Tomographic Reconstruction - Univie.ac.at

Tomographic Reconstruction3D Image ProcessingTorsten Möller Torsten Möller

Reading Gonzales Woods, Chapter 5.11 Torsten Möller2

Overview PhysicsHistoryReconstruction — basic ideaRadon transformFourier-Slice theorem(Parallel-beam) filtered backprojectionFan-beam filtered backprojectionAlgebraic reconstruction technique (ART) Torsten Möller3

X-Rays photonsproducedby anelectronbeam similar tovisible light,but higherenergy! Torsten Möller4

X-Rays - Physics associated with inner shell electrons as the electrons decelerate in the targetthrough interaction, they emitelectromagnetic radiation in the form of Xrays. patient between an X-ray source and afilm - radiograph cheap and relatively easy to use potentially damaging to biological tissue Torsten Möller5

X-Rays - Visibility bones contain heavy atoms - with manyelectrons, which act as an absorber of Xrays commonly used to image gross bonestructure and lungs excellent for detecting foreign metal objects main disadvantage - lack of anatomicalstructure all other tissue has very similar absorptioncoefficient for X-rays Torsten Möller6

X-Rays - Angiography inject contrast medium (electron densedye) used to image vasculature (bloodvessels) angiocardiography (heart) cholecystography (gall bladder) myelography (spinal cord) urography (urinary tract) Torsten Möller7

X-Rays - Images Torsten Möller8

CT or CAT - Principles Computerized (Axial) Tomography introduced in 1963/1972 by Hounsfield andCormack (1979 Noble prize in medicine) natural progression from X-rays based on the principle that a threedimensional object can be reconstructedfrom its two dimensional projections based on the Radon transform (a map froman n-dimensional space to an (n-1)dimensional space) Torsten Möller9

CT or CAT - Methods measures the attenuation of X-rays frommany different angles a computer reconstructs the organ understudy in a series of cross sections orplanes combine X-ray picturesfrom various angles toreconstruct 3D structuresvideo Torsten Möller10

CAT Torsten Möller11

Overview PhysicsHistoryReconstruction — basic ideaRadon transformFourier-Slice theorem(Parallel-beam) filtered backprojectionFan-beam filtered backprojectionAlgebraic reconstruction technique (ART) Torsten Möller12

History G1 (first gen) CT:– employ “pencil” X-ray beam– single detector– linear translation ofsource/detector pair G2 CT:– same as G1, but– beam is shaped as a fan– allows multiple detectors Torsten Möller13

History G3 CT:– great improvement withbank of detectors( 1000 detectors)– no need for translation G4 CT:– circular ring of detectors( 5000 detectors) G3 G4:– higher speed– higher dose and higher cost Torsten Möller14

Modern scanners G5 CT (EBCT — electron beam CT)– eliminate mechanical motion– beams controlled electromagnetically G1-G5: one image at a time, then patientis moved. Patient must hold breath for lung CT Torsten Möller15

Modern scanners G6 CT (helical CT):– continuous movement of patient andsource/detector G7 CT (multi-slice CT):– thick fan-beams, collecting multiple slices atonce– reducing cost dosage Torsten Möller16

CT - 2D vs. 3D Linear advancement (slice byslice)– typical method– tumor might fall between ‘cracks’– takes long time helical movement– 5-8 times faster– under-utilization of cone beam– heart synchronization difficult 2D projections– enhanced speed Torsten Möller17

CT - Beating Heart? Noise if body parts move! Heart - synchronize imagingwith heart beat– can’t capture beating well– need faster techniquesUniversity Of Iowa Dynamic Spatial Reconstructor– has 14 X-ray/camera pairs– but turns slower– 2D projections seem moreplausible– and cheaper Torsten Möller18

CT or CAT - Advantages significantly more data is collected superior to single X-ray scans far easier to separate soft tissues other thanbone from one another (e.g. liver, kidney) data exist in digital form - can be analyzedquantitatively adds enormously to the diagnostic information used in many large hospitals and medicalcenters throughout the world Torsten Möller19

CT or CAT - Disadvantages significantly more data is collected soft tissue X-ray absorption still relativelysimilar still a health risk MRI is used for a detailed imaging ofanatomy Torsten Möller20

Overview PhysicsHistoryReconstruction — basic ideaRadon transformFourier-Slice theorem(Parallel-beam) filtered backprojectionFan-beam filtered backprojectionAlgebraic reconstruction technique (ART) Torsten Möller21

