A Novel Sidelobe Reduction Algorithm Based on Two ... - MDPI

sensors Article

A Novel Sidelobe Reduction Algorithm Based on Two-Dimensional Sidelobe Correction Using D-SVA for Squint SAR Images Min Liu 1 1 2

*

ID

, Zhou Li 2, * and Lu Liu 1

Qian Xuesen Laboratory of Space Technology, Beijing 100094, China; [email protected] (M.L.); [email protected] (L.L.) Beijing Institute of Remote Sensing Information, Beijing 100192, China Correspondence: [email protected]; Tel.: +86-13811011901

Received: 30 January 2018; Accepted: 3 March 2018; Published: 5 March 2018

Abstract: Sidelobe reduction is a very primary task for synthetic aperture radar (SAR) images. Various methods have been proposed for broadside SAR, which can suppress the sidelobes effectively while maintaining high image resolution at the same time. Alternatively, squint SAR, especially highly squint SAR, has emerged as an important tool that provides more mobility and flexibility and has become a focus of recent research studies. One of the research challenges for squint SAR is how to resolve the severe range-azimuth coupling of echo signals. Unlike broadside SAR images, the range and azimuth sidelobes of the squint SAR images no longer locate on the principal axes with high probability. Thus the spatially variant apodization (SVA) filters could hardly get all the sidelobe information, and hence the sidelobe reduction process is not optimal. In this paper, we present an improved algorithm called double spatially variant apodization (D-SVA) for better sidelobe suppression. Satisfactory sidelobe reduction results are achieved with the proposed algorithm by comparing the squint SAR images to the broadside SAR images. Simulation results also demonstrate the reliability and efficiency of the proposed method. Keywords: squint SAR; sidelobe reduction; double spatially variant apodization (D-SVA); non-integer Nyquist sampled imagery; impulse response

1. Introduction Synthetic aperture radar (SAR) transmits linear frequency modulation (LFM) signals and receives the backscattered echoes from Earth observation scenes. Typically, SAR images can be obtained with high spatial resolution by different image formation techniques when SAR works in broadside or slightly squint modes. However for some special applications, only squint mode SAR can satisfy such demands. In comparison, the squint SAR provides more mobility and flexibility. However, their severe echo signal range-azimuth coupling issues make squint SAR imaging algorithms more complicated than those for broadside SAR [1,2]. The impulse response of a point target in one dimension can be written as a sinc function (sin(πx)/(πx)) in the image domain [3]. For a strong scatterer, its sidelobe energy falls off gradually away from the mainlobe. If a weak scatterer happens to locate in the sidelobe regions of the strong scatterer, it can be masked or distorted easily when the sidelobe energy of the strong scatterer is much higher than the mainlobe energy of the weak scatterer. Therefore, it is very important to find an effective sidelobe reduction algorithm. Various methods have been presented to control the sidelobe levels for SAR imagery [3–9]. Bidimensional linear apodization has been applied to data in frequency domain, which removes sidelobes for SAR images at the expense of some resolution degradation [3]. Sensors 2018, 18, 783; doi:10.3390/s18030783

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Alternatively, a spatially variant apodization (SVA) algorithm can suppress sidelobes and preserve image resolution at the same time [3–9]. Depending on the Nyquist sampling rate, SVA algorithm complexity might be very different. The simplest form is the three point convolution, which requires the data to be sampled at an integer multiple of the Nyquist frequency [3]. For those with sampling rate that is not the integer multiple of the Nyquist frequency, several modified algorithms have been proposed, but all at the expense of much larger computational complexity [5,6]. Up to present, most SVA algorithms are limited to broadside SAR images. Because the sidelobes of squint SAR images may not locate on the image principal axes (rows and columns), bi-dimensional SVA filters can hardly fully extract sidelobes information [9]. Hence for squint SAR images, such algorithms could obtain poor sidelobe reduction results. To address this issue, Castillo-Rubio et al. proposed an SVA algorithm for squint SAR images [9], in which the authors implemented the range migration algorithm (RMA) for image formation. By taking advantages of sampling parameters recalculation and nearest interpolation computation, the authors were able to extract most of the sidelobe information for the squint SAR images. However, this algorithm is complex and time consuming as it used 5-tabs SVA methods. In this paper, we propose a novel sidelobe reduction algorithm using double spatially variant apodization (D-SVA) for squint SAR images. The first step is to pre-process the squint SAR images by aligning the sidelobe directions parallel or vertical to the principal axes (row and column). Then, the sidelobe suppression is achieved by applying the D-SVA algorithm to the pre-processed data. Finally, an inverse transform is performed as a post-processing for the squint SAR images. Simulation results demonstrate that this algorithm is reliable and efficient. 2. Sidelobe Control Algorithms for SAR Images 2.1. Traditional Linear Windowing There are many classic windowing functions to suppress the impulse response (IPR) sidelobes for SAR images, such as Hamming, Taylor, Kaiser, and Blackman. Among all these windows, Hanning (cosine-on-pedestal) window is widely used to achieve low sidelobes, though at the cost of doubling the mainlobe width compared to the uniform (unweighted) signal data. Linear windows have the undesired side effect of broadening the IPR mainlobe width, which decreases the image resolution. If there is a high demand for SAR image resolution, linear windows could not meet the requirement to suppress sidelobe level whiling maintaining high image resolution at the same time. 2.2. Nonlinear Apodization (Windowing) Nonlinear apodization (windowing) methods include spatially variant apodization (SVA), adaptive sidelobe reduction (ASR) [10], parametric windows, adaptive Kaiser window [11], etc. The most commonly used method is SVA or improved (modified) SVA, which applies a particular signal-domain windowing function to filter each pixel in SAR images. Whether the pixel is the mainlobe or sidelobe depends on the properties of the neighboring image pixels. Nonlinear apodization methods can keep high image resolution and suppress sidelobes at the same time. Furthermore, several super-resolution/bandwidth-extrapolation algorithms are presented based on Super-SVA [12–14]. 2.3. Dual-Delta Factorization Dual-delta factorization algorithm is performed based on the point spread function (PSF), which divides the PSF into a group of different simple subsystems (dual-Delta operators) [8]. It then factorizes the PSF as a group of dual-Delta operators and passes the observed signal through their inverses. The greedy strategy is used to suppress sidelobes in this algorithm. 2.4. Iterative Adaptive Approach (IAA) IAA is an estimation algorithm based on iterative adaptive approach, which estimates the target’s backscatter coefficient to achieve low sidelobe levels for SAR images [15]. IAA is not a post-processing

