Theory of image reconstruction in DOT

DOT - Lecture 2Theory of Image ReconstructionDiffuse Optical TomographyBrian W. Pogue, Ph.D.Thayer School of EngineeringDartmouth Collegehttp://www.thayer.dartmouth.edu/~bpogue/

Inverse Problem - x-rays versus light(Beer’s Law vs. Diffusion) (linear vs. PDE)Linear Image reconstruction(eg. x-ray computedtomography)Non-Linear Image reconstruction(eg. SPECT, Optical Tomo, )x-rayspatientfan beamdetectorslaser360 0detectors“guess” the image and calculate projections[measurements] = [attenuation op.] [object ][image] = [attenuation op.] T [filter] [measurements][Error ] = [measurements] - [calculated]must be solved through Taylor’s expansion

Inverse Problem - x-raysy idetectory i = x 1 µ 1 + x 2 µ 2 + …x M µ My=ln(I o /I)wherey i = ∑x ij µ jy = J µ (matrix equation)

Inverse Problem - reconstructionfrom projection measurementsy idetectorµ = J -1 y (or µ = [J T J] -1 J T y)J is matrix describing the projection geometry in (x,y)µ is the image of attenuation coefficients to be calculated

Linear image Reconstruction –concept of backprojectionRef: Bushberg

Image Reconstruction –Algebraic Reconstruction technique (ART)1 1 0 01 0 1 01 0 0 10 1 1 00 1 0 10 0 1 1ABCD=765987Over-determined non-square matrix J•Multiply by J T first•Invert square matrixLarger problems must be solved iterativelyusing standard methods for solving largematrix operation problems.J µ = bJ T J µ = J T bµ = [J T J] -1 J T bRef: Bushberg

Non-linear Inverse Problemreconstruction from projection measurementsy idetectory = Φ(µ) + rΦ(µ) is the solution to the diffusion equationµ is the image of attenuation coefficients to be calculatedr is the residual due to measurement error

Interrogating Tissue withFrequency-domain tomographicProjectionsDiffuse light fieldas source rotatesProjections from sourceto each detector⎛ iω⎞∇⋅D( r)∇Φ(r,ω)−⎜µa(r)+ ⎟Φ(r,ω)= −So(ω)δ(r −ro)⎝ c ⎠r d∫ ⎡Φ(, r ω )( ∆µ a) Γ(, r ω)⎤drr ⎣⎦s

Inverse Problem - reconstruction fromprojection measurements2χ=NMC M 2∑( Φ −Φ)i ii=1Taylor’s seriesexpansion of derivative2∂χ∂µ=2∂χ∂µ2d ⎛ ∂χdµ⎜⎝ ∂µ⎞⎟ +⎠( µ ) + ( µ − µ ) ⎜ ( µ ) ⎟ ...000Rearrange for Newton’siterative method solutionµi+1⎡= µi+ ⎢⎣d ⎛dµ⎜⎝2∂χ∂µ( µ ) ⎟ ( µ )i⎞⎤⎟⎥⎠⎦−12∂χ∂µiWhat is the derivative?What is the2 nd derivative?2⎛ ∂χµ⎜⎝ ∂µ∂∂2∂χ∂µC⎛ ∂Φ⎞C M( µ ) = 2⎜⎟ ( Φ −Φ)i⎜⎝∂µ⎟⎠TTCC 2 C⎞ ⎛ ∂Φ⎞ ∂Φ⎛ ∂ Φ ⎞C M( µ ) ⎟ = 2⎜⎟ + 2⎜⎟ ( Φ −Φ)i⎟⎠⎜⎝∂µ⎟⎠∂µ⎜⎝2∂µ⎟⎠TCTCCT⎡⎛∂Φ⎞ ⎛ ∂Φ⎞⎤⎛ ∂Φ⎞µiµi⎢⎜⎥−µ⎟⎜µ⎟⎜µ⎟+ 1= +⎢⎥ ⎝ ∂⎣⎝∂ ⎠ ⎝ ∂ ⎠⎦⎠−1C M( Φ Φ )

