PACE: An Autofocus Algorithm For SAR
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Transcript of PACE: An Autofocus Algorithm For SAR
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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.
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Focusing Spotlight Mode Data
ekunf NnkjN
kii
/21
0
)()(
)(nf iwhere focused (azimuth compressed) image
unfocused (range compressed) data
i = range bin
n = image azimuth position
k = aperture position
)(kui
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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
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Causes of Phase Error
• Non-planar terrain
• Platform motion deviations
• Atmospheric effects
• Hardware characteristics
• Software approximations
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Phase Error Model
)(kui
)(k
ekuku kjii
)()(~)(
)(~ ku i
, where
Uncorrupted azimuth phase history
Data corrupted by phase error
Phase correction
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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
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Contrast Definition
1
0
1 M
i i
i
μσ
MC
1
0
)(1 N
nii nf
N
1
0
2)(
1 N
niii nf
N
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
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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
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Gradient of Contrast
1
0
* )()(Im)(
M
iii kqku
kd
dC
i
i
ii MN
11
enf
nfkq Nnkj
N
n i
i /21
0 )(
)()(
, where
and
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Computational Efficiency
• Bulk of calculations are FFTs
• Algorithm is parallelizable
• Adjustable tradeoff between speed and accuracy
– Number of iterations
– Fraction of range bins used
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High Order Phase Error
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Image Blurred by High Order Phase Error
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Image Restored Using PACE
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SAR Image Before and After PACE
Contrast = 0.626 Contrast = 0.759
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Contrast vs. Iteration
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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
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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
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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
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Accuracy vs. Speed for PGA and PACE
AlgorithmCPU time (seconds)
RMS Error (degrees)
PGA 13.4 6.0
PACE 13.1 3.2
PACE 6.4 5.8
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Advantages of PACE
• Nonparametric
• Highly accurate
• Computationally efficient
• Robust in the presence of noise
• Virtually independent of scene content
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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
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Quadratic Phase Error Equations
• Model for phase error
• Derivative of contrast with respect to parameter a
kkak 0)(
1
0
*21
0
)()()(Im 0
N
ni
M
ii kqkukk
da
dC
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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
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Accuracy vs. Function Calls for Quadratic PACE
Absolute Error Function Calls
0.777504 3
0.033050 5
0.001378 7
0.000307 10
0.000058 14
0.000021 16
0.000009 18
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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