and the Existence of True Models

Transcription

1 Statistical Image Reconstruction and the Existence of True Models Joseph A. O Sullivan Samuel C. Sachs Professor Dean, UMSL/WU Joint Undergraduate Engineering Program Professor of Electrical and Systems Engineering, Biomedical Engineering, and Radiology jao@wustl.edu Supported by Washington University and NIH-NCI grant R01CA75371 (J. F. Williamson, VCU, PI),NIH-NCI (CA110011) (Y.-C. Tai, PI), Susan G. Komen for the Cure (BCTR ), Washington University Internal Fund

2 Collaborators Faculty Richard Laforest David G. Politte Chrysanthe Preza, U Memphis Donald L. Snyder Yuan-Chuan Tai Bruce R. Whiting Jeffrey F. Williamson, VCU Heyu Wu Chemical Engineers M. Al-Dahhan, Mo. U. S&T M. Dudukovic R. Varma Students Norbert Agbeko Alan Kaplan Po-Hsiang Lai Aswin John Mathews Ikenna Odinaka Jasenka Benac Josh Evans, VCU Daniel Keesing Giovanni Lasio, VCU Debashish Pal Liangjunn Xie Post-Docs Ju-Chieh Kevin Cheng (Laforest lab) Sergey Komarov (Tai s lab) Tae Yong Song (Tai s lab) 2

3 Selected Work Dual energy X-ray CT imaging with radiation oncology applications Positron emission tomography: virtual pinhole PET; novel radionuclides Dual energy gamma ray imaging of chemical columns Emerging interest in X-ray imaging for baggage scanning Differential interference contrast microscopy 3

4 Bottom Line Message of Talk Model-based imaging works well There are limits to what is achievable There exist tools to quantify limits For imaging, i there are two types of errors Approximation error Modeling errors, systematic errors (limits in data space) Function approximation errors (limits in image space) Estimation error Noise and quantization errors Total error = approximation error + estimation error 4

5 Existence of a True Model Jorma Rissanen: There is no true model Consider Given imaging scanner Phantom whose physics are known exactly Repeated measurements of that phantom The statistics of the measurements given this known phantom can be measured as accurately as desired Does this yield a true model? What is a true model? Sampling and quantization Function space models pd ( c) Rods are brass, iron, aluminum, and teflon Revisit later 5

6 Computational Imaging for Microscopy Physics of Specimens Light Propagation Physics Abstraction Barrier p( θ ) Specimen Model or simply Light Propagation Model p( s θ ) max log θ Θ θ Θ p( d θ ) + log p( θ ) max log p( d θ ) θ Θ Algorithm Development Raw data Physics of Microscopes Data Processor C. Preza, CI2008 Microscope Model p( d s) J (θ ), bias, variance, Processed data contrast, noise,... Performance Analysis

7 Computational Imaging Physics of Scenes Abstraction Barrier Scene Model Scene to Sensor Propagation Model Algorithm Development Sensors Sensor Model Raw data Processed data Data Processor C. Preza, CI2008 Performance Analysis

8 Computational Imaging Physics of Scenes Abstraction Barrier Scene Model Scene to Sensor Propagation Model Algorithm Development SOMATOM Definition CT Scanner ccir.wustl.edu Raw data Sensors Sensor Model Data Processor Processed data Performance Analysis Data from R. Laforest and M. Mintun, MIR.

9 Computational Imaging Physics of Scenes Abstraction Barrier Scene Model Scene to Sensor Propagation Model Algorithm Development This Sensor Sensors approach works well. Model SOMATOM Definition CT Scanner ccir.wustl.edu What is the limit? Raw data When is there enough physics? Data Processor Processed data Performance Analysis

10 X-Ray Transmission Physics for CT Source Spectrum, Energy E Beer s Law and Attenuation l SEe ( ) μ ( x, E) dx SE ( ) Mean Photons/Detector I 0 μ ( x, E) dx l ISEe 0 ( ) Beam-hardening Detector Sensitivity Spectrum Mean (Photon) Flux DE ( ), Φ ( E ) = SEDE ( ) ( ) ( xe, ) dx l I Φ( E) e μ 0 J. L. Prince and J. Links, Medical Imaging Signals and Systems. Pearson Education: Upper Saddle River, NJ, Φ Mean photon model kvp ( xe, ) dx l I0 ( E) e μ Φ de 0 10

11 Linear Image Reconstruction Normalize relative to an air scan Beam-hardening correction to target energy Negative log to estimate line integrals at target energy Linear inversion, normalize (e.g. water-equivalent) equivalent) l( y) μ( xedx, ) log kvp Φ μ ( xe, ) dx l( y) I ( E) e de BH kvp ( μ ) 0 I 0 Φ ( E ) de ˆ( μ xe, ) ˆ l( y) μ water ( E ) ˆ μ( xe, ) = FBP ( xedx, ), cx ( ) = dy () Normalize & correct Negative logarithm FBP normalize cx () 0 11

