mujdatcetin sar12 public.ppthelper.ipam.ucla.edu/publications/sar2012/sar2012_10407.pdf ·...

S it D i R d I iSparsity‐Driven Radar Imaging

Müjdat Ç[email protected]

Faculty of Engineering and Natural Sciences, Sabancı Universityİstanbul Turkeyİstanbul, Turkey

Contributors: Özben Önhon, Sadegh Samadi, W. Clem Karl, Kush Varshney,

Müjdat Çetin – Challenges in Synthetic Aperture Radar, IPAM, UCLA, February 8, 2012.

Randy Moses, Lee Potter, Özge Batu, Ivana Stojanovic, Alan S. Willsky.

Sparsity and Compressibility of SignalsSparsity and Compressibility of Signals


Involvement in Sparsity• Feature‐enhanced SAR imaging [1999‐2002]• Evaluation of the impact of sparsity‐driven SAR on classification [2000‐2003]• Array signal processing – acoustic source localization [2001‐2004]

HRR fil t ti [2002]• HRR profile reconstruction [2002]• Optimal sparse representations (l 1‐l0 & l p‐l0 equivalence conds.) [2002‐2004]• Homotopy continuation for sparse signal representation [2004‐2005]• Ultrasound imaging for non destructive evaluation [2003 2006]• Ultrasound imaging for non‐destructive evaluation [2003‐2006]• Wide‐angle SAR imaging [2003‐2005]• Passive radar imaging [2004‐2005]• Sparse communication channel estimation [2004‐2005]Sparse communication channel estimation [2004 2005]• Anisotropic SAR imaging using structured dictionaries [2004‐2008]• Automatic hyperparameter choice [2007‐2009]• Multiple feature‐enhanced imaging [2008‐2011]p g g [ ]• Compressed sensing in radar imaging [2008‐2009]• Joint imaging and model error correction [2008‐ongoing]• SAR despeckling & its parallel implementation [2011‐ongoing]


p g p p g g• Moving target imaging [2011‐ongoing]

Linear Inverse Problems

Observation Matrix=

L non-zero elements

y αΨA

Matrix

Observed d t NM ×:Adata

fN M L> >

NM ×:A

• A simple example:– : “band‐limited”

DFT operatorConventional

signal processingA

DFT operator– : identity matrix

signal processing resultΨ


Underdetermined Linear Inverse Problems and SparsityProblems and Sparsity

• Basic problem: find an estimate of , where

[ ]• Underdetermined ‐‐ non‐uniqueness of solutions• Additional information/constraints needed for a unique

[ ]

qsolution

• Typical approach:• If we know is sparse (i.e. has few non‐zero elements)?

Number of non-zero elements in f

• Intractable combinatorial optimization problem• Recent work on sparse signal representation has produced

Number of non-zero elements in f


p g p pprincipled and feasible alternatives

SAR Ground‐plane GeometrySAR Ground‐plane Geometry

S l l fl f ld• Scalar 2‐D complex reflectivity field • Transmitted chirp signal:• Received, demodulated return from circular patch:


SAR Obser ation ModelSAR Observation Model• Observations are related to projections of the field:

• SAR observations are band‐limited slices from the 2‐D Fourier transform of the reflectivity field:

• Range profiles:• Range profiles: • Discrete tomographic SAR observation model:( bi i ll t )

Observed data

SAR Projection

Noise


(combining all measurements) SAR ProjectionOperator Unknown field

Conventional Image FormationConventional Image Formation• Given SAR returns, create an estimate of the reflectivity field f

��

Support of observed data in the spatial frequency domain Sample Conventional Image

��

Polar format algorithm:• Each pulse gives slice of 2‐D Fourier transform of field• Polar to rectangular resampling


Polar to rectangular resampling• 2‐D inverse FFT

Outline1) Sparsity‐driven, feature‐enhanced radar imaging

– An analysis‐based formulation for point and region‐enhanced imagingenhanced imaging

