BWH test pattern. Google “BWH SMPTE Judy” for jpeg copies of BWH & SMPTE test patterns

SMPTE test pattern

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

Projection Single Slice Axial What’s a projection? This is what mathematicians say. Given object and angle, it is integrals through the object.

Single Axial Slice: 3600 collimator Ignoring attenuation, SPECT data are projections Gamma camera rotates 360o around object. Data it collects is projections (ignoring attenuation)

Attenuation: 180o = 360o x Tc-99m keV 150 100 80 60 50 htl(140 keV) ≈ 4 cm This is a large patient, 47 cm 140 keV Tc-99m photons are highly attenuated, 0.1% Cardiac 180o collection is special. In general, 360o

Cardiac Perfusion Data Collection Special Case - 180o Axial Coronal / Sagittal Multiple simultaneous axial slices One horizontal line from each of these images, makes one axial slice in the last slide. Notice arms are moving; heart is fairly still.

Dual-Head General-Purpose Gamma Camera: 900 “Cardiac” Position 2 heads: 900 rotation = 1800 data 1 2 Phillips nee ADAC SPECT camera with attenuation correction Simultaneous emission and transmission

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.

Original data Original data gives a much better reconstruction. Some people on read bulls eye -> tech essential reads study

Single Axial Slice: 3600 One line from each projection image. More usual 360o rotation. Blob becomes 1-D function. Marked zero so you can follow what happens with rotation.

Sinogram: Projections Single Axial Slice 60 projection angle x Display projections as lines in an image. First line projection to top. Blob follows a sine wave. Called sinogram; commonly used.

Uniformity & Motion on Sinogram 1 head 24 min 2 heads 12 min Can use sinogram to detect motion and uniformity. I don’t think sinogram is particularly insightful.

Reconstruction by Backprojection Backprojection tails We 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.

Backprojection 2 projections 2 objects If we have two blobs and only two projections, backprojection gives us 4 blobs.

projection tails merge resulting in blurring More projections give us a better image, but in is a blurred out representation of the original.

Projection -> Backprojection of a Point lines add at the point tails spread point out If 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.)

Projection -> Backprojection more backprojection angles, more like object

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.

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) 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.

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

Spatial Frequency Basis Functions f(u,v) ≠ 0, single u,0 f(u,v) ≠ 0, single 0,v f(u,v) ≠ 0, single u = v High school: you learned about basis functions for vector spaces f(u) not zero for some u, f(v) not zero for some v Sum, f(u,v) not zero for some u = v MRI samples f(u,v) directly in k-space

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

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 Ramp filter increases high frequencies

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

Negative Side Lobes: Sharpening Negative side-lobes enhance edges / increase high frequencies

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.

Filtered Backprojection Here is filtered backprojection. Final image is equal to the object. No blurry edges. No background.

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.

Effect of Noise Reconstruction 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.

Shows the whole process.

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 slide and filtering (sharpening) the smooth projection->backprojection image. Mathematically, these are equivalent.

Low Pass Times Ramp Filter Butterworth – noise Ramp – reconstruct Red 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.

SPECT Filter: Ramp - Reconstruction Low Pass Filter - Noise Reduction Noise and resolution Exact reconstruction SPECT filter (A or B) are the combination of a ramp (reconstruction) filter and a LPF.

Reconstruction from Projections Projection: data collection Backprojection: 1st 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

Reconstruction Kernel with LPF (Fourier transform of filter) LPF makes hot blobs fatter hot blobs eat adjacent objects LPF stretches the reconstruction kernel. The center becomes fatter. The negative side lobes move out farther. Side effect is that hot blobs eat adjacent objects.

Inferior wall defect ? Is the hot liver eating the inferior wall or is there a real defect?

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.

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.

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: $$$, energy resolution scatter rejection, dual isotope Pixelated detector: count rate & potential high resolution poorer uniformity Non-uniform sampling: sensitivity potential for artifacts Special purpose design closer to patient: system resolution upright: ameliorates diaphragmatic attenuation SPECT typically not count rate limited; intrinsic resolution typically not major issue in system resolution Also small foot print may help sitting. For me, major issues are improved sensitivity and improved system resolution. $$$ hidden in expensive RFID

D-SPECT - initial fast scan to find heart region-of-interest, ROI Heart ROI sampled for longer time. Hi (sic) Lo (sic)

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 Single photon imaging requires collimators for image formation -> sensitivity / resolution trade off Hype: high resolution can be recovered with image processing. Reality: ameliorated Steve Moore says that collimation should be at resolution of the object being imaged D-SPECT uses low resolution for quick localization imaging

Dual Head: Non-Uniform Sampling

Activity Measurement: Attenuation keV 150 100 80 60 50 htl(140 keV) ≈ 4 cm CT measures attenuation SPECT measures activity, attenuation complicates SPECT Surprising how well we can do without AC

