Nonlinear Modeling and Processing Using Empirical Intrinsic Geometry with Application to Biomedical Imaging

28/08/2013
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Nonlinear Modeling and Processing Using Empirical Intrinsic Geometry with Application to Biomedical Imaging

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.

Nonlinear
 Modeling
 and
 Processing
  Using
 Empirical
 Intrinsic
 Geometry
  with
 Application
 to
 Biomedical
 Imaging
  Ronen
 Talmon1,
 Yoel
 Shkolnisky2,
 and
 Ronald
 Coifman1
  1Mathematics
 Department,
 Yale
 University
  2Applied
 Mathematics
 Department,
 Tel
 Aviv
 University
 
 
 
 
  Geometric
 Science
 of
 Information
 (GSI
 2013)
  August
 28-­‐30,
 2013,
 Paris
  Introduc)on
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  2
  •  Example
 for
 Intrinsic
 Modeling
 I
  "   Molecular
 Dynamics
  "   Consider
 a
 molecule
 oscilla)ng
 stochas)cally
 in
 water
 
  –  For
 example,
 Alanine
 Dipep)de
  "   Due
 to
 the
 coherent
 structure
 of
 molecular
 mo)on,
 we
 assume
 that
 the
  configura)on
 at
 any
 given
 )me
 is
 essen)ally
 described
 by
 a
 small
 number
  of
 structural
 variables
  – 
 In
 the
 Alanine
 case,
 we
 will
 discover
 two
 factors,
 
 corresponding
 to
 the
 dihedral
 angles
  1 8 7 4 3 9 5 6 2 10 Introduc)on
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  3
  •  Example
 for
 Intrinsic
 Modeling
 I
  "   We
 observe
 three
 atoms
 of
 the
 molecule
 for
 a
 certain
 period,
 three
 other
  atoms
 for
 a
 second
 period,
 and
 the
 rest
 in
 the
 last
 period
  "   The
 task
 is
 to
 describe
 the
 posi)ons
 of
 all
 atoms
 at
 all
 )mes
  –  More
 precisely,
 derive
 intrinsic
 variables
 that
 correspond
 to
 the
 dihedral
 angles
  and
 describe
 their
 rela)on
 to
 the
 posi)ons
 of
 all
 atoms
  –  We
 always
 derive
 the
 same
 intrinsic
 variables
 (angles)
 from
 par)al
 observa)ons
  (independently
 of
 the
 specific
 atoms
 we
 observe)
 
  –  If
 we
 learn
 the
 model,
 we
 can
 describe
 the
 
  posi)ons
 of
 all
 atoms
  1 8 7 4 3 9 5 6 2 10 Introduc)on
  Talmon,
 Shkolnisky,
 and
 Coifman
  4
  •  Example
 for
 Intrinsic
 Modeling
 II
  "   PredicBng
 EpilepBc
 Seizures
  " Goal:
 to
 warn
 the
 pa)ent
 prior
 to
 the
 seizure
  (when
 medica)on
 or
 surgery
 are
 not
 viable)
  " Data:
 intracranial
 EEG
 recordings
  8/28/13
  0 0.5 1 1.5 2 x 10 5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Samples icEEGRecording Time Frames Frequency[Hz] 50 100 150 200 250 300 350 400 112 96 80 64 48 32 16 Introduc)on
  Talmon,
 Shkolnisky,
 and
 Coifman
  5
  •  Example
 for
 Intrinsic
 Modeling
 II
  " Our
 assump)on:
 the
 measurements
 are
 controlled
 by
 underlying
 processes
 that
  represent
 the
 brain
 ac)vity
  " Main
 Idea:
 predict
 seizures
 based
 on
 the
 “brain
 ac)vity
 processes”
  " Challenges:
 Noisy
 data,
 unknown
 model,
 and
 no
 available
 examples
  8/28/13
  0 0.5 1 1.5 2 x 10 5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Samples icEEGRecording Time Frames Frequency[Hz] 50 100 150 200 250 300 350 400 112 96 80 64 48 32 16 Introduc)on
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  6
  •  Manifold
 Learning
"   Represent
 the
 data
 as
 points
 in
 a
 high
 dimensional
  space
  "   The
 points
 lie
 on
 a
 low
 dimensional
 structure
  (manifold)
 that
 is
 governed
 by
 latent
 factors
  "   For
 example,
 atom
 trajectories
 and
 the
 dihedral
  angles
  Introduc)on
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  7
  •  Manifold
 Learning
"   Tradi)onal
 manifold
 learning
 techniques:
  –  Laplacian
 eigenmaps
 [Belkin
 &
 Niyogi,
 03’]
  –  Diffusion
 maps
 [Coifman
 &
 Lafon,
 05’;
 Singer
 &
 Coifman,
 08’]
  Manifold
  Learning
  Parameteriza)on
 of
  the
 manifold
  Empirical
 Intrinsic
 Geometry
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  8
  •  Formula)on
 –
 “State”
 Space
" Dynamical
 model:
 let
 
