1. PACE: An Autofocus Algorithm For SAR
Tuesday, September 6, 3:40 PM
Jesse Kolman, PhD
Lockheed Martin IS&S
Some preparation of this material was done under US Government Contract.

2. Focusing Spotlight Mode Data
where focused (azimuth compressed) image
unfocused (range compressed) data
i = range bin
n = image azimuth position
k = aperture position

3. SAR Autofocus Image corrupted by phase error
Multiplicative error in azimuth phase history domain
Independent of range
Results in blurring due to wider impulse response
Benefits of estimating phase error
Image quality improvement
Phase error of intrinsic value in some applications

4. Causes of Phase Error Non-planar terrain Platform motion deviations
Atmospheric effects
Hardware characteristics
Software approximations

5. Phase Error Model , where Uncorrupted azimuth phase history
Data corrupted by phase error
Phase correction

6. Phase Adjustment by Contrast Enhancement (PACE)
Maximizes contrast
Uses gradient-based optimization algorithm
Versions exist for both strip-mapping and spotlight mode SAR
Fast quadratic version exists

7. Contrast Definition
Contrast of image is average of contrast of range bins
Contrast of range bin is ratio of standard deviation of pixel magnitudes to mean of pixel magnitudes

8. Optimization Algorithm
Contrast is maximized using conjugate gradients or quasi-Newton algorithm
Requires explicit formula for gradient of contrast with respect to phase corrections
Iterative
Typically requires 10 – 100 iterations
Each iteration is itself iterative, requiring 2 – 3 function and gradient calculations

9. Gradient of Contrast
, where
and

10. Computational Efficiency
Bulk of calculations are FFTs
Algorithm is parallelizable
Adjustable tradeoff between speed and accuracy
Number of iterations
Fraction of range bins used

11. High Order Phase Error

12. Image Blurred by High Order Phase Error

13. Image Restored Using PACE

14. SAR Image Before and After PACE
Contrast = 0.626
Contrast = 0.759

15. Contrast vs. Iteration

16. Assessment of Phase Estimate Accuracy
Real SAR image fully focused using algorithm to be tested
Phase error incorporated into azimuth phase history data
Additive, white, Gaussian noise applied in measurement domain
Autofocus performed
Result compared to applied phase error

17. RMS Errors for PGA and PACE
SNR
PGA
PACE
No Noise
6.0
0.0064
10 dB
6.2
2.0
3dB
7.2
4.9
0dB
8.2
8.1
Residual RMS Errors in Degrees

18. Accuracy vs. Speed Comparison
Phase Gradient Algorithm (PGA) run to convergence, time and RMS error recorded
PACE run for maximum number of iterations possible in less time than PGA required
PACE run for minimum number of iterations required to produce lower RMS error than PGA

19. Accuracy vs. Speed for PGA and PACE
Algorithm
CPU time (seconds)
RMS Error (degrees)
PGA
13.4
6.0
PACE
13.1
3.2
6.4
5.8

20. Advantages of PACE Nonparametric Highly accurate
Computationally efficient
Robust in the presence of noise
Virtually independent of scene content

21. Quadratic Version of PACE
Common causes result in quadratic phase error
Constant terrain height error
Azimuth velocity discrepancy
Range acceleration
Single parameter problem reduces optimization algorithm to line search
Derivative of contrast with respect to parameter still needed for efficient maximization

22. Quadratic Phase Error Equations
Model for phase error
Derivative of contrast with respect to parameter a

23. Quadratic Autofocus Algorithm Comparison
Standard algorithms (Mapdrift, Phase Difference) have nominal accuracy of 90 degrees
One function call to PACE takes about twice as long as these algorithms
Standard algorithms can achieve improved accuracy proportional to increased processing time

24. Accuracy vs. Function Calls for Quadratic PACE
Absolute Error
Function Calls 3 5 7 10 14 16 18

25. Conclusions
PACE is an accurate and efficient nonparametric autofocus algorithm
Produces maximum contrast
Performs well in the presence of noise
Does not depend explicitly on scene content
Quadratic version achieves accurate results with fast one-dimensional search