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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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Presentation on theme: "Image Reconstruction from Projections J. Anthony Parker, MD PhD Beth Israel Deaconess Medical Center Boston, Massachusetts Caveat Lector"— Presentation transcript:

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3 Image Reconstruction from Projections J. Anthony Parker, MD PhD Beth Israel Deaconess Medical Center Boston, Massachusetts Caveat Lector Tony_Parker@BIDMC.Harvard.edu

4 Projection Single Slice Axial

5 Single Axial Slice: 360 0 collimator Ignoring attenuation, SPECT data are projections

6 Attenuation: 180 o = 360 o keV 150 100 80 60 50 x Tc-99m htl(140 keV) ≈ 4 cm

7 Cardiac Perfusion Data Collection Special Case - 180 o AxialCoronal / Sagittal Multiple simultaneous axial slices

8 Dual-Head General-Purpose Gamma Camera: 90 0 “Cardiac” Position 2 heads: 90 0 rotation = 180 0 data 1 2

9 Inconsistent projections “motion corrected”

10 Original data

11 0 0 0 0 0 0 0 0 Single Axial Slice: 360 0

12 Sinogram: Projections Single Axial Slice 060 0 projection angle x x x

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

14 Reconstruction by Backprojection Backprojection tails

15 Backprojection 2 projections 2 objects

16 projection tails merge resulting in blurring

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

18 Projection -> Backprojection

19 Projection -> Backprojection Shepp-Logan Phantom

20 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)

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

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

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

24 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

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

26 Negative Side Lobes: Sharpening

27 Filtered Backprojection: Negative Side Lobes Eat Projection Tails

28 Filtered Backprojection

29 Effect of Limited Projections

30 Effect of Noise

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

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

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

35 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

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

37 Inferior wall defect ?

38 Iterative (OSEM) Reconstruction

39 3 hour delay shows defect Better data always wins

40 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

41 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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43 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

44 Dual Head: Non-Uniform Sampling

45 Activity Measurement: Attenuation keV 150 100 80 60 50 htl(140 keV) ≈ 4 cm

46 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 = 180 0 data

47 Semi-erect: Ameliorates Attenuation

48 Leaning Forward, < 500 Pounds

49 Digirad: Patient Rotates X-ray Attenuation Correction

50 CT: Polychromatic Beam -> Dose keV 150 100 80 60 50

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

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

53 Digirad X-ray Spectrum

54 Detectors D-SPECTDigirad CrystalsCZTCsI(Tl) PhotodetectorNAPD Energy resolution6%10% Count rate1.35 x 10 6 15 x 10 6 Heads93 FOV (cm)15.7 x 3.915.8 x 21.2 Total cm 2 5581005 X-rays (blank)none20 x 10 6

55 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

56 Soft Tissue Attenuation: Supine breast lung

57 Soft Tissue Attenuation: Prone breast

58 Soft Tissue Attenuation: Digirad Erect breast post

59 Sequential Tidal-Breathing Emission and Average-Transmission Alignment

60 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

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

62 (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

63 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

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

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

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

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

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

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

70 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)

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

72 f p A is projection

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

74 f ^ H is backprojection

75 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 ^ ^ ^ ^

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

77 f ^ p ^ A ^

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

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

80 e ^

81 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

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

83 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

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

85 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 ^ ^

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

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

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

89 Ill Conditioned: Need Stopping Heuristic EM Iterations

90 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

91 Noise: FBP vs EM SD/mean

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

93 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

94 Iterative Reconstruction Noise is “Blobby”

95 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))

96 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

97 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

98 Applause

99 Projection

100 Backprojection

101 A ^


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