 
 
 
 
 
 be
 a
 
 
 
 -­‐dimensional
 underlying
 process
 (the
 state)
 in
  )me
 index
 
 
 
 
 that
 evolves
 according
 to
 
 
  where
 
 
 
 
 
 
 are
 unknown
 dri`
 coefficients
 and
 
 
 
 
 
 
 
 are
 independent
 white
 noises
 
  " Measurement
 modality:
 let
 
 
 
 
 
 be
 an
 
 
 
 -­‐dimensional
 measured
 signal,
 given
 by
  – 
 
 
 
 
 
 
 is
 the
 clean
 observa)on
 component
 drawn
 from
 the
 )me-­‐varying
 pdf
  – 
 
 
 
 
 
 
 is
 a
 corrup)ng
 noise
 (independent
 of
 
 
 
 
 
 )
  – 
 
 
 
 
 
 
 is
 an
 arbitrary
 measurement
 func)on
  " The
 goal:
 recover
 and
 track
 
 
 
 
 
 
 
 given
  zt = g(yt, vt) f(y; ✓) d✓i t = ai (✓t)dt + dwi t, i = 1, . . . , d Empirical
 Intrinsic
 Geometry
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  9
  •  Manifold
 Learning
 for
 Time
 Series
" The
 general
 outline:
  –  Construct
 an
 affinity
 matrix
 (kernel)
 between
 the
 measurements
 
 
 
 
 
 ,
 e.g.,
  –  Normalize
 the
 kernel
 to
 obtain
 a
 Laplace
 operator
 [Chung,
 97’]
  –  The
 spectral
 decomposi)on
 (eigenvectors)
 represents
 the
 underlying
 factors
  Manifold
  Learning
  k(zt, zs) = exp ⇢ kzt zsk2 " 'i 2 RN $ ✓i t 2 RN Measurement
  Modality
  N N N N N Empirical
 Intrinsic
 Geometry
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  10
  •  Intrinsic
 Modeling
  "   The
 mapping
 between
 the
 observable
 data
 and
 the
 underlying
 processes
  is
 o`en
 stochas)c
 and
 contains
 measurement
 noise
  –  Repeated
 observa)ons
 of
 the
 same
 phenomenon
 usually
 yield
 different
  measurement
 realiza)ons
  –  The
 measurements
 may
 be
 performed
 using
 different
 instruments/sensors
 
  "   Each
 set
 of
 related
 measurements
 of
 the
 same
 phenomenon
 will
 have
 a
  different
 geometric
 structure
  –  Depending
 on
 the
 instrument
 and
 the
 specific
 realiza)on
  –  Poses
 a
 problem
 for
 standard
 manifold
 learning
 methods
  Empirical
 Intrinsic
 Geometry
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  11
  •  Intrinsic
 Modeling
  Intrinsic(Embedding( Observable(Domain(II(Observable(Domain(I( Par6al(Observa6on(I7A( Par6al(Observa6on(I7B( Empirical
 Intrinsic
 Geometry
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  12
  •  How
 to
 Obtain
 an
 Intrinsic
 Model?
"   Q:
 Does
 the
 Euclidean
 distance
 between
 the
 measurements
 convey
 the
 informa)on?
  Realiza)ons
 of
 a
 random
 process
 and
 measurement
 noise
  "   A:
 We
 propose
 a
 new
 paradigm
 -­‐
 Empirical
 Intrinsic
 Geometry
 (EIG)
 
