Скачать презентацию Image Reconstruction from Projections J Anthony Parker MD Скачать презентацию Image Reconstruction from Projections J Anthony Parker MD

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

Projection Single Slice Axial Projection Single Slice Axial

Single Axial Slice: 3600 collimator Ignoring attenuation, SPECT data are projections Single Axial Slice: 3600 collimator Ignoring attenuation, SPECT data are projections

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

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

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

Inconsistent projections “motion corrected” Inconsistent projections “motion corrected”

Original data Original data

Single Axial Slice: 3600 0 0 0 0 Single Axial Slice: 3600 0 0 0 0

0 60 x projection angle Sinogram: Projections Single Axial Slice x 60 0 x 0 60 x projection angle Sinogram: Projections Single Axial Slice x 60 0 x

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

Reconstruction by Backprojection tails Reconstruction by Backprojection tails

Backprojection 2 projections 2 objects Backprojection 2 projections 2 objects

projection tails merge resulting in blurring projection tails merge resulting in blurring

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

Projection -> Backprojection Projection -> Backprojection

Projection->Backprojection Smooths or “blurs” the image (Low pass filter) ((Convolution with 1/r)) Nuclear Medicine Projection->Backprojection 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-Slice Theorem) “k-space (k, )” detail low frequency spatial domain 2 D Fourier transform (Projection-Slice Theorem) “k-space (k, )” detail low frequency spatial domain 2 D Fourier transform spatial frequency domain

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

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

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

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

What’s Good about FPB Ramp filter exactly reconstructs projection Efficient (Linear shift invariant) (FFT 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

New Cardiac Cameras Solid state - CZT: $$$, energy resolution scatter rejection, dual isotope New Cardiac Cameras Solid state - CZT: $$$, 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

Collimator Resolution* Single photon imaging (i. e. not PET) Collimators: image formation Sensitivity / 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

Dual Head: Non-Uniform Sampling Dual Head: Non-Uniform Sampling

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

Attenuation Correction: Simultaneous Emission (90%) and Transmission (10%) Gd-153 rods T 1/2 240 d Attenuation Correction: Simultaneous Emission (90%) and Transmission (10%) Gd-153 rods T 1/2 240 d e. c. 100% 97 ke. V 29% 103 ke. V 21% 2 heads: 900 rotation = 1800 data

Semi-erect: Ameliorates Attenuation Semi-erect: Ameliorates Attenuation

Leaning Forward, < 500 Pounds Leaning Forward, < 500 Pounds

Digirad: Patient Rotates X-ray Attenuation Correction Digirad: Patient Rotates X-ray Attenuation Correction

CT: Polychromatic Beam -> Dose ke. V 150 100 80 60 50 CT: Polychromatic Beam -> Dose ke. V 150 100 80 60 50

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

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

Digirad X-ray Spectrum Digirad X-ray Spectrum

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

Soft Tissue Attenuation: Supine breast lung Soft Tissue Attenuation: Supine breast lung

Soft Tissue Attenuation: Prone breast Soft Tissue Attenuation: Prone breast

Soft Tissue Attenuation: Digirad Erect breast post Soft Tissue Attenuation: Digirad Erect breast post

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

Sensitivity / Resolution Trade-Off Non-uniform sampling -> sensitivity Special purpose design -> resolution Advantages 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

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

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

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

Iterative Backprojection Reconstruction projection n object f backprojection estimate data A + p ^ Iterative Backprojection Reconstruction projection n object f backprojection estimate data A + p ^ f n-1 ^ A estimated data ^ pn-1 0 r model estimate ^ f H error - ^n-1 e H x estimate + backprojected error + ^ f n

Reconstruction, H, can be Approximate n f A + p ^ f H 0 Reconstruction, H, can be Approximate n f A + p ^ f H 0 r ^ f n-1 ^ A ^ pn-1 - ^n-1 e H x + ^ f n

^ is Key Accuracy of Model, A, n f A + p ^ f ^ is Key Accuracy of Model, A, n f A + p ^ f H 0 r ^ f n-1 ^ A ^ pn-1 - ^n-1 e H x + ^ f n

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

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

Iterative Reconstruction Noise is “Blobby” Iterative Reconstruction Noise is “Blobby”

What’s Good About Iterative Reconstruction Able to model: Data collection, including new geometries Attenuation 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/nsubsets of EM iterations))

What’s Wrong with Iterative Reconstruction (Complicated by ill conditioned model) ((Estimating projections not object)) 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

Conclusions Filtered backprojection, FBP Efficient (Models noise) “Easy” to understand Iterative reconstruction, OSEM Moderately 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

Applause Applause