part 3 ‐ applications ofyazici/icassptutorial/icassp... · birsen yazıcı & venky krishnan...
TRANSCRIPT
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Part 3Part 3 ‐‐ Applications ofApplications ofPart 3 Part 3 Applications of Applications of MicrolocalMicrolocal Techniques to ImagingTechniques to Imagingq g gq g g
Birsen Yazıcı & Venky KrishnanBirsen Yazıcı & Venky KrishnanRensselaer Polytechnic Institute
Electrical, Computer and Systems Engineering, p y g g
March 15th, 2010
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan
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Outline
• Synthetic Aperture Imaging (SAI) modalities• SAI forward model• SAI image formation via microlocal techniquesg q
– Bistatic SAI– Monostatic SAI– Multistatic SAI– Synthetic Aperture Hitchhiker– Doppler Synthetic Aperture Hitchhiker
• Conclusions
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 2
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Synthetic Aperture Imagingy p g g
Applicable to –
• Synthetic aperture radar• Synthetic aperture sonary p• Seismic imaging• Microwave imaging• Microwave imaging• Ground penetrating radar and sonar• Synthetic aperture imaging in dispersive• Synthetic aperture imaging in dispersive medium
• Diffu e o ti al i a i
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• Diffuse optical imaging
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Synthetic Aperture Imaging Modalitiesy p g g
• SAI systems form images from backscattered waves measured along the source and detector trajectory(ies)measured along the source and detector trajectory(ies)– Mono‐static‐SAI– Bi‐static‐SAIMulti static SAI
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– Multi‐static SAI – Passive SAI
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Monostatic Synthetic Aperture Imaging
• Monostatic SAI – transmitter and receiver are colocated
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Mono‐static SAI – Flat Topography
• Iso‐range surfaces = Spheres• Iso‐range contours = Intersection of spheres with topography
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Iso range contours Intersection of spheres with topography• Flat topography Circles
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Mono‐SAI – Plane Wave Approximationpp
• Approximate spherical wavefronts with planar wavefronts• Intersection of planes with flat surface = lines
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f p f f• Spotlight Mode – Radon Transform
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Bi‐static Synthetic Aperture Imaging
• Bi‐static SAI – Transmitter and receiver are sufficiently far apart
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p
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BiSAI – Flat Topography
• Iso‐range surfaces = Ellipsoid• I t I t ti f lli id ith t h
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• Iso‐range contours = Intersection of ellipsoids with topography• Flat topography Ellipses
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Multi‐static Synthetic Aperture Imaging
• Multi‐static SAI – Sufficiently apart multiple antennas that can either receive, transmit or both
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Passive Synthetic Aperture Imagingy p g g
• Passive SAI –– Scene illuminated by transmitters of opportunity: TV, cell‐phone stations, communication satellites; ambient noise etc.Receive only antennas use backscattered measurements to
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– Receive only antennas use backscattered measurements to make an image of the scene
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Hitchhiker – Flat Topography
• Iso‐range surfaces = Hyperboloids• Iso range contours = Intersection of hyperpoloids with topography
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• Iso‐range contours = Intersection of hyperpoloids with topography• Flat topography Hyperbolas
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Outline
• Synthetic Aperture Imaging (SAI) modalities• SAI forward model• SAI image formation via microlocal techniquesg q
– Bistatic SAI– Monostatic SAI– Multistatic SAI– Synthetic Aperture Hitchhiker– Doppler Synthetic Aperture Hitchhiker
• Conclusions
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 13
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Model for the Received Signal
• Measurement Model Transmitterlocation
Receiverlocation
Reflectivity
Phase TermTransmitter and Receiver
Reflectivity fnc
• This model allows for:Amplitude Term
Receiverbeampatterns
– Arbitrary waveforms, trajectories; spatially distributed antennas– Array antennas in which different elements are activated with
different waveforms– Applicable to both spotlight and strip‐map SAI mode
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Construction of the Imaging Operator
• The imaging operator has the same phase as(L2 adjoint)(L adjoint)– Compare with inverse Fourier transform and inverse Radon transform
• Forward map
e e