  [Talmon
 &
 Coifman,
 PNAS,
 13’]
  –  Find
 a
 proper
 high
 dimensional
 representa)on
  –  Find
 an
 intrinsic
 distance
 measure:
 robust
 to
 measurement
 noise
 and
 modality
 
 
 
  k(zt, zs) = exp ⇢ kzt zsk2 " Empirical
 Intrinsic
 Geometry
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  13
  •  Geometric
 Interpreta)on
"   Exploit
 perturba)ons
 to
 explore
 and
 learn
 the
 tangent
 plane
  "   Compare
 the
 points
 based
 on
 the
 principal
 direc)ons
 of
 the
 tangent
  planes
 (“local
 PCA”)
  Underlying*Process* Measurement*1* Measurement*2* Empirical
 Intrinsic
 Geometry
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  14
  •  The
 Mahalanobis
 Distance
"   We
 view
 the
 local
 histograms
 as
 feature
 vectors
 for
 each
 measurement
 
  "   For
 each
 feature
 vector,
 we
 compute
 the
 local
 covariance
 matrix
 in
 a
 temporal
  neighborhood
 of
 length
 
 
 
  where
 
 
 
 
 
 
 is
 the
 local
 mean
  "   Define
 a
 symmetric
 
 
 
 
 -­‐dependent
 distance
 between
 feature
 vectors
Defini)on
 –
 Mahalanobis
 Distance
  zt ! ht L C d2 C(zt, zs) = 1 2 (ht hs)T (C 1 t + C 1 s )(ht hs) Ct = 1 L tX s=t L+1 (hs µt)(hs µt)T Empirical
 Intrinsic
 Geometry
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  15
  •  Results

 
  "   Each
 histogram
 bin
 can
 be
 expressed
 as
 
 
  where
 
 
 
 
 
 
 
 are
 the
 histogram
 bins
 
 
  "   By
 relying
 on
 the
 independence
 of
 the
 processes:
  Assump)on
  "   The
 histograms
 are
 linear
 transforma)ons
 of
 the
 pdf
 
 
 
 
 
 
 
 
 
 
 
 
 
 
p(z; ✓) = Z g(y,v)=z f(y; ✓)q(v)dydv Lemma
  "   In
 the
 histograms
 domain,
 any
 sta)onary
 noise
 is
 a
 linear
 transforma)on
p(z; ✓) Hj hj t = Z z2Hj p(z; ✓)dz Empirical
 Intrinsic
 Geometry
  Assump)on
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  16
  •  Results
The
 Mahalanobis
 distance:
  "   Is
 invariant
 under
 linear
 transforma)ons,
 thus
 by
 lemma,
 noise
 resilient
  "   Approximates
 the
 Euclidean
 distance
 between
 samples
 of
 the
 underlying
 process,
 i.e.,
 
 
  "   The
 process
 
 
 
 
 
 
 can
 be
 described
 as
 a
 (possibly
 nonlinear)
 bi-­‐Lipschitz
 func)on
 of
 the
  underlying
 process
  "   We
 rely
 on
 a
 first
 order
 approxima)on
 of
 the
 measurement
 func)on:
 
  where
 
 
 
 
 
 
 
 
 is
 the
 Jacobian,
 defined
 as
  k✓t ✓sk2 = d2 C(zt, zs) + O(kht hsk4 ) Theorem
 [Talmon
 &
 Coifman,
 PNAS,
 13’]
  ht Jt ht = JT t ✓t + ✏t Jji t = @hj @✓i Empirical
 Intrinsic
 Geometry
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  17
  •  Rela)onship
 to
 Informa)on
 Geometry
Q:
 Does
 the
 structure
 of
 the
 measurements
 convey
 the
 informa)on?
  A:
 The
 local
 densi)es
 of
 the
 measurements
 do
 and
 not
 par)cular
 realiza)ons
 
  " Informa)on
 Geometry
 [Amari
 &
 Nagaoka,
 00’]:
  –  Use
 the
 Kullback-­‐Liebler
 divergence
 approximated
 by
 the
 Fisher
 metric
 
  where
 
 
 