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algorithm, and it can also be viewed as an imaging algorithm. First, the transition matrix for every pixel’s echo is built and the covariance matrix is calculated, and then the process is iteratively applied to calculate all the points’ backscatter coefficient. This approach gets low sidelobe levels by constant iterations and updates, which is rather complicated and time-consuming. The major merit for this algorithm is that it can recover weak targets. 3. Non-Integer Nyquist SVA Algorithm SVA is a very effective technique to reduce the sidelobes of a SAR image sampled at an integer multiple of the Nyquist frequency. When processing the image with non-integer sampling rate by SVA algorithm, the sidelobes can not be completely suppressed [4]. In order to suppress sidelobes effectively of SAR image sampled at any case of Nyquist sampling rate, the filter is extended from 3-taps to 5-taps [6]. However, these algorithms rely on constrained optimizations, which introduced substantial computational cost. The frequency domain weighting function can be written as: W ( f ) = a + 2ω1 · cos(2πl

f ) fs

(1)

i h f f where fs is the sampling frequency, f is the frequency whose support region is − 20 , 20 , f 0 is the signal bandwidth, a is a constraint parameter which guarantees the unit gain at the center of the aperture, and ω 1 is the parameter which can vary according to the weighting function. The corresponding IPR of Equation (1) can be expressed as: w(m) = aδ(m) + ω1 · [δ(m − l ) + δ(m + l )] δ(m) = sinc(ws m) =

sin(πws m) πws m

(2) (3)

where w(m) is the Fourier transform (FT) of W(f ), m is a sample point in discrete domain, f l = f s (when l is a positive integer, the Nyquist sampling rate is integer. Otherwise, the Nyquist 0

f

sampling rate is non-integer), ws = f0s represents the oversampling of the signal. Typically, SAR images are oversampled at non-integer Nyquist sampling rate. Therefore, in this section only the images with non-integer Nyquist sampling rate are considered. Therefore, Equation (2) can be rewritten as: fs fs w(m) = a · sinc(ws m) + ω1 sinc ws m − +sinc ws m + f0 f0

(4)

The sampling points for SAR and ISAR images must be integers with non-integer Nyquist sampling rate. So Equation (4) can be rewritten as: fs fs I (m) = a · sinc(ws m) + ω1 sinc ws m − +sinc ws m + f0 f0

(5)

where the sign b·c represents rounding down. In order to avoid invalid window functions, three constraints are given by [6]: I (0) = 1

(6)

W ( f 0 /2) ≥ 0

(7)

W (0) ≥ W ( f 0 /2)

(8)

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The above three constraints can be simplified as: Sensors 2018, 18, x

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W (0)  W ( f 0 / 2)

(8)

1 (f 0 / 2) j k i (8) (9) The above three constraints k  Was: jW (0) 0 ≤ ω1 ≤canh be simplified f fs 2 sinc f · ws − cos π f s ws 0 0 The above three constraints can be simplified as: 1 0  1     f s  1   f   (9) 2  sinc  cos fs  s  ws     ws  w 0 a 1= 1 − 2ω 1 f·0sinc (10) f s    f  f  f0   (9) 2  sinc   s   ws   cos 0  s  ws   f        f 0 sampling   f rate   0may The SVA algorithm for non-integer Nyquist be seen as a filter with 3-point a  1  21  sinc  ws  s   (10)   f convolver. Define g(m) to be either the real or imaginary of the image, and m is an index  f0    component s 21  sinc  ws image  1 SVA-filtered (10)  number running over the image pixel. Soathe  f  can be expressed as:

