# A Review of Synthetic-Aperture Radar Image Formation Algorithms and Implementations: A Computational Perspective

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## Abstract

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## 1. Introduction

## 2. Synthetic-Aperture Radar

## 3. Synthetic-Aperture Radar Image Formation Algorithms

#### 3.1. Range–Doppler Algorithm

- 1.
- A range compression is performed along the range direction, with a fast convolution. This means that a range FFT is performed, a matched filter multiplication and, lastly, a range inverse fast Fourier transform. Using the received demodulation signal given by Equation (4), assuming ${S}_{0}({f}_{\tau},\eta )$ is the range FFT of ${s}_{r}$ and $G\left({f}_{\tau}\right)$ is the frequency domain matched filter, the output of this step of the Range–Doppler algorithm is given by$$\begin{array}{cc}\hfill {s}_{rc}(\tau ,\eta )& =IFF{T}_{\tau}\left[{S}_{0}({f}_{\tau},\eta )G\left({f}_{\tau}\right)\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={A}_{0}{p}_{r}\left[\tau -\frac{2{R}_{t}\left(\eta \right)}{c}\right]{w}_{a}(\eta -{\eta}_{c})\times exp\left[-j\frac{4\pi {f}_{0}{R}_{t}\left(\eta \right)}{c}\right]\hfill \end{array}$$
- 2.
- The data are transformed into the Range–Doppler domain with an azimuth FFT. Since the first exponential in Equation (5) is constant for each target and with ${f}_{\eta}=-{K}_{a}\eta $, where ${K}_{a}$ is the azimuth FM rate of point target signal, the output after the azimuth FFT is given by$$\begin{array}{cc}\hfill {s}_{1}(\tau ,{f}_{\eta})& =FF{T}_{\eta}\left[{S}_{rc}({f}_{\tau},\eta )\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={A}_{0}{p}_{r}\left[\tau -\frac{2{R}_{rd}\left({f}_{\eta}\right)}{c}\right]{W}_{a}({f}_{\eta}-{f}_{{\eta}_{c}})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[-j\frac{4\pi {f}_{0}{R}_{t}\left(\eta \right)}{c}\right]exp\left[j\pi \frac{{f}_{\eta}^{2}}{{K}_{a}}\right]\hfill \end{array}$$
- 3.
- The platform movement causes range variations in the data, a phenomenon called range migration, and hence, a correction is performed to rearrange the data in memory, and straighten the trajectory. This way, it is possible to perform azimuth compression along each parallel azimuth line. This step is called range cell migration correction, and is given by$$\Delta R\left({f}_{\eta}\right)=\frac{{\lambda}^{2}{R}_{t}{f}_{\eta}^{2}}{8{V}_{r}^{2}}$$$${s}_{2}(\tau ,{f}_{\eta})={A}_{0}{p}_{r}\left[\tau -\frac{2{R}_{t}}{c}\right]{W}_{a}({f}_{\eta}-{f}_{{\eta}_{c}})\times exp\left[-j\frac{4\pi {f}_{0}{R}_{t}\left(\eta \right)}{c}\right]exp\left[j\pi \frac{{f}_{\eta}^{2}}{{K}_{a}}\right]$$
- 4.
- Azimuth compression is performed to compress the energy in the trajectory to a single cell in the azimuth direction. A matched filter is applied to the data after RCMC and, lastly, an IFFT is performed.The frequency domain matched filter is given by$${H}_{az}\left({f}_{\eta}\right)=exp\left[-j\pi \frac{{f}_{\eta}^{2}}{{K}_{a}}\right]$$After azimuth compression, the resulting signal is given by$$\begin{array}{cc}\hfill {s}_{3}(\tau ,{f}_{\eta})& ={S}_{2}(\tau ,\eta ){H}_{az}{f}_{\eta}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={A}_{0}{p}_{r}(\tau -2{R}_{t}/c){W}_{a}({f}_{\eta}-{f}_{{\eta}_{c}})\times exp\left[-j\frac{4\pi {f}_{0}{R}_{t}}{c}\right]\hfill \end{array}$$
- 5.
- Lastly, an azimuth IFFT transforms the data into the time domain, resulting in a compressed complex image. After this step, the compressed image is given by$$\begin{array}{cc}\hfill {s}_{ac}(\tau ,{f}_{\eta})& =IFF{T}_{\eta}\left[{S}_{3}(\tau ,{f}_{\eta})\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={A}_{0}{p}_{r}(\tau -2{R}_{t}/c){p}_{a}\left(\eta \right)\times exp\left[-j\frac{4\pi {f}_{0}{R}_{t}}{c}\right]exp\left[j2\pi {f}_{{\eta}_{c}}\eta \right]\hfill \end{array}$$

