A Continuous-Coordinate Image Reconstruction Method for List-Mode ...

A continuous-coordinate image reconstruction method for list-mode time-of-flight position emission tomography Chien-Min Kao, Senior Member, IEEE, Heejong Kim, Member, IEEE, Chang-Han Huang, Cheng-Ying Chou, Weichung Wang, and Chin-Tu Chen

Abstract—We present a new formulation for reconstructing list-mode time-of-flight (TOF) positron emission tomography (PET) data onto the continuous-coordinate event space based on maximizing the likelihood function. With this formulation, the spatial resolution is not limited by the a priori choice of the image voxels. We derive an iterative reconstruction algorithm and propose a serial processor (SP) for practical use. This SP reconstruction may support concurrent imaging, which is a new feature for PET imaging. The SP reconstruction approach is examined by using computer-simulated TOF PET data having a 600 ps FWHM concidence time resolution. Preliminary results show that the proposed approach is very promising.

I. I NTRODUCTION Image reconstruction for positron emission tomography (PET), including time-of-flight (TOF) PET, is often achieved by solving a matrix equation that describes the imaging process. This equation is typically derived from the continuous, linear model of the detection physics and statistics while modeling the image as consisting of non-overlapping voxels containing uniform radioactivity within [1]. More sophisicated basis functions for representing the image function have also been proposed [2,3]. Typically, the matrix equation is solved by optimizing a certain criterion, such as the maximum likelihood (ML), in order to account for data noise and other modeling errors. The a priori information about the solution can also be introduced. Typically, iterative algorithms are employed for finding the optimization solution. This approach has proved successful but it has an inherent limitation: The chosen voxel size will define the achievable image resolution. Apparently, smaller voxels are desirable in order not to artificially limit the resolution capability of the imaging system. However, using small voxels will drastically increase the complexity of the already compute-intensive and memory-demanding reconstruction task. Moreover, it can lead to greatly amplified image Chien-Min Kao is with the Department of Radiology, The University of Chicago, Illinois, USA. Email: [email protected]. This work was support in part by CTSA UL1 TR000430. Heejong Kim is with the Department of Radiology, The University of Chicago, Chicago, Illinois, USA. Email: [email protected]. Chang-Han Huang is with the Department of Bio-Industrial Mechatronics Engineering, National Taiwan University, Taipei, Taiawn. Cheng-Ying Chou is with the Department of Bio-Industrial Mechatronics Engineering, National Taiwan University, Taipei, Taiwan. Email: [email protected]. Weichung Wang is with the Department of Mathematics, National Taiwan University, Taipei, Taiwan. Email: [email protected] Chin-Tu Chen is with the Department of Radiology, The University of Chicago, Chicago, Illinois, USA. Email: [email protected].

978-1-4799-0534-8/13/$31.00 ©2013 IEEE

noise that in effect obscures small and low-contrast structures. Therefore, the voxel size shall be recognized as a priori condition imposed on the solution and it needs to be carefully determined to balance the above considerations. To date, most iterative reconstruction algorithms in PET have been using uniform voxels that are heuristically chosen. Recently, there are reported efforts in using variable image voxels in x-ray computed tomography (XCT) to boost the resolution in a certain region by using smaller voxels for this region [4]. For non-TOF PET, Sitek et al has recently proposed the Original Ensemble (OE) method in which continuous-coordinate events are drawn from the data likelihood function by using the Markov Chains Monte Carlo (MCMC) method and the result approximates the ML solution [5,6]. In this approach, image voxelization is applied post reconstruction for displaying the reconstruction result. Motivated by the OE method, in this work we present a formulation for reconstructing list-mode TOF PET data onto the continuous-coordinate event space. We consider listmode TOF PET data because of the increasing prevalence of TOF PET imaging and list-mode format. Our formulation is however based on a theoretically different concept from the OE method: Rather than sampling from a probability function, we explicitly seek the ML solutions of the original emission positions of the TOF PET events. In addition, we propose a serial processor that has the potential of supporting concurrent imaging. The remainder of this paper is organized as follows. In Sec. II, we describe the theoretical basis for the proposed formulation. In Sec. III, we derive a reconstruction algorithm based on the formulation and propose a serial processor for practical use. In Sec. IV, we apply the serial process to simulated list-mode 2D TOF PET data. Concluding remarks are given in Sec. V. II. T HEORECTICAL FORMULATION A. Maximum-likelihood solution for list-mode TOF PET data Consider a TOF PET system consisting of N detection channels, numerically label by using a discrete variable c. The ith annihilation event detected by the system is denoted by d~i = (ci , ti ),