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azimuth azimuth

  may be seen as a filter with 3-point The SVA algorithm for non-integer Nyquist sampling   0rate or imaginary convolver. Define g(m) to be either the real component of the image, and m is an index fs s The SVA algorithm for non-integer Nyquist sampling rate may be fseen as a filter with 3-point(11) 0 g ( m ) = ag ( m ) + ω g m − + g m + number running over the image pixel. So 1 the SVA-filtered image can be expressed as: f 0 component of the f 0 image, and m is an index convolver. Define g(m) to be either the real or imaginary  number running over the image pixel. So the SVA-filtered as:   can  f   image  f be  expressed m )  the ag ( mfollowing g  m   s    independently on (11) )  1  g  msteps   s are   implemented In order to get the value ofgg'(0 (m), both the       ff0    ff0    s s real and imaginary parts of the gimage [4]. '( m )  ag ( m )  1  g  m      g  m      (11)   f 0   areimplemented steps  f 0    In order to get the value of g’(m), the following independently on both   1. Calculate theimaginary value of parts g0 (m)offor the real and theωimage [4]. ω 1 = ω 1_max (the upper limit of Equation (9)). 1 = 0 and In order to get the value of g’(m), the following steps are implemented independently on both 0 2. If1. thereal two values g (m) sign, output equals to zero at pixel(9)). m. Calculate theof value of are g’(m) forimage ω1 = 0in and ω1 =the ω1_max (the upper limit of Equation the and imaginary parts ofopposite the [4]. 3. Otherwise, the output equals to the lowest magnitude. 2. If the two values of g’(m) are opposite in sign, the output equals to zero at pixel m. 1. Calculate the value of g’(m) for ω1 = 0 and ω1 = ω1_max (the upper limit of Equation (9)). 3. Otherwise, the output equals to the lowest magnitude. 2. If 1 the two2 values g’(m) target are opposite sign, the SVA output equals toimages zero at pixel m. Figures and show aofpoint beforeinand after processed for broadside SAR and 1 and show aequals point to target before magnitude. and after SVA processed images for broadside SAR 3. Figures Otherwise, the2output the lowest squint SAR, respectively. The broadside SAR image’s sidelobes are suppressed efficiently as shown in and squint SAR, respectively. The broadside SAR image’s sidelobes are suppressed efficiently as Figures and 2 SAR showimage’s a point target before and after suppressed. SVA processed images for broadside SAR Figureshown 1b, but the 1squint sidelobes are barely in Figure 1b, but the squint SAR image’s sidelobes are barely suppressed. and squint SAR, respectively. The broadside SAR image’s sidelobes are suppressed efficiently as shown in Figure 1b, but the squint SAR image’s sidelobes are barely suppressed.

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Figure 1. Impulse response for broadside SAR images (a) Original image; (b)(b) SVA processed image. (a) Figure 1. Impulse response for broadside SAR images (a) Original image; (b) SVA processed image. 20

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azimuth azimuth

Figure 1. Impulse response for broadside SAR images (a) Original image; (b) SVA processed image.

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Figure 2. Impulse response SVA processed image. (a) for Squint SAR images (40°). (a) Original image; (b)(b) Figure 2. Impulse response for Squint SAR images (40°). (a) Original image; (b) SVA processed image.

Figure 2. Impulse response for Squint SAR images (40◦ ). (a) Original image; (b) SVA processed image.

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4. Two-Dimensional Sidelobe Correction Using D-SVA for Squint SAR Images 4. Two-Dimensional Sidelobe Correction Using D-SVA for Squint SAR Images 4.1. D-SVA for Non-Integer Nyquist Sampled Imagery 4.1. D-SVA for Non-Integer Nyquist Sampled Imagery This section introduces a double-SVA (D-SVA) algorithm for non-integer Nyquist sampling rate, This section introduces a double-SVA (D-SVA) algorithm for non-integer Nyquist sampling rate, which has much lower sidelobe level than SVA and costs less computational complexity as well as which has much lower sidelobe level than SVA and costs less computational complexity as well as time than 5-taps filter algorithm. time than 5-taps filter algorithm. 4.1.1. Algorithm 4.1.1. Additional Additional SVA SVA Algorithm From above section, observed that that Equation Equation (5) special form form of of Equation Equation (4). (4). If the From the the above section, it it is is observed (5) is is aa special If the sampling points take the integers that are rounded up, it is given by: sampling points take the integers that are rounded up, it is given by:        f   f ss     sinc  w  m   f s  fs     I m a sinc w m sinc w m ( ) ( )       I (m) = a · sinc(ws m) s+ ω1 1sinc ws s m  −  f   +sinc sw s m +  f 00     f 0  f0     

(12) (12)

where rounding up.up. where d·e means rounding means Similarly, Equations (9)–(11) can be rewritten as: Similarly, Equations (9)–(11) can be rewritten as: , f       fss  · w  − cos   πf s  f s w 0 ≤ ω010 ≤ '1 1 22 sinc   sinc   f 0   ws s  cos     fw0s   s 1 f f   0 0             fs 0 0  a = 1 − 2ω1' sinc ws f s   a '  1  21 sinc  ws  f 0      f0   The new output signal in image domain can be expressed as: The new output signal in image domain as: can be expressed fs f s 0 0 g00 (m) = a g(m) + ω1 g  m −  f   + g m +  f s   f ' s0 f g ''(m)  a ' g (m)    g  m   g m  0 1

  

   f0  

(13) (13)

(14) (14)

(15) (15)

   f 0   

 

