3 Image Reconstruction from Projections J. Anthony Parker, MD PhDBeth Israel Deaconess Medical CenterBoston, MassachusettsTalk is a bit dry, basic sciency, but Tom keeps asking me to do it.There are a small number of useful pearls, watch for them.Caveat Lector
4 Projection Single Slice Axial What’s a projection? This is what mathematicians say.Given object and angle, it is integrals through the object.
5 Single Axial Slice: 3600 collimator Ignoring attenuation, SPECT data are projectionsGamma camera rotates 360o around object.Data it collects is projections (ignoring attenuation)
6 Attenuation: 180o = 360o x Tc-99m keV 150 100 80 60 50 htl(140 keV) ≈ 4 cmThis is a large patient, 47 cm140 keV Tc-99m photons are highly attenuated, 0.1%Cardiac 180o collection is special. In general, 360o
7 Cardiac Perfusion Data Collection Special Case - 180o AxialCoronal / SagittalMultiple simultaneous axial slicesOne horizontal line from each of these images, makes one axial slice in the last slide.Notice arms are moving; heart is fairly still.
8 Dual-Head General-Purpose Gamma Camera: 900 “Cardiac” Position 2 heads: 900 rotation = 1800 data12Phillips nee ADAC SPECT camera with attenuation correctionSimultaneous emission and transmission
9 Inconsistent projections “motion corrected” Tech did motion correction wrong.Note walls don’t line up, hurricane.Tall is about how technology can improve imaging, but remember it can introduce new errors.
10 Original data Original data gives a much better reconstruction. Some people on read bulls eye -> tech essential reads study
11 Single Axial Slice: 3600One line from each projection image. More usual 360o rotation.Blob becomes 1-D function.Marked zero so you can follow what happens with rotation.
12 Sinogram: Projections Single Axial Slice 60projection anglexDisplay projections as lines in an image.First line projection to top.Blob follows a sine wave. Called sinogram; commonly used.
13 Uniformity & Motion on Sinogram 1 head24 min2 heads12 minCan use sinogram to detect motion and uniformity.I don’t think sinogram is particularly insightful.
14 Reconstruction by Backprojection Backprojection tailsWe are going to make backprojection the 1st step in reconstruction.Projections are 1, 2, and 3.No information in the projection about where the data came from, so put it back uniformly in all the possible sources.Artifact from projection lines.
15 Backprojection 2 projections 2 objects If we have two blobs and only two projections, backprojection gives us 4 blobs.
16 projection tails merge resulting in blurring More projections give us a better image, but in is a blurred out representation of the original.
17 Projection -> Backprojection of a Point lines add atthe pointtails spread point outIf we start out with an object which consists of a single point, projection followed by backprojection will make 1/r.Infinite number of projections.Each point in the image is replaced with this function. (Convolution; linear shift invariance.)
18 Projection -> Backprojection more backprojection angles, more like object
19 Projection -> Backprojection Shepp-Logan Phantom This is a typical brain type of mathematical phantom.Ignore limited gray scale in back projection.Again, backprojection makes a blurs the original.
20 Projection->Backprojection Smooths Smooths or “blurs” the image(Low pass filter)((Convolution with 1/r))Nuclear Medicine physicsSquare law detector adds pixels-> always blursDifferent from MRI (phase)Key point about nuclear medicine physics is that you always sum the magnitude of pixels.Detection always smoothes.Different in MRI: pixels add and subtract. Doesn’t smooth.
21 (Projection-Slice Theorem) “k-space (k,)”detaillowfrequencyspatial frequency domainspatial domain2D FouriertransformObject is defined in spatial domain. Equivalent representation as spatial frequencies.Polar coordinates in spatial frequency domain often k and theta. So, spatial frequency domain often called k-space, especially in MRI.Projection in the object domain is mathematically equivalent to a line of data in k-space, often called a slice.Any engineers? Parenthesis are for the engineers. This is called the projection-slice theorem.
22 Spatial Frequency Basis Functions f(u,v) ≠ 0, single u,0f(u,v) ≠ 0, single 0,vf(u,v) ≠ 0, single u = vHigh school: you learned about basis functions for vector spacesf(u) not zero for some u, f(v) not zero for some vSum, f(u,v) not zero for some u = vMRI samples f(u,v) directly in k-space
23 Projection -> Backprojection: k-space (Density ofslices is 1/k)(Fourier Transform of 1/r <-> 1/k)one projectionmultiple projectionsIn k-space, a projection is equivalent to a line of data.If we have an infinite number of projections we will have lines which all cross at zero. Over sample low frequencies <- square law detector.The result in k-space is just like the result in object space. People often skip over the fact that this is a special case.