• Inverse mapPhase of L2 adjoint
scene
To be determinedimage
Backprojection Filter
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To be determined
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Outline
• Synthetic Aperture Imaging (SAI) modalities• SAI forward model• SAI image formation via microlocal techniquesg q
– Bistatic SAI– Monostatic SAI– Multistatic SAI– Synthetic Aperture Hitchhiker– Doppler Synthetic Aperture Hitchhiker
• Conclusions
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Bi‐static Synthetic Aperture Imaging
receiverS th ti A t
transmitter
receiverSynthetic Aperture:
• Bi‐static SAI – Forward Model
antenna beam pattern, geometrical spreading factors etc
reflectivityfast‐timegeometrical spreading factors, etc.slow‐time: trajectory
C.E. Yarman, B. Yazıcı, and M. Cheney, “Bistatic synthetic aperture radar imaging with arbitrary trajectories,” IEEE Transaction on Image Processing, Vol. 17, No: 1, pp: 84‐93, 2008.ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan
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Isorange Contours of BiSAR
• Main contributions come from isorange curves40
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isorange curves
Ellipsoids of revolution with foci and0
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isorange curvesintersection of ellipsoids with topography
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at topog ap y a d i ed a titude
Backproject onto isorange curves …ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan
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FBP Image Formation for BiSAI
Filtered backprojection onto isorange curves …I
filterInverse map
image
How do we determine the filter?image
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How to Determine the FBP Filter
• Study the Point Spread Function (PSF) of the imaging b h h d f i hoperator by the method of stationary phase
• PSF ‐Mapping from the scene/medium to the imageH fi d PSF? Pl i h d d l i h• How to find PSF? Plug in the data model into the inverse map and do the t integration
• Ideal PSF functionPoint Spread Function (z,x)
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Forming an Ideal Point Spread Function
Phase of the point spread function‐
• Use stationary phase theorem to determine where the main contributions to a point in the image come from – Gradient with respect to omega and slow‐time variables
Defines iso‐range and iso‐Doppler curves
• Determine conditions so that the main contributions• Determine conditions so that the main contributions are when
• Perform linearization in the phase of the PSF Taylor• Perform linearization in the phase of the PSF – Taylor series expansion around the main contributions
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Stationary Phase of BiSAI PSF
Stationary Phase ‐IsoDoppler contours (Red)+ I oRa e o tou (Blue)
Bistatic Range
+ IsoRange contours (Blue)
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Flat topography and fixed altitude
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Bistatic SAI Reconstruction
• Bistatic PSF ‐ Perform linearization in the phase of PSF
• Perform change of variablesg
L di d t ib ti• Leading order contribution Jacobian from CV
Choose Filter
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Bistatic SAI Resolution
• Leading order contribution to the PSF Indicator function
• Data collection manifold ‐ Set of Fourier vectors contributing to the reconstruction of the point z
• Band‐limited Fourier reconstruction ‐ The frequency content hence resolution determined by the data collection manifold
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Data Collection Manifold for Bistatic SAI
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Data Collection Manifold & Visibility
Data Collection Manifold:
edge visible in
Example: Isotropic transmitters and receivers & Flat Topography
visible edge not a visible edgeg g
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Numerical Simulations
• 22 x 22 km2 scene• Rectangular targets at the scene• Circular flight trajectory at 6.5 km height
Ult id b d f• Ultra‐wideband waveforms
Transmitter 1
Receiver 1
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Numerical Simulations
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Receiver trajectoryReceiver trajectoryTransmitter location/trajectoryTransmitter location/trajectoryICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan
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Outline
• Synthetic Aperture Imaging (SAI) modalities• SAI forward model• SAI image formation via microlocal techniquesg q
– Bistatic SAI– Monostatic SAI– Multistatic SAI– Synthetic Aperture Hitchhiker– Doppler Synthetic Aperture Hitchhiker
• Conclusions
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Synthetic Aperture Imaging
receivertransmitter
Synthetic Aperture:Synthetic Aperture:
• SAI – Forward Model
antenna beam pattern, reflectivityfast‐time pgeometrical spreading factors, etc.