(1)

where ci identifies the channel making the detection and ti is the detected position of the event on the line-of-response (LOR) associated with channel ci . In deriving the theoretical

formulation and reconstruction algorithm, ti is assumed to be a continuous variable given by ti = v0 τi /2, where v0 is the speed of light in air and τi is the differential time measured for the two detected gamma rays arriving at the detectors. Let pc (t|~r) denote the probability density for a positron released at location ~r to give rise to detection d~ = (c, t). Then, ˆ λc (t; f ) = d~r pc (t|~r)f (~r) (2)

by using a shift-invariant Gaussian function independent of channel c : ) 0 0 (10) p(τ c (t|t ) = g1 (t − t ; σt ) where g1 (x; σ) = √ and

 σt =

is the mean event-rate density for detecting annihilation events at t by channel c when the underlying image function is f (~r). Given the list-mode TOF PET data DM = {d~1 , · · · , d~M }, we show in Appendix A that, up to a non-zero scaling factor the ML estimate of f (~r) is given by   M X λci (ti |f ) , (3) log fˆ(~r) = argmaxf : φ(f )=1 λ(f ) i=1 where λ(f ) =

Xˆ c

ˆ

∞

dt λc (t|f ) =

d~r s(~r)f (~r)

(4)

−∞

is the total mean event-rate density of the system given f (~r), Xˆ ∞ dt pc (t|~r) (5) s (~r) = c

−∞

is the overall detection sensitivity of the system to a positron released at ~r, and ˆ φ(f ) = d~r m(~r)f (~r) (6) for some desirable function m(~r). B. Channel coordinate system and response functions Consider Lc , the LOR of channel c. It can be expressed as Lc : {t~e(t) e(u) e(v) c + uc~ c + vc~ c : −∞ < t < ∞}, (t)

(u)

(7)

2 2 1 e−x /2σ 2πσ

1 v0 rτ 2

(11)

 /2.35

(12)

with rτ being the coincidence resolving time (CRT) of the (d) system, in FWHM. On the other hand, pc (~r) is usually given by r) = Ωc (~r · ~e(t) r · ~e(u) r · ~e(v) p(d) c (~ c )δ(uc − ~ c , vc − ~ c ),

(13)

(d)

where the delta function restricts the support of pc (~r) to Lc and Ωc (t) is the solid angle of channel c to the point at coordinate t on Lc . For mathematical tractability, in this paper we will sassume     r) = Ωc ~r · ~e(t) r · ~e(v) g2 uc − ~r · ~e(u) p(d) c (~ c c , vc − ~ c ; σd , (14) where (15) g2 (x, y; σ) = g1 (x; σ)g1 (y; σ), with σd  1 so that g2 (x, y; σ) ≈ δ(x, y). In summary, we have (16) pc (t|~r) = c Ωc (t)hc (t|~r), where r · ~e(u) r · ~e(v) hc (t|~r) = g1 (t − ~r · ~e(t) c ; σt )g2 (uc − ~ c , vc − ~ c ; σd ). (17) In above discussion and in this paper, we ignore subject attenuation, positron range, photon acolinearity, and nontrivial spatially variant detector blurring (e.g., the depth-ofinteraction (DOI) blurring). In theory, these factors can be (d) included in pc (~r).

