design in imaging from compressive to comprehensive sensing
DESCRIPTION
Design in Imaging From Compressive to Comprehensive Sensing . Lior Horesh Joint work with E. Haber, L. Tenorio , A. Conn, U. Mello, J. Fohring. IBM TJ Watson Research Center SIAM Imaging Science 2012, Philadelphia, PA - May 2012. Introduction. - PowerPoint PPT PresentationTRANSCRIPT
DESIGN IN IMAGING FROM COMPRESSIVE TO COMPREHENSIVE
SENSING
Lior HoreshJoint work with E. Haber, L. Tenorio,
A. Conn, U. Mello, J. Fohring
IBM TJ Watson Research CenterSIAM Imaging Science 2012, Philadelphia, PA - May 2012
INTRODUCTION
• Aim: infer model
• Given • Experimental design y• Measurements• Observation model
• Ill - posedness Naïve inversion ...
Fails...
• Need to fill in the missing information
EXPOSITION - ILL - POSED INVERSE PROBLEMS
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m F m y d y= -14444444244444443
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HOW TO IMPROVE MODEL RECOVERY ?
• How can we ...
• Improve observation model ?
• Incorporate more meaningful a-priori information ?
• Extract more information in the measurement procedure ?
• Provide more efficient optimization schemes ?
• Define appropriate distance measures / noise model ?
• Forward problem (simulation, description)
• Given: model m & observation model
• Simulate: data d
• Inverse problem (estimation, prediction)
• Given: data d & observation model
• Infer: model m (and uncertainties)
• Design (prescription)
• Given: inversion scheme & observation model
• Find: the ‘best’ experimental settings y, regularization S, …
EVOLUTION OF INVERSIONFROM SIMULATION TO DESIGN
WHY DESIGN ?
WHICH EXPERIMENTAL DESIGN IS BEST ?
Design 1 Design 3Design 2
• Where light sources and optodes should be placed ?
• In what sequence they should be activated ?
• What laser source intensities should be used ?
DESIGN QUESTIONS – DIFFUSE OPTICAL TOMOGRAPHY
Arridge 1999
• What frequencies should be used ?
• What trajectory of acquisition should be considered ?
DESIGN QUESTIONS –ELECTROMAGNETIC INVERSION
Newman 1996, Haber & Ascher 2000
DESIGN QUESTIONS – SEISMIC INVERSION
Clerbout 2000
• How simultaneous sources can be used ?
• How many projections should we collect and at what angles ?
• How accurate should the projection data be ?
DESIGN QUESTIONS –LIMITED ANGLE TOMOGRAPHY
WHAT CAN WE DESIGN ?
DESIGN EXPERIMENTAL LAYOUT
Stonehenge 2500 B.C.
DESIGN EXPERIMENTAL PROCESS
Galileo Galilei 1564-1642
RESPECT EXPERIMENTAL CONSTRAINTS…
French nuclear test, Mururoa, 1970
HOW TO DESIGN ?
• Previous work
• Well-posed problems - well established [Fedorov 1997, Pukelsheim 2006 ]
• Ill-posed problems - under-researched [Curtis 1999, Bardow 2008 ]
• Many practical problems in engineering and sciences are ill-posed
What makes ill-posed problems so special ?
ILL VS. WELL - POSED OPTIMAL EXPERIMENTAL DESIGN
OPTIMALITY CRITERIA IN WELL-POSED PROBLEMS
• For linear inversion, employ Tikhonov regularized least squares solution
• Bias - variance decomposition
• For over-determined problems
• A-optimal design problem
0a =
OPTIMALITY CRITERIA IN WELL-POSED PROBLEMS• Optimality criteria of the information matrix
• A-optimal design average variance
• D-optimality uncertainty ellipsoid
• E-optimality minimax
• Almost a complete alphabet…
( )1
C y-
THE PROBLEMS...
• Ill-posedness controlling variance alone reduces the error mildly [Johansen 1996 ]
• Non-linearity bias-variance decomposition is impossible
What strategy can be used ?
Proposition 1 - Common practice so far
Trial and Error…
EXPERIMENTAL DESIGN BY TRIAL AND ERROR
• Pick a model
• Run observation model of different experimental designs, and get data
• Invert and compare recovered models
• Choose the experimental design that provides the best model recovery
THE PROBLEMS...
