Abstract:
The use of the ordinary Poisson iterative reconstruction algorithm in PET requires the estimation of expected random coincidences. In a clinical environment, random coincidences are often acquired with a delayed coincidence technique, and expected randoms are estimated through variance reduction (VR) of measured delayed coincidences. In this paper we present iterative VR algorithms for random compressed sonograms, when previously known methods are not applicable. Iterative methods have the advantage of easy adaptation to any acquisition geometry and of allowing the estimation of singles rates at the crystal level when the number of crystals is relatively small. Two types of sonogram compression are considered: axial (span) rebinning and transaxial mashing. A monotonic sequential coordinate descent algorithm, which optimizes the Least Squares objective function, is investigated. A simultaneous update algorithm, which possesses the advantage of easy parallelization, is also derived for both cases of the Least Squares and Poisson Likelihood objective function.

Description:
BACKGROUND OF THE INVENTION 
     1. Technical Field 
     In general, the present disclosure relates to nuclear medical imaging. More particularly, the disclosure relates to Positron Emission Tomography (PET) imaging and accurate estimation of expected random coincidences through variance reduction (VR). 
     2. General Background of the Invention 
     Nuclear medicine is a unique specialty wherein radiation emission is used to acquire images that show the function and physiology of organs, bones or tissues of the body. The technique of acquiring nuclear medicine images entails first introducing radiopharmaceuticals into the body—either by injection or ingestion. These radiopharmaceuticals are attracted to specific organs, bones, or tissues of interest. Upon arriving at their specified area of interest, the radiopharmaceuticals produce gamma photon emissions, which emanate from the body and are then captured by a scintillation crystal. The interaction of the gamma photons with the scintillation crystal produces flashes of light, which are referred to as “events.” Events are detected by an array of photo detectors (such as photomultiplier tubes), and their spatial locations or positions are then calculated and stored. In this way, an image of the organ or tissue under study is created from detection of the distribution of the radioisotopes in the body. 
     One particular nuclear medicine imaging technique is known as positron emission tomography, or PET. PET is used to produce images for diagnosing the biochemistry or physiology of a specific organ, tumor or other metabolically active site. The measurement of tissue concentration using a positron emitting radionuclide is based on coincidence detection of the two gamma photons arising from a positron annihilation. When a positron is annihilated by an electron, two 511 keV gamma photons are simultaneously produced and travel in approximately opposite directions. Gamma photons produced by an annihilation event can be detected by a pair of oppositely disposed radiation detectors capable of producing a signal in response to the interaction of the gamma photons with a scintillation crystal. Annihilation events are typically identified by a time coincidence between the detection of the two 511 keV gamma photons in the two oppositely disposed detectors; i.e., the gamma photon emissions are detected virtually simultaneously by each detector. When two oppositely disposed gamma photons each strike an oppositely disposed detector to produce a time coincidence event, they also identify a line-of-response (LOR) along which the annihilation event has occurred. An example of a PET method and apparatus is described in U.S. Pat. No. 6,858,847, which patent is incorporated herein by reference in its entirety. 
       FIG. 1  is a graphic representation of a line of response. An annihilation event  140  occurring in imaged object mass  130  may emit two simultaneous gamma rays (not shown) traveling substantially 180° apart. The gamma rays may travel out of scanned mass  130  and may be detected by block detectors  110 A and  110 B, where the detection area of the block detector defines the minimum area or maximum resolution within which the position of an incident gamma ray may be determined. Since block detectors  110 A and  110 B are unable to determine precisely where the gamma rays were detected within this finite area, the LOR  120  connecting block detectors  110 A and  10 B may actually be a tube with its radius equal to the radius of block detectors  110 A and  110 B. Similar spatial resolution constraints are applicable to other types of detectors, such as photomultiplier tubes. 
     To minimize data storage requirements, clinical projection data are axially compressed, or mashed, to within a predetermined span. With a cylindrical scanner, which has translational symmetry, the geometrical blurring resulting from axial compression may be modeled by projecting a blurred image into LOR space, followed by axial compression. This eliminates the storage of the axial components, and special algorithms have been developed to incorporate system response. The system response modeling then will allow the use of standard reconstruction algorithms such as Joseph&#39;s Method, and a reduction of data storage requirements. 
     The LOR defined by the coincidence events are used to reconstruct a three-dimensional distribution of the positron-emitting radionuclide within the patient. In two-dimensional PET, each 2D transverse section or “slice” of the radionuclide distribution is reconstructed independently of adjacent sections. In fully three-dimensional PET, the data are sorted into sets of LOR, where each set is parallel to a particular detector angle, and therefore represents a two dimensional parallel projection p(s, Φ) of the three dimensional radionuclide distribution within the patient, where “s” corresponds to the displacement of the imaging plane perpendicular to the scanner axis from the center of the gantry, and “Φ” corresponds to the angle of the detector plane with respect to the x axis in (x, y) coordinate space (in other words, Φ corresponds to a particular LOR direction). 
     Coincidence events are integrated or collected for each LOR and stored in a sonogram. In this format, a single fixed point in f(x, y) traces a sinusoid in the sonogram. In each sonogram, there is one row containing the LOR for a particular azimuthal angle Φ; each such row corresponds to a one-dimensional parallel projection of the tracer distribution at a different coordinate along the scanner axis. This is shown conceptually in  FIG. 2 . 
       FIG. 3  depicts mashing and axial rebinning of LOR into sonograms. Five of six LORs are combined in one axial span complex. Four LORs are combined together transaxially. 
     The ordinary Poisson iterative reconstruction algorithm is now widely-used in the PET imaging field to produce clinically useful data from the sonograms. The model behind the algorithm considers prompt coincidences as independent Poisson variables, with expected values equal to the forward projection of the reconstructed image added to the expected scattered and random contributions. The expected random contribution of line-of-response (LOR) with indexes ij, (where i and j denote crystals in coincidence), can be estimated from a crystal singles rate, according to the classical random rate equation:
 