#### 3.2. Chirp Scaling Algorithm

- 1.
- The data are transformed into the complex Doppler domain using an azimuth FFT.
- 2.
- Chip scaling is applied, employing a phase multiply, in order to adjust the range migration of the trajectories. Assuming a linear frequency-modulated (FM) pulse, a range invariant radar velocity and a range invariant modified pulse FM rate, ${K}_{m}$ in the Range–Doppler domain, the scaling function [7] is given by$${s}_{sc}({\tau}^{\prime},{f}_{\eta})=exp\left[j\pi {K}_{m}\left(\frac{D({f}_{\eta {}_{r}ef},{V}_{{r}_{ref}})}{D({f}_{\eta},{V}_{{r}_{ref}})}\right){\left({\tau}^{\prime}\right)}^{2}\right]$$$${s}_{1}(\tau ,{f}_{\eta})={s}_{sc}({\tau}^{\prime},{f}_{\eta}){S}_{rd}(\tau ,{f}_{\eta})$$$$\begin{array}{cc}\hfill {s}_{rd}(\tau ,{f}_{\eta})& =A{w}_{r}\left[\tau -\frac{2{R}_{0}}{cD({f}_{\eta},{V}_{r})}\right]{W}_{a}({f}_{\eta}-{f}_{{\eta}_{c}})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[-j\frac{4\pi {f}_{0}{R}_{0}D({f}_{\eta},{V}_{r})}{c}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[j\pi {K}_{m}{\left(\tau -\frac{2{R}_{0}}{cD({f}_{\eta},{V}_{r})}\right)}^{2}\right]\hfill \end{array}$$
- 3.
- The data are transformed into the two-dimensional frequency domain with a range FFT, resulting in the signal given by$$\begin{array}{cc}\hfill {s}_{2}({f}_{\tau},{f}_{\eta})& ={A}_{1}{W}_{r}\left({f}_{r}\right){W}_{a}({f}_{\eta}-{f}_{{\eta}_{c}})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[-j\frac{4\pi {f}_{0}{R}_{0}D({f}_{\eta},{V}_{r})}{c}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[-j\frac{\pi D({f}_{\eta},{V}_{r})}{{K}_{m}D\left({f}_{{\eta}_{ref},{V}_{r}}\right)}{f}_{\tau}^{2}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[-j\frac{4\pi {R}_{0}}{cD({f}_{{\eta}_{ref}},{V}_{{r}_{ref}})}{f}_{\tau}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[-j\frac{4\pi}{c}\left(\frac{1}{D({f}_{\eta},{V}_{{r}_{ref}})}\right.\right.\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left.\left.-\frac{1}{D({f}_{{\eta}_{ref}},{V}_{{r}_{ref}})}\right){R}_{ref}{f}_{\tau}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[j\frac{4\pi {K}_{m}}{{c}^{2}}\left(\frac{D({f}_{\eta},{V}_{{r}_{ref}})}{D({f}_{{\eta}_{ref}},{V}_{{r}_{ref}})}\right)\right.\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left.\times {\left(\frac{{R}_{0}}{D({f}_{\eta},{V}_{r})}-\frac{{R}_{ref}}{D({f}_{\eta},{V}_{r})}\right)}^{2}\right]\hfill \end{array}$$
- 4.
- Range compression, secondary range compression (SRC), and bulk RCMC are applied using a phase multiply with a reference function. This step compensates the second and fourth exponentials from Equation (15), resulting in$$\begin{array}{cc}\hfill {s}_{3}({f}_{\tau},{f}_{\eta})& ={A}_{1}{W}_{r}\left({f}_{r}\right){W}_{a}({f}_{\eta}-{f}_{{\eta}_{c}})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[-j\frac{4\pi {f}_{0}{R}_{0}D({f}_{\eta},{V}_{r})}{c}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[-j\frac{4\pi {R}_{0}}{cD({f}_{{\eta}_{ref}},{V}_{{r}_{ref}})}{f}_{\tau}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[-j\frac{4\pi}{c}\left(\frac{1}{D({f}_{\eta},{V}_{{r}_{ref}})}\right.\right.\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left.\left.-\frac{1}{D({f}_{{\eta}_{ref}},{V}_{{r}_{ref}})}\right){R}_{ref}{f}_{\tau}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[j\frac{4\pi {K}_{m}}{{c}^{2}}\left(\frac{D({f}_{\eta},{V}_{{r}_{ref}})}{D({f}_{{\eta}_{ref}},{V}_{{r}_{ref}})}\right)\times \right.\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left.{\left(\frac{{R}_{0}}{D({f}_{\eta},{V}_{r})}-\frac{{R}_{ref}}{D({f}_{\eta},{V}_{r})}\right)}^{2}\right]\hfill \end{array}$$
- 5.
- Data are converted to the Range–Doppler domain using an IFFT, resulting in a signal in the Range–Doppler domain given by$$\begin{array}{cc}\hfill {s}_{4}(\tau ,{f}_{\eta})& ={A}_{2}{p}_{r}\left(\tau -\frac{2{R}_{0}}{cD({f}_{{\eta}_{ref}},{V}_{{r}_{ref}})}\right){W}_{a}({f}_{\eta},{f}_{{\eta}_{c}})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[-j\frac{4\pi {R}_{0}{f}_{0}D({f}_{\eta},{V}_{r})}{c}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times exp\left[j\frac{4\pi {K}_{m}}{{c}^{2}}\left(1-\frac{D({f}_{\eta},{V}_{{r}_{ref}})}{D({f}_{{\eta}_{ref}},{V}_{{r}_{ref}})}\right)\right.\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left.\times {\left(\frac{{R}_{0}}{D({f}_{\eta},{V}_{r})}-\frac{{R}_{ref}}{D({f}_{\eta},{V}_{r})}\right)}^{2}\right]\hfill \end{array}$$
- 6.
- This step consists of an azimuth compression with a range-varying matched filter, followed by a phase correction and an azimuth IFFT. The matched filter is the complex conjugate of the first exponential of Equation (17). The phase correction is given by the complex conjugate of the second exponential of Equation (17) for linear FM signals. After this step, including azimuth-matched filtering, phase correction and azimuth, the compressed signal at point target is given by$${s}_{5}(\tau ,\eta )={A}_{4}{p}_{r}\left[\tau -\frac{2{R}_{0}}{cD({f}_{{\eta}_{ref}},{V}_{{r}_{ref}})}\right]{P}_{a}(\eta -{\eta}_{c})\times exp\left[j\theta (\tau ,\eta )\right]$$