– Making the representation dictionary explicit: multiple feature‐enhanced imaging

2) Wid l i i2) Wide‐angle imaging– Composite image formation– Structured overcomplete dictionaries for anisotropic radar

imaging

3) Phase errors– Joint imaging and autofocusingJoint imaging and autofocusing– Moving target imaging

• Automatic hyperparameter selectionC d i i d i i


• Compressed sensing in radar imaging

Initial motivation for our workInitial motivation for our work

Some challenges for automatic decision‐making from SAR images:

• Accurate localization of dominant scatterers– Limited resolution

Cl d if

Some challenges for automatic decision making from SAR images:

– Clutter and artifact energy

• Region separabilitySpeckle– Speckle

– Object boundaries• Low SNR, limited apertures


Sparsity‐Driven Radar Imagingp y g g

C l l d d d i• Complex‐valued data and image• Phase of field spatially uncorrelated• Magnitude of complex‐valued field admits sparseMagnitude of complex valued field admits sparse

representation• Typical choices for L:

identity (point enhanced imaging)– identity (point‐enhanced imaging) – gradient (region‐enhanced imaging)

• Optimization problem structure is not identical to i blcommon sparse representation problems

• Bayesian interpretation: MAP estimation problem with heavy‐tailed priors


y p[Çetin and Karl, 2001]

Efficient Solution of the Optimization Problem• Cost functional

• Challenging optimization problem(non‐quadratic, non‐convex cost, constraint on |f|, large size)

• Developed a half‐quadratic regularization technique for complex‐valued random‐phase fieldsfor complex valued, random phase fields

• Could be viewed as a quasi‐Newton algorithm with a special Hessian approximationp pp

• Non‐quadratic constraints other than lp‐norms have also been used.


[Çetin and Karl, 2001, 2002]

Efficient Solution of the Optimization Problem• Quasi‐Newton‐based iterative scheme:

E h l l d bl• Each iteration equivalent to solving a quadratic problem:

• Can interpret as sequence of non‐stationary Gaussian problems

Spatially Adaptive WeightsFeature Preservation


Point‐Enhanced Imaging

Synthetic sceneSynthetic scene

Original Conventional Point‐Enhanced


Point‐Enhanced Imaging

MIT Lincoln Laboratory ADTS dataMIT Lincoln Laboratory ADTS data

Conventional Point‐Enhanced


Region‐Enhanced Imaging

MIT Lincoln Laboratory ADTS dataMIT Lincoln Laboratory ADTS data

Conventional Region‐Enhanced


Making the representation dictionary explicit: multiple feature‐enhanced imagingmultiple feature enhanced imaging

• Original (analysis‐based) imaging formulation:pJ ||)( 2 fLAfyf λ+−=

• Let us use the representation N h i fl i i h

pJ ||)(

2fLAfyf λ+−=

Ψαf =

ff Φ Φ• Note: , where contains reflectivity phases• Alternate (synthesis‐based) formulation:

ff Φ=

pJ λΦΨΦ 2)( A

Φ

• Joint optimization over phase and representation of

p

pJ αλαα +ΦΨ−=Φ

2),( Ay its ii ∀=Φ 1..

Joi op i i a io o e p a e a ep e e a io omagnitude

• Opens up the possibility of various dictionaries Ψ


[Samadi, Çetin, and Shirazi, 2011]

Multiple Feature‐Enhanced SAR Imaging:Use of Various DictionariesUse of Various Dictionaries

Original Conventional Analysis sparse (point+region)

Spikes + squares Wavelet Spikes + waveletProposed approach with various dictionaries

p q p


Multiple Feature‐Enhanced SAR Imaging:Use of Various DictionariesUse of Various Dictionaries

Original Conventional Analysis sparse (point+region)

Proposed approach with various dictionariesp ppSpikes + squares Wavelet Spikes + wavelet






imaging





Non‐traditional apertures: pwide‐angle data

• I l PSF• Irregular PSF• Anisotropic scattering

• Conventional imaging produces Co e io a i agi g p o u einaccurate reflectivities