(v)

where ~ec , ~ec , and ~ec are three orthonormal vectors that (t) span the 3d space with ~ec pointing in the direction of Lc and (uc , vc ) is the 2d coordinate on the t = 0 plane where Lc intersects. Using this channel-dependent coordinates, the 3d position of d~i can be written as ~ri = ti~e(t) e(u) e(v) ci + uci ~ ci + vci ~ ci .

(8)

Rather than using the popular voxel representation for the image function, in spirit of Monte-Carlo sampling we propose to represent an image by using point processes. That is, we ~j , j = consider a sampler for generating M 0 point samples ρ 1, · · · , M 0 , from f (~r) and use them to construct

(9)

q(~r) X f˜ (~r; ρ ~1 , · · · , ρ ~M 0 ) = δ (~r − ρ ~j ) M 0 j=1

M0

In addition, we can model
) pc (t|~r) = c p(d) t|~r · ~tc , r) p(τ c (~ c

C. Sampling representation of the image function

where c is the coincidence detection efficiency at channel c, i.e., the probability for a pair of annihilation photons, given that they hit the detectors of channel c, to register a coinci(d) dence event; pc (~r) is the probability for a positron released at ~r to give rise to a pair of annihilation photons hitting the (τ ) detectors of channel c; and pc (t|t0 ) is the probability for a detected positron that is released at coordinate t0 on Lc to be measured at coordinate t due to the finite TOF resolution of (τ ) the system. In the literature, pc (t|t0 ) is typically modelled

(18)

in the sense o n ~M 0 ) = f (~r). E f˜ (~r; ρ ~1 , · · · , ρ

(19)

To determine q(~r), assume that the sampler has the sampling sensitivity s˜(~r) so that the probability for it to generate from f (~r) a point at ~r is proportional to f (~r)˜ s(~r). Thus, given a sample the probability for it to occur at ~r is equal to ˆ ps (~r) = s˜(~r)f (~r)/ d~r s˜(~r)f (~r). (20)

For independently-generated point samples, we then have ~M 0 |M } = Pr{~ ρ1 , · · · , ρ

M Y

ˆ

ˆ

0

0

with m(f ) = q −1 (~r) :

ps (~ ρj ),

~1 , · · · , ρ ~M 0 ) = d~r q −1 (~r)f˜(~r; ρ

(21)

M0

d~r

j=1

s˜(~r) X δ(~r − ρ ~j ) M 0 j=1 s˜(~ ρj )

M ˆ s˜(~r) 1 X d~r δ(~r − ρ ~j ) = 1. = 0 M j=1 s˜(~ ρj ) 0

and consequently n

~M 0 ) E f˜ (~r; ρ ~1 , · · · , ρ =

q(~r) M0

o

ˆ

0

d~ ρ1 · · · d~ ρM 0

M X

So far, M 0 is not specifically defined. When applied to listmode TOF PET data, it is not unreasonable to set M 0 = M and tie ρ ~j to ~ri by

0

M Y

δ(~r − ρ ~j )

j=1

ps (~ ρk )

k=1

ρ ~j = ~rj + uj ~e(t) cj , j = 1, · · · , M.

M0

q(~r) X ps (~r) = q(~r)ps (~r). = M 0 j=1

(22)

To satisfy Eq. (19), we would therefore require s˜(~r) = 6 0 whenever f (~r) 6= 0 and ´ ( d~ r s˜(~ r )f (~ r) f (~ r) = f (~r) 6= 0 p (~ r ) s˜(~ r) s q(~r) = (23) 0 f (~r) = 0. Consequently, we have the sampling representation ´ f˜ (~r; ρ ~1 , · · · , ρ ~M 0 ) =

´ d~r θ(~r)f (~r) ≈

M0

d~r s˜(~r)f (~r) X θ(~ ρj ) . 0 M s˜(~ ρj ) j=1

~ˆM 0 = argmaxρ~1 ,··· ,~ρM 0 ρ ~ˆ1 , · · · , ρ

M X i=1

(25)

 log

λi λ

 ,

(26)

where ´ ´ λ=

j=1

j=1

(32) (24)