1

1

0.8

0.8

0.6

0.6

Amplitude

Amplitude

g”(m) can be derived by executing the following steps: g”(m) can be derived by executing the following steps: 1. Calculate the value of g”(m) for ω 0 1 = 0 and ω 0 1 = ω 0 1_max (the upper limit of Equation (13)). 1. Calculate the value of g”(m) for ω’1 = 0 and ω’1 = ω’1_max (the upper limit of Equation (13)). 2. If the two values of g”(m) are opposite in sign, the output equals to zero at pixel m. 2. If the two values of g”(m) are opposite in sign, the output equals to zero at pixel m. 3. Otherwise, the output equals to the lowest magnitude. 3. Otherwise, the output equals to the lowest magnitude. As shown shownin inFigure Figure3,3,the the over sampled signal a sinc function in continuous time domain As over sampled signal is aissinc function in continuous time domain (blue (blue line), and the red asterisks are the original sampling points. The green dots in Figure are line), and the red asterisks are the original sampling points. The green dots in Figure 3a are3aSVA SVA processed result and the black triangles in Figure 3b are additional SVA processed result. It can processed result and the black triangles in Figure 3b are additional SVA processed result. It can be be observed Figure3a,b 3a,bhave havethe thesame samesampling samplingamplitude amplitudeof of the the mainlobe, mainlobe, but but the the sampled observed thatthat Figures sampled amplitudes of their sidelobes are different. amplitudes of their sidelobes are different.

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(a)

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Figure 3. SVA and Additional SVA processed results ( ws  0.6 ). (a) SVA; (b) Additional SVA. Figure 3. SVA and Additional SVA processed results (ws = 0.6). (a) SVA; (b) Additional SVA.

4.1.2. D-SVA Algorithm

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4.1.2. D-SVA Algorithm

The The outputs outputs of of D-SVA D-SVA are the results selected selected from from the the SVA SVA and additional additional SVA. SVA. By comparing comparing ) atpixel the SVA processed processedresult resultg0g’(m) andthe the additional SVA processed result any pixel m, if the SVA (m) and additional SVA processed result g00 (mg) ''(atmany m, if either 0 g (m) or g”(m) zero,isthe pixel to zero;tootherwise, the pixel minimum value. value. either g’(m) or is g”(m) zero, theequals pixel equals zero; otherwise, thetakes pixelthe takes the minimum Figure 44 shows shows the the D-SVA processed result compared to other algorithms. In In Figure Figure 4, 4, the the red red Figure asterisks are the original sampling points, the green dots are the SVA processed result, the black asterisks are the original sampling points, the green dots are the SVA the black triangles are the and thethe redred circles areare thethe final outputs using D-SVA triangles the additional additionalSVA SVAprocessed processedresult, result, and circles final outputs using Dfor corresponding pixels. SVA for corresponding pixels. 1

Amplitude

0.8 0.6 0.4 0.2 0 -0.2 -10

-5

0 Samples

5

10

ws =0.6 Figure Figure4.4.Comparison Comparisonofofdifferent differentprocessing processingalgorithms algorithms((w 0.6).). s 4.2. 4.2. Specturm Specturm Correction Correction for for Squint Squint SAR SAR Images Images At At present, present, various various SVA SVA algorithms algorithms have have been been applied applied to to broadside broadside SAR SAR images images [3–9]. [3–9]. But But for for squint SAR images, there are only a few papers proposed for sidelobe reduction [9]. When SAR squint SAR images, there are only a few papers proposed for sidelobe reduction [9]. When SAR is is in in squint squint mode, mode, its its antenna antenna is is not not perpendicular perpendicular to to the the flight flight path. path. The The range range and and azimuth azimuth sidelobes sidelobes are are no no longer longer parallel parallel to to the the image image principal principal axes axes (rows (rows and and columns). columns). As As shown shown in in Figure Figure 2, 2, the the impulse impulse response for a point target has poor results after SAR signal processing and imaging response for a point target has poor results after SAR signal processing and imaging processing processing (RMA). Therefore, the thetraditional traditionalSVA SVAalgorithms algorithms could hardly sidelobe information, (RMA). Therefore, could hardly getget allall thethe sidelobe information, andand the the sidelobe level is still unbearable. sidelobe level is still unbearable. It It should should be be noted noted that that the the support support region region for for squint squint SAR SAR images images in in 2-D 2-D frequency frequency domain domain is is not not the same as the broadside SAR. This results in rotated sidelobe directions for the point targets that the same as the broadside SAR. This results in rotated sidelobe directions for the point targets that are are parallel to the principal axes. By applying phase term, squint SAR’s spectral shape not not parallel to the principal axes. By applying the the phase shiftshift term, the the squint SAR’s spectral shape can can be transformed similarly to the broadside SAR whose range and azimuth sidelobes are vertical. be transformed similarly to the broadside SAR whose range and azimuth sidelobes are vertical. The The Fourier Fourier transform transform pair pair can can be be written written as: as: 

G( )Z ∞  g ( x)exp j 2 xdx

G (ξ ) =



−∞

g( x ) exp{− j2πξx }dx

g ( x)   G( ) exp  j 2 xd  Z ∞

(17)

g( x ) = G (ξ ) exp{ j2πξx }dξ Their two-dimensional expressions are−given by: ∞   Their two-dimensional expressions by: j 2  x   y  dxdy G( , )  areggiven ( x, y) exp

