Image Reconstruction from Projections

3. Image Reconstructionfrom 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: 3600 collimator Ignoring attenuation, SPECT data are projections

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

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

8. 1 2 2 heads: 900 rotation = 1800 data Dual-Head General-Purpose Gamma Camera: 900 “Cardiac” Position

9. Inconsistent projections “motion corrected”

10. Original data

11. Single Axial Slice: 3600 0 0 0 0 0 0 0 0

12. 0 60 x projection angle 60 0 x x Sinogram: ProjectionsSingle Axial Slice

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

14. Backprojection tails Reconstruction by Backprojection

15. Backprojection 2 projections 2 objects

16. projection tails merge resulting in blurring

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

18. Projection -> Backprojection

19. 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) Projection->Backprojection Smooths

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

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

22. 1/k (Density of slices is 1/k) one projection multiple projections (Fourier Transform of 1/r <-> 1/k) Projection -> Backprojection: k-space

23. 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 Image Reconstruction: Ramp Filter

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

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

26. 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

28. 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 Collimator Resolution*

29. Dual Head: Non-Uniform Sampling

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

31. Gd-153 rods T1/2 240 d e.c. 100% 97 keV 29% 103 keV 21% 2 heads: 900 rotation = 1800 data Attenuation Correction: Simultaneous Emission (90%) and Transmission (10%)

32. Semi-erect: Ameliorates Attenuation

33. Leaning Forward, < 500 Pounds

34. Digirad: Patient RotatesX-ray Attenuation Correction

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

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

37. Digirad X-ray Source: X-rays on Lead 74W 82Pb X-rays interaction - ionization - no 10 bremsstrahlung

38. Digirad X-ray Spectrum

39. New Cardiac Cameras D-SPECT CardiArc Digirad GE Detector CZT* NaI(Tl) CsI(Tl) CZT* Electronics SS* PMT PD*? SS* Pixelated Y N Y Y Collimation holes slits*? holes pinholes Non-uniform Y* Y* ~N Y* Limited angle Y Y N ~N Closer to pt Y Y Y ~N AC N CT? CT* CT Position ~semi semi erect supine

40. breast lung Soft Tissue Attenuation: Supine

41. breast Soft Tissue Attenuation: Prone

42. breast post Soft Tissue Attenuation: Digirad Erect

43. Sequential Tidal-Breathing Emission and Average-Transmission Alignment

44. 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 180o sampling - success Sequential emission transmission Sensitivity / Resolution Trade-Off

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

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

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

48. n projection backprojection estimate object data f p H - + + r model estimated data estimate error x ^ ^ ^ ^ fn f0 fn-1 ^ H A pn-1 A estimate + backprojected error ^ en-1 Iterative Backprojection Reconstruction

49. n f p H - + + r x ^ ^ ^ ^ f0 fn fn-1 ^ H A pn-1 A ^ en-1 Reconstruction, H, can be Approximate

50. n f p H - + + r x ^ ^ ^ ^ f0 fn fn-1 ^ H A pn-1 A ^ en-1 ^ Accuracy of Model, A, is Key