12 X-Ray Transmission Tomography Detector Sensitivity D(E) Back to Basics Probability yphoton of energy E is detected Mean response to a photon of energy E: photon counting response = 1 energy integrating response = E Detector Statistics: photon Poisson distribution Photon counting, Beer s Law as survival probability Poisson Energy integrating compound Poisson Poisson mean λ: k λ λ PN ( = k) = e, k 0 k! Log-likelihood function: l( λ) = kln λ λ I-divergence: k Ik ( λ ) = kln k+ λ λ Ik ( λ) = l( λ) + klnk k ML equivalent to min I-divergence ( ) 12

13 X-Ray Transmission Tomography Reconstruction Principles Deterministic Model Data equal a function of the desired image Approximately invert that function to reconstruct image (minimize a measure of error between the data and the model) 2 min d g( : μ) μ Random Model Find the log-likelihood function for the data Maximize the (possibly penalized) log-likelihood function over possible images max ld ( g( : μ)) μ 13

14 Statistical Image Reconstruction Source-detector pairs indexed by y; voxels indexed by x Data d j (y) Poisson, means g j (y:μ), log-likelihood likelihood function ld ( g( : μ)) = d( y)ln g( y: μ) g( y: μ) j j j j j y Y g j( y: μ) = I jφ j( y, E)exp h( y, x) μ( x, E) + β j( y) E x X Mean unattenuated counts I j, mean background β j Attenuation function μ(x,e),, E energies I μ( x, E) = ci ( x) μi ( i= 1 E) Maximize over μ or c i 15

15 Maximum Likelihood/Minimum I-Divergence Original Estimation Problem: ( ( i ) ) ( i ) ( ) max l d q, E min I d q, E μ E Convex Decomposition Lemma: π m ln qm = min π mln π m P m qm Lifted (Variational) Estimation Problem: μ pye (, ) minmin I( p q) = minmin p( y, E)ln p( ye, ) qye (, ) q E p L q E p L qy (, E ) + L E ( ) { p: p y, E d( y) } = = E y = q: q( y, E) = I jφ j( y, E)exp h( y, x) μ( x, E) x X E E 16

16 Aside: Connection Between Maximum Likelihood and Maximum Entropy ( θ) (, θ) p y = q y x dx ln p ( y θ ) = ln q( y, x θ ) dx π ( x y) = min π ( x y ) ln dx π ( xy ) P q( y, x θ ) Reformulated ML Estimation Problem: ( ) ( ) maxln p y θ = max max π( x y)ln q y, x θ dx π( x y)ln π( x y) dx θ θ π( xy ) P Maximum entropy problem arises naturally within the ML problem. Double maximization leads naturally to an alternating maximization (EM) alg. DLR 77, Snyder and Miller 89, O Sullivan 98 17

17 Statistical Iterative Reconstruction AM Algorithm Alternating Minimization Algorithm Several papers from our group (O S & Benac, TMI 2007) Monotonically increasing log-likelihood I g j( y: μ) I j j( y, E)exp = Φ h( y, x) μi( E) ci( x) + β j( y), E x X i= max ld ( j g j( : μ)) = d j( y)ln g j( y: μ) g j( y: μ) i j= 1 j= 1 y Y { c ( x) } g j( y) = qj( y, E) E Compare using backprojection of data, estimated means ( k ) bi, j( x) ( k+ 1) ( k+ 1) 1 j cˆ ˆ i = c i ( x ) ln ˆ ( k ) Zi ( x) bi, j( x) j ˆ ( k) ( k) b ( x) = h( y, x) μ ( E) q ( y, E) i, j i j ye, ( k ) d ( ) j( y) qj ( y, E) k b i, j( x) = h( y, x) μi( E) ( k ) ye, q (, ') E ' j y E d Compare g Forward Model Update c 18

18 Additional Approximation Error min min I( p q) q E p L L L A smaller linear family has more * 1 2, pi = argmin I( p q) p L I( p q ) = I( p p ) + I( p q ) i constraints. E E * 1 2, qi = argmin I( p q) q E I ( p q ) = I ( p q ) + I ( q q ) i A smaller exponential family has fewer parameters. Additional Approximation Error = Iq ( q )

19 Estimation Error min min I( p q) q E p L Given true p and random p, in L, define T q = arg min I( p q), q = arg min I( p q). * * T T q E q E * * * * Then I( pt q ) = I( pt qt) + I( qt q ). * * Estimation error = ET I( qt q ). Pythagorean-type equality within an exponential family if the minimum is achieved. 20