D. Formulation for image reconstruction ~M 0 ) as the output of the We propose to set f˜(~r; ρ ~1 , · · · , ρ reconstruction process. That is, we envision that the PET imager and reconstruction process together to behave as a sampler for f (~r). By using this representation in Eqs. (2)-(6), the unknowns to solve for become ρ ~j , j = 1, · · · , M 0 , and their ML solutions are given by

λi = ci Ωci (ti ) ×

It follows that ρ ~j can be interpreted as the original emission position of the ith event and hence the proposed formulation seeks the ML estimates of the continuous coordinates of the original emission positions of the detected events. Using Eq. (31) and ignoring terms that do not depend on ~u = (u1 , · · · , uM )t , Eqs. (26) becomes     M M  N  X X X h s ij j ˆ = argmax ~u log − M log , ~ u  s˜j s˜j  i=1

With this representation, we have the following approximation for evaluating a linear functional on f (~r) : For any θ(~r), ˆ

(31)

0

M d~r s˜(~r)f (~r) X δ(~r − ρ ~j ) . 0 M s ˜ (~ ρ ) j j=1

M0

d~r s˜(~r)f (~r) X hij × M0 s˜ j=1 j M

(27)

0

d~r s˜(~r)f (~r) X sj × M0 s˜ j=1 j

(28)

where hij , sj and mj are treated as functions of uj . III. R ECONSTRUCTION METHODS A. ML solution ˆ, we take the derivative of the term in braces in To find ~u Eq. (32) with respect to uk , k = 1, · · · , M, and equate it to zero to obtain     M X ∂ hik sk ∂ 0= yik log log −M , (33) ∂u s ˜ ∂u s˜k k k k i=1 where

hik /˜ sk . yik = PM h sj j=1 ij /˜

(34)

~εik = ~ri − ρ ~k = δ~rik − uk~e(t) ck ,

(35)

Next, by writing

where δ~rik = ~ri − ~rk is the difference between the detected positions of events i and k, we have 2 2   (t) (t) 2 ~εik · ~eci k~εik k − ~εik · ~eci log hik = − − (36) 2σt2 2σd2 and (t)

(29)

The constraint φ(f ) = 1 in Eq. (3) is absent from Eq. (26) because it is implicitly satisfied by the sampling representation

(t)

(t)

~ec − γci ck ~eci γci ck ~eci + k σt2 σd2

∂ log hik = ∂uk (t)

with ρj ), sj = s(~ ρj ), s˜j = s˜(~ ρj ). hij = hci (ti |~

(30)

(t)

! · ~εik ,

(37)

where γci ck = ~eci · ~eck is the cosine of the LORs of the detected events i and k. Substituting Eq. (37) into Eq. (33) PM and dividing the result by i=1 yik , we obtain ( ) Nd (t) (t) (t) X ~eck − γci ck ~eci γci ck ~eci · ~εik − βk , (38) ωik + 0= σt2 σd2 i=1

where ωik = yik /

M X

yik

(39)

i=1

and ∂ M zk ∂ log s˜k + PM log ∂uk ∂uk i=1 yik

βk =



sk s˜k

 ,

(40)

where sk /˜ sk zik = PM . sj j=1 sj /˜

(41)

PM It is noted that i=1 ωik = 1. We assume σd2  σt2 , which applies in practice. Under this condition, for Eq. n o(38) to be valid we shall have PNd (t) (t) ω − γ ~ e ~ e · ~εik ≈ 0. Consequently, we are c c ik c c i k i k i=1 motivated to require

0=

2) Compute [n+1]

Nd 1 X 0= 2 ωik γci ck ~e(t) εik − βk ci · ~ σt i=1 Nd X

Figure 1. A serial processor of size L.

uk

(42)

(43)

M X

[n]

ωik

  ~e(t) · δ~rik e(t) ck − γci ck ~ ci

i=1

+

n o ωik ~e(t) e(t) · ~εik . ck − γci ck ~ ci

=

M X

[n] ωik γc2i ck

!( M X

i=1

[n] ωik

  [n] ~e(t) rik − σt2 βk ck · δ~

,

i=1

(48)

i=1

By using Eqs. (35) and (43), we can rewrite Eq. (42) as 0=

)

