on the half-circles are used to efficiently calculate the field projections using 1D inverse Fourier transform.

In numerical implementations, the binary image in the spatial domain is sampled on a grid, and 2D DFT is calculated to obtain the frequency domain representation of the object. Jagged edges of the sampled object cause significant errors in the 2D spectrum. Especially, spectral values in high frequency regions may be completely corrupted. This leads to intolerable errors in the derived scattered sound values. To solve the problem, an effective method is needed to reduce the sampling-caused spectral artifacts in order to improve accuracy of the generalized projections derived from the 2D spectrum.

Research has been done to correct spectral errors in the discrete Fourier transform domain for various purposes. Zhao et al. [3] study a time reversal method and propose a de-trending algorithm to reduce phase errors at different noise levels. Zhu et al. [4] use the Parseval’s theorem to derive two expressions for line spectra, and correct amplitudes by averaging multiple points in the DFT sequence. Ding and Ming [5] develop a phase-difference method to correct frequencies and phases of spectral peaks. The frequency and phase are corrected using the phase difference between spectral lines, and amplitude modified using the spectrum of a window function.

To improve the spectrum of cross-sectional images in the acoustic scattering applications, one might consider increasing spatial resolution by up-sampling the object in the space domain, or using zero-padding to calculate 2D-FFT. In [6], Mueller and Nguyen integrate several interpolation methods and use region classification to choose a suitable method for a particular region. Zhang and Wu [7] propose an edge-guided non-linear interpolation approach by directional filtering and data fusion based on minimum MSE estimation to reduce computation cost. Interpolation of binary images has also been addressed. For example, Ledda et al. use interpolation and morphological hit-miss transform to reduce jagged edges of binary images to magnify logos, diagrams, texts and cartoons [8]. In these methods, jagged image edges are smoothed in an enlarged image. However, using these methods, the spectra often contain artifacts such as ringing, aliasing, and blocking, and therefore are unsuitable for space-domain projection reconstruction in our applications.

Up-sampling and zero-padding either increase computation cost and memory requirements with some remaining errors, or simply cannot eliminate high-frequency artifacts as will be shown in Section 4.2. In this paper, we propose an alternative method based on interpolation of Radon projections to keep a low space-domain sampling rate and provide much improved high frequency performance.

In Section 2, an operation of repeated interpolation/smoothing is described. Interpolation is performed on the projection of the object obtained from the discrete Radon transform, hence the name Radon projection, followed by low-pass filtering and decimation to produce a smoother projection. This process is repeated several times to achieve adequate accuracy with the same spatial sampling rate. One-dimensional Fourier transform of the smoothed projections gives improved frequency domain samples on radials of the polar coordinates in the frequency domain. In Section 3, two-dimensional conversion in the Fourier domain is carried out to map the polar samples onto a Cartesian grid so as to obtain improved 2D discrete Fourier transform of the jagged binary image. Results of numerical computation are presented in Section 4 to show effectiveness of the method. Section 5 concludes the paper.