#### 3.3. Omega-K Algorithm

- 1.
- Transforming the data into the two-dimensional frequency domain using a 2D FFT, resulted in the baseband uncompressed signal given by$${S}_{2df}({f}_{\tau},{f}_{\eta})=A{W}_{r}\left({f}_{\tau}\right){W}_{a}({f}_{\eta}-{f}_{{\eta}_{c}})\times exp\left[j{\theta}_{2df}({f}_{\tau},{f}_{\eta})\right]$$
- 2.
- Computing the reference function multiply, which is usually computed for the midswath range. Assuming the range pulse is an up chirp with an hyperbolic equation, the phase is given by$${\theta}_{2df}({f}_{\tau},{f}_{\eta})=-\frac{4\pi {R}_{0}}{c}\sqrt{{({f}_{0}+{f}_{\tau})}^{2}-\frac{{c}^{2}{f}_{\tau}^{2}}{4{V}_{r}^{2}}}-\frac{\pi {f}_{\tau}^{2}}{{K}_{r}}$$By setting the range and effective radar velocity to their midrange or reference values, the phase of the reference function multiplier (RFM) filter is$${\theta}_{ref}=\frac{4\pi {R}_{ref}}{c}\sqrt{{({f}_{0}+{f}_{\tau})}^{2}-\frac{{c}^{2}{f}_{\eta}^{2}}{4{V}_{{r}_{ref}}^{2}}}+\frac{\pi {f}_{\tau}^{2}}{{K}_{r}}$$After applying the filter, the phase remaining is given by$${\theta}_{RFM}\approx \frac{4\pi ({R}_{0}-{R}_{ref})}{c}\sqrt{{({f}_{0}+{f}_{\tau})}^{2}-\frac{{c}^{2}{f}_{\eta}^{2}}{4{V}_{r}^{2}}}$$The approximation comes from the assumption that ${V}_{r}$ is range-invariant. This step is called bulk compression.
- 3.
- After the previous step, the data are focused at reference range, and are thus necessary to focus the objects at other ranges. This can be done using the Stolt interpolation, which consists of the mapping of the range frequency axis. This interpolation performs the steps seen in the algorithms presented above, RCMC, SRC, and azimuth compression. The idea of this interpolation is to modify the range frequency axis, replacing the square root in Equation (22) with the shifted and scaled variable, ${f}_{0}+{f}_{\tau}^{\prime}$, so that$$\sqrt{{\left({f}_{0}p{f}_{\tau}\right)}^{2}-\frac{{c}^{2}{f}_{\eta}^{2}}{4{V}_{r}^{2}}}={f}_{0}+{f}_{\tau}^{\prime}$$This results map the original variable, ${f}_{\tau}$, into a new one, ${f}_{\tau}^{\prime}$. After the Stolt interpolation, the phase function is given by$${\theta}_{stolt}({f}_{\tau}^{\prime},{f}_{\eta})=-\frac{4\pi ({R}_{0}-{R}_{ref})}{c}({f}_{0}+{f}_{\tau}^{\prime})$$
- 4.
- The last step of this algorithm is a two-dimensional IFFT, transforming the data back into the time domain, and resulting in a compressed complex image.

#### 3.4. Polar Format Algorithm

- 1.
- Map the phase history, or received data, to the correct coordinate of the spatial Fourier transform.
- 2.
- Perform the two-stage interpolation on the K-space data, as described above. This step is going to interpolate the data in a keystone shape to a rectangular grid.
- 3.
- A two-dimensional inverse FFT is performed in the interpolated data, converting the data from the K-space to the Euclidean space, resulting in the final image.

