minimum entropy sar autofocus
TRANSCRIPT
L
Minimum Entropy SAR Autofocus
Ali F. YegulalpMIT Lincoln Laboratory
10 March 1999
Presented at ASAP ’99
MIT Lincoln Laboratory
L Outline
� Int roductio n to SAR and the auto focu s problem
� Auto focu s algorithms
– Mini mum entropy auto focus
– Phase gradien t autofocu s (PGA)
� Properties of ent ropy
� Example s of the ent ropy curve
� Application s and compariso n wit h PGA
� Concludin g remarks
MIT Lin coln Laboratory
L Introduction to SAR
Phase History Data SAR Image
MatchedFilterBank
Data Collection
Range
Ape
rtur
e
Range
Cro
ss-r
ange
MIT Lincoln Laboratory
L The SAR Autofocus Problem
Error Sources
- Off-track motion- Terrain height variation- Index of refraction- Antenna pattern
Blurring Filter
B(�1; �2; : : : ; �p)
Autofocus Filter
H(^�1; ^�2; : : : ; ^�p)
H � B�1
AutofocusAlgorithm
Estimate
^�1; ^�2; : : : ; ^�p
Ideal Image Blurred Image Autofocused Image
MIT Lincoln Laboratory
L Outline
� Int roductio n to SAR and the auto focu s problem
) Auto focu s algorithms
– Mini mum entropy auto focus
– Phase gradien t autofocu s (PGA)
� Properties of ent ropy
� Example s of the ent ropy curve
� Application s and compariso n wit h PGA
� Concludin g remarks
MIT Lin coln Laboratory
LReason s for Usin g Entropy
� Sensitiv e measur e of image focu s quality
� Smoot h dependenc e on auto focu s paramete rs facilitate s numericalminimization
� No specifi c target or clutte r mode l required
� Extensiv e literatur e on blin d deco nvolutio n usin g ent ropy
– Wiggins ’ mini mum entropy deco nvolutio n (MED) for seismi c reflectiondata (1977)
– Shalv i & Weinstei n demonstrat e ent ropy-base d deco nvolutionconverges to correc t solutio n unde r fair ly general condition s (1990)
Data must be non-G aussian
Transfer fun ctio n of blu rr ing filt er must not have zeros
– Cafforio , Prati , and Rocc a conside r mini mum entropy for Seasat SARautofocu s (1991)
MIT Lin coln Laboratory
LMinimum Entropy Algorithm
�i = 0 H(�i) S;@S
�i
Input Test Output
NumericalMinimizer��i
MIT Lincoln Laboratory
LDefinition of Image Entropy
� For image X with complex-valued pixels xnm, the Shannon entropy is
S(X) = �
Xnm
pnm log(pnm);
where
pnm =
jxnmj2
P
andP = total power =
Xnm
jxnmj2
� Renyi entropy generalizes Shannon entropy to family of entropyfunctions smoothly parameterized by r > 0:
Sr(X) =
11� rlog(X
nm
prnm)
� As r ! 1, Renyi ! Shannon
� No obvious information-theoretic interpretation
MIT Lincoln Laboratory
MIT Lincoln LaboratoryASAP99-2AFY 4/2/99
Entropy and Gradient Calculation
nx
pα
kx~ ky~ ny 2log nnn yyz =
kz~
( ) { }*~~Im2
kkk
pp
zykfN
S ∑−=∂∂α
*n
nn yzS ∑−=
( )∑= p
pp kfi
k ehα~
FFT FFT
FFT
MIT Lincoln LaboratoryASAP99-1AFY 4/2/99
Phase Gradient Autofocus
Z
ZZR H=reigenvecto top=v
ri
ii v
vh =
Input Image Find BrightestPoints Center
FFT
FFT
IFFT
Output Image
Phase Correction
Data Matrix
LPhilosophica l Compariso n of PGA and
Mini mum Entropy
� PGA
– Makes st rong assumption s abou t clutte r (poin t scattere rs)
– Makes weak assumption s abou t filte r (constan t modulu s transferfunction)
� Mini mum Entropy
– Makes weak assumption s abou t clutte r (non-Gaussian)
– Makes st rong assumption s abou t filte r (parameteri zed filte r based onphas e error model)
MIT Lin coln Laboratory
L Outline
� Int roductio n to SAR and the auto focu s problem
� Auto focu s algorithms
– Mini mum entropy auto focus
– Phase gradien t autofocu s (PGA)
) Properties of ent ropy
� Example s of the ent ropy curve
� Application s and compariso n wit h PGA
� Concludin g remarks
MIT Lin coln Laboratory
MIT Lincoln LaboratoryASAP99-3AFY 4/2/99
Examples of Image Entropy
S = 0 S = log(3) S = log(7)
S = 10.585 S = 11.245 S = 11.625
MIT Lincoln LaboratoryASAP99-4AFY 4/2/99
Invariance Properties of Entropy
• Scale invariance
• Permutation invariance
1 2 3
4 5 6 1
2
34
56
MIT Lincoln LaboratoryASAP99-5AFY 4/2/99
1 2 3
4 5 6
klog ρρρ ∑∑ −=k
kkk
k SS
=kS Entropy of region k
=kρ Fraction of total power in region k
Subadditivity of Entropy
• First term is weighted average of subimage entropies