 
 
 is
 the
 Fisher
 InformaBon
 matrix
" EIG:
 a
 similar
 data-­‐driven
 metric:
 consider
 the
 following
 features
  It D(p(zt; ✓)||p(zt0 ; ✓)) = ✓T t It ✓t lj t = ↵j log ⇣ hj t ⌘ Theorem
  "  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (underlying
 manifold
 dimensionality)
  "  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (feature
 vectors
 dimensionality)
 
It = JT t Jt Ct = JtJT t Empirical
 Intrinsic
 Geometry
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  18
  •  Anisotropic
 Kernel
"   Let
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 be
 a
 set
 of
 measurements
  –  For
 each
 measurement,
 we
 compute
 the
 local
 histogram
 and
 covariance
  "   Construct
 an
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  symmetric
 affinity
 matrix
  –  Approximates
 the
 Euclidean
 distances
 between
 the
 underlying
 process
  –  Invariant
 to
 the
 measurement
 modality
 and
 resilient
 to
 noise
  "   The
 corresponding
 Laplace
 operator
 
 
 
 
 
 
 can
 recover
 the
 underlying
 process
  "   Compute
 the
 eigenvalues
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 and
 eigenvectors
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 of
 
 
 
 
 
 
 
 
 
  "   The
 leading
 eigenvectors
 represent
 the
 underlying
 process
  L Wts = c exp ⇢ d2 C(zt, zs) " L Applica)ons
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  19
  •  Molecular
 Dynamics
" Task:
 track
 Alanine
 Dipep)de
 in
 water
 from
 par)al
 observa)ons
  [Dsilva,
 Talmon,
 Rabin,
 Coifman
 &
 Kevrekidis,
 13’]
  Odd
 Atoms
  Even
 Atoms
  EIG:
  DM:
  1 8 7 4 3 9 5 6 2 10 Applica)ons
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  20
  •  Molecular
 Dynamics
"   3
 Snapshots
 of
 the
 true
 and
 reconstructed
 trajectories
  –  Using
 a
 mul)scale
 method
 (Laplacian
 Pyramid
 [Rabin
 &
 Coifman,
 SDM,
 12’])
 
  [Dsilva,
 Talmon,
 Rabin,
 Coifman
 &
 Kevrekidis,
 13’]
  Applica)ons
  " Results:
 
 
 
  –  3D
 points
 -­‐
 the
 3
 leading
 eigenvectors
 (each
 point
 –
 an
 EEG
 )me
 frame)
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  21
  •  Predic)ng
 Epilep)c
 Seizures
0 0.5 1 1.5 2 x 10 5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Samples icEEGRecording Input:
  −0.04 −0.02 0 0.02 0.04 0.06 −0.04 −0.02 0 0.02 0.04 0.06 −0.05 0 0.05 Training, preseizure Training, interictal Testing, preseizure Testing, interictal Nonlinear
 Processing
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  22
  •  Bayesian
 Filtering
"   How
 to
 do
 processing
 in
 the
 inferred
 parametric
 domain?
  –  Combine
 the
 inferred
 geometry
 and
 the
 )me
 series
 dynamics
  " Main
 idea:
  –  Define
 a
 pseudo-­‐likelihood
 funcBon
 based
 on
 the
 inferred
 intrinsic
 model
  –  Empirically
 define
 a
 prior
 funcBon
 based
 on
 past
 observa)ons
  –  Combine
 the
 two
 using
 a
 Bayesian
 framework
  " Assump)on:
  –  Locally,
 the
 distribu)on
 in
 the
 embedded
 domain
 is
 a
 good
 approxima)on
 of
 the
  distribu)on
 in
 the
 underlying
 process
 original
 domain
  –  The
 posterior
 pdf
 of
 the
 underlying
 process
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 can
 be
 es)mated
 based
  on
 the
 embedding
  Nonlinear
 Processing
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  23
  •  Pseudo-­‐likelihood
Parametric
 Domain
Measurement
 Domain
Inaccuracy
 of
 the
 representa)on
(zt)|✓t ⇠ N (✓t, C✓,t) Nonlinear
 Processing
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  24
  •  Empirical
 Prior
Parametric
 Domain
Measurement
 Domain
Explicit
 use
 of
 the
  chronological
 order
 to
 obtain
  empirical
 dynamical
 model
✓t|✓t 1 ⇠ N ⇣ ✓f t 1, Cf ✓,t 1 ⌘ Nonlinear
 Processing
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  25
  •  Monte
 Carlo
 Approach
"   Represent
 the
 posterior
 pdf
 by
 a
 set
 of
 (random)
 samples
  –  Let
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 be
 a
 set
 of
 support
 samples
 (“parBcles”)
 that
 characterizes
 the
  posterior
 pdf
 given
 the
 previous
 stage
 and
 the
 new
 measurement
  –  Let
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 be
 a
 set
 of
 weights
 associated
 with
 the
 par)cles
  p(✓t|✓t 1, zt) ⇡ PX k=1 w (k) t (✓t ✓ (k) t ) Nonlinear
 Processing
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  Talmon,
 Shkolnisky,
 and
 Coifman
  26
  •  Monte
 Carlo
 Approach
"   Where
 the
 weights
 are
 denoted
 as
 