Inverse Problem - reconstruction fromprojection measurementsInverse Problem - reconstruction fromprojection measurementsµ i µ iµ∆−= +1µ i µ iµ∆−= +1[ ] ( )MCTTJIJJΦΦλµ∆−+=−1⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡=aNNNMaNMaNMaNNNMaNMaNMNNNMNMNMNNNMNMNMaNNaaaNNaaaNNaaaNNaaNNNNNNNNIIIDDDDIDIDIIIIIIIDDDDIDIDIDDDDIDIDIJδµδθδµδθδµδθδµδδµδδµδδδθδδθδδθδδδδδδδµδθδµδθδµδθδµδδµδδµδδµδθδµδθδµδθδµδδµδδµδδδθδδθδδθδδδδδδδδθδδθδδθδδδδδδ212121212221222212121111211122212222121211112111lnlnln;;lnlnlnlnlnlnlnlnln;;lnlnln;;lnlnlnCJµ∂Φ= ∂

NIRFAST package - ShareWareAuthor: Hamid Dehghani, Dartmouthhamid@dartmouth.eduhttp://biolight.thayer.dartmouth.edu

Finite Element Meshing ToolsNETGET – Shareware!by Joachim Schoberl, Univercity Lintz, Austria,http://www.hpfem.iku.at/netgen/Many Many 2-D meshing programs on the webCommercial Solutions:MATLAB – PDE Toolbox - 2D meshes ($$)FEMLAB – PDE Solver – 2D/3D ($$$)ANSYS – High end PDE Solver ($$$$)FLUENT – Professional PDE Solver ($$$$$)

Finite Element MeshingLow resolution High resolutionMRISegmented Tissue MeshRat Cranium Mesh

3D Finite Element MeshingPartial volume meshDeformed breast mesh from MRIFull volume meshDeformation Volume mesh

Inverse Problem - reconstruction fromprojection measurementsCalculated by:1)PerturbationMethod2)Direct AnalyticJacobian3) Adjoint method

Inclusion of Prior Information: Hard Prior Info~J =JK~J =JKJ⎡1 ......... NN ⎤⎢2 ........ NN⎥⎢⎥⎢ .⎥= ⎢ ⎥⎢ .⎥⎢ .⎥⎢⎣NM .... NN ⎥⎦KR1R2 Rn⎡ k1,1k1,2 k1,n ⎤⎢⎥⎢k2,1k2,2 k2,n⎥⎢ ⎥⎢⎥⎢ kj,1kj,2 kj,n=⎥⎢− − − − − − − − − − − −⎥⎢ k1,1k1,2 k1,n ⎥⎢⎥⎢k2,1~ k2,2 k2,nJ = JK ⎥⎢ ⎥⎢⎥⎢ kj,1kj,2 kj,n⎣⎥⎦kζ , η⎧1,= ⎨⎩0,ζ ∈ Rηζ ∉ Rη⎫⎬⎭∆µ=⎡1.........NN ⎤⎢2........ NN⎥⎢ ⎥⎢.⎥J = ⎢ ⎥⎢.⎥⎢.⎥⎢⎣NM....NN ⎥⎦~ ~ 1[ ]T T(C MJ J J Φ −Φ)− ~⎡1 .........2 2 1⎤⎢2 ........2 2 1⎥⎢⎥⎢ .⎥J = ⎢ ⎥⎢ .⎥⎢ .⎥⎢⎣2 4 0 ....2 2 1 ⎥⎦J⎡1, 2 ⎤⎢2, 2⎥⎢ ⎥⎢ 3 ⎥= ⎢ ⎥⎢ . ⎥⎢ . ⎥⎢⎣240,2⎥⎦

Inclusion of Prior Information: Soft Prior Info~J =JK~J =JK⎡ l1,⎢⎢⎢lNN⎢L = ⎢ −⎢⎢⎢⎢⎣1, 1−0ll1,NNNN , NN−|||−|||ll−1,1NN , 10−ll−1,NNNN , NN⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Prior MR Info⎡1.........NN ⎤⎢2........ NN⎥⎢ ⎥⎢.⎥J = ⎢ ⎥⎢.⎥⎢.⎥⎢⎣NM....NN ⎥⎦Structured mesh Hard Prior Soft Priorli,j=⎧⎪⎨−⎪⎩110NRRiii==≠jRRjj⎫⎪⎬⎪⎭

MR/NIR Patient imaging3D SPGR T1 MRI: approximately 50 coronal slices (1.5 mm thickness, s, nogap)Six NIR wavelengths measuredExam time Patient positioning: 5 minutes Data acquisition: 10 minutesPost processing time: 30 minutes (2D meshing and image reconstruction)ction)

Indirect reconstruction examplei. No priors ii. Spatial priors2D FEM meshindicating adipose andfibroglandular tissuelocations, and opticalfiber positionsi.ii.