f yslow‐time
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan
B. Yazıcı, M. Cheney, C.E. Yarman “Synthetic aperture inversion in the presence of noise and clutter,” Inverse Problems, Vol. 22, pp: 1705‐1729, 2006
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SAI in the Presence of Clutter & Noise
Obj f iObject of interestClutter (Unwanted scatterers)
Data model:Object of interest Clutter noise
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B. Yazıcı, M. Cheney, C.E. Yarman “Synthetic aperture inversion in the presence of noise and clutter,” Inverse Problems, Vol. 22, pp: 1705‐1729, 2006
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SAI in the Presence of Noise and Clutter
• FBP ReconstructionBackproject onto isorange curves– Backproject onto isorange curves
– Choose filter wrt to a figure of merit that takes into account noise, clutter and target statisticsSt d th bi d i f th lti ti t b th d– Study the bias and variance of the resulting estimate by method of stationary phase Mono‐static inversion map
• Figure of merit – Minimum mean square error
D i th FBP filt h th t i i i i d!
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Design the FBP filter such that is minimized!
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Optimal Filter in the Presence of Noise and Clutter
• Explicit expression for the FBP filter
Target spectrumBeylkin determinant
• Space varying filter because belongs to the MS dataClutter spectrum noise
spectrum
• Space varying filter, because belongs to the MS data collection manifold and varies for each pixel reconstructed
• In the absence of clutter and noise
Filter:
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Filter:
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Numerical Simulations
ProjectionsBeylkin Recon Wiener Recon
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Clutter Suppression
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Outline
• Synthetic Aperture Imaging (SAI) modalities• SAI forward model• SAI image formation via microlocal techniquesg q
– Bistatic SAI– Monostatic SAI– Multistatic SAI– Synthetic Aperture Hitchhiker– Doppler Synthetic Aperture Hitchhiker
• Conclusions
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Multi‐static Synthetic Aperture Imaging
• Multiple transmitters traversing arbitrary trajectories, and transmitting arbitrary waveforms simultaneouslytransmitting arbitrary waveforms simultaneously.
• Interference causes artifacts in reconstructed images.
• Multistatic SAI – Sufficiently apart multiple antennas that can either receive transmit or both
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 37
that can either receive, transmit or bothV. Krishnan, J. Swoboda, C.E. Yarman, B. Yazıcı, ʺMulti‐static Synthetic Aperture Radar Image Formation,ʺ to appear in IEEE Transaction on Image Processing.
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Multi‐static Synthetic Aperture Imaging
• Artifacts (additional singularities) due to interference if bi static reconstruction is usedbi‐static reconstruction is used
• Three transmitters on linear trajectories & one receiver on a circular trajectoryj y
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 38Original phantom Image reconstructed
using a bi‐static method
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Bi‐static Received Signal Model
• M transmitters and N receiverFast‐time
S bSlow‐time Scene to be reconstructed
Received data at the qth A b
• Bi static distance
Received data at the qth receiver due to pth
transmitterAntenna beampatterns
• Bi‐static distance
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 39
qth receiver trajectory
pth transmitter trajectory
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Multi‐static Synthetic Aperture Imaging
• Multi‐static data model – Measurement at the qth receiver due toBi t ti SAI qth receiver due to pth transmitter
Bistatic SAI forward model
Scene to be
Measurements at the qth receiver
Bistatic delay• Plus additive noise – reconstructedBistatic delay
for the p‐q pair
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Multi‐static Image Formationg
• Idea – Backproject with respect to a particular transmitter suppress the data due to other transmitterstransmitter, suppress the data due to other transmitters by a suitably designed filter
• Filter Suppress the Fourier components due to all• Filter – Suppress the Fourier components pq due to all the transmitters, but the one data is backprojected to.
• This converts the multi‐static data to bi‐static data• This converts the multi‐static data to bi‐static data
• Do this for each transmitter and coherently sum up the images to form the final imageimages to form the final image
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 41Fourier vector pq
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Multi‐static Image Formation
• Backprojection operator
B k j ti t Filter
Image
Backprojection wrt to the pth transmitter to be determined
greconstructed with the qth received signal
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 42Artifacts to be suppressed
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Design of FBP Filters
• Best possible image we can form
• The Fourier space data collection manifold• The Fourier space data collection manifold
• Choose the FPB filters so that
Projection ontothe tangent plane at z
Bi‐static bisector
• Choose the FPB filters so that
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 43minimized
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Design of FBP Filters
• Reconstructed image due to (p,q)th pair
Bi static recon due
• Mean Square Error
Bi‐static recon due to (p,q)th pair Artifact images Image due to noise
Images to be suppressed
I1
I2
I3
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I4
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FBP Filter
• Filter for the p‐q pair
Scene power spectral
density
Noise power spectral
density
JacobianDue to change of
variables densityvariables
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Multi‐static Image Formation
• When backprojecting wrt 1st (2nd) transmitter suppress the Fourier component ξ (ξ )the Fourier component ξ2 (ξ1).
