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Image Reconstruction from Projections J. Anthony Parker, MD PhD Beth Israel Deaconess Medical Center Boston, Massachusetts Caveat Lector

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Projection Single Slice Axial

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Single Axial Slice: collimator Ignoring attenuation, SPECT data are projections

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Attenuation: 180 o = 360 o keV x Tc-99m htl(140 keV) ≈ 4 cm

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Cardiac Perfusion Data Collection Special Case o AxialCoronal / Sagittal Multiple simultaneous axial slices

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Dual-Head General-Purpose Gamma Camera: 90 0 “Cardiac” Position 2 heads: 90 0 rotation = data 1 2

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Inconsistent projections “motion corrected”

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Original data

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Single Axial Slice: 360 0

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Sinogram: Projections Single Axial Slice projection angle x x x

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Uniformity & Motion on Sinogram 1 head 24 min 2 heads 12 min

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Reconstruction by Backprojection Backprojection tails

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Backprojection 2 projections 2 objects

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projection tails merge resulting in blurring

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Projection -> Backprojection of a Point (1/r) backprojection lines add at the point tails spread point out

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Projection -> Backprojection

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Projection -> Backprojection Shepp-Logan Phantom

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Projection->Backprojection Smooths Smooths or “blurs” the image (Low pass filter) ((Convolution with 1/r)) Nuclear Medicine physics Square law detector adds pixels -> always blurs Different from MRI (phase)

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(Projection-Slice Theorem) “k-space (k, )” detail low frequency spatial frequency domainspatial domain 2D Fourier transform

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Spatial Frequency Basis Functions f(u,v) ≠ 0, single u,0f(u,v) ≠ 0, single 0,v f(u,v) ≠ 0, single u = v

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Projection -> Backprojection: k-space 1/k (Density of slices is 1/k) (Fourier Transform of 1/r 1/k) one projection multiple projections

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Image Reconstruction: Ramp Filter Projection -> Backprojection blurs with 1/r in object space k-space 1/k ( 1/r 1/k) Ramp filter sharpen with k (windowed at Nyquist frequency) k k

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Ramp Filter in the Image Domain (Fourier transform of windowed ramp) Replace each point with this kernel Negative side lobe

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Negative Side Lobes: Sharpening

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Filtered Backprojection: Negative Side Lobes Eat Projection Tails

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Filtered Backprojection

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Effect of Limited Projections

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Effect of Noise

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Mathematically Equivalent Implementations Backproject filtered projections “Filtered backprojection” Filter backprojected projections “Fourier method”

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Low Pass Times Ramp Filter Low pass, Butterworth – noise Ramp – reconstruct

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SPECT Filter: Ramp - Reconstruction Low Pass Filter - Noise Reduction Noise and resolutionExact reconstruction

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Reconstruction from Projections Projection: data collection Backprojection: 1 st step in reconstruction projection->backprojection: blurs 1/k Sharpen: ramp filter multiplies by k Noise reduction: low pass filter (LPF) LPF also decreases image detail

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Reconstruction Kernel with LPF (Fourier transform of filter) LPF makes hot blobs fatter hot blobs eat adjacent objects

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Inferior wall defect ?

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Iterative (OSEM) Reconstruction

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3 hour delay shows defect Better data always wins

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What’s Good about FPB Ramp filter exactly reconstructs projection Efficient (Linear shift invariant) (FFT is order of n log(n) n = number of pixels) “Easily” understood

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New Cardiac Cameras Solid state - CZT: $$$, energy resolution scatter rejection, dual isotope Pixelated detector: count rate & potential high resolution poorer uniformity Non-uniform sampling: sensitivity potential for artifacts Special purpose design closer to patient: system resolution upright: ameliorates diaphragmatic attenuation