24 Image Reconstruction: Ramp Filter Projection -> Backprojectionblurs with 1/r in object spacek-space 1/k ( 1/r<-> 1/k)Ramp filtersharpen with k(windowed at Nyquist frequency)kRamp filter increases high frequencies
25 Ramp Filter in the Image Domain (Fourier transform ofwindowed ramp)Replace each point with this kernelNegative side lobeIn the image domain each point in the projection-backprojection image is replaced with this kernel.(Convolve the projection-backprojection image with this function.)Negative side lobe, typical of sharpening function. Unsharp masking in Photoshop.
26 Negative Side Lobes: Sharpening Negative side-lobes enhance edges / increase high frequencies
27 Filtered Backprojection: Negative Side Lobes Eat Projection Tails Replace each point in the projections with the kernel, you get the red functions.Negative side lobes eat away the trails left by the other projections. Filtered projections all reinforce at the location of the blob.
28 Filtered Backprojection Here is filtered backprojection.Final image is equal to the object. No blurry edges. No background.
29 Effect of Limited Projections This shows the effects of too few projections.In the object, and especially around the outside you see projections which have not be completely eaten by negative side lobes.
30 Effect of NoiseReconstruction noise is high spatial frequency, i.e. detail. Looks wormy. Especially see the first image.As we reduce the noise (more counts) first see large structures, then fine structures.Remember we over sample low spatial frequencies <- square law detector.
32 Mathematically Equivalent Implementations Backproject filtered projections“Filtered backprojection”Filter backprojected projections“Fourier method”Some people try to make a distinction between filtered backprojection as on the last slideand filtering (sharpening) the smooth projection->backprojection image.Mathematically, these are equivalent.
33 Low Pass Times Ramp Filter Butterworth– noiseRamp –reconstructRed shows a low pass filter, a so called Butterworth filter.Butterworth filter has two parameters, a cut off defined as the 50% point, and an order which determines the rate of transition.The ramp (reconstruction) filter modified by the LPF follows the ramp at low frequencies, but smoothly becomes zero at high frequencies.
34 SPECT Filter: Ramp - Reconstruction Low Pass Filter - Noise Reduction Noise and resolutionExact reconstructionSPECT filter (A or B) are the combination of a ramp (reconstruction) filter and a LPF.
35 Reconstruction from Projections Projection: data collectionBackprojection: 1st step in reconstructionprojection->backprojection:blurs <-> 1/kSharpen: ramp filter multiplies by kNoise reduction: low pass filter (LPF)LPF also decreases image detail
36 Reconstruction Kernel with LPF (Fourier transformof filter)LPF makes hotblobs fatterhot blobseat adjacentobjectsLPF stretches the reconstruction kernel.The center becomes fatter.The negative side lobes move out farther. Side effect is that hot blobs eat adjacent objects.
37 Inferior wall defect ?Is the hot liver eating the inferior wall or is there a real defect?
38 Iterative (OSEM) Reconstruction Iterative reconstruction doesn’t have as much of a problem with hot objects eating adjacent.But, still can’t tell what is going on. The hot liver is smeared into the inferior wall.
39 3 hour delay shows defect Better data always wins 3 hour delay with decreased liver activity shows there is a reversible inferior wall defect.Pearl: Better data always wins over processing.
40 What’s Good about FPBRamp filter exactly reconstructs projectionEfficient(Linear shift invariant)(FFT is order of n log(n)n = number of pixels)“Easily” understood
41 New Cardiac Cameras Solid state - CZT: $$$, energy resolution scatter rejection, dual isotopePixelated detector: count rate &potential high resolutionpoorer uniformityNon-uniform sampling: sensitivitypotential for artifactsSpecial purpose designcloser to patient: system resolutionupright: ameliorates diaphragmatic attenuationSPECT typically not count rate limited; intrinsic resolution typically not major issue in system resolutionAlso small foot print may help sitting.For me, major issues are improved sensitivity and improved system resolution.$$$ hidden in expensive RFID
42 D-SPECT - initial fast scan to find heart region-of-interest, ROI Heart ROI sampled for longer time.Hi (sic) Lo (sic)
43 Collimator Resolution* Single photon imaging (i.e. not PET)Collimators: image formationSensitivity / resolution trade-offResolution recovery hype“Low resolution, high sensitivity ->image processing = high resolution”Reality - ameliorates low resolutionSteve Moore: “Resolution: data = target object”Can do quick, low resolution image* not resolution from reduced distance due to designSingle photon imaging requires collimators for image formation -> sensitivity / resolution trade offHype: high resolution can be recovered with image processing. Reality: amelioratedSteve Moore says that collimation should be at resolution of the object being imagedD-SPECT uses low resolution for quick localization imaging
