Two‐dimensional sparse fractional Fourier transform and its applications

Highlights

• Efficient estimation of sampled 2D DFRFT utilizing the sparse nature of the signal.

• Analysis of the effect of noise on the estimated positions of significant frequencies.

• A robust positions estimation method to upgrade 2D SFRFT to robust 2D SFRFT.

• Verify performance in numerical simulations and applications of image and SAR radars.

Abstract

The discrete fractional Fourier transform is an excellent tool in non-stationary signal processing. And an efficient and accurate computation is important for the two-dimensional discrete fractional Fourier transform (2D DFRFT). Inspired by the sparse Fourier transform algorithm, we propose a two-dimensional sparse fractional Fourier transform (2D SFRFT) algorithm to estimate the fractional Fourier spectrum efficiently. Compared with existing methods, we have achieved the lowest runtime and sample complexity. Moreover, by analyzing the errors due to noises, the 2D SFRFT algorithm is improved to be robust. The applications in image fusion, parameter estimation of multicomponent 2D chirp signal and complex maneuvering targets in SAR radar demonstrate the effectiveness of the proposed algorithms.

Introduction

The Fourier transform (FT) is an important tool in communication, signal processing, and other fields [1], [2], [3], [4]. However, for non-stationary signals, the FT cannot analyze their time-frequency characteristics completely. The non-stationary signals, especially the chirp signals, are common in radar [5], [6], gravitational waves [7], etc. As a generalized FT, the fractional Fourier transform (FRFT) [8], [9], [10], [11] has the advantage of processing non-stationary signals [12], [13], [14], [15]. It decomposes the signals by chirp orthonormal basis instead of sine basis, and the chirp signals are energy aggregated in the FRFT domain. The FRFT has been applied to optical systems [16], communication [17], signal and image processing [18], [19], [20], and other fields [21], [22], [23], [24] successfully.

Efficiently numerical algorithms for FRFT are necessary for engineering. Many definitions [25], [26], [27], [28], [29], [30], [31] of discrete fractional Fourier transform (DFRFT) have been proposed and can be classified as sampled DFRFT, eigendecomposition DFRFT, and linearly weighted DFRFT. The linearly weighted DFRFT is used rarely since it cannot approximate the continuous FRFT. The eigendecomposition DFRFT is not closed and there is no effective fast algorithm. So, it is difficult to be used in real-time engineering. For applications that do not require rotational additivity, the sampled DFRFT is the favorite and corresponding fast algorithms [30], [31], [32], [33], [34], [35], [36], [37] are studied. Pei [31] proposes currently the most efficient definition of DFRFT with the runtime complexity $O\left(N\log N\right)$. Based on Pei-type DFRFT and the sparse nature of some real signals, Liu [32] proposes the SFRFT algorithm and reduces the runtime complexity to $O(N+\sqrt{kN\log N}\log N)$. Where $k$ denotes the signals’ sparsity. Subsequently, variants of SFRFT are investigated and applied to radar [33], [34], [35], [36]. However, the study of SFRFT is limited to one dimension. The two-dimensional (2D) DFRFT [38] is important and more computational. Thus, fast algorithms for 2D DFRFT are needed.

Based on the advantages of FRFT, 2D DFRFT has been applied to image processing [39], [40], [41], [42], [43], remote sensing [44], radar [18], artificial intelligence [45], and so on. However, little literature is devoted to the efficient implementation of 2D DFRFT. Generally, the 2D DFRFT is obtained by cascading the 1D DFRFT of the rows and columns of the signal. Utilizing the fast algorithm of 1D DFRFT, the runtime complexity of 2D DFRFT is $N_1+N_2$ times that of 1D DFRFT. It is worth noting that SFRFT requires the signal to be sparse after the transformation. Thus, the SFRFT algorithm with low runtime complexity cannot replace two sets of 1D DFRFTs. To summarize, 2D DFRFT is time-consuming. Moreover, all discrete points of the signal are used, making data acquisition, storage, and transmission challenges. It is necessary to study the SFRFT algorithm for two dimensions signals.

With low computational and sampling costs, the SFT [46], [47], [48], [49], [50] are efficient algorithms to estimate the discrete Fourier transform (DFT). And the 2D SFT algorithms [51], [52], [53], [54], [55], [56] are proposed. Among them, the downsampled method [51] performs only a few 1D DFTs. The algorithm has a simple structure and low complexity for noiseless signals. However, it may be locked if the signals are not appropriate. To address this limitation, different schemes [52], [53], [54] are investigated. And by randomly downsampled slices, the algorithm in Wang et al. [54] has the strongest degree of freedom and the lowest runtime complexity. Moreover, the sparsity does not have to be pre-given, which fits the real applications. However, 2D SFT cannot be used for non-stationary signals. Therefore, we are willing to propose algorithms to estimate the 2D DFRFT based on sparsity and randomly downsampled slices. In addition, for the case of noise, the voting method in Wang et al. [54] is only suitable for high SNR. We expect to extend the range of SNR in the novel algorithm estimating 2D DFRFT.

To estimate 2D DFRFT with low sample and runtime complexity, we will propose a two-dimensional sparse fractional Fourier transform (2D SFRFT) algorithm. The signals’ 2D FRFT domain are assumed to be sparse, which is common in radar, magnetic resonance images, optical images, and other engineering. The 2D SFRFT algorithm will be implemented by chirp products and downsampled 2D SFT. The sparsity can be unknown and only a few slices of data will be leveraged. Thus the 2D SFRFT algorithm can achieve an unprecedented low sample and runtime complexity. The localization errors caused by noises will be analyzed and the significant frequency position estimation method of the 2D DFRFT algorithm will be improved to robust. The robust 2D SFRFT algorithm can achieve 2D DFRFT estimation at low SNR with great probability. The specific contributions of this paper are as follows.

• Firstly, based on the Pei-type DFRFT, a definition of 2D DFRFT is given, which is the target to be estimated.

• Secondly, based on randomly downsampled slices, we establish the 2D SFRFT algorithm for noiseless signals. The 2D DFRFT can be fully estimated with sample complexity $O\left(T\right)$ and runtime complexity $O(L\log L+k)$. Where $L=LCM(N_1,N_2)$ and $T$ is the number of iterations.

• Thirdly, we analyze the effect of noises on the 2D SFRFT and upgrade 2D SFRFT to be robust. With $s$ inner loops, the sample and runtime complexity are $O\left(L\right(T+sk\left)\right),O\left(\right(T+sk\left)L\log L\right)$, respectively. The correctly estimated probabilities are also analyzed.

• Finally, we verify the performance and convergence of the proposed algorithms by simulation. Multi-component 2D chirp signal parameter estimation is achieved and precision is far superior to that of 2D SFT. In addition, the robust 2D SFRFT algorithm is applied to image fusion and SAR radar successfully.

The rest of this paper is organized as follows. Section 2 describes the definition of Pei-type DFRFT and 2D DFRFT. In Section 3, the 2D SFRFT algorithm is introduced. Robust improvements are provided in Section 4. The relevant simulations and applications are presented in Sections 5 and 6. Section 7 concludes the paper.