Applying multiple priors in patient imagingT1 axial and oblique coronal MRINopriorsSpatialpriors2D FEM mesh defines imaginggeometry and spatial priorSpectralpriorsSpectralandspatialpriorsBrooksby, Srinivasan, et al. Optics Letters. 2005, in press

Representative patient imagesAxial Oblique coronal FEM meshesReconstructed NIR parameters show MR-like resolution andquantify tissue properties unavailable via conventional imagingtechniques

Summary of component breast tissue propertiesestimated for (N=11) subjects10080604020Adipose tissueGlandular tissue32.521.510.5Adipose tissueGlandular tissue0[HbT](µM)StO2(%)Water(%)[HbT](µM)StO2(%)Water(%)0ScatteringamplitudeScatteringpowerScatteringamplitudeScatteringpowerGlandular tissue shows significantly elevated [HbT], water, and b, andreduced A, relative to adipose tissue (p-values

Imaging Geometry AnalysisPogue et al., Opt. Express. 4, 270-86 (1999)

Imaging Geometry AnalysisCompressed slab - high noise- accurate shape and optical propertiesCircular tomography - low noise- accurate shape and optical propertiesDepth profiling - irregular shaped images- high noise- inaccurate optical properties

In a regularized situation, you canalways recover an image!2 Sources 2 Detectors86807570656055504540350.02200.02000.01500.01000.00803025201510500 10 20 30 40 50 60 70 8086

4 Sources 4 Detectors86807570656055504540350.02200.02000.01500.01000.00803025201510500 10 20 30 40 50 60 70 8086

8 Sources 8 Detectors86807570656055504540350.02200.02000.01500.01000.00803025201510500 10 20 30 40 50 60 70 8086

Images for different numbers of sources & detectors862 x 2864 x 4 8 x 8868075700.02200.02008075700.02200.02008075700.02200.020065656560550.015060550.015060550.0150504540350.01000.0080504540350.01000.0080504540350.01000.00803030302525252020201515151010105550000 10 20 30 40 50 60 70 80 860 10 20 30 40 50 60 70 80 860 10 20 30 40 50 60 70 80 8612 x 12 16 x 16 24 x 248686868075700.02200.02008075700.02200.02008075700.02200.020065656560550.015060550.015060550.0150504540350.01000.0080504540350.01000.0080504540350.01000.00803030302525252020201515151010105550000 10 20 30 40 50 60 70 80 8632 x 32860 10 20 30 40 50 60 70 80 860 10 20 30 40 50 60 70 80 868075700.02200.02006560550.0150504540350.01000.00803025201510500 10 20 30 40 50 60 70 80 86Pogue et al, Optics Exp. 1999

Fullreflectance &transmittancetissueFanbeamtissue100Circular tomography 16x16 imaging10010010010090800.02200.020090800.02200.020090800.02200.020090800.02200.020090800.02200.0200700.0175700.0175700.0175700.0175700.0175605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.00602020202020101010101000 10 20 30 40 50 60 70 80 90 10000 10 20 30 40 50 60 70 80 90 10000 10 20 30 40 50 60 70 80 90 10000 10 20 30 40 50 60 70 80 90 10000 10 20 30 40 50 60 70 80 90 1001008060402000 20 40 60 80 100Fan Beam tomography0.022 1000.020 800.0150.0100.00860402000 20 40 60 80 1000.0220.0200.0150.0100.0081008060402000 20 40 60 80 1000.0220.0200.0150.0100.0081008060402000 20 40 60 80 1000.0220.0200.0150.0100.008Pogue et al, Optics Exp. 1999