   r     ij =2τ   s     i     s     j ,  (1)
 
where  s   i  is the mean single rate for a given crystal i, and 2τ is the coincidence time window. This approach requires the acquisition of singles rates at the crystal level. A potential problem with this method is that real random coincidences may have additional dead time when compared to singles (due e.g. to multiplexing).
 
     Alternatively, random projection data can be acquired separately, since random events are part of the list mode data. In the latest case, random data are noisy and mean values for the random need to be estimated. We redefine the singles rate as s i =√{square root over (2τ  s   i )}, so one does not need to keep track of the multiplicative factor. The problem is the estimation of these single rates from noisy random data. 
     There are a number of methods that may be used to estimate singles from uncompressed LOR or list mode delayed coincidence data. Analytical methods use equation (1), which is averaged over selected groups of crystals in full coincidence. A measured, mashed group of random data, serves as a good approximation of the mean value. Then the problem is reduced to finding values for the smaller set singles over group of crystals. The Casey smoothing method does not estimate singles rates, but rather estimates expected random data based on large groups of random coincidences. The method from Defrise estimates group singles rates from a set of nonlinear equations using only part of the available data. Both methods are exact and useful for list mode data processing, where large groups of crystals can be defined by the user. The fan sum method is not exact and leads to bias with non-uniformly distributed activity. An analytical method allows for fast pre-processing of data. This is essential in the clinical environment, where image reconstruction consumes most of the available processing time and VR must be performed for each patient bed acquisition. However, the choice of the crystal group in full coincidence can be a complicated task. For example, a group of crystals is commonly chosen to be a detector head. However, on most clinical scanners, only data up to a maximum ring difference are passed to list mode, since the very oblique LORs are not used in reconstruction. Therefore, the detector heads are not in full coincidence and the choice of crystal groups becomes cumbersome. In addition to this, the group size influences the final result. 
     On the other hand, crystal singles can be estimated by the Maximum Likelihood (ML) approach, where random coincidences are Poisson variables with expected values provided by equation (1). The iterative approach has the advantage of versatility, where all available data are easily accommodated. It is truly a statistical method. This approach results in solving a large system of nonlinear equations iteratively, and is consequently potentially resource consuming. By examining the structure of the objective function in case of list mode data acquisition it was shown that random data have sufficient statistics. In order to estimate singles rate only fan sum of random data are needed. This interesting property has the potential to save histogramming bandwidth when the fan sums are computed at coincidence processor level. The coordinate ascent and surrogate functions monotonic algorithm were developed. Such algorithms are quite fast since there is no need to model mean random data in iterative process, only their fan sum. 
     In a clinical environment, PET data are acquired with some type of compression to reduce data size. While it is important for prompt data, delayed coincidences are processed by hardware rebinners in a similar way. Two types of data compression are considered in  FIG. 3 . The first compression is axial (span) compression, where the number of oblique planes is significantly reduced by combining the LOR of similar polar angles. The second compression is transaxial mashing, where the number of sonogram views is reduced by combining the LOR of similar azimuthal angles. Data compression does change the basic random equation (1) and causes the mentioned methodologies to no longer be applicable. 
     The goal behind of this paper was development of simple update equation iterative algorithms to estimate singles from compressed random data. The first algorithm is Coordinate Descent (CD), which possesses an advantage of fast convergence. Unfortunately, a closed form updated equation can be designed only for a Least Squares objective function. A simultaneous update (SU) algorithm is derived by a surrogate function construction. We consider two SU algorithms. One optimizes the same objective function as a CD algorithm and another optimizes the Poisson Likelihood function. 
     Unfortunately, in case of compressed data, iterative algorithms need to compare measured random data against modeled random data. Construction of mean random data takes significant time, so algorithms are not computationally fast, comparing with list mode data acquisition, where only fan sum are needed to be constructed. 
     SUMMARY OF THE INVENTION 
     An object of embodiments of the present disclosure is to provide simple update iterative algorithms for estimating random coincidences expected during a PET scan. Previously known methods for estimating random coincidences are not directly applicable since compression procedures render previously known methods non-applicable. The iterative methods presented herein have the advantages of easy adaptation to any acquisition geometry, and of allowing estimation expected random coincidences. A monotonic sequential coordinate descent algorithm, which optimizes the Least Squares objective function, is disclosed. Two simultaneous update algorithms, which possess the advantage of easy parallelization, are also disclosed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows a conceptual representation of a LOR in PET imaging; 
         FIG. 2  is an illustration of the development of a sonogram; 
         FIG. 3  shows two types of data compression of data compression of LOR data in PET imaging; 
         FIG. 4  is an illustration of an object used for random sonogram study; 
         FIG. 5  illustrates reconstruction of singles from random data in case of 6 hours scan by three embodiments; 
         FIG. 6  illustrates reconstruction of singles from random data and VR randoms in case of 1 min scan by three exemplary embodiments. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Data Compression and Objective Functions 
     Assuming that LOR random data r ij  is Poisson variable, the basic random equation for mean value of compressed data is 
                       r   _       ρ   ,   θ   ,   ζ       =         1   2     ⁢       ∑     i   ,   j       ⁢       r   _     ij         =       1   2     ⁢       ∑     i   ,   j       ⁢       ω     ρθζ   ,   ij       ⁢     s   i     ⁢       s   j     .                     (   2   )               
Measured data r ρθζ  is Poisson variable too. Data are parameterized by ρ,θ,ζ, where ρ is the radial index, θ is the azimuthal index, and ζ is the axial plane index (including oblique and direct planes). r and s are counts, rather than rates, and are measured and estimated for a given time frame. ω is the ij LOR contribution factor:
 