 



 

(16) (16)



Z ∞ Z ∞   G (ξ, ηg)(= ) exp x, y)    Gg((x, , )yexp j 2 j2π  y+  d)}  dxdy  x (ξx dηy  {− −∞ −∞

Z ∞ Z ∞ According the Fourier transform property, the following two equations are given by: g( x, y) = G (ξ, η ) exp{+ j2π (ξx + ηy)}dξdη  −∞ −∞ H FT ( , y)   g ( x, y) exp  j 2 xdx  According the Fourier transform property, the following two equations are given by: 

HTF ( x, ) Z  ∞ g ( x, y) exp  j 2 ydy 

HFT (ξ, y) = g( x, y) exp{− j2πξx }dx −∞ The definition of time shifting is given by: Z ∞

(17)

(18) (18) (19) (19)

(20) (21) (20)

 ) {− j2πηy}dy HTF ( x, η ) = g ( xg) ( x, yG)(exp

(22) (21)

g ( x  x0 )  G( ) exp  j 2 x0 

(23)

−∞

The effect of a time shift on a signal is to introduce a phase shift 2 ft0 in its transform.

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The definition of time shifting is given by: g( x ) ↔ G (ξ )

(22)

g( x − x0 ) ↔ G (ξ ) exp{− j2πξx0 }

(23)

The effect of a time shift on a signal is to introduce a phase shift −2π f t0 in its transform. SAR image data is recorded as g(x,y), in which x and y represent the azimuth and range of the image, respectively. As shown in Figure 5, define α as the range sidelobe correction angle between the original range sidelobe and the corrected range sidelobe, and the corresponding time delay in time domain as k1 y: RFT

HFT (ξ, y) · exp{ j2πξk1 y} ←→ G (ξ, η − k1 ξ )

(24)

Similarly, in Figure 5, define the azimuth sidelobe correction angle as β, and the corresponding time delay is k2 x: AFT

HTF ( x, η ) · exp{ j2πηk2 x } ←→ G (ξ − k2 η, η )

(25)

The squint SAR spectrum is: 2DFT

g( x + k1 y, y + k2 x ) ←→ G (ξ − k2 η, η − k1 ξ )

(26)

when k2 6= 0 and k1 6= 0, from Equations (24) and (26): RIFT

G (ξ − k2 η, η − k1 ξ ) ←→ HFT (ξ − k2 η, y) · exp{ j2πξk1 y}

(27)

Therefore, the range corrected spectrum is given by: 0 0 HFT (ξ − k2 η, y) · exp{ j2πξk1 y} · exp − j2πξk1 y = HFT (ξ − k2 η, y) exp − j2πξ k1 − k1 y

(28)

where exp − j2πξk01 y is the phase shift term determined by the range sidelobe correction angle, k01 = tanα0 , and α0 is the estimate angle for range sidelobes. When k01 = k1 , the range sidelobes are corrected after azimuth Inverse Fourier Transform (IFT). The azimuth sidelobes are corrected after the range sidelobe correction, and the expression is given by: 2DFT

g( x, y + k2 x ) ←→ G (ξ − k2 η, η )

(29)

From Equations (25) and (27), the following transformation holds AIFT

G (ξ − k2 η, η ) ←→ HTF ( x, η ) · exp{ j2πηk2 x }

(30)

Therefore, the azimuth corrected spectrum is given by: HTF ( x, η ) · exp{ j2πηk2 x } · exp − j2πηk02 x = HTF ( x, η ) · exp − j2πη k02 − k2 x

(31)

where exp{− j2πηk02 x } is the phase shift term determined by the azimuth sidelobe correction angle, k01 = tanβ0 , and β0 is the estimate angle for azimuth sidelobes. When k02 = k2 , the azimuth sidelobes are corrected after range IFT.

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HTF ( x, )  exp  j 2 k2 x  exp  j 2 k2' x  HTF  x,   exp  j 2  k2'  k2  x

(31)

where exp  j 2 k2' x is the phase shift term determined by the azimuth sidelobe correction angle, =Sensors 2018, , and 18, β’ 783is the estimate angle for azimuth sidelobes. When are corrected after range IFT.

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, the azimuth sidelobes 8 of 15

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Figure 5. The schematic diagram azimuthsidelobes sidelobes squint SAR image. Figure 5. The schematic diagramofofrange range and and azimuth forfor squint SAR image.