• Ill-posedness controlling variance alone reduces the error mildly [Johansen 1996 ]
• Non-linearity bias-variance decomposition is impossible
What other strategy can be used ?
Proposition 2 - Minimize bias and variance altogether
How to define the optimality criterion ?
OPTIMALITY CRITERIA FOR DESIGN
• Loss
• Mean Square Error
• Bayes risk
Depends on the noise Depends on unknown modelT
m
Depends on unknown model
Computationally infeasible
Tm
OPTIMALITY CRITERIA FOR DESIGN
• Bayes empirical risk
• Assume a set of authentic model examples is available
• Discrepancy between training and recovered models [Vapnik 1998 ]
How can y and S be regularized ?
• Regularized Bayesian empirical risk
{ }1,...,
mnm m
OTHER DESIGNERS / KEY PLAYERS
• This is one doctrine
• Other interesting choices were developed by
• Y. Marzouk et. al, MIT (2011, 2012)
• A. Curtis et. al, University of Edinburgh (1999, 2010)
• D. Coles & M. Prange, Schlumberger (2008, 2012)
• S. Körkel et. al, Heidelberg University (2011)
• A. Bardow, RWTH Aachen University (2008, 2009)
DIFFERENTIABLE OBSERVATIONSPARSITY CONTROLLED DESIGN
• Assume: fixed number of observations
• Design preference: small number of sources / receivers is activated
• The observation model
• Regularized risk
Horesh, Haber & Tenorio 2011
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DIFFERENTIABLE OBSERVATION SPARSITY CONTROLLED DESIGN
• Total number of observations may be large
• Derivatives of the forward operator w.r.t. y
• Effective when activation of each source and receiver is expensive
Horesh, Haber & Tenorio 2011
Difficult…
WEIGHTS FORMULATION SPARSITY CONTROLLED DESIGN
• Assume: a predefined set of candidate experimental setups is given
• Design preference: small number of observations
( )1;F m y( )2;F m y( )999;F m y ( )1
;F m y( )2;F m y( )3;F m y( )1000
;F m y
Haber, Horesh & Tenorio 2010
WEIGHTS FORMULATION SPARSITY CONTROLLED DESIGN
• Let be discretization of the space [Pukelsheim 1994]
• Let
• The (candidates) observation operator is weighted
• w inverse of standard deviation
• - reasonable standard deviation - conduct the experiment
• - infinite standard deviation - do not conduct the experiment
( )( ) ( ) ( )1 1 1
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Haber, Horesh & Tenorio 2010
WEIGHTS FORMULATION SPARSITY CONTROLLED DESIGN
• Solution - more observations give better recovery
• Desired solution many w‘s are 0
• Add penalty to promote sparsity in observations selection
• Less degrees of freedom
• No explicit access to the observation operator needed
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THE OPTIMIZATION PROBLEMS
• Leads to a stochastic bi-level optimization problem
• Direct formulation
• Weights formulation
Haber, Horesh & Tenorio 2010, 2011
THE OPTIMIZATION PROBLEM(S)
• In general (multi-level) optimization problem
• Difficult to solve (although not impossible) [Alexaderov &
Denis 1994 ]
• To simplify make assumptions on
• F - linear, nonlinear
• S - quadratic, matrix form
• y - discrete, continuous
• Important - sufficient improvement can be “good enough”
HOW MANY SAMPLES ARE NEEDED ?