Nd   1 X ωik ~e(t) rik − uk − βk , ck · δ~ 2 σt i=1

[n]

(44)

[n]

where ωik and λk are ωik and λk evaluated by using [n] [n] {~ ρ1 , · · · , ρ ~M }. 3) Let n = n + 1 and repeat step 2 until ~u[n] converges or when n reaches a pre-determined limit.

yielding uk =

Nd X

  ωik ~e(t) rik − σt2 βk . ck · δ~

(45)

i=1

Similarly, by using Eqs. (35) we can reqrite Eq. (43) as uk =

Nd X

Nd   X (t) ωik ~e(t) − γ ~ e +u ωik γc2i ck . (46) ·δ~ r ci ck ci ik k ck

i=1

i=1

Combination of Eqs. (45) and (47) then yields uk =

M X

  ωik ~e(t) e(t) · δ~rik ck − γci ck ~ ci

i=1

+

M X

ωik γc2i ck

!( M X

i=1

  − σt2 βk ωik ~e(t) · δ~ r ik ck

) ,

(47)

i=1

for k = 1, · · · , M. Note that ωik and λk on the right-hand side of this equation are nonlinear functions of ~u. B. A fixed-point algorithm for image reconstruction From Eq. (47), we can derive a fixed-point algorithm for [n] [n] [n] (t) finding the ML solution of ~u. Let uk and ρ ~k = ~rk + uk ~eck denote the estimates obtained for uk and ρ ~k at the nth iteration. We have the following algorithm: 1) Set n = 0 and k = 1, · · · , M.

[0] uk

= 0, and hence

[0] ρ ~k

= ~rk , for

C. A serial processor for image reconstruction [n]

The computation of ωik has a complexity of M 2 . In practice, the number of events M is a large number, on the order of 106 to 108 , and therefore the above algorithm is difficult to implement. To deal with this challenge, we propose to apply Eq. (48) to the most recent L events only, where L  M, to obtain a serial processor (SP) as illustrated by Fig. 1. As shown, it contains a first-in-first-out (FIFO) input buffer for holding L events. Another L-event output FIFO stores the original-coordinate estimates of the events in the input FIFO. The input and output FIFOs are initially empty. When a new event d~i is acquired, the following actions take place. First, this event is pushed into the FIFOs to push out the oldest elements to the output of the processor. Second, the process applies Eq. (48) for m times to update the original-coordinate estimates, which are stored in the output FIFO, for the events in the input FIFO. Consequently, this processor serially outputs the original-coordinate estimates of the detected events with an L-event delay. The two design parameters L and m of the SP are chosen to balance the speed of the computation and accuracy of the resulting event coordinates. IV. S IMULATION R ESULTS A. Data Generation To evaluate the proposed SP, we employ computer-simulated 2D TOF PET data. We consider a 2D scanner that consists of

640 detectors, each of which is 4 mm wide, on a 81.5 cmdiameter ring. The sources are made of pixels (voxels in 2D) and within each pixels the radioactivity distribution is assumed uniform. The simulation data is generated as follows. Given a source, a pixel is selected for creating an annihilation event by sampling from a multinomial distribution with parameters given by the intensities of the image pixels. A point within the pixel is then randomly selected as the annihilation position and two opposite gamma-rays are generated at the point with a randomly determined emission direction sampled from a uniform distribution over [0, π). The two detectors intersecting with the two gamma rays, and the difference in the arrival times of the gamma rays at the detectors, are determined. The detected event position assuming ideal TOF resolution is then assigned to the LOR of the intersecting detectors, which is defined as the line connecting the centers of the front faces of the detectors, at the location corresponding to the determined differential arrival time. Then, the location on the LOR is displaced by a random amount in accordance with the assumed CRT of the scanner. The simulated data are stored in listmode format. In practice, a TOF PET scanner has a finite binsize for TOF measurement. Therefore, a user-specified TOF binsize is applied to the stored data before reconstruction. In our simulation, positron range, photon acolinearity, scatter, randoms and subject attenuation are ignored. In this paper, we will consider a CRT of 600 ps FWHM and a TOF binsize of 25 ps. B. Reconstruction Results For the SP, we consider L = 2048 and m = 1. We also assume s˜(~r) = 1. In addition, it is not difficult to see that s(~r) = constant for the simulation process described above. The update equation given in Sec. III therefore becomes [n+1]

uk

=

M X

[n]