According to the time shifting property, the squint SAR image can be corrected as the broadside According to the time shifting property, the squint SAR image can be corrected as the broadside SAR SAR image. TheThe corrected data can sidelobesuppression suppression applying D-SVA image. corrected data canthen thenaccomplish accomplish sidelobe byby applying the the D-SVA algorithm. Since the the range and azimuth bothchanged changedafter after correction, while the space algorithm. Since range and azimuthsidelobes sidelobes are are both correction, while the space geometrical relation of the SAR image is also changed [16], it is necessary to recover the original space geometrical relation Sensors 2018, 18, x of the SAR image is also changed [16], it is necessary to recover the original 8 of 15space geometrical location of the image after the D-SVA processing. Thanks to the time shifting property, geometrical location of thethe image afterand theazimuth D-SVA processing. to theare time shifting property, this can be achieved because range direction Thanks corrections invertible. ◦ this can be achieved because the range and azimuth direction corrections are invertible. As shown in Figure 6, Figure 6a is a point target for 40 squint SAR image; Figure 6b is the shownfor inFigure Figure 6a 6, Figure 6a is asidelobe point target for 40° squint image; Figure 6b is the for processedAsimage after range correction; FigureSAR 6c is the processed image processed image for Figure 6a after range sidelobe correction; Figure 6c is the processed image for Figure 6a after range and azimuth sidelobe correction; (d~f) are the corresponding 2-D spectrum Figure 6a after range and azimuth sidelobe correction; (d~f) are the corresponding 2-D spectrum images for (a~c), respectively.

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Figure 6. The squint SAR images (a) Squint angle 40° SAR image; (b)Range sidelobes corrected;

Figure 6. The squint SAR images (a) Squint angle 40◦ SAR image; (b)Range sidelobes corrected; (c)Range and azimuth sidelobes corrected; (d~f) are the 2-D spectrum images for (a~c), respectively. (c)Range and azimuth sidelobes corrected; (d~f) are the 2-D spectrum images for (a~c), respectively.

4.3. Sidelobe Reduction for Squint SAR Images

4.3. Sidelobe Reduction for Squint SAR Images

Sidelobe reduction for squint SAR images includes the following steps.

Sidelobe imagesEincludes the following steps. 1. First,reduction transform for the squint squint SAR SAR images 0 into frequency domain and shift the center of the 1.

2.

frequency spectrum to the data’s center. Then by applying an inverse IFT to the data in time

First,domain, transform thedata squint images E0 into frequency domain and shift the center of the the new E1 isSAR recorded. frequency spectrum to the data’s center. Then by applying an inverse IFT to the data in time 2. Range sidelobe correction domain, the new data E1 is recorded. Apply FT to correction the azimuth of E1. According to the sidelobe directions of the strong scatterer, Range sidelobe estimate the range sidelobe angle αy, so:

dyri  dri  tan( y )

(32)

dYr   dy r 1 , dy r 2 ,  , dy rN r 

(33)

E 2  ift A  ft A  E1   exp  j 2    dYr 

(34)

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Apply FT to the azimuth of E1 . According to the sidelobe directions of the strong scatterer, estimate the range sidelobe angle αy , so: dyri = dri · tan(αy )

(32)

dYr = [dyr1 , dyr2 , · · · , dyrNr ]

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E2 = i f t A [ f t A ( E1 ) · exp{− j2π · ξ · dYr }] h where dri = N2r − i , i = 1, 2, L, Nr , ξ is a column vector, ξ = −0.5 : 1 : 0.5 − iftA (·) are the azimuth FT and azimuth IFT, respectively.

3.

(34) 1 Na

i T

, ftA (·) and

Azimuth sidelobe correction

Apply FT to the range of E2 . According to the sidelobe directions of the strong scatterer, estimate the azimuth sidelobe angle αx , so: dx aj = da j · tan(α x ) (35) Sensors 2018, 18, x

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dXa = [dx a1 , dx a2 , · · · , dx aN ] Ta

T

(36)

dX a   dxa1 , dxa 2 , , dxaNa 

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E3 = i f t R [ f t R ( E2 ) · exp{− j2π · dXa · η }] (37) i E 3  ift R  ft R ( E 2 )  exp  j 2  dXha   (37) where da j = N2a − j , j = 1, 2, L, Na , η is a row vector , η = −0.5 : 1 : 0.5 − N1r , ftR (·) and iftR (·) where = and−range , =IFT, 1,2,respectively. , ,  is a row vector , = −0.5: 1: 0.5 − , ftR(·) and iftR(·) are are the range FT

the range FT and range IFT, respectively.

4. 5.

Use D-SVA to process the range and azimuth, obtaining E4 . Use D-SVA to process the range and azimuth, obtaining E4. Use azimuth phase shift term to rotate the azimuth sidelobes for E4 :

4.

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Use azimuth phase shift term to rotate the azimuth sidelobes for E4:

E5 = i f t R [ f t R ( E4 ) · exp{ j2π · dXa · η }] E5  ift R  ft R ( E 4 )  exp  j 2  dX a   6.

Use range phase shift term to rotate the range sidelobes for E5 :

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Use range phase shift term to rotate the range sidelobes for E5:

 ifti fA tAft[ Af t A E5(E5exp j 2{j2π   dY EE66 = · rξ· dYr }] ) · exp

7.

(38) (38)

7.

(39)

(39)

At each spatial location, select the minimum value between E6 and E1 as output.

At each spatial location, select the minimum value between E6 and E1 as output. To summarize, Figure 7 is the flowchart of the sidelobe reduction algorithm for squint SAR

To summarize, Figure 7 is the flowchart of the sidelobe reduction algorithm for squint SAR images. images.

Figure 7. The flowchart of sidelobe reduction algorithm for squint SAR images.

Figure 7. The flowchart of sidelobe reduction algorithm for squint SAR images.