50 CT: Polychromatic Beam -> Dose keV150100806050This is a large patient, 47 cmLower energy X-rays have provide very little informationIncrease dose without benefit: Ideal mono-energetic X-rays
51 X-ray Tube Spectra X-ray tube: electrons on Tungsten or Molybdenum bremsstrahlungcharacteristic X-rayse- interaction:- ionization- deflectionX-ray tube: electrons on Tungsten or MolybdenumCharacteristic X-rays from ionization from charge particlesBremsstrahlung from electron slowing in tissue
52 Digirad X-ray Source: X-rays on Lead 74W82PbX-rays interaction- ionization- no 10 bremsstrahlungX-rays not electrons hit lead. No direct direct bremsstrahlungLead high Z than tungsten, higher energy characteristic X-rays
54 Detectors D-SPECT Digirad Crystals CZT CsI(Tl) Photodetector NA PD Energy resolution 6% 10%Count rate 1.35 x x 106Heads 9 3FOV (cm) 15.7 x x 21.2Total cmX-rays (blank) none 20 x 106
55 New Cardiac Cameras D-SPECT CardiArc Digirad GE Detector CZT* NaI(Tl) CsI(Tl) CZT*Electronics SS* PMT PD*? SS*Pixelated Y N Y YCollimation holes slits*? holes pinholesNon-uniform Y* Y* ~N Y*Limited angle Y Y N ~NCloser to pt Y Y Y ~NAC N CT? CT* CTPosition ~semi semi erect supineD-SPECT BWH, Digirad BIDMCAdvantages: CZT, PD?, slits?, non-uniform sampling, Digirad CTPD versus APD, Digirad, modestly converging collimators
59 Sequential Tidal-Breathing Emission and Average-Transmission Alignment Sequential emission and transmission images suffer from respiratory or gross patient motionTidal breathing average CT better than instantaneous transmission
60 Sensitivity / Resolution Trade-Off Non-uniform sampling -> sensitivitySpecial purpose design -> resolutionAdvantagesThroughput at same noisePatient motion - Hx: 1 head -> 2 headCostNon-uniform sampling -> artifactsHistory: 7-pinhole - failed180o sampling - successSequential emission transmissionFor me, the key advantage of new cameras is sensitivity from non-uniform sampling and resolution from design.Can design collimator to trade-off sensitivity / resolution
61 What’s Wrong with Ramp-Filtered Backprojection for SPECT NoiseAttenuationScatterDepth dependant resolutionNew imaging geometries
62 (Noise is White (Poisson) in Projection Space not in Object / Image Space) fnp+^Hkf = objectA = imaging systemn = noisep = projectionsH = image reconstruction, FBPf = estimate of object^
63 Spatial Frequency of Signal and Noise in Reconstructed Image Signal energy equal at all spatial frequencieskNoise energy greater at high spatial frequency
64 Decreasing Signal-to-Noise kNo problem when there is a lot more signal than noise. With increased amounts of noise, the noise swamps the signal.First at very high frequencies, then at low frequencies.Noise affects high frequencies first,then lower frequencies
65 What’s Wrong with Filtered Backprojection, FBP, for SPECT Can’t model:AttenuationScatterDepth dependant resolutionNew imaging geometries(Linear shift invariant model)
66 Attenuation and Scatter Correction Increased density (sub-diaphragmatic)-> increase in bothattenuationscatterOpposite effects on count rateNeed to correct for both simultaneously
67 SolutionIterative reconstructionUses:Simultaneous linear equationsMatrix algebraCan model image physics(Linear model)
68 Projections as Simultaneous Equations (Linear Model) But, exact solution for a largenumber of equations isn’t practicalProjection data can be seen to be equivalent to a set of simultaneous linear equations.Solution of equations is equivalent to reconstruction.
69 ((Simultaneous Equations)) Under-determined -> extra samplesIll conditioned -> a priori constraintsOver-determined -> inconsistentNoise -> more inconsistentNormal Equations of Linear Algebra
70 Large Set of Simultaneous Equations Image modeled as a vector, not a matrix128 by 128 image -> 16,384 pixelsSinogram: about the same16,384 by 16,384 matrix is impossibleSolution: iterative reconstruction(Sparse matrix)Matrix algebra becomes difficult for small sizes, 6x6. Double precision gets to maybe 10x10.Solution
71 + Image Acquisition A f n p f = object A = imaging system n = noise p = projections (sinogram)f is the object, the patients heart.A is data collection.p is the projections, the sinogram.
72 f p A is projection f is the object A is the the projection operation, the red arrowsp is the projection operationp
73 ^ + Image Reconstruction A f n p H ^ H = image reconstruction, e.g. BP, FBP etc.f = image (estimate) of object, f^H is the backprojection operation.f hat is the estimate of the object, f.