• Conventional imaging does not characterize aspect dependencep p

Main pieces of proposed approach (first version):p p p pp ( )•Sparsity‐driven subaperture image reconstruction•Composite wide‐angle image formation


[Moses, Potter, and Çetin 2004]

Composite Image FormationComposite Image Formation• Many scatterers won’t persist over wide angles• Form a composite wide‐angle image from narrow‐Form a composite wide angle image from narrow

angle subaperture images:Subaperture

index

Pixel indices

k-th subaperture image

Composite image

• Preservation of anisotropic scatterers with short persistence• Partial characterization of aspect dependence


Wide‐angle SAR Imagingg g g

• Angular range = 110°• Composite images

Reference image Conventional Sparsity drivenReference image Conventional Sparsity‐driven


[Moses, Potter, and Çetin 2004]

Wide‐angle SAR Imaging with F b d O i i

• A ula a e 110°

Frequency‐band Omissions

• Angular range = 110°• Composite images• 70% of the band available

Reference image Conventional Sparsity‐driven


[Çetin and Moses 2005]

Visualization of aspect dependenceVisualization of aspect dependencein wide‐angle imaging70% band availability70% band availability

Bandwidth = 4GHz Bandwidth = 1GHz


Scattering ModelsScattering Models• Model for data collected from P isotropicscatterers:

( ) ( ) ( )∑= ⎭

⎬⎫

⎩⎨⎧ +−=

P

mmmmm yx

cjyxfr

1

sincos2exp,, θθωθω

– Image formation: Recovering f(x,y)• When we consider anisotropy :py

( ) ( ) ( )∑= ⎭

⎬⎫

⎩⎨⎧ +−=

P

mmmmm yx

cjyxfr

1

sincos2exp,,, θθωθθω

– Joint imaging and anisotropy characterization:Recovering f(x,y,θ)


• The problem becomes much more underdetermined!

Joint Imaging and g gSparsity‐Driven Anisotropty Characterization

• Joint Imaging and Anisotropy Characterizationg g py– Modeling anisotropy using an overcomplete dictionary– Structured overcomplete dictionaries (hierarchy of atoms within

molecules)o ecu es)– Solution by approximate, graphical algorithm

• Outcomes:– Accurate reflectivity estimation for scatterers that do not persist

over wide angular apertures– Characterization of scattering direction and angular width

• Potentially useful feature for automatic target recognition


[Varshney, Çetin, Fisher, and Willsky, 2008]

Modeling angular anisotropy• Model f(x,y,θ) by the dictionary {b1(θ), b2(θ), …, bM(θ)}y y y M

( ) ( ) ( )∑= ⎭

⎬⎫

⎩⎨⎧ +−=

P

mmmmm yx

cjyxfr

1

sincos2exp,,, θθωθθω

( ) ( ) ( )∑∑⎭⎬⎫

⎩⎨⎧ +−=

P M

mmmmiim yxc

jyxbar , sincos2exp,,, θθωθθω

= ⎭⎩m 1

• Problem: determine unknown coefficients am I

= = ⎭⎩m i c1 1

Problem: determine unknown coefficients am,I• How do we choose the bi(θ)?


Dictionary selectiony• Want to choose dictionary to allow parsimonious

representation• Observation: Non‐zero scattering usually in one contiguous

interval of θ• Dictionary elements: contiguous intervals of anisotropy – all y g py

widths and all shifts

s(θ)

||s


Aspect Angle

Dictionary selectiony• Incorporates some prior information about scattering structure• Consider rectangular pulse shape (Hamming, triangular, sinc,

etc are other possibilities)etc. are other possibilities)

θ1

b1(θ) b2(θ) bM(θ)