Basics: Projection vs.backprojectionBackprojectionProjection Torsten Möller22

3 Torsten Möller1243223

4 Torsten Möller12326424

Overview PhysicsHistoryReconstruction — basic ideaRadon transformFourier-Slice theorem(Parallel-beam) filtered backprojectionFan-beam filtered backprojectionAlgebraic reconstruction technique (ART) Torsten Möller25

Math. principles Expression of a simple line:x cos y sin computingprojections:Z Zg( j , k ) f (x, y) (x cos k y sin k j )dxdy Also known asRadon Transform Torsten Möller26

Sinogram Radon transformwritten as simpleimages Torsten Möller27

Simple reconstruction mathematically “smearing” is:f k (x, y) g( , k ) g(x cos k y sin k , k ) simply summing it all up:f (x, y) Z f (x, y)d 0 Torsten Möller28

Simple reconstruction leads to lots of blurring proper reconstruction — projection slice Torsten Möller29

Overview PhysicsHistoryReconstruction — basic ideaRadon transformFourier-Slice theorem(Parallel-beam) filtered backprojectionFan-beam filtered backprojectionAlgebraic reconstruction technique (ART) Torsten Möller30

Projection-slice theoremG(!, ) Z11g( , )e Torsten Möllerj2 ! d 31

Projection-slice theorem mathematicallyZ Z Zf (x, y) (x cos y sin )e j2 ! dxdyd ZZ Z f (x, y)(x cos y sin )e j2 ! d dxdyZ Z f (x, y)e j2 !(x cos y sin ) dxdyG(!, ) meaning:G(!, ) Z Zf (x, y)ej2 (ux vy)dxdyu ! cos ;v ! sin F (! cos , ! sin ) Torsten Möller32

Overview PhysicsHistoryReconstruction — basic ideaRadon transformFourier-Slice theorem(Parallel-beam) filtered backprojectionFan-beam filtered backprojectionAlgebraic reconstruction technique (ART) Torsten Möller33

some mathf (x, y) Z Zj2 (ux vy)dudv !d!d G(!, ) G( !, )F (x, y)edudvZ 2 Z 1 F (! cos , ! sin )ej2 !(x cos y sin ) !d!d Z02 ZZ0 ZZ000101 ZG(!, )ej2 !(x cos y sin ) !d!d ! G(!, )ej2 !(x cos y sin ) d!d 111 ! G(!, )ej2 ! d! Torsten Möllerd x cos y sin 34

Multiplication with a ramp sharp cut-off, yields ringing! Torsten Möller35

Multiplication with a ramp Torsten Möller36

Multiplication with a ramp Torsten Möller37

Overview PhysicsHistoryReconstruction — basic ideaRadon transformFourier-Slice theorem(Parallel-beam) filtered backprojectionFan-beam filtered backprojectionAlgebraic reconstruction technique (ART) Torsten Möller38

Principle idea simple: just untangle allthe beams from the faninto the right parallel beamreconstruction more direct: D sin Torsten Möller39

Principle idea simple: just untangle allthe beams from the faninto the right parallel beamreconstruction more direct: D sin f (r, ) Z2 01R2 Z mq( , )h( 0 m Torsten Möller )d d1 2h( ) s( )402 sin

Results typically need more projections! Torsten Möller41

Results typically need more projections! Torsten Möller42

Overview PhysicsHistoryReconstruction — basic ideaRadon transformFourier-Slice theorem(Parallel-beam) filtered backprojectionFan-beam filtered backprojectionAlgebraic reconstruction technique (ART) Torsten Möller43

CT - Reconstruction: ART/EM Algebraic Reconstruction TechniqueExpectation Maximization (EM)iterative techniqueInitial Guessattributed to tual DataSlices Torsten Möller44

CT - Reconstruction: ART (2)pi 1v3 v2 v1pipi-1wj Torsten Möller45

CT - Reconstruction: ART (3) object reconstructed on a discrete grid by asequence of alternating grid projections andcorrection back-projections. Projection: measures how close the currentstate of the reconstructed object matchesone of the scanner projections Back-projection: corrective factor isdistributed back onto the grid many projection/back-projection stepsneeded for a certain tolerance margin Torsten Möller46

CT - FBP vs. ARTFBPART Computationallycheap Clinically usually500 projections perslice problematic fornoisy projections Still slow better quality forfewer projections better quality fornon-uniform project. “guided” reconstruct.(initial guess!) Torsten Möller47

Tomographic Reconstruction 3D Image Processing Torsten Möller . History Reconstruction — basic idea Radon transform Fourier-Slice theorem (Parallel-beam) filtered backprojection Fan-beam filtered backprojection . reconstruction more direct: 39