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Collimator Resolution* Single photon imaging (i.e. not PET) Collimators: image formation Sensitivity / resolution trade-off Resolution recovery hype “Low resolution, high sensitivity -> image processing = high resolution” Reality - ameliorates low resolution Steve Moore: “Resolution: data = target object” Can do quick, low resolution image * not resolution from reduced distance due to design

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Dual Head: Non-Uniform Sampling

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Activity Measurement: Attenuation keV htl(140 keV) ≈ 4 cm

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Attenuation Correction: Simultaneous Emission (90%) and Transmission (10%) Gd-153 rods T 1/2 240 d e.c. 100% 97 keV 29% 103 keV 21% 2 heads: 90 0 rotation = data

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Semi-erect: Ameliorates Attenuation

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Leaning Forward, < 500 Pounds

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Digirad: Patient Rotates X-ray Attenuation Correction

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CT: Polychromatic Beam -> Dose keV

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X-ray Tube Spectra bremsstrahlung characteristic X-rays e - interaction: - ionization - deflection X-ray tube: electrons on Tungsten or Molybdenum

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Digirad X-ray Source: X-rays on Lead 74 W 82 Pb X-rays interaction - ionization - no 1 0 bremsstrahlung

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Digirad X-ray Spectrum

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Detectors D-SPECTDigirad CrystalsCZTCsI(Tl) PhotodetectorNAPD Energy resolution6%10% Count rate1.35 x x 10 6 Heads93 FOV (cm)15.7 x x 21.2 Total cm X-rays (blank)none20 x 10 6

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New Cardiac Cameras D-SPECTCardiArcDigiradGE DetectorCZT*NaI(Tl)CsI(Tl)CZT* ElectronicsSS*PMTPD* ? SS* PixelatedYNYY Collimationholesslits* ? holespinholes Non-uniformY*Y*~NY* Limited angleYYN~N Closer to ptYYY~N ACNCT?CT*CT Position~semisemierectsupine

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Soft Tissue Attenuation: Supine breast lung

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Soft Tissue Attenuation: Prone breast

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Soft Tissue Attenuation: Digirad Erect breast post

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Sequential Tidal-Breathing Emission and Average-Transmission Alignment

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Sensitivity / Resolution Trade-Off Non-uniform sampling -> sensitivity Special purpose design -> resolution Advantages Throughput at same noise Patient motion - Hx: 1 head -> 2 head Cost Non-uniform sampling -> artifacts History: 7-pinhole - failed 180 o sampling - success Sequential emission transmission

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What’s Wrong with Ramp-Filtered Backprojection for SPECT Noise Attenuation Scatter Depth dependant resolution New imaging geometries

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(Noise is White (Poisson) in Projection Space not in Object / Image Space) Af n p + f ^ H f = object A = imaging system n = noise p = projections H = image reconstruction, FBP f = estimate of object ^ k

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Spatial Frequency of Signal and Noise in Reconstructed Image Signal energy equal at all spatial frequencies k Noise energy greater at high spatial frequency k

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Decreasing Signal-to-Noise kk kk Noise affects high frequencies first, then lower frequencies

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What’s Wrong with Filtered Backprojection, FBP, for SPECT Can’t model: Attenuation Scatter Depth dependant resolution New imaging geometries (Linear shift invariant model)

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Attenuation and Scatter Correction Increased density (sub-diaphragmatic) -> increase in both attenuation scatter Opposite effects on count rate Need to correct for both simultaneously

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Solution Iterative reconstruction Uses: Simultaneous linear equations Matrix algebra Can model image physics (Linear model)

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Projections as Simultaneous Equations (Linear Model) But, exact solution for a large number of equations isn’t practical

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((Simultaneous Equations)) Under-determined -> extra samples Ill conditioned -> a priori constraints Over-determined -> inconsistent Noise -> more inconsistent Normal Equations of Linear Algebra

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Large Set of Simultaneous Equations Image modeled as a vector, not a matrix 128 by 128 image -> 16,384 pixels Sinogram: about the same 16,384 by 16,384 matrix is impossible Solution: iterative reconstruction (Sparse matrix)