ωik

  e(t) ~e(t) · δ~rik ck − γci ck ~ ci

i=1

+

M X i=1

[n] ωik γc2i ck

!( M X

[n] ωik

)   ~e(t) rik ck · δ~

(49)

i=1

PM PM where ωik = yik / i=1 yik with yik = hik / j=1 hij . The results generated by the proposed SP are compared with the original emission positions, i.e., the annihilation positions of the simulated events, and with the most-likely position (MLP) reconstruction, which estimates the emission positions of the events by simply using their detected positions. However, when applying this simplest approach the MLP result will exhibit concentric rings due to the digitization effect of the 25 ps TOF binsize. To remove such ring artifacts, we randomly displace the measured event positions along their LORs by an amount sampled from a uniform distribution in (−d, d), where d is the distance corresponding to 25 ps TOF measurement. The original emissions, the SP output, and the MLP result all contain continuous-coordinate events in listmode format. When displaying, these results are binned onto image matrices using user-specified matrix and pixel sizes. In the first experiment, we consider a 1×1 mm2 square source in air. Figure 2 shows the results obtained from data

Figure 2. Results obtained by the proposed SP in comparison with the original emissions and MLP reconstructions from data containing 1K, 10K, and 500K events simulated for a 1×1 mm2 source in air. The results are displayed onto 256×256 image matrices containing 1×1 mm2 pixels.

containing 1K, 10K and 500K events generated for this source, displayed onto 256×256 image matrices containing 1×1 mm2 pixels. Figure 3 shows the intensity profiles on the horizontal line through the center of the source. Examining the MLP images and profiles obtained for 1K data, it is evident that the detected events are distributed over a much larger area than the extent of the source due to the uncertainty in TOF measurement, with few events being actually detected within the source. In contrast, the proposed SP can accurately relocate the events to occur within the source. Having more counts in the case of 10K and 500K data, on the MLP results the source can be identified on top of a diffuse background. The substantially lower amplitudes of the MLP profiles than the original-emission profiles (about 1/100) again reflect the substantially wider event distributions in the MLP result. In comparison, at all three count levels examined the SP results agree with the original-emission results well. In particular, the SP profiles match with the original-emission profiles but do show low-amplitude tails. Because of the tails, the amplitudes of SP profiles are about half of the original-emission profiles. In the second experiment, we consider a chest image derived from the Zubal phantom [8] consisting of 4×4 mm2 pixels. Figures 4 and 5 show the reconstruction results and their sample intensity profiles. Examining the images, in comparison with the MLP results the proposed SP reconstruction can significantly enhance the resolution and contrast. Also, more details are revealed in the SP result as more events are available. Examining the sample profiles, the SP result similarly shows improved contrast in comparison with the MLP result. However, the SP profiles still show uncorrected backgrounds when compared with the original-emission profiles. This is

Figure 4. Results obtained from simulated data containing 100K, 1M, and 1.9M events for a chest image derived from the Zubal phantom. The source image is made of 4×4 mm2 pixels and the results are displayed onto 360×240 image matrices containing 2×2 mm2 pixels.