#### 3.5. Matched Filter Algorithm

#### 3.6. Backprojection Algorithm

#### 3.7. Comparison Between Algorithms

## 4. Synthetic-Aperture Radar Imaging Implementations

#### 4.1. Software-Only Implementations

#### 4.2. Comparison Between Software-Only Implementations

#### 4.3. GPU Accelerators for SAR

#### 4.4. Hardware Accelerators

#### 4.5. Comparison Between GPU and Hardware Implementations

#### 4.6. Precision Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AFRL | Air Force Research Laboratory |

ASIC | Application-Specific Integrated Circuit |

BRAM | Block Random-Access Memory |

CMOS | Complementary Metal–Oxide–Semiconductor |

CORDIC | COordinate Rotation DIgital Computer |

CPU | Central Processing Unit |

CUDA | Compute Unified Device Architecture |

DSP | Digital-Signal Processing |

EDMA3 | Enhanced Direct Memory Access |

ESA | European Space Agency |

FFT | Fast Fourier Transform |

FLOP | Floating-Point Operation |

FM | Frequency-Modulated |

FMCW | Frequency-Modulated Continuous-Wave |

FPGA | Field-Programmable Gate Array |

GFLOP | Giga-Floating-Point Operation |

GPU | Graphical Processing Unit |

HPC | High-Performance Computing |

IFFT | Inverse Fast Fourier Transform |

LUT | Look-Up Table |

MPI | Message Passing Interface |

MSE | Mean Squared Error |

NASA | National Aeronautics and Space Administration |

OCM | On-Chip Memory |

OS | Operating System |

PC | Program Counter |

PL | Programmable Logic |

PLR | Process-Level Redundancy |

PS | Processing System |

PSLR | Peak Side Lobe Ratio |

PSNR | Peak Signal-to-Noise Ratio |

RCMC | Range Cell Migration Correction |

SAR | Synthetic-Aperture Radar |

SNR | Signal-to-Noise Ratio |

SoC | System-on-Chip |

SRC | Secondary Range Compression |

SSIM | Structural Similarity |

UAV | Unmanned Aerial Vehicle |

UEMU | Unified EMUlation Framework |

## Appendix A. Mathematical Notation

Symbol | Meaning | Units |
---|---|---|

A | Complex constant | — |

${A}_{0}$ | Complex constant, ${A}_{0}={A}_{0}^{{}^{\prime}}exp\left(i\varphi \right)$ | — |

${A}_{0}^{{}^{\prime}}$ | Magnitude | — |

c | Speed of light | m/s |

${d}_{a}\left({\tau}_{n}\right)$ | Range to the scene center, also referred to as ${r}_{0}$ | m |

${d}_{{a}_{0}\left({\tau}_{n}\right)}$ | Distance between the radar and the pixel | m |

$D\left(\right)$ | Migration factor in Range–Doppler domain | — |

${D}_{y}$ | Diameter of the radar in the azimuth domain | m |

${f}_{0}$ | Carrier frequency | Hz |

${f}_{1}$ | Minimum frequency for every pulse | Hz |

${f}_{k}$ | Frequency sample per pulse | Hz |

${f}_{\eta}$ | Azimuth frequency | Hz |

${f}_{{\eta}_{c}}$ | Azimuth FM rate of point target signal | Hz |

${f}_{{\eta}_{ref}}$ | Reference azimuth frequency | Hz |

${f}_{\tau}$ | Range frequency | Hz |

${f}_{\tau}^{{}^{\prime}}$ | Range frequency after Stolt mapping | Hz |

${g}_{r}$ | Ground reflectivity | — |

$G\left({f}_{r}\right)$ | Frequency domain matched filter | — |

${H}_{az}$ | Second frequency domain matched filter | — |

${k}_{c}$ | Wavenumber at carrier frequency | m |

K | Number of frequency samples per pulse | — |

${K}_{a}$ | Azimuth FM of the point target signal in Range–Doppler domain | Hz/s |

${K}_{m}$ | Range FM of the point target signal in Range–Doppler domain | Hz/s |

${K}_{r}$ | Chirp rate | Hz/s |

L | Half size of the aperture | m |

m | Range bin | — |

${N}_{fft}$ | FFT length | — |

${N}_{p}$ | Number of pulses | — |

${p}_{a}\left(\eta \right)$ | Amplitude of the azimuth impulse response | m |

${p}_{r}\left(\tau \right)$ | Pulse envelope | — |

${P}_{a}\left(\eta \right)$ | IFFT of the window ${W}_{a}\left({f}_{\eta}\right)$ | — |

