Minimum Entropy SAR Autofocus

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


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


L Outline


) Auto focu s algorithms








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)


LMinimum Entropy Algorithm

�i = 0 H(�i) S;@S

�i

Input Test Output

NumericalMinimizer��i


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 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


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)


L Outline





) Properties of ent ropy






Examples of Image Entropy

S = 0 S = log(3) S = log(7)

S = 10.585 S = 11.245 S = 11.625


Invariance Properties of Entropy

• Scale invariance

• Permutation invariance

1 2 3

4 5 6 1

2

34

56


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


L Outline






) Example s of the ent ropy curve




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


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


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


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


L Outline







) Application s and compariso n wit h PGA



LAutofocus on Low-Contrast Image

ADTS Stockbridge Data

Original ImageImage Blurred with

Crossrange Quadratic Phase

Minimum-Entropy Autofocus PGA


LAutofocus with High-Order Phase Errors

CARABAS Keystone Data (Mission 2, Pass 2)

Original Image Image with High-Order Phase Error

Minimum-Entropy PGA


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)


LMinimum-Entropy on 2D Phase Errors

ADTS Stockbridge Data

Original ImageImage Blurred with 2DQuadratic Phase Error Minimum-Entropy Autofocus


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


Minimum Entropy SAR Autofocus

Documents

makes strong assumptions

mit lincoln laboratory

lautofocus algorithms properties

pga concluding remarks

entropy curve applications

original image

point scatterers

image blurred