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Image Acquisition Af n p + f = object A = imaging system n = noise p = projections (sinogram)

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f p A is projection

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Image Reconstruction Af n p + f ^ H H = image reconstruction, e.g. BP, FBP etc. f = image (estimate) of object, f ^

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f ^ H is backprojection

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How Good is the Projection-> Backprojection Estimate? Can’t compare f to f (f is unknown) Project f to make p, estimate of projections Need model of system, A ^ ^ ^ ^

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Produce Estimated Projections Af n p f ^ p ^ + f ^ H A ^

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f ^ p ^ A ^

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Compare Projections to Estimate Af n p f ^ p ^ e ^ + - f ^ H A ^ |e| ^ < |p| e is the error in estimate, f ^ ^

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Corrections: Backproject Errors Af n p f ^ p ^ e ^ + - f ^ H H A ^ corrections

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e ^

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Iterative Backprojection Reconstruction Af n p f n-1 ^ p n-1 ^ e n-1 ^ fnfn ^ + - x + f0f0 ^ r H H A ^ objectdata projectionbackprojection estimate model error estimate estimated data estimate + backprojected error

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Iterative Reconstruction Af n p f n-1 ^ p n-1 ^ e n-1 ^ fnfn ^ + - x + f0f0 ^ r H H A ^

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Iterative Reconstruction Af n p f n-1 ^ p n-1 ^ e n-1 ^ fnfn ^ + - x + f0f0 ^ r H H A ^ relaxation factor 0-1

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Reconstruction, H, can be Approximate Af n p f n-1 ^ p n-1 ^ e n-1 ^ fnfn ^ + - x + f0f0 ^ r H H A ^

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Accuracy of Model, A, is Key Af n p f n-1 ^ p n-1 ^ e n-1 ^ fnfn ^ + - x + f0f0 ^ r H H A ^ ^

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Model, A, is Well-known Physics Problem: Model of the Body ^ Tc-99m half-tissue layer: 4 cm

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Attenuation Map Gd-153 Transmission Map adds noise to reconstruction and can introduce artifacts

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(Ill Conditioned) Af n p f n-1 ^ p n-1 ^ e n-1 ^ fnfn ^ + - x + f0f0 ^ r H H A ^ Estimating

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Ill Conditioned: Need Stopping Heuristic EM Iterations

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Iterative Reconstruction is Ill Conditioned Stop after N iterations ((The ill conditioned portion of the solution will be similar to starting conditions Starting condition often 0)) A priori constraints Often smoothness

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Noise: FBP vs EM SD/mean

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Filtered Back Projection (FBP) vs Ordered Subset Estimation Maximization (OSEM) FDG PET

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Ordered Subset Estimation Maximization (OSEM) Estimation Maximization (EM) is an iterative reconstruction method Ordered subset EM is a variation of EM (Update estimate for each subset) Advantage of OSEM: faster (OSEM has about 1/n subsets of EM iterations) Currently it is the principle iterative method for both SPECT and PET

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Iterative Reconstruction Noise is “Blobby”

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What’s Good About Iterative Reconstruction Able to model: Data collection, including new geometries Attenuation Scatter Depth dependant resolution Fairly efficient given current computers (Iterative solution, e.g. EM, reasonable) (OSEM is even better) ((OSEM has about 1/n subsets of EM iterations))

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What’s Wrong with Iterative Reconstruction (Complicated by ill conditioned model) ((Estimating projections not object)) Noise character bad for oncology To model attenuation & scatter - need to measure attenuation - adds noise

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Conclusions Filtered backprojection, FBP Efficient (Models noise) “Easy” to understand Iterative reconstruction, OSEM Moderately efficient Models noise, attenuation, scatter, depth dependant resolution, and new cameras

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Applause

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Projection

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Backprojection

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A ^

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