Sub-surfaceImagingtissuetissue100Sub-surface tomography 8x8 imaging10010010010090800.02200.020090800.02200.020090800.02200.020090800.02200.020090800.02200.0200700.0175700.0175700.0175700.0175700.0175605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.00602020202020101010101000000 10 20 30 40 50 60 70 80 90 1000 10 20 30 40 50 60 70 80 90 1000 10 20 30 40 50 60 70 80 90 1000 10 20 30 40 50 60 70 80 90 100Sub-surface tomography 4x4 imaging10010010010000 10 20 30 40 50 60 70 80 90 10010090800.02200.020090800.02200.020090800.02200.020090800.02200.020090800.02200.0200700.0175700.0175700.0175700.0175700.0175605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.0060605040300.01500.01250.01000.00750.00602020202020101010101000 10 20 30 40 50 60 70 80 90 10000 10 20 30 40 50 60 70 80 90 10000 10 20 30 40 50 60 70 80 90 10000 10 20 30 40 50 60 70 80 90 10000 10 20 30 40 50 60 70 80 90 100Pogue et al, Optics Exp. 1999

Sub-surface tomography1100.020 1100.0201100.020756010 40 60 801100.0150.0100.006756010 40 60 801100.0150.0100.006756010 40 60 801100.0150.0100.0061100.0201100.020756010 40 60 801100.0150.0100.006756010 40 60 801100.0150.0100.006Pogue et al, Optics Exp. 1999

Full reflectance andtransmittance imaging1201101009080700.02000.01800.01600.01400.01200.01000.00800.006060120500 10 20 30 40 50 60 70 80 90 100 1101201201101000.02000.01800.0160110100900.02000.01800.01600.01400.01209080700.01400.01200.01000.00800.00608070600.01000.00800.006060500 10 20 30 40 50 60 70 80 90 100 110120120500 10 20 30 40 50 60 70 80 90 100 1101201201101000.02000.01800.01601101009080700.02000.01800.01600.01400.01200.01000.00800.006090807060500 10 20 30 40 50 60 70 80 90 100 1101200.01400.01200.01000.00800.006060500 10 20 30 40 50 60 70 80 90 100 110120Pogue et al, Optics Exp. 1999

Slab Fan-beam geometry imaging1100.0201100.020756010 40 60 801101100.0150.0100.0060.020756010 40 60 801101100.0150.0100.0060.020756010 40 60 801101100.0150.0100.0060.020756010 40 60 801101100.0150.0100.0060.020756010 40 60 801101100.0150.0100.0060.020756010 40 60 801100.0150.0100.006756010 40 60 801100.0150.0100.006Pogue et al, Optics Exp. 1999

Slab Fan-beam imaging1100.0201100.020756010 40 60 801101100.0150.0100.0060.020756010 40 60 801101100.0150.0100.0060.02075601011040 60 801100.0150.0100.0060.020756010 40 60 801101100.0150.0100.0060.02075601011040 60 801100.0150.0100.0060.020756010 40 60 801101100.0150.0100.0060.020756010 40 60 801101100.0150.0100.0060.020756010 40 60 801101100.0150.0100.0060.020756010 40 60 801100.0150.0100.006756010 40 60 801100.0150.0100.006Pogue et al, Optics Exp. 1999

Finite element mesh generationMRI scan FEM mesh Absorption coeff. mapFEM nodes 0.02560.050.040.030.020.010.00.0-10.0-20.0-30.0-40.0-50.0-60.0-50.0 -40.0 -20.0 0.0 20.0 40.0 50.011010080604020101020 40 60 801100.0200.0150.008Pogue et al, Optics Exp. 1999

Preliminary tests for cerebral imagingTest field 1 iteration, initial guess µ a =0.01120100806040200.0230.0200.0150.008120100806040200.0130.0120.0110.0100.00900 25 50 75 10012500 25 50 75 100125Pogue et al, Optics Exp. 1999

Cerebral imaging- characterization of an increased blood concentrationTest fieldReflectance & Transmittance110100Fan-beam0.0280.025Sub-surface imaging800.020600.015400.00820101020 40 60 80110Pogue et al, Optics Exp. 1999

References:http://biolight.thayer.dartmouth.eduNIRFAST - MATLAB FEM Diffusion/Reconstruction packageAuthor: Hamid Dehghani, Dartmouthhamid@dartmouth.eduhttp://www-nml.thayer.dartmouth.edu/NIRA Priori Info Work: Ben Brooksby Ph.D. Thesis (2005, Dartmouth)A priori & spectral: Heng Xu, Ph.D. Thesis (2005, Dartmouth)Spectral Reconstr.: Subhadra Srinivasan, Ph.D. Thesis(2005, Dartmouth)