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Numerical Simulations
• Two transmitters and a one receiver• 22 x 22 km2 scene• Rectangular targets at the scene• Circular flight trajectory at 6 5 km height• Circular flight trajectory at 6.5 km height• Ultra‐wideband waveforms • π/6 degrees apart
Transmitter 1
Transmitter 2Receiver 1
B. Yazıcı & V. KrishnanICASSP 2010, Dallas, TX47
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Numerical Simulations
Multi‐static ReconBi‐SAI Recon• Two transmitters and a one receiver on circular flight trajectory
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Numerical Simulation
• Three transmitters on linear trajectories & on receiver on a circular trajectorya circular trajectory
Multi‐static ReconBi‐SAR Recon
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Multi static ReconBi SAR Recon
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Outline
• Synthetic Aperture Imaging (SAI) modalities• SAI forward model• SAI image formation via microlocal techniquesg q
– Bistatic SAI– Monostatic SAI– Multistatic SAI– Synthetic Aperture Hitchhiker– Doppler Synthetic Aperture Hitchhiker
• Conclusions
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Passive Synthetic Aperture Radary p
• Synthetic Aperture Hitchhiker– Scene illuminated by wideband transmitters of opportunity: cell‐phone stations; ambient noise etc.
C.E. Yarman, B. Yazıcı, ʺSynthetic aperture hitchhiker imaging,ʺ IEEE Transaction on Image Processing, Vol. 17, No. 11, pp: 2156‐2173, 2008.
• Doppler Synthetic Aperture Hitchhiker– Scene illuminated by ultra‐narrowband transmitters of
dopportunity: TV, radio stations etc.
C.E. Yarman, L. Wang, B. Yazıcı, ʺDoppler Synthetic Aperture Hitchhiker Imaging ʺ accepted to Inverse Problems October 2009
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Hitchhiker Imaging, accepted to Inverse Problems, October 2009.
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Why Passive Imaging?
• Rapid growth in the number of broadcasting stations, mobile phone base stations, terrestrial and space based communication and navigation satellites Passive SAI offers a viable approach to imagingoffers a viable approach to imaging
• Can provide wide area coverage in urban environments
C l t ti t t i f• Can complement active systems to increase frequency and angular diversity
• O ly i lude e ei e i e e i e obile e atile• Only includes receivers inexpensive, mobile, versatile, and suitable for rapid deployment
• Less vulnerable
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 52
• Less vulnerable
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Outline of the Approachpp
• Novel synthetic aperture imaging modalities based on correlation of received signal and microlocal analysiscorrelation of received signal and microlocal analysis.
• Two types of correlation ‐– Hitchhiker – Delay based correlation– Doppler Hitchhiker – Scaling and delayed based correlation
• Correlation‐based processing removes transmitter related terms from the phase of the resulting signalterms from the phase of the resulting signal.
• Resulting forward models involve FIOs where the scene radiance is projected onto certain curved manifolds.p j– SAH – projection onto hitchhiker iso‐range curves– DSAH – projection onto hitchhiker iso‐Doppler curves
• Image Formation: Invert SAH and DSAH FIOs using
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 53
• Image Formation: Invert SAH and DSAH FIOs using microlocal techniques.
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Synthetic Aperture Hitchhiker
• Hitchhiker –T i b i i b h– Transmitters can be stationary or moving or both
– Transmitters can be cooperative or non‐cooperative– Receive‐transmit pair data association may not be possible
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 54
Receive transmit pair data association may not be possibleC.E. Yarman, B. Yazıcı, ʺSynthetic aperture hitchhiker imaging,ʺ IEEE Transaction on Image Processing, Vol. 17, No. 11, pp: 2156‐2173, 2008.