Attenuation Correction: Simultaneous Emission (90%) and Transmission (10%) Gd-153 rods T1/2 240 d e.c. 100% 97 keV 29% 103 keV 21% 2 heads: 900 rotation = 1800 data Phillips nee ADAC SPECT camera with attenuation correction Dale Bailey spent 6 mo at BIH Simultaneous emission and transmission

Semi-erect: Ameliorates Attenuation

Leaning Forward, < 500 Pounds Digirad shows large patient Chair takes large patients Nukes doesn’t like large patients

Digirad: Patient Rotates X-ray Attenuation Correction

CT: Polychromatic Beam -> Dose keV 150 100 80 60 50 This is a large patient, 47 cm Lower energy X-rays have provide very little information Increase dose without benefit: Ideal mono-energetic X-rays

X-ray Tube Spectra X-ray tube: electrons on Tungsten or Molybdenum bremsstrahlung characteristic X-rays e- interaction: - ionization - deflection X-ray tube: electrons on Tungsten or Molybdenum Characteristic X-rays from ionization from charge particles Bremsstrahlung from electron slowing in tissue

Digirad X-ray Source: X-rays on Lead 74W 82Pb X-rays interaction - ionization - no 10 bremsstrahlung X-rays not electrons hit lead. No direct direct bremsstrahlung Lead high Z than tungsten, higher energy characteristic X-rays

Digirad X-ray Spectrum

Detectors D-SPECT Digirad Crystals CZT CsI(Tl) Photodetector NA PD Energy resolution 6% 10% Count rate 1.35 x 106 15 x 106 Heads 9 3 FOV (cm) 15.7 x 3.9 15.8 x 21.2 Total cm2 558 1005 X-rays (blank) none 20 x 106

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 D-SPECT BWH, Digirad BIDMC Advantages: CZT, PD?, slits?, non-uniform sampling, Digirad CT PD versus APD, Digirad, modestly converging collimators

Soft Tissue Attenuation: Supine breast lung

Soft Tissue Attenuation: Prone breast

Soft Tissue Attenuation: Digirad Erect breast post

Sequential Tidal-Breathing Emission and Average-Transmission Alignment Sequential emission and transmission images suffer from respiratory or gross patient motion Tidal breathing average CT better than instantaneous transmission

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 180o sampling - success Sequential emission transmission For 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

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

(Noise is White (Poisson) in Projection Space not in Object / Image Space) f n p + ^ H k f = object A = imaging system n = noise p = projections H = image reconstruction, FBP f = estimate of object ^

Spatial Frequency of Signal and Noise in Reconstructed Image Signal energy equal at all spatial frequencies k Noise energy greater at high spatial frequency

Decreasing Signal-to-Noise k No 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

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

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

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 equations isn’t practical Projection data can be seen to be equivalent to a set of simultaneous linear equations. Solution of equations is equivalent to reconstruction.

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

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) Matrix algebra becomes difficult for small sizes, 6x6. Double precision gets to maybe 10x10. Solution

+ 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.

f p A is projection f is the object A is the the projection operation, the red arrows p is the projection operation p

^ + 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.

^ 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.

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 ^ 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.

Produce Estimated Projections f n p ^ + H A hat is just physics which is pretty well known.

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 ^

Compare Projections to Estimate f n p ^ e + - H If 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

Corrections: Backproject Errors f n p ^ e + - H corrections What should we do with the errors? We can backproject (reconstruct) the errors to give us correction factors.

corrections e ^

Iterative Backprojection Reconstruction f n p fn-1 ^ pn-1 en-1 fn + - x f0 r H object data projection backprojection estimate model error estimated estimate + backprojected Instead of exact solution, use approximate iterative solution Then add the errors to the image (estimate) of the object to obtain a refined estimate. Subscripts show the whole iterative process.

Iterative Reconstruction f n p fn-1 ^ pn-1 en-1 fn + - x f0 r H Then add the errors to the image (estimate) of the object to obtain a refined estimate. Subscripts show the whole iterative process.

Iterative Reconstruction f n p fn-1 ^ pn-1 en-1 fn + - x f0 r H relaxation factor 0-1

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

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

Model, A, is Well-known Physics Problem: Model of the Body ^ Model, A, is Well-known Physics Problem: Model of the Body The 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

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

(Ill Conditioned) A f n p fn-1 ^ pn-1 en-1 fn + - x f0 r H Estimating

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

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

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

Filtered Back Projection (FBP) vs Ordered Subset Estimation Maximization (OSEM) FDG PET Iterative reconstruction can be used for PET scanners.

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/nsubsets of EM iterations) Currently it is the principle iterative method for both SPECT and PET

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.

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)) 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 efficient Models noise, attenuation, scatter, depth dependant resolution, and new cameras

Applause

Projection Projection: 1, 2, 3, and 4 are projection data.

Backprojection Put the projection data back

A ^ Since the system is just projection, we will use projection as our estimate of the system. In this case the model is exactly equal to the system.