Figure 3. Intensity profiles, on the horizontal line through the center of the source, of the images shown in Figure 2 obtained from 1K-, 10K- and 500Kevent data. For the 1K case, the MLP profile is zero but at a few positions, reflecting the fact that the small number of detected events are distributed into a larger space than the size of the source by the uncertainty of the TOF measurement. In comparison, the events in the SP result are concerntrated onto the source. With more events in the 10K case, the source can be identified in the MLP result; however, its low amplitude reflects the wider event distribution in space. For the 500K case, the MLP and SP profiles are scaled by a factor of 100 and 2 respectively to match the amplitude of the original-emission result. It is evident that the MLP profile is much wider that the SP and originalemission profiles. At all count levels, the SP profiles are in good agreement with the original-emission profiles, showing slightly wider distributions with low-amplitude tails.

not surprising because, as discussed in the above section the SP reconstruction is not exact due to the use of only a very limited subset of the detected events at each update step. The background profiles may be reduced by increasing the SP buffer size to include more events (i.e., a larger L), by using more iterations (i.e., a larger m), or by introducing other techniques. In the third experiment, we reduce the pixel size of the chest phantom in the above experiment to 2×2 mm2 . Figure 6 shows the resulting images and sample profiles obtained from simulated data containing 3M events. In comparison with the results shown in Figures 4 and 5, the MLP result has reduced resolution and contrast because the structures are now half in size. Comparing the images and profiles in Figure 6 to those in Figures 4 and 5, the SP reconstruction appears to provide better resolution and contrast recovery from the MLP reconstruction than in the second experiment in which the source image is

Figure 5. Sample vertical and horizontal profiles of the 1.9 M-event results shown in Figure 4.

made of larger pixels. V. C ONCLUSIONS AND D ISCUSSION We have proposed an ML formulation and developed an iterative algorithm for reconstructing list-mode TOF PET data by representing the image as consisting of continuouscoordinate events rathter than by using voxels or other basis functions. The resulting algorithm, given by Eq. (48), has a simple form. It is worthy of noting that this algorithm is automatically count conserving and, because it estimates event coordinates, the resulting image is always non-negative. The

one another. Given M detections, the probability for the ith detection to be made by channel ci and its TOF value to be in the interval [ti , ti + dti ) is then given by M o Y n λci (ti |f )dti M . (51) Pr {ci , [ti , ti + dti )}i=1 |M, f = λ(f ) i=1

It follows that, given the image function f (~r), the probability density for having the list-mode data DM = {d~1 , · · · , d~M }, where d~i = (ci , ti ), is given by pr(DM |M, f ) = Figure 6. Reconstruction images and sample profiles obtained from simulated data containing 3M events for the chest image when assuming 2×2 mm2 pixels.

algorithm has a complexity of M 2 , where M is the number of detected events, and therefore is extremely difficult to use in practice. In this paper, we propose a serial processor to address this challenge. By considering only a very limited subsets of the detected events at each update step, this processor is not theoretically exactly but it may generate images concurrent to data acquisition. Also, our numerical results demonstrate that the contrast and resolution of the resulting image can be useful for detection tasks. Our method is similar to the OE method recently developed by Sitek [5,6] in that both methods estimate the continuous coordinates of the events instead of image intensities within heuristically-defined voxels. In comparison, our method is derived by explicitly maximizing the likelihood function while the OE method employs the MCMC algorithm to sample the likelihood function (or other probability functions). Also, our method applied only to list-mode TOF PET data whereas the OE method is applicable to all emission tomography. While our initial results are promising, there are many open theoretical questions and practical issues in the proposed formulation for continuous-coordinate reconstruction and the SP approach that need to be answered and further investigated. We will extend the SP reconstruction to 3d imaging. We will investigate approaches to improve the reconstruction accuracy of the SP reconstruction. We will extend the imaging model to include more physical factors, including subject attenuation and non-uniformity in detection efficiency. Scattered and random events will be included as well. We will evaluate the performance properties of the reconstruction approaches under various imaging conditions. A PPENDIX A. L IST- MODE LIKELIHOOD FUNCTIONS The derivation below extends the derivation of Parra and Barrett [7]. Consider a detected event in TOF PET. The probability for this event to be made by channel c and for its TOF value to be in the interval [t, t + dt) is equal to Pr(c, [t, t + dt)|1, f ) = λc (t; f )dt/λ(f ),