74 ^ f H is backprojection H is the backprojection operation. f hat is the estimate of the object.Since this is backprojection our estimate isn’t very good.
75 How Good is the Projection-> Backprojection Estimate? Can’t compare f to f (f is unknown)Project f to make p, estimate of projectionsNeed model of system, A^Can’t compare f hat to f because f is unknown.Solution: project f hat to and make a sinogram p hat.But, we need to model data collection, A hat.
76 Produce Estimated Projections fnp^+HA hat is just physics which is pretty well known.
77 f^A^Projection backprojection is not exact, so the image (estimate) of the object is not exact.Thus, the estimate of the projections will not be equal to the projection data.p^
78 Compare Projections to Estimate fnp^e+-HIf the reconstruction (backprojection) is somewhat accurate, then the errors should be small compared to the original data.|e|^< |p|e is the error in estimate, f
79 Corrections: Backproject Errors fnp^e+-HcorrectionsWhat should we do with the errors?We can backproject (reconstruct) the errors to give us correction factors.
81 Iterative Backprojection Reconstruction fnpfn-1^pn-1en-1fn+-xf0rHobjectdataprojectionbackprojectionestimatemodelerrorestimatedestimate +backprojectedInstead of exact solution, use approximate iterative solutionThen add the errors to the image (estimate) of the object to obtain a refined estimate.Subscripts show the whole iterative process.
82 Iterative Reconstruction fnpfn-1^pn-1en-1fn+-xf0rHThen add the errors to the image (estimate) of the object to obtain a refined estimate.Subscripts show the whole iterative process.
84 Reconstruction, H, can be Approximate fnpfn-1^pn-1en-1fn+-xf0rHOur reconstruction operation, backprojection, is not very accurate. It makes a blurry image of the object.In general, the reconstruction does not have to be very good.As long as the image is somewhat like the object, that is good enough.
85 Accuracy of Model, A, is Key ^Accuracy of Model, A, is KeyAfnpfn-1^pn-1en-1fn+-xf0rHWhat is important is that the model is an accurate representation of the data collection process.Then we get an accurate comparison between the data and the estimated data.
86 Model, A, is Well-known Physics Problem: Model of the Body ^Model, A, is Well-known Physics Problem: Model of the BodyThe model of the imaging system is well known physics.You can accurately model attenuation and scatter if you know the shape of body.It’s doable, but it’s not easy. My perception is that the various vendors do this with more or less accuracy.Tc-99m half-tissue layer: 4 cm
87 Attenuation Map Gd-153 Transmission Map adds noise to reconstructionand can introduce artifactsAs part of the data collection process we obtain transmission images at the same time as the perfusion images.Left: woman, prone - heart anterior, considerable chest wall soft tissue.Right: man, arms down, supine - heart posterior,down in diaphragm.
89 Ill Conditioned: Need Stopping Heuristic Iterative reconstruction is ill conditioned.Reconstruction is fairly good after about 10 iterations, but then it gets worse; it goods to the ill conditioned portion of the solution.EM Iterations
90 Iterative Reconstruction is Ill Conditioned Stop after N iterations((The ill conditioned portion of thesolution will be similar tostarting conditionsStarting condition often 0))A priori constraintsOften smoothness
91 Noise: FBP vs EM SD/mean Note that iterative reconstruction is much less noisy than FBP.Number in the corner is standard deviation over the mean.SD/mean
92 Filtered Back Projection (FBP) vs Ordered Subset Estimation Maximization (OSEM) FDG PETIterative reconstruction can be used for PET scanners.
93 Ordered Subset Estimation Maximization (OSEM) Estimation Maximization (EM) is an iterativereconstruction methodOrdered subset EM is a variation of EM(Update estimate for each subset)Advantage of OSEM: faster(OSEM has about 1/nsubsets of EM iterations)Currently it is the principle iterative method forboth SPECT and PET
94 Iterative Reconstruction Noise is “Blobby” Notice that the noise character of iterative reconstruction is different.Noise makes blobs. Recall that FBP noise was “wormy”.Blobs are a problem in oncology where what you are looking for is blobs.
95 What’s Good About Iterative Reconstruction Able to model:Data collection, including new geometriesAttenuationScatterDepth dependant resolutionFairly efficient given current computers(Iterative solution, e.g. EM, reasonable)(OSEM is even better)((OSEM has about 1/nsubsets of EM iterations))
96 What’s Wrong with Iterative Reconstruction (Complicated by ill conditioned model)((Estimating projections not object))Noise character bad for oncologyTo model attenuation & scatter- need to measure attenuation- adds noise