${r}_{n}$ | Target radial distance from center of aperture | m |

${R}_{0}$ | Slant range of closest approach | m |

${R}_{ref}$ | Reference range | m |

${R}_{t}$ | Instantaneous slant range | m |

${s}_{r}$ | Reflected SAR signal | — |

${s}_{t}$ | Emitted SAR signal | — |

$S({f}_{k},{\tau}_{n})$ | Phase history | — |

${T}_{r}$ | Pulse duration | s |

${V}_{r}$ | Effective radar velocity | m/s |

${V}_{{r}_{ref}}$ | Effective radar velocity at reference range | m/s |

${w}_{a}$ | Azimuth envelope (a sinc-squared function) | — |

${w}_{r}$ | Range envelope (a rectangular function) | — |

${W}_{a}$ | Envelope of the Doppler spectrum of antenna beam pattern | — |

${W}_{r}$ | Envelope of range spectrum of radar data | — |

$({x}_{a}\left(\eta \right),{y}_{a}\left(\eta \right),{z}_{a}\left(\eta \right))$ | Position of the radar | — |

${\Delta}_{a}$ | Azimuth resolution | m |

$\Delta f$ | Frequency step size | Hz |

${\Delta}_{r}$ | Range resolution | m |

$\Delta R$ | Differential range | m |

$\eta $ | Azimuth time | s |

${\eta}_{c}$ | Beam center crossing time relative to the time of closest approach | s |

${\theta}_{n}\left(0\right)$ | Aspect angle of the nth target when radar is at (0, 0) | rad |

${\theta}_{z}$ | Average depression angle of target area | rad |

$\theta (\tau ,\eta )$ | Target phase | — |

$\lambda $ | Wavelength of carrier frequency, ${f}_{0}$ | m |

${\lambda}_{c}$ | Wavelength at carrier fast-time frequency | m |

${\rho}_{max}$ | Maximum polar radius in spatial frequency domain for support of a target at center of the spotlighted area | m |

${\rho}_{min}$ | Minimum polar radius in spatial frequency domain for support of a target at center of the spotlighted area | m |

$\tau $ | Range time | s |

${\tau}_{n}$ | Transmission time of each pulse | s |

$\varphi $ | Phase change resulting from the scattering process | — |

${\varphi}_{0}$ | Polar angle in spatial frequency domain | rad |

${\omega}_{0}$ | Radar signal half bandwidth | rad |

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**Figure 1.**Schematics of an airborne SAR system. The airplane with the SAR system moves along the azimuth direction, illuminating a region called swath. The direction of the antenna is the range direction.

**Figure 2.**Types of reflections generated by different surfaces. From left to right, (

**a**) a flat surface, such as water, makes the wave reflect forward without any reflection back; (

**b**) forest and vegetation generates multiple signal reflections but highly attenuated due to penetration in the trees; (

**c**) cultivation fields are similar to the forest, but have less attenuation; (

**d**) the inclination of mountains generates direct reflections to the sensor on the illuminated side, but no reflection at all on the other; (

**e**) irregular terrain produces scattered reflections; (

**f**) urban buildings tend to create reflections with high intensity, but small streets represent a complete absense of reflections.

**Figure 3.**Illustration of the different operating SAR modes: Stripmap, Spotlight, and Circular SAR. (

**a**) Stripmap SAR, where the platform movement allows for a larger ground cover, as the swath moves along with it. The resolution is lower than other SAR modes, however, the covered area is larger. (

**b**) Spotlight SAR, where the antenna moves along with the platform, illuminating the same region at every instance of time, allowing for higher resolution images. (

**c**) Circular SAR, where the platform moves in a circular motion, illuminating the same region at every instance of time, allowing for higher resolution of images due to the multiangular data collection.

**Figure 4.**Block diagram of the Range–Doppler Algorithm, from [7].

**Figure 5.**Block diagram of the chirp scaling algorithm, from [7].

**Figure 6.**Block diagram of the omega-K algorithm, from [7].

**Figure 8.**Block diagram of the backprojection algorithm, from [27].

**Figure 9.**Images generated using the matched filter algorithm and the Gotcha Volumetric SAR dataset at different azimuth angles: 39°, 1–10° and 1–50°. Due to the computational complexity of the algorithm, larger ranges were not generated, since they would take days. (

**a**) GOTCHA Volumetric dataset image generated using the matched filter algorithm at 39° azimuth. (

**b**) GOTCHA Volumetric dataset image generated using the matched filter algorithm from 1° to 10° azimuth. (

**c**) GOTCHA Volumetric dataset image generated using the matched filter algorithm from 1° to 50° azimuth.

**Figure 10.**Images from the GMTI dataset generated using the backprojection algorithm, matched filter algorithm, and fast-factorized backprojection algorithm. (

**a**) GMTI dataset image generated using the backprojection algorithm. (

**b**) GMTI dataset image generated using the matched-filter algorithm. (

**c**) GMTI dataset image generated using the fast-factorized backprojection algorithm.

**Figure 11.**Images from the point target dataset generated using the backprojection algorithm, matched filter algorithm and fast-factorized backprojection algorithm. (

