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.
Focusing Spotlight Mode Data
N 1
f i ( n)   u i ( k ) e
j 2nk / N
k 0
where f i (n)  focused (azimuth compressed) image
u i (k )  unfocused (range compressed) data
i =
range bin
n=
image azimuth position
k=
aperture position
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
Causes of Phase Error
• Non-planar terrain
• Platform motion deviations
• Atmospheric effects
• Hardware characteristics
• Software approximations
Phase Error Model
j ( k )
~
ui (k )  u i (k ) e ,
u i (k ) 
~
u i (k ) 
 (k ) 
where
Uncorrupted azimuth phase history
Data corrupted by phase error
Phase correction
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
Contrast Definition
1
C
M
M 1
1
i 
N
N 1
i 
σi

i 0 μi
f
n 0
i
Contrast of image is average
of contrast of range bins
( n)
2
1 N 1
 f i ( n)   i 

N n 0
Contrast of range
bin is ratio of
standard deviation
of pixel magnitudes
to mean of pixel
magnitudes
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
Gradient of Contrast
M 1
dC
*
   i Imu i (k )q(k ) , where
d (k ) i 0
1  1  i 
   and
i 
MN   i  i 
N 1
q(k )  
n 0
f i (n)  j 2nk / N
e
f i ( n)
Computational Efficiency
• Bulk of calculations are FFTs
• Algorithm is parallelizable
• Adjustable tradeoff between speed and accuracy
– Number of iterations
– Fraction of range bins used
High Order Phase Error
Image Blurred by High Order Phase Error
Image Restored Using PACE
SAR Image Before and After PACE
Contrast = 0.626
Contrast = 0.759
Contrast vs. Iteration
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
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
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
Accuracy vs. Speed for PGA and PACE
Algorithm
CPU time
(seconds)
RMS Error
(degrees)
PGA
13.4
6.0
PACE
13.1
3.2
PACE
6.4
5.8
Advantages of PACE
• Nonparametric
• Highly accurate
• Computationally efficient
• Robust in the presence of noise
• Virtually independent of scene content
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
Quadratic Phase Error Equations
• Model for phase error
 (k )  ak  k 
0
• Derivative of contrast with respect to parameter a
dC M 1
 N 1

2 *
   i Im  (k  k 0) ui (k )q(k )
da i 0
 n 0

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
Accuracy vs. Function Calls for Quadratic PACE
Absolute Error
0.777504
0.033050
0.001378
0.000307
0.000058
0.000021
0.000009
Function Calls
3
5
7
10
14
16
18
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