1θ2θ3θ4θ55θ6θ7θ8

N M t i M N(N+1)/2NxM matrix M=N(N+1)/2

•Can solve the resulting sparse representation problem in principle•Exact solution is memory intensive


y•Use structure to develop less expensive approximate algorithm

Illustration of graph‐structured algorithm


Quick ExampleQuick Example on Anisotropy Characterization

li d i i l ti

0 5

0

0.5

0 5

0

0.5

generalized inversion sparse solution

0 500 1000-0.5

a1

-0.5

0

0.5

0 500 1000-0.5

a3

-0.5

0

0.5

coeffi-cients

0

0.8

s( x

1, y 1, θ

)|

0

0.8

s( x

3, y 3, θ

)|

0

0.8

s( x

1, y 1, θ

)|

0

0.8

s( x

3, y 3, θ

)|

0 500 1000a

0 500 1000a

0θ

| 0

0.8

| s( x

4, y 4, θ

)|

0θ

|

0

0.8

| s( x

2, y 2, θ

)|

0θ

|

0

0.8

| s( x

4, y 4, θ

)|

0θ

|

0

0.8

| s( x

2, y 2, θ

)|

f(θ)


θθθθ

Anisotropic Imaging Results

-5

Anisotropic Imaging Resultson the Backhoe Data

ange

(m)

0

Ra

Cross-range (m)-5 0 55


Cross-range (m)

S l lt f l i b d f l tiSample result of analysis‐based formulation for anisotropy characterization

θ θ

θ θ

y

Aspect dependent scattering behavior in indicated sub-area

x


[Stojanovic, Çetin, and Karl, 2008]





imaging





Phase Errors and Defocusing

• Phase errors in the SAR data Defocusing of the reconstructed imagereconstructed image

Phase errorsInexact knowledge of the round‐tripInexact knowledge of the round trip

propagation time of the transmitted signal

Uncertainties on the SAR platform position

space‐invariant defocusingthe amount of defocusing is same for

all points in the scene

Delays in the transmitted signal due to atmospheric effects

space‐variant defocusingdefocusing in the corresponding spatial

Moving targets in the sceneg p g p

region in the reconstructed image


1D phase errors due to demodulation time errorsdue to demodulation time errors

• Measurement errors on the round‐trip propagation time of the transmitted signal

• After certain approximations, the relationship between the erroneous and error‐free discrete data can be expressed as

mmj

mj rerer Dm )(10 φωε

ε ==• 1D phase errors cause defocusing in the reconstructed image in

the cross‐range direction

mmmε

j )(φ

( ) ( ) ( )[ ]TDDDD M1111 ,........,2,1 φφφφ =

MmforfAefA mmj

DmD ,...,2,1)( )(

11 == φφ

The observation model matrix which t k h i t t


takes phase errors into account

Joint imaging and model error correction: Sparsity‐Driven Autofocus (SDA)

• SAR observation model may not be known perfectly, due to e g uncertainties in platform locationdue to, e.g., uncertainties in platform location

• This leads to phase errors in observed data• Proposed approach: joint optimization over the

reflectivities and model parameters:

2)()( ffAyf λφφ +J12

)(),( ffAyf λφφ +−=J

• Solution through coordinate descent− Closed‐form solution for the phase error estimation step


[Önhon and Çetin, 2009]

SDA on a synthetic scene

Without phaseerrors

Original scene Conventional Sparsity-driven imaging

errors

WithPhase error Conventional Proposed SDA

With phaseerrors


Phase error Conventional Proposed SDA

SDA on slicy

Without phaseerrors

Picture of slicy Conventional Sparsity-driven imaging

errors

Quadratic phase error

Conventional Sparsity-driven imaging Proposed SDA

R dRandom, uniform

phase error


Conventional Sparsity-driven imaging Proposed SDA

SDA on backhoe

Without phasephaseerrors

C ti l i i S it d i i iConventional imaging Sparsity-driven imaging

Random, uniform phaseperror

Conventional imaging Sparsity-driven imaging Proposed SDA


1D Phase error uniform. dist. in ⎥⎦⎤

⎢⎣⎡ +−

2,

2ππ

Existing Autofocus TechniquesPhase Gradient Autofocus (PGA)

PGA estimates phase errors using the data obtained by i l ti i l d f d t t i tisolating many single defocused targets via center‐shifting and windowing operations.