• The optimist - choose a single m and design for it
• The pessimist - assume that m is the worst it can be
• The realist - choose an ensemble of m’s and average
• A Bayesian estimate of the frequentist risk [Rubin 1984 ]
OPTIMAL DESIGN IN PRACTICE
LINEAR DESIGN
LINEAR DESIGN
• Assume
• Then
LINEAR DESIGN
• Assume m has second moment
• Then
• Best linear design
CASE STUDY ILINEAR EXPERIMENTAL DESIGN
THE BEST (LINEAR) EXPERIMENT
• Considering discretization of the design space
• We obtain
BOREHOLE RAY TOMOGRAPHY -OBSERVATION MODEL
• Use 322 × 3 = 3072 rays
• Goal: choose 500 optimal observations
BOREHOLE RAY TOMOGRAPHY EXPERIMENTAL DESIGN ASSESSMENT
• Assess design performance upon unseen models (cross validation)
Test model Optimal designNon-optimal design
NON - LINEAR DESIGN
NON - LINEAR DESIGN
• Use Stochastic Optimization - approximate by sampling [Shapiro, Dentcheva & Ruszczynski 2009 ]
NON - LINEAR DESIGN
• Solution through sensitivity calculation
and therefore
• The sensitivities
NON - LINEAR DESIGN
• The reduced gradient
• Can use steepest descent/LBFGS/Truncated Gauss-Newton to solve the problem
CASE STUDY IINON-LINEAR EXPERIMENTAL DESIGN
• Governing equations
• Following Finite Element discretization
• Given model and design settings
• Find data , n < k
• Design: find optimal source-receiver configuration
IMPEDANCE TOMOGRAPHY – OBSERVATION MODEL
IMPEDANCE TOMOGRAPHY – DESIGNS COMPARISON
True modelNaive design Optimized design
Horesh, Haber & Tenorio 2011
MAGNETO - TULLERICS TOMOGRAPHY – OBSERVATION MODEL
• Governing equations
• Following Finite Volume discretization
• Given: model and design settings (frequency )
• Find: data ,
• Design: find an optimal set of frequencies
( ) {1; ( ; )
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Ñ ´ ´ = ¶W
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MAGNETO - TELLURICS TOMOGRAPHY – DESIGNS COMPARISON
Test model Naive design
Optimized non-linear designOptimal linearized design
Haber, Horesh & Tenorio 2008, 2010
THE PARETO CURVE – A DECISION MAKING TOOL
Haber, Horesh & Tenorio 2010
10
6
8
0 20
2
4
40 60 80 1000
Ris
k
1w
• To drill or not to drill ?
Shakespeare 1564-1616
CASE STUDY IIIREGULARIZATION DICTIONARY
DESIGN
OPTIMAL OVER-COMPLETE DICTIONARY DESIGN
• Use popular L1 to get a sparse solution - but what dictionary should be used ?
• Accounts for the model space, observation operator and noise characteristics of the problem
• Requires more sophisticated algorithms to compute
• Non-smooth optimization framework Modified L-BFGS [Overton 2003 ] Horesh & Haber 2009
OPTIMAL DICTIONARY DESIGN - NUMERICAL RESULTS
Tm ( )T
F m d+ =ò ( )0 0D u D
Horesh & Haber 2009
TVm( )ˆ ˆDu D
OPTIMAL DICTIONARY DESIGN - ASSESSMENT WITH NOISE
( )TF m d+ =ò ( )0 0
D u D
Horesh & Haber 2009
( )ˆ ˆDu D
0.1%
1%
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TVm
OPTIMAL DICTIONARY DESIGN GEOPHYSICAL ASSESSMENT
( )TF m d+ =ò
Horesh & Haber 2009
Tm ( )0 0
D u D ( )ˆ ˆDu D
SUMMARY
THE QUOTE CORNER
• What I see in nature is a grand design that we can comprehend only imperfectly, and that must fill a thinking person with a feeling of humility — A. Einstein
• Don’t design for everyone. It’s impossible. All you end up doing is designing something that makes everyone unhappy — L. Reichelt
• It’s art if can’t be explainedIt’s fashion if no one asks for an explanationIt’s design if it doesn’t need explanation — W. Stokkel
TAKE HOME MESSAGES
• Only two (important) elements in the big (inversion) puzzle...
• Experimental design
• Regularization design
• Design in ill-posed inverse problems is an important topic which requires more attention
• New frontiers in inverse problems and optimization
EPILOGUE
DESIGN IN INVERSION – OPEN COLLABORATIVE RESEARCH
• IBM Research
• MITACS
• University of British Columbia
www.wix.com/ocrdesign/home
Andrew Conn
Eldad Haber
Raya Horesh
Jim Nagy
Luis Tenorio
Andreas Wächter
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.
.
ACKNOWLEDGMENTS
Michele Benzi
Jennifer Fohring
Hui Haung
Ulisses Mello
David Nahamoo
Alessandro Veneziani
DOE NSF
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Questions?Thank you