5. Simulation Results and Analysis Table 1 presents the simulation parameters for a squint SAR system. Figure 8 shows the original 40° squint SAR image, which has nine point targets. The sidelobes of the nine targets are seen clearly in Figure 8. Table 1. Simulation parameters. Parameters Wavelength

Value 0.05657

Units m

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5. Simulation Results and Analysis Table 1 presents the simulation parameters for a squint SAR system. Figure 8 shows the original 40◦ squint SAR image, which has nine point targets. The sidelobes of the nine targets are seen clearly in Figure 8. Table 1. Simulation parameters. Parameters

Value

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Wavelength Pulse bandwidth Sampling rate Velocity Pulse duration Pulse repetition frequency Squint angle Synthetic aperture time Central slant range

0.05657 100 120 250 1 500 40 2.5136 22.057

m MHz MHz m/s µs Hz deg sec Km

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Figure 8. The original squint SAR Image. Figure 8. The original squint SAR Image. 8. The original squint SAR the Image. Figures 9 and 10 compare the Figure processed images after traditional SVA squint SAR image processing andFigures the proposed algorithm. In Figure 9, the nine point targets’ sidelobes dim slightly but 9 and 10 compare the processed images after the traditional SVA squint SAR image are still observable, which means algorithm. that the sidelobe is suppressed to some extentbut but not all processing and the proposed In Figure 9,energy the nine point targets’ sidelobes dim slightly are still observable, which means that the sidelobe energy is suppressed to some extent but not all by by using traditional SVA algorithm. In comparison, Figure 10 indicates that the squint SAR image using traditional SVA algorithm. In comparison, Figure 10 indicates that the squint SAR image sidelobes are suppressed effectively by using the proposed framework.

sidelobes are suppressed effectively by using the proposed framework. 50 100 150

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Figure 9. The squint SAR processed image using the traditional SVA algorithm. 50 100 150

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Figures 11 11 and and 12 12 show show the the range range and and azimuth azimuth profiles profiles of of the the nine nine point point targets. targets. The The blue blue line line Figures corresponds to the original result of the squint SAR image (Figure 8), the green line is the result corresponds to the original result of the squint SAR image (Figure 8), the green line is the result processed by by the the traditional traditional SVA SVA (Figure (Figure 9), 9), and and the the red red line line is is the the result result processed processed by by the the proposed proposed processed framework (Figure 10). It is apparent that the traditional SVA algorithm is ineffective for squint SAR framework (Figure 10). It is apparent that the traditional SVA algorithm is ineffective for squint SAR image sidelobe sidelobe reduction reduction while while the the proposed proposed framework framework can can suppress suppress sidelobes sidelobes well well and and keep keep the the image resolution at the same time. resolution at the same time. range sidelobe reduction

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Figure 11. Range profiles for the nine points. (a11~a33) are the nine points’ range profiles, respectively. azimuth sidelobe reduction



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Table 2 presents the sidelobe reduction results for the nine point targets. It indicates that the Table 2SVA presents the sidelobe reduction results forSAR the nine targets. It indicates that the traditional algorithm is ineffective for the squint imagepoint sidelobe reduction, in which the traditional SVA algorithm is ineffective for the squint SAR image sidelobe reduction, in which the peak sidelobe ratio (PSLR) of the nine point targets’ range and azimuth only decreased slightly. On peak sidelobe ratio of the nine point onlythe decreased slightly. On the the other hand, the(PSLR) nine point targets’ PSLRtargets’ all fell range belowand −30azimuth dB by using proposed framework. other hand, the nine point targets’ PSLR all fell below −30 dB by using the proposed framework. The proposed algorithm further reduction comparedresults with the method in reference Table 2.isSidelobe for the nine point targets. [9]. Figure 13 presents the sidelobe reduction results of the center point (a22) as shown in Figure 8. Figure 13a is the processed Performance Index Traditional SVA Proposed Framework result by using the method of Reference [9]. From this figure, it can be observed that the point Range PSLR −15.38dB −33.13dB a11 are suppressed to some extent, but they are still traceable in the processed image. target’s sidelobes Azimuth PSLR −14.42dB −30.63dB Figure 13b is the processed result target’s sidelobes are Range PSLR by using the proposed −14.73dB method. The point −30.64dB a12 Azimuth PSLR be seen in −14.67dB −30.51dB suppressed very well and can hardly this figure. Figure 14 is the comparison results for Range PSLR −39.42dB the range anda13 azimuth profiles between the two −14.95dB algorithms, in which Figure 14a presents the range Azimuth PSLR −14.54dB −30.88dB profiles, and Figure 14b presents the azimuth profiles. In Figure 14, the line with red asterisks is Range PSLR −15.36dB −33.36dB processed by a21 our proposed method the line with black triangles Azimuth PSLR (method 1) and −14.42dB −30.64dBis processed by the method of [9] (method 2).Range As shown it indicates that our proposed method has a better PSLR in Figure 14,−14.72dB −30.80dB a22 Azimuth PSLR −30.44dBit should be noted sidelobe reduction result for both the range and −14.63dB azimuth directions. In addition, Range PSLR −15.04dB −39.62dB that the method a23 of [9] costs much more time by using 5-taps SVA method. a31 a32 a33

Azimuth PSLR Range PSLR Azimuth PSLR Range PSLR Azimuth PSLR Range PSLR Azimuth PSLR