Techniques based on the Optimization of the Sharpness Metrics of the Image Intensity

These techniques optimize different sharpness metrics of the conventionally reconstructed defocused imageof the conventionally reconstructed defocused image intensity.

Multi‐Channel Autofocus (MCA)MCA i b d it ti l ith hi h fi dMCA is based on a non‐iterative algorithm which finds the focused image in terms of a basis formed from thedefocused image, relying on a condition on the image

b l


support to obtain a unique solution.

Comparison of autofocus methodsth tion a synthetic scene

Phase error uniform. dist. in [ ]ππ +− ,

Original scene Conventional imaging fromnoisy data with phase error

Conventional imaging fromnoisy data without phase error o sy data t p ase e oo sy data t out p ase e o


PGA Entropy Minimization Proposed SDA

Quantitative analysis

2ˆ1 ffMSE = ⎟⎟⎟⎞

⎜⎜⎜⎛

=∈ iTi f

TBR1

ˆmaxlog20 10

φφφ ˆ−=e

12

ffI

MSE −=⎟⎟

⎠⎜⎜

⎝∑∈Bj

jB

fI

ˆ1g10

2

211

ePE MMSE φ∇

−=


Comparison of autofocus methods on backhoeautofocus methods on backhoe

1D Phase error uniform. dist. in [ ]ππ +− ,

Conventional imaging fromdata without phase error

Sparsity-driven imaging fromdata without phase error

Conventional imaging fromdata with phase errorp p


PGA Entropy Minimization Proposed SDA

Moving‐target imaging• SAR platform position uncertainties cause space‐invariant

defocusing of the reconstructed image, i.e., the amount of defocusing is the same for all points in the scene.

• Motion of a target in the scene can also be modeled as a phase error over the phase history data corresponding to a stationary scene.

• Moving targets in the scene cause artifacts including defocusing around the spatial neighborhood of the target in the scene.

Space‐variant defocusingNeed to keep an account of the contributions from each spatial location to the phase error at each aperture position


Sparsity‐Driven M i T t I iMoving‐Target Imaging

ffAyfJff

−++−= )(minarg),(minarg1211

2

2βλλφβ

ββ1

iits

ffyfff

∀= 1)(..

)(g)(g12112,,

β

βφβββ

[ ]Tmjmjmj Ieee )()()( 21 φφφβ = ⎤⎡ β[ ]m eee ,.......,,β =

⎥⎥⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎢⎢⎡

=

ββ

β..

2

1

• Involves sparsity constraints both on the

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ Mβ.

Involves sparsity constraints both on the reflectivity field and on the motion field.

• Have also developed a more efficient and potentially robust version based onpotentially robust version based on constructing ROIs and performing space‐invariant focusing within them.


Moving‐target imaging resultsOriginal scene Conventional imaging

smvcr

/81 =

smvcr

/52 =

S it d i i iSparsity-driven imaging Proposed method


Moving‐target imaging resultsOriginal scene Conventional imaging

smv cr /8=

S it d i i i P d th dSparsity-driven imaging Proposed method


Summaryy• Radar imaging presents interesting sparse representation

problems involving some variations of basic sparse representation formulationsformulations

• Significant amount of work exists on sparsity‐driven algorithms as well as associated results with potential impact on applications

• We have examined and contributed to• We have examined and contributed to– various sparse representation formulations and algorithms– stationary and moving target imaging– wide‐angle and passive radar imagingwide angle and passive radar imaging– handling model errors– hyperparameter choice– impact on target recognition – compressed sensing for monostatic and multistatic SAR

• Empirical successes in regime where sufficient sparsity/CS conditions not satisfied


mujdatcetin sar12 public.ppthelper.ipam.ucla.edu/publications/sar2012/sar2012_10407.pdf ·...

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