• Second term is entropy among subimages
• The Shannon entropy is the only image function withsubadditivity, scale invariance, and permutation invariance
LEntropy of Noise
� Assume pixels are I.I.D. random variables
� Expected entropy of pure noise image with N pixels isEfSg � logN � Efjxj2 log(jxj2)g
� For zero-mean complex Gaussian noise, the entropy is
EfSg � logN � � logN � 0:422784
� The expected entropy of Gaussian noise is invariant under imagefiltering
MIT Lincoln Laboratory
L Outline
� Int roductio n to SAR and the auto focu s problem
� Auto focu s algorithms
– Mini mum entropy auto focus
– Phase gradien t autofocu s (PGA)
� Properties of ent ropy
) Example s of the ent ropy curve
� Application s and compariso n wit h PGA
� Concludin g remarks
MIT Lin coln Laboratory
LEntropy of Point Scatterers in Gaussian Noise
� Simulate one-dimensional data with randomly located point scatteringcenters and complex Gaussian noise
� Plot Shannon entropy as a function of quadratic phase error
−3000 −2000 −1000 0 1000 2000 30003.6
3.8
4
4.2
4.4
4.6
4.8
5
5.2
Quadratic Phase Error (degrees)
Sh
an
no
n E
ntr
op
y
50 100 150 200 250
−40
−35
−30
−25
−20
−15
−10
−5
0
Pixel Number
Po
we
r (d
B)
Point Scatterers in Gaussian Noise Shannon Entropy vs. Quadratic Phase
MIT Lincoln Laboratory
L Closed Form Solution for Point Scatterers� Assume point scatterer centered at t0, with quadratic phase error � and
Gaussian spectral window:
x(t) =Z1
�1
ei!(t�t0)�i�!2��!2d!;
jx(t)j2 =
�r�2 + �2e��(t�t0)2
2(�2+�2)
� The entropy is
S =
12�1
2log(�
2�) +1
2log(�2 + �2)
� Entropy is minimized at � = 0, as expected
� Multiple point scatterers can be approximated using subadditivity
MIT Lincoln Laboratory
LHeight-of-Focus Example
CARABAS VHF SAR� Use image from CARABAS VHF SAR
� Apply height-of-focus correction filter based on known aircraft motion
� Plot Shannon entropy as a function of terrain height parameter
Focused Image Entropy vs. Height-of-Focus
−300 −200 −100 0 100 200 30010.9
10.92
10.94
10.96
10.98
11
11.02
Height−of−Focus Error (m)
Sh
an
no
n E
ntr
op
y
MIT Lincoln Laboratory
LEntropy of Random Clutter
� Generate random clutter image where each pixel is drawn independentlyfrom a log-normal distribution
� Plot entropy as a function of quadratic phase error
−3000 −2000 −1000 0 1000 2000 30004
4.5
5
5.5
6
6.5
7
7.5
Quadratic Phase Error (degrees)
Sha
nnon
Ent
ropy
Random Clutter Image
Entropy vs. Quadratic Phase Error
MIT Lincoln Laboratory
L Outline
� Int roductio n to SAR and the auto focu s problem
� Auto focu s algorithms
– Mini mum entropy auto focus
– Phase gradien t autofocu s (PGA)
� Properties of ent ropy
� Example s of the ent ropy curve
) Application s and compariso n wit h PGA
� Concludin g remarks
MIT Lin coln Laboratory
LAutofocus on Low-Contrast Image
ADTS Stockbridge Data
Original ImageImage Blurred with
Crossrange Quadratic Phase
Minimum-Entropy Autofocus PGA
MIT Lincoln Laboratory
LAutofocus with High-Order Phase Errors
CARABAS Keystone Data (Mission 2, Pass 2)
Original Image Image with High-Order Phase Error
Minimum-Entropy PGA
MIT Lincoln Laboratory
LHigh-Order Phase Error Function
0 100 200 300 400 500 600−1200
−1000
−800
−600
−400
−200
0
Frequency bin
Pha
se e
rror
(de
gree
s)
MIT Lincoln Laboratory
LMinimum-Entropy on 2D Phase Errors
ADTS Stockbridge Data
Original ImageImage Blurred with 2DQuadratic Phase Error Minimum-Entropy Autofocus
MIT Lincoln Laboratory
LConcludin g Remarks
� New component s develope d for mini mum-ent ropy method
– Parameteri zed filte r design
Exploit s knowledge of blurr ing filt er structure
Drasticall y reduces dimension of space for minimizati on procedure
– Gradien t for mula for numerica l minimization
� Mini mum entropy can outper form PGA at expens e of greatercomputation
� Mini mum-ent ropy has the mos t benefi t over PGA unde r certainci rcumstances:
– Low-contras t clutter
– Well-modele d phas e errors
– Low-dimensiona l paramete r spac e for phas e errors
– Severe phas e errors
MIT Lin coln Laboratory