(50)

where λc (t; f ) and λ(f ) are given by Eqs. (2) and (4), respectively. Assume the detection of each event is independent of

M Y λci (ti |f ) . λ(f ) i=1

(52)

This is the likelihood function for f (~r) given DM ; hence, the log-likelihood function is   M X λci (ti |f ) l(f |DM , M ) = (53) log λ(f ) i=1 It is easy to check that this equation is invariant to applying a non-zero scaling to f (~r). Consequently, the ML estimate of f (~r),   M X λci (ti |f ) fˆM (~r) = argmaxf , (54) log λ(f ) i=1 is determined only up to a scaling factor. The above derivation considers the preset-count scenario in which scanning is terminated upon acquiring exactly M events. In the preset-time scenario, scanning is carried out for a pre-determined time period T and the number of detected events M becomes a Poisson random variable having the probability distribution ¯ ¯M e −M M ¯ = E{M } = λ(f )T. (55) Pr(M |T, f ) = , M M! The likelihood function in this case becomes L(f |DM , T ) = p(DM |T, f ) = Pr(M |T, f )p(DM |M, f ) = Pr(M |T, f )L(f |DM , M ).

(56)

Observing that Pr(M |T, f ) depends on f (~r) through λ(f ) and L(f |DM , M ) is invariant to a scaling of f, it is not difficult to see that the ML estimate for the preset-time case is fˆT (~r) = ˆ α ˆ fˆM (~r) where α ˆ is the maximizer   of Pr(M |T, αfM ). It is ˆ trivial to show that α ˆ = M/λ fM T, yielding fˆT (~r) =

M  × fˆM (~r). ˆ λ fM T 

(57)

Consequently, we have observed that for both scenarios our task is to find fˆM (~r). Since fˆM (~r) is determined up to a scaling factor, we can also impose a constraint on the magnitude of f (~r) when maximizing the log-likelihood function. This constraint can be generally given in the form ˆ φ(f ) = d~r m(~r)f (~r) = 1 (58) for some function m(~r). The ML estimate is then given by   M X λci (ti |f ) . (59) log fˆ(~r) = argmaxf :φ(f )=1 λ(f ) i=1

It is possible to choose φ(f ) = λ(f ) and obtain fˆ(~r) = argmaxf :λ(f )=1

M X

log λci (ti |f ).

(60)

i=1

R EFERENCES [1] A.J. Reader and H. Zaidi, “Advances in PET image reconstruction,” PET Clin, 2, pp. 173-190, 2007. [2] R.M. Lewitt, “Multidimensional digital image representations using generalized Kaiser-Bessel window functions,” Journal of the Optical Society of America A, 7, pp. 1834–1846, 1990. [3] A. Sitek, R.H. Huesman and G.T. Gullberg, “Tomographic reconstruction using an adaptive tetrahedral mesh defined by a point cloud,” IEEE Trans Med. Imag., 25, pp. 1172-1179, 2006. [4] Z. Zhang, J. Bian, X. Han, D. Shi, A. Zamyatin, E. Y. Sidky and X. Pan “Iterative image reconstruction with variable resolution in diagnostic CT,” 2nd International Meeting on Image Formation in X-Ray Computed Tomography, pp 254-258, 2012. [5] A. Sitek, “Representation of photon limited data in emission tomography using origin ensembles,” Phys Med Biol. 53, pp. 3201–3216, 2008. [6] A. Sitek, “Reconstruction of emission tomography data using origin ensembles,” IEEE Trans. Med. Imag., 30, pp. 946-956, 2011. [7] L. Parra and H.H. Barrett, “List-mode likelihood: EM algorithm and image quality estimation demonstrated on 2D PET,” IEEE Trans. Med. Imag., 17, pp. 228-235, 1998. [8] I. Zubal, C. Harrell, E. Smith, Z. Rattner, G. Gindi and P. Hoffer, “Computerized three-dimensional segmented human anatomy,” Medical Physics, 21, pp.299-302, 1994.