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Bi‐static Received Signal Model
slow‐timefast‐time reflectivity
Bistatic range:
Trans. location Receiver trajectory
Receiver Antenna :
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Transmitter Antenna :
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Hitchhiker Received Signal Model
Integrate over all transmitters:
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 56
ith receiver
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SAH ‐ Spatio‐temporal Correlation of Received DataReceived Data
• Perform spatio‐temporal correlation of the received data
fast‐timeslow‐time
ith receive antenna data
jth receive antenna data
• Eliminates the transmitter related information from the phase of the forward map Passive detection & imagingphase of the forward map Passive detection & imaging
• Takes into account multiple scattering from the transmitter to target
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to target
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Hitchhiker FIO Model
• SAH FIO ‐Scene radiance –Image to be formed
Phase Term –Antenna beampatterns + geometric spreading factorsPhase Term
Hitchhiker Range – Main contributions to SAH FIO
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Iso‐Range Contours of Hitchhiker for Flat TopographyFlat Topography
• Iso‐range contours = Intersection of hyperboloids with topography
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Backproject onto isorange curves …
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Inversion of Hitchhiker FIO
• Filtered‐backproject onto hitchhiker iso‐range curves
Filter to be determined
• Irrespective of the choice of filter, backprojection puts the singularities of the scene at the correct location and correct orientation
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 60
correct orientation
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How to Determine the FBP Filter
• Point Spread Function (PSF) ‐Mapping from the scene h ito the image
hh k S h
Point Spread Function (z,x)
• Hitchhiker PSF phase
• Ideal PSF functionHitchhiker range
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Iso‐Range and Iso‐Doppler Contours of HitchhikerHitchhiker
40Hitchhiker Range
IsoDoppler contours (Red)+ IsoRange contours (Blue)
30
40
Intersection of hyperboloids and the ground topography
10
20
−10
0Hitchhiker Doppler
The difference between the radial
−20 −10 0 10 20 30 40−20
ffvelocities at two different points in the trajectories.
Flat topographyd fi d
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 62
and fixed altitude
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Hitchhiker Resolution Analysis ‐ Fourier Domain Data Collection ManifoldDomain Data Collection Manifold
Fourier vector
Fourier components contributing to the reconstruction of the point z – A vector in the direction of the difference of the unit vectors in the look directions of the two receivers with its length proportional to the
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 63
the look directions of the two receivers with its length proportional to the bandwidth of the transmitted waveform.
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Numerical Simulations
• 22 x 22 km2 scene• Multiple point targets• Circular flight trajectory at 6.5 km height• Infinite bandwidth waveform
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Infinite bandwidth waveform • 128 x 128 pixel image
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Data Collection Manifolds for the Linear and Parabolic Trajectoriesfor the Linear and Parabolic Trajectories
Auto‐correlation of Linear ‐ 1
Auto‐correlation of Parabolic ‐ 2
Cross‐correlation of Linear & Parabolic ‐ 3
1+2+3
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Numerical Simulations –BiSAR and SAH Recon ComparisonBiSAR and SAH Recon Comparison
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 66ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan
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Outline
• Synthetic Aperture Imaging (SAI) modalities• SAI forward model• SAI image formation via microlocal techniquesg q
– Bistatic SAI– Monostatic SAI– Multistatic SAI– Synthetic Aperture Hitchhiker– Doppler Synthetic Aperture Hitchhiker
• Conclusions
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Doppler Synthetic Aperture Hitchhiker
• Most sources of opportunity are single frequencySAH d ll i if ld i ll f l• SAH data collection manifold is small for ultra‐narrow band sources of opportunity
• New approach Doppler SAH
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pp ppC.E. Yarman, L. Wang, B. Yazıcı, ʺDoppler Synthetic Aperture Hitchhiker Imaging,ʺ accepted to Inverse Problems, October 2009.