**a**) Point target dataset image generated using the backprojection algorithm. (

**b**) Point target dataset image generated using the matched-filter algorithm. (

**c**) Point target dataset image generated using the fast-factorized backprojection algorithm.

**Figure 12.**Images generated using the backprojection algorithm and the Gotcha Volumetric SAR dataset at different azimuth angles: 39°, 1–10°, 1–50°, 1–100° and 1–360°. (

**a**) GOTCHA Volumetric dataset image generated using the backprojection algorithm at 39° azimuth. (

**b**) GOTCHA Volumetric dataset image generated using the backprojection algorithm from 1° to 10° azimuth. (

**c**) GOTCHA Volumetric dataset image generated using the backprojection algorithm from 1° to 50° azimuth. (

**d**) GOTCHA Volumetric dataset image generated using the backprojection algorithm from 1° to 360° azimuth. (

**e**) GOTCHA Volumetric dataset image generated using the backprojection algorithm from 1° to 100° azimuth.

**Figure 13.**Images generated using the fast-factorized backprojection algorithm and the Gotcha Volumetric SAR dataset at different azimuth angles: 39°, 1–10°, 1–50°, 1–100° and 1–360°. (

**a**) GOTCHA Volumetric dataset image generated using the fast-factorized backprojection algorithm at 39° azimuth. (

**b**) GOTCHA Volumetric dataset image generated using the fast-factorized backprojection algorithm from 1° to 10° azimuth. (

**c**) GOTCHA Volumetric dataset image generated using the fast-factorized backprojection algorithm from 1° to 50° azimuth. (

**d**) GOTCHA Volumetric dataset image generated using the fast-factorized backprojection algorithm from 1° to 100° azimuth. (

**e**) GOTCHA Volumetric dataset image generated using the fast-factorized backprojection algorithm from 1° to 360° azimuth.

**Figure 14.**Images generated using the backprojection algorithm with the PERFECT dataset (sizes small, medium and large). (

**a**) Perfect dataset image (size small) generated using the backprojection algorithm. (

**b**) Perfect dataset image (size medium) generated using the backprojection algorithm. (

**c**) Perfect dataset image (size large) generated using the backprojection algorithm.

**Figure 15.**Images generated using the polar format algorithm with range interpolation with the PERFECT dataset (sizes small, medium and large). (

**a**) Perfect dataset image (size small) generated using the polar format algorithm with range interpolation. (

**b**) Perfect dataset image (size medium) generated using the polar format algorithm with range interpolation. (

**c**) Perfect dataset image (size large) generated using the polar format algorithm with range interpolation.

**Figure 16.**Images generated using the polar format algorithm with azimuth interpolation with the PERFECT dataset (sizes small, medium and large). (

**a**) Perfect dataset image (size small) generated using the polar format algorithm with azimuth interpolation. (

**b**) Perfect dataset image (size medium) generated using the polar format algorithm with azimuth interpolation. (

**c**) Perfect dataset image (size large) generated using the polar format algorithm with azimuth interpolation.

**Figure 17.**Small dataset of the PERFECT suite in three versions: golden reference, original implementation and single-precision only. (

**a**) Golden reference of the small dataset of the PERFECT suite. (

**b**) Small dataset of the PERFECT suite generated using the original provided code, with variables in double and single-precision. (

**c**) Small dataset of the PERFECT suite generated using the original provided code, with variables in single-precision only.

**Table 1.**Most used frequency bands in SAR and respective wavelengths [3].

Frequency Band | Ka | Ku | X | C | S | L | P |
---|---|---|---|---|---|---|---|

Frequency [GHz] | 40–25 | 17.6–12 | 12–7.5 | 7.5–3.75 | 3.75–2 | 2–1 | 0.5–0.25 |

Wavelength [cm] | 0.75–1.2 | 1.7–2.5 | 2.5–4 | 4–8 | 8–15 | 15–30 | 60–120 |

**Table 2.**Range and azimuth resolution of Stripmap, Spotlight, and Circular SAR [1].

Range Resolution | Azimuth Resolution | |
---|---|---|

Stripmap SAR | ${\Delta}_{r}=\frac{c\pi}{2{\omega}_{0}}$ | ${\Delta}_{a}=\frac{{D}_{y}}{2}$ |

Spotlight SAR | ${\Delta}_{r}=\frac{c\pi}{2{\omega}_{0}}$ | ${\Delta}_{a}=\frac{{r}_{n}{\lambda}_{c}}{4Lcos{\theta}_{n}\left(0\right)}$ |

Circular SAR | ${\Delta}_{r}=\frac{\pi}{{\rho}_{max}-{\rho}_{min}}$ | ${\Delta}_{a}=\frac{\pi}{2{k}_{c}cos{\theta}_{z}sin{\varphi}_{0}}$ |