−14.59dB −15.33dB −14.43dB −14.75dB −14.61dB −15.07dB −14.59dB

30.43dB −33.58dB −30.84dB −30.85dB −30.26dB −41.15dB −30.34dB

The proposed algorithm is further compared with the method in reference [9]. Figure 13 presents the sidelobe reduction results of the center point (a22) as shown in Figure 8. Figure 13a is the processed result by using the method of Reference [9]. From this figure, it can be observed that the

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Table 2. Sidelobe reduction results for the nine point targets. Performance Index

Traditional SVA

Proposed Framework

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Figure 14. Range and azimuth profiles comparison between the two methods (a) Range profiles; Azimuth profiles. (b) Azimuth profiles. Figure 15 further illustrates an unprocessed and processed squint SAR image for a real scene. Figure 15a is the unprocessed image Figure 15b is the processed image. This image farm Figure 15 further illustrates an and unprocessed and processed squint SAR image forshows a real ascene. near Tianjin with a corner reflector in the center of the test scene. It’s a C-band SAR system, whose Figure 15a is the unprocessed image and Figure 15b is the processed image. This image shows a farm resolution is 3with m and the equivalent angle isof40°. backscatter the corner reflector near Tianjin a corner reflector squint in the center theAs test scene. It’sintensity a C-bandofSAR system, whose far exceeds the background image, the sidelobes of the corner reflector in the image appears resolution is 3 m and the equivalent squint angle is 40°. As backscatter intensity of the corner reflector obviously. D-SVA algorithm theofsidelobes of reflector the corner reflector been far exceedsAfter the background image, processing, the sidelobes the corner in the imagehas appears significantly suppressed. The corner reflector is common calibration equipment for SAR system. The obviously. After D-SVA algorithm processing, the sidelobes of the corner reflector has been

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Figure Figure 15 15 further further illustrates illustrates an an unprocessed unprocessed and and processed processed squint squint SAR SAR image image for for aa real real scene. scene. Figure 15a is the unprocessed image and Figure 15b is the processed image. This image Figure 15a is the unprocessed image and Figure 15b is the processed image. This image showsshows a farm anear farmTianjin near Tianjin a corner reflector in the center of the test scene. It’s a C-band SAR system, with a with corner reflector in the center of the test scene. It’s a C-band SAR system, whose ◦ . As backscatter intensity of the corner whose resolution is 3 m and the equivalent squint angle is 40 resolution is 3 m and the equivalent squint angle is 40°. As backscatter intensity of the corner reflector reflector far exceeds the background the sidelobes of the corner reflectorininthe the image image appears far exceeds the background image,image, the sidelobes of the corner reflector appears obviously. After D-SVA algorithm processing, the sidelobes of the corner reflector has been significantly obviously. After D-SVA algorithm processing, the sidelobes of the corner reflector has been suppressed. corner reflector is common calibration equipment forequipment SAR system. The result based significantlyThe suppressed. The corner reflector is common calibration for SAR system. The on realbased scene on canreal effectively verify the experimental conclusion of conclusion this paper. of this paper. result scene can effectively verify the experimental

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Figure 15. The unprocessed and processed squint SAR images (a) Unprocessed image; (b) Processed image by using our proposed method.

6. Conclusions A novel two-dimensional sidelobe suppression algorithm using D-SVA is proposed for squint SAR images. In this paper, the sidelobe distribution of squint SAR image is approximately corrected to broadside SAR image by introducing the shift phase term, and the D-SVA algorithm can accurately process the sidelobe energy of any squints angle. This algorithm’s computational efficiency is twice as fast as SVA. It can reduce the sidelobe energy efficiently without degrading resolution. This algorithm is independent of imaging algorithm and is capable to reduce sidelobes for all squint SAR images. It simplifies the sidelobe reduction process by skipping frequency domain interpolation and sampling parameters recalculation. The D-SVA algorithm can also be applied directly to broadside SAR images for sidelobe reduction, which can achieve better sidelobe suppression results than the SVA algorithm. Acknowledgments: This work was supported by the National Basic Research Program of China (Grant No. 613258XXX). Author Contributions: Min Liu proposed the algorithm, performed the experiments and wrote the paper; Zhou Li provided important suggestions to improve the algorithm; Lu Liu provided consultation and advices to this paper. Conflicts of Interest: The authors declare no conflict of interest.

References 1.

2. 3.

Sun, Z.C.; Wu, J.J.; Li, Z.Y.; Huang, Y.L.; Yang, J.Y. Highly Squint SAR Data Focusing Based on Keystone Transform and Azimuth Extended Nonlinear Chirp Scaling. IEEE Trans. Geosci. Remote Sens. Lett. 2015, 12, 145–149. An, D.X.; Huang, X.T.; Jin, T.; Zhou, Z.M. Extended Nonlinear Chirp Scaling Algorithm for High-Resolution Squint SAR Data Focusing. IEEE Trans. Geosci. Remote Sens. 2012, 50, 3595–3609. [CrossRef] Stankwitz, H.C.; Dallaire, R.J.; Fienup, J.R. Nonlinear apodization for sidelobe control in SAR imagery. IEEE Trans. Aerosp. Electron. Syst. 1995, 31, 267–279. [CrossRef]

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