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Doppler Synthetic Aperture Hitchhiker
• Spread of the hyperbolas determined by the range ambiguity
Range ambiguity function of the transmitted waveform
Hyperbola formed by correlating received signals
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p f yp y g g yfunction of the transmitted waveforms
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Doppler Synthetic Aperture Hitchhiker
• Window measurements, then scale& correlate…
Scaling parameterShift
parameters
window: ith received data: filter to be determined
• Results in a new FIO• Larger data collection manifold…
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g
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Doppler Hitchhiker FIO Model
• DSAH FIO ‐S diScene radiance –Image to be formed
Phase Term –hase erm
Hitchhiker‐Doppler Scale Factor –
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DSAH FIO Main Contributions –Hitchhiker Iso‐Doppler ContoursHitchhiker Iso‐Doppler Contours
• Iso‐Doppler surface ‐ Intersection of two constant Doppler cones
Constant Doppler Cone 1 :Constant Doppler Cone 1 :
Constant Doppler Cone 2 :
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DSAH Iso‐Doppler Contours
• Iso‐Doppler contours for two receivers and flat topographytopography
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Backproject onto iso‐Doppler curves …
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Inversion of DSAH FIO
• Weighted‐backprojection onto iso‐Doppler curves
Weight to be determined
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Inversion of DSAH FIO
• Weighted‐backprojection onto iso‐Doppler curves
Weight to be determined
Filter –
Weight – Jacobian that comes from a certain change of variables
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DSAH Iso‐Doppler and Iso‐Doppler‐Rate ContoursContours
Iso‐Doppler curves Iso‐Doppler‐rate curves
~ radial acceleration in
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Iso‐Doppler‐rate surface –~ radial acceleration in the look direction of the radar
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DSAH and SAH Contours
40
30
40
10
20
−10
0
DSAH contoursSAH contours−20 −10 0 10 20 30 40
−20
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 77
DSAH contoursSAH contours
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Resolution Analysis ‐ Fourier Domain Data Collection Manifold for DSAH
Projections of the platform velocity onto the planes perpendicular to the radar line of sightperpendicular to the radar line of sight
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 78Fourier vector
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Rasolution of DSAH
• Larger the Fourier vector higher the resolution
Fourier vector
• t : Support of the windowing function
Fourier vector
pp g– Longer the support of the windowing function Higher the
resolution
• 2/ : Frequency• 2/ : Frequency – Larger the transmit frequency Higher the resolution
• ’ : ~ Number of segments correlated
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 79
g– Higher sampling of ’ Higher the resolution
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Numerical Simulations
A single stationary transmitter, two moving receivers
• 22 x 22 km2 scene• A point target at [16, 11, 0] km• Circular flight trajectory at 6 5 km height• Circular flight trajectory at 6.5 km height• A fixed‐frequency waveform • 128 x 128 pixel image
Transmitter
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 80Image X‐profile Y‐profile
f0 = 0.4 MHz, L = 3.657 s.
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Numerical Simulations f0 = 0.4 MHz, L = 3.657 s.Shorteer w
indow s
Image X‐profile Y‐profile
support
f0 = 0.4 MHz, L = 14.629 s. Longer window
su
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 81X‐profile Y‐profileImage
Longer window support improves resolution
upport
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Numerical Simulation f0 = 0.4 MHz, L = 3.657 s.
Smmaller frequ
Image X‐profile Y‐profile
ency
f0 = 4 MHz, L = 3.657 s. Largerr frequency
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 82X‐profile Y‐profileImage
Higher transmit frequency improves resolution
y
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Numerical Simulation
16 τ′ values uniformly spaced in [0, 248.258]s
f0 = 4 MHz, L = 3.657 s.
ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan 83’ : ~ Larger the number of segments correlated, better the resolution
Partial data collection manifold τ′= 165.5054s
Partial data collection manifold τ′=33.1011s
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Conclusions
• Microlocal analysis provides a unified mathematical f k f id f bl i i dframework for a wide‐range of problems in sensing and imaging problems
P id f l t i i i ht i t i i• Provides a powerful geometric insight into imaging
• Leads to novel modalities
h l d• Techniques result in adaptive processing
• Many open problems…l l A l I k h I• Microlocal Analysis in Imaging Workshop at RPI in
August 2010.• Microlocal Analysis in Imaging tutorial for IEEE Trans
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Microlocal Analysis in Imaging tutorial for IEEE Trans. on Image Processing www.ecse.rpi.edu/~yazici
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References
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• [Bleistein‐Cohen‐Stockwell] N. Bleistein, J. K. Cohen and J. Stockwell ``The Mathematics of Multidimensional Seismic Inversion’’, Springer, New York, 2000.
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ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan
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ICASSP 2010, Dallas, TX B. Yazıcı & V. Krishnan