**Table 3.**Comparison between the Range–Doppler, chirp scaling, omega-K, polar format, and backprojection algorithms.

Algorithm | Main Features |
---|---|

Range–Doppler | Frequency domain for range and azimuth; uses block processing; range cell migration correction between range and azimuth; simple one-dimensional operations; not good for high-squint angles. |

Chirp Scaling | Offers a good trade-off in terms of simplicity, efficiency, and accuracy; high computing load; limited accuracy for high squint, and wide-aperture uses. |

Omega-K | Commonly used for processing raw stripmap SAR in frequency domain; good results for high-squint angles. |

Polar Format | good for cases where resolution is close to the nominal wavelength of the radar. |

Backprojection | Time-domain processing; most complex; better image. |

**Table 4.**Computational load of the Range–Doppler, chirp scaling and omega-K algorithms in GFLOP. These values were calculated and published in [7].

Algorithm | GFLOPs |
---|---|

Range–Doppler | 5.61 |

Chirp scaling | 4.05 |

Omega-K | 4.38 |

**Table 5.**Execution times and SSIM values of the images generated by the backprojection algorithm, matched filter algorithm, and fast-factorized backprojection algorithm. The SSIM values were obtained in comparison with the backprojection algorithm images, which is why the algorithm does not have a value. The asterisk in the fast-factorized backprojection image indicates that the SSIM value of the GMTI was obtained compared with the image the matched filter algorithm generated, instead of the backprojection. This is due to the differences in the algorithm implementations, where the backprojection leaves dark triangles in the corners, while the other two algorithms leave an extremely unfocused area. Since the unfocused is more similar, the comparison is assumed to be fairer this way.

Backprojection Algorithm | Matched-Filter Algorithm | Fast-Factorized Backprojection Algorithm | ||||
---|---|---|---|---|---|---|

Time | Time | SSIM | Time | SSIM | ||

Gotcha | 39° | 2.13 s | 364.30 s | 0.999101 | 1.14 s | 0.861199 |

1–10° | 20.49 s | 3755.68 s | 0.999100 | 7.85 s | 0.824825 | |

1–50° | 104.36 s | 18625.73 s | 0.999117 | 36.60 s | 0.832565 | |

1–100° | 208.74 s | — | — | 73.10 s | 0.817611 | |

1–360° | 759.96 s | — | — | 266.99 s | 0.817061 | |

GMTI | 387.86 s | 24791.79 s | 0.843764 | 71.91 s | 0.946678 * | |

Point Target | 2.04 s | 107.34 s | 0.995989 | 1.05 s | 0.904601 |

**Table 6.**Execution times, and the average pixels per second, of the formation of the PERFECT suite dataset image using the backprojection algorithm and the polar format algorithm with two different interpolations: range and azimuth.

Backprojection Algorithm | Polar Format Algorithm (Range) | Polar Format Algorithm (Azimuth) | |
---|---|---|---|

Small | 5.844 s | 0.056 s | 0.065 s |

(512 × 512) | 44.8 k PPS | 4681 k PPS | 4032 k PPS |

Medium | 57.913 s | 0.205 s | 0.238 s |

(1024 × 1024) | 18.1 k PPS | 5115 k PPS | 4405 k PPS |

Large | 565 s | 0.798 s | 1.054 s |

(2048 × 2048) | 7.4 k PPS | 5256 k PPS | 3979 k PPS |

**Table 7.**Comparison between the different GPU/many-core implementations of SAR image formation algorithms described in Section 4.3. This table includes the implemented algorithm, execution time and average pixels per second (PPS), whether is real-time, device used and number of cores, image dimension, speedups and additional notes.

Alg. | PPS | Execution Time | Device | Image Dimension | Notes | |
---|---|---|---|---|---|---|

[34] | BP | n/a | 315 s 79 s 51 s | NVIDIA GeForce GT 650M (384 cores) NVIDIA GeForce GTX 660 Ti (1344 cores) NVIDIA Tesla K20c (2496 cores) | — | — |

[36] | BP | n/a | Real-time | NVIDIA GTX 285 (240 cores) | — | — |

[38] | BP | n/a | 40× to 60× of speedup, depending on block size | NVIDIA Quadro FX 5600 (128 cores) NVIDIA Tesla C1060 (240 cores) | 501 × 501 px 8 K × 384 px | SNR > 30 dB. |

[41] | BP | 1747 k | 0.15 s (real-time) | 4 nodes, each with NVIDIA Quadro 5600 (128 cores) | 512 × 512 px * | Speedup of 31×. * More image sizes were tested, with different speedups. For more details, check Section 4.3. |

[40] | BP | n/a | Real-time | Node with dual-socket Intel Xeon E5-2670 (8 cores) processors and two Intel Knights Corner co-processor (60 cores) | 3 K × 3 K px * | * Using 16 nodes, 13 K × 13 K images can be generated in real-time. |

[39] | BP | 63,337 k | 1.0 s (real-time) | 4 nodes, each with two Intel Xeon E5-2690v3 (12 cores) and 4 NVIDIA Tesla M60 (2048 cores) | 8965 × 7065 px | Speedup of 11.5× compared to 1 node (4 GPUs). |

[42] | RD | 64,000 k | 0.25 s (real-time) | TMX320C6678 DSP (8 cores) | 4 K × 4 K px | 10 Watts of power consumption. |

**Table 8.**Comparison between the different hardware implementations of SAR image formation algorithms, described in Section 4.4. This table includes the algorithm, execution time and average pixels per second (PPS), whether the work is real-time or not, image dimensions and quality metrics. Notes marked with an asterisk are displayed in the last column of Table 9, named Notes.