  with
  "   By
 Bayes’
 theorem
 
 
  where
 the
 densi)es
 can
 be
 es)mated
 in
 the
 embedded
 domain
  Nonlinear
 Processing
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  27
  •  MMSE
 Es)mator
"   For
 example,
 based
 on
 the
 es)mated
 posterior
 pdf,
 the
 MMSE
 esBmator
 of
 the
  factors
 at
 
 
 
 can
 be
 computed
 by
  –  Requires
 few
 ini)al
 values
 of
 the
 original
 factors
 for
 alignment
  b✓t = E[✓t|✓t 1, zt] = Z ✓tp(✓t|✓t 1, zt)d✓t ⇡ PX k=1 p(✓ (k) t |✓t 1, zt)✓ (k) t = PX k=1 w (k) t ✓ (k) t Applica)ons
 
 
 
  "   Imaging
 model
 -­‐
 consider
 a
 2D
 shape
 measured
 by
 a
 1D
 linear
 sensor
 array
  –  Rigid
 biological
 material
 that
 vibrates
 over
 )me
  –  Emission
 of
 radia)on
 or
 light
  –  Noisy
 measurements
 of
 the
 instantaneous
 amount
 of
 radia)on
 
  that
 travelled
 through
 the
 object
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 Shkolnisky,
 and
 Coifman
  28
  •  Biomedical
 Imaging
Applica)ons
 
 
  "   ObjecBve:
 to
 track
 the
 object
 based
 on
 the
 measurements
  "   We
 simulated
 a
 diffusion
 process
 and
 output
 signals
 of
 15
 sensors
 with
  Gaussian
 noise:
  "   Note:
 the
 simulated
 model
 was
 not
 used
 for
 the
 inference
 and
 the
 tracking
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  29
  •  Biomedical
 Imaging
zi t = f(pi ✓t) + vi t Applica)ons
 
 
  "   Tracking
 the
 center
 posi)on
 of
 the
 shape:
  –  The
 yellow
 curve
 is
 the
 true
 posi)on
 of
 the
 center
  –  The
 ver)cal
 gray
 strips
 represent
 the
 posterior
 pdf
 es)mate
  –  The
 solid
 black
 curve
 is
 the
 expected
 value
 (MMSE
 es)mator)
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  30
  •  Biomedical
 Imaging
Time CenterPosition 50 100 150 200 250 300 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 MMSE Ground Truth Conclusions
  8/28/13
  Talmon,
 Shkolnisky,
 and
 Coifman
  31
  •  Summary
  "   The
 no)on
 of
 empirical
 intrinsic
 modeling
  –  Empirical
 geometry
 of
 local
 distribu)ons
  "   Nonlinear
 processing
 framework
 in
 the
 low
 dimensional
 intrinsic
 domain
  "   Used
 in
 a
 wide
 variety
 of
 applica)ons
  –  Nonlinear
 problems
 without
 exis)ng
 defini)ve
 models
  –  In
 par)cular,
 biomedical
 imaging
 
  (based,
 for
 example,
 on
 photon
 counter
 sensors)
 
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 Shkolnisky,
 and
 Coifman
  32