20 Model Refinement Criterion (Likelihood Term) Consider a hierarchical model Refine a model if the refined estimation error is lower than the original estimation error plus the additional approximation error * * * * * * E T Iq ( 2T q 2) Iq ( 2T q1 T) + E T Iq ( 1T q 1) Interpretation as description length: Change in the number of bits for the likelihood term Estimation error can be computed from Fisher information and is approximately one half the number of parameters The approximation error term must be computed Repeated experiments yield empirical statistics 21

21 The estimation error is orthogonal to the approximation i error q 2 * * q 1 2 * q 1T * q 2T E 2 E 1 22

22 Simulations: Post-reconstruction vs. Joint Statistical Image Reconstruction Large Phantom: 20cm in diameter; thin outer lucite shell; water; four rods in inner 60mm lucite cylinder. Calibration rods: calcium chloride, ethanol, teflon and polystyrene in the 12, 3, 6, and 10 o clock positions, resp. Test phantom rods: muscle, ethanol, teflon and substance X (a bonelike material) in the 12, 3, 6, and 10 o clock positions, resp. Small Phantom: 60mm diameter lucite cylinder with rods FBP-BVM (Basis Vector Method; JF Williamson, et al. 2006) Water-equivalent equivalent beam hardening correction Requires calibration data to estimate linear transformation that generates the component images FBP uses a ramp filter AM-DE (AM Dual Energy Algorithm) NO pre-correction of data NO calibration data NO regularization 23

23 Materials Used - Fractions Substance Styrene Ca. Chloride Fraction Fraction Styrene 1 0 Ca. Chloride 0 1 I μ( x, E) = c ( x) μ ( E) i= 1 i i Ethanol Lucite Teflon Water Muscle Substance X

24 Mini CT Small Phantom 360 Source positions 92 detectors Based on Siemens Somatom Plus 4 scanner 64 by 64 image 25

25 Small Phantom Component Images Styrene Calcium Chloride Truth FBP-BVM AM-DE 26

26 Small Phantom Synthesized Images 20 kev 65 kev Truth FBP-BVM AM-DE 27

27 Small Phantom: Relative Errors I 0 = 100, FBP-BVM AM-DE 28

28 Fisher Information: Within a Component Image Calcium chloride Maximum 1.2E5 Styrene Maximum 1.6E4 29

29 Fisher Information: Across Images Maximum 4.2E4 30

30 CRLB images for 20keV (above) and 65keV (below) images x CRLB FBP-BVM AM-DE 31 Var / CRLB

31 CRLB Images: Diagonals of Inverse of Fisher Information Ratio of variance images to CRLB. Different display windows. Styrene Calcium Chloride CRLB FBP-BVM AM-DE 32

32 CRLB Images: Diagonals of Inverse of Fisher Information Ratio of variance images to CRLB. SAME display windows. Styrene Calcium Chloride CRLB FBP-BVM AM-DE 33

33 Small Phantom Ensemble mean and variance (from 15 samples) I 0 = 100,000 20keV Image AM-DE FBP-BVM Truth Mean Variance 34

34 Small Phantom Ensemble mean and variance (from 15 samples) I 0 = 100,000 65keV Image AM-DE FBP-BVM Truth Mean Variance 35

35 Typical Computation * * * * * * ET Iq ( 2T q 2) Iq ( 2T q1 T) + ET Iq ( 1T q 1) Computed from 20 independent samples (normalized): * * E I T ( q2t q2) * * E T I( q1 T q 1) Iq * * ( 2T q1 T ) * * * * Iq ( 2T q1 T) + E T Iq ( 1T q 1) Choose the more complex model with two components. 36

36 Physics/Data Model Complexity Algorithm Complexity Regularization Energy integrating g detectors EdN ( y,, E ) Finite detector size, better source model Finite pixel, voxel size Average integral or average exponential (arithmetic vs. geometric average) Partial volume effects Motion Scattering Limited angle tomography Region of interest Scanner implementations: beam hardening correction, sampling, etc.

37 Enough Physics? Physical models are always approximate Real-world issues Detector variability Pixelization as function representation Energy dependence as function representation Forward data models as function representation Detector models: efficiency, uniformity, geometry, underlying physics Energy integrating detectors Quantization

38 Enough Physics? Physical models are always approximate Cost of model mismatch? Modeling error via log-likelihood function Estimation error through information geometry and Fisher information Is there a true model? There exists true mean data (repeated measurements) The important cost of increasing model complexity is estimation error Physical models are always approximate Tools exist for optimizing representation (model)