Ref | Alg. | PPS | Image Dimension | Execution Time | Real-Time | Quality Metrics |
---|---|---|---|---|---|---|

[43] | BP | 60 k | $1500\times 40$ px | 1.0 s | ✓ | — |

[27] | BP | n/a | $501\times 501$ px | — | — | SSIM > 0.99 |

[45] | BP | 2184 k 1456 k 728 k 383 k | $256\times 256$ px, $512\times 512$ px, $1024\times 1024$ px, $2048\times 2048$ px | 0.03 s 0.18 s 1.44 s 10.94 s | ✗ | — |

[26] | BP | 2085 k | $501\times 501$ px | 120.34 ms | ✓ | — |

[46] | RD | 697 k | $2048\times 4096$ px | 12.03 s | ✗ | PSNR and MSE almost identical |

[47] | PF | 16777 k | $4096\times 4096$ px | 1.0 s | ✓ | Range PSLR: −28 dB Range resolution: 2.65 m Azimuth PSLR: −40 dB Azimuth resolution: 1.03 m |

[48] | BP | n/a | — | 146 ms to 351 ms | ✓ | Depending on the dataset, error percentage ranges from 0.9% to 5.6%. |

[49] | RD | 13530 k | $2048\times 2048$ px | 0.31 s | ✓ | Range PLSR of −44.11 dB and azimuth PSLR between −46.44 dB and −39.40 dB |

**Table 9.**Comparison between the different hardware implementations of SAR image formation algorithms, described in Section 4.4. This table includes the device reference, device frequency, hardware resources, power consumption and additional comments.

Ref | Device | Device Frequency | Hardware Resource Occupation | Power Consumption | Notes |
---|---|---|---|---|---|

[43] | Xilinx Virtex-7 | 300 MHz | 60% | 10W | This work was simulated. |

[27] | ASIC | 1.2 GHz | — | — | — |

[45] | Xilinx ML605 | 100 MHz | 78% LUTs 62% BRAMs 40% DSPs | — | Speedup of 68 with a parallelization factor of 8 compared to the execution on a quad-core Intel i5 3.2 GHz. |

[26] | Altera Arria-V SoC | 133 MHz | — * | 26.55 W | * 20 BP cores fitted into the device. |

[46] | 4 Xilinx Virtex-6-550T | 130 MHz | 67% LUTs | 85 W | * The paper did not specify values of quality metrics. |

[47] | Xilinx Kintex-7 | 200 MHz | 68% LUTs 48% registers 42% BRAM 96% DSPs | — | — |

[48] | Dual 2.2 GHz Intel Xeon PC and Anapolis Microsystems WildStar II FPGA board with two Virtex-II FPGAs | 133 MHz | — | — | — |

[49] | DE2-115 Terasic Development kit with Cyclone E IV | 50 MHz | 57% LUTs 31% registers 21% BRAMs 56% DSPs. | — | — |

**Table 10.**Comparison of image quality and execution time for the small dataset of the PERFECT suite between the original version and the float-only. These images are compared to the golden reference, which is provided with the dataset.

SNR | SSIM | Time | |
---|---|---|---|

Original | 139.16 dB | 1.000000 | 270.80 s |

Single-Precision | 15.02 dB | 0.893861 | 259.75 s |

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**MDPI and ACS Style**

Cruz, H.; Véstias, M.; Monteiro, J.; Neto, H.; Duarte, R.P. A Review of Synthetic-Aperture Radar Image Formation Algorithms and Implementations: A Computational Perspective. *Remote Sens.* **2022**, *14*, 1258.
https://doi.org/10.3390/rs14051258

**AMA Style**

Cruz H, Véstias M, Monteiro J, Neto H, Duarte RP. A Review of Synthetic-Aperture Radar Image Formation Algorithms and Implementations: A Computational Perspective. *Remote Sensing*. 2022; 14(5):1258.
https://doi.org/10.3390/rs14051258

**Chicago/Turabian Style**

Cruz, Helena, Mário Véstias, José Monteiro, Horácio Neto, and Rui Policarpo Duarte. 2022. "A Review of Synthetic-Aperture Radar Image Formation Algorithms and Implementations: A Computational Perspective" *Remote Sensing* 14, no. 5: 1258.
https://doi.org/10.3390/rs14051258