     
       
         
           
             
               
                 
                   
                     ω 
                     
                       ρθζ 
                       , 
                       ij 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             1 
                             , 
                           
                         
                         
                           
                             if 
                             ⁢ 
                             
                               
                                   
                               
                               ⁢ 
                               
                                   
                               
                             
                             ⁢ 
                             ij 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             contribute 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             to 
                             ⁢ 
                             
                               
                                   
                               
                               ⁢ 
                               
                                   
                               
                             
                             ⁢ 
                             sinogram 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             bin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 ρ 
                                 , 
                                 θ 
                                 , 
                                 ζ 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             0 
                             , 
                           
                         
                         
                           
                             otherwise 
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     A factor of ½ in (2) is due to the double summation over crystal indices. 
     The Least Squares objective function may be minimized to diminish differences between measured data and the modeled mean of random data: 
     
       
         
           
             
               
                 
                   LS 
                   = 
                   
                     
                       ∑ 
                       
                         ρ 
                         , 
                         θ 
                         , 
                         ζ 
                       
                     
                     ⁢ 
                     
                       
                         
                           ( 
                           
                             
                               r 
                               ρθζ 
                             
                             - 
                             
                               
                                 r 
                                 _ 
                               
                               ρθζ 
                             
                           
                           ) 
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Since random data are typically uniform in appearance we consider the non-weighted Least Squares objective function. Nevertheless, the weights inclusion is straightforward. 
     One good example of an objective function from statistical point of view will be a log of Poisson Likelihood function: 
                   L   =       ∑     ρ   ,   θ   ,   ζ       ⁢       (         r   ρθζ     ⁢   ln   ⁢           ⁢       r   _     ρθζ       -       r   _     ρθζ       )     .               (   5   )               
Coordinate Descent Algorithm
 
     In one embodiment, a simple update equation of the CD method is derived where the objective function is minimized with respect to one parameter, while the rest are fixed. It can be done easily in the case of the Least Squares (Gaussian model) objective function. The Poisson Likelihood model does not provide a simple update equation in the case of mashing/rebinning, when more than one LOR contributes to the sonogram bin. Then a line search should be exploited; however, this significantly complicates the algorithm. The CD method has at least two advantages: fast convergence, and easy implementation of the non-negativity constraint. 
     If one minimizes (4) only with respect to one parameter, say ε k , then this equation should be solved: 
     
       
         
           
             
               
                 
                   
                     
                       
                         0 
                         = 
                           
                         ⁢ 
                         
                           
                             ∂ 
                             
                               ∂ 
                               
                                 ɛ 
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                         = 
                           
                         ⁢ 
                         
                           
                             ∑ 
                             
                               ρ 
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                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   r 
                                   ρθζ 
                                 
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                                     1 
                                     2 
                                   
                                   ⁢ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         , 
                                         j 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         ω 
                                         
                                           ρθζ 
                                           , 
                                           ij 
                                         
                                       
                                       ⁢ 
                                       
                                         s 
                                         i 
                                       
                                       ⁢ 
                                       
                                         s 
                                         j 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             × 
                             
                               
                                 ( 
                                 
                                   - 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         , 
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                                     ⁢ 
                                     
                                       
                                         ω 
                                         
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                                           , 
                                           ij 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             
                                               δ 
                                               
                                                 i 
                                                 , 
                                                 k 
                                               
                                             
                                             ⁢ 
                                             
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                                           + 
                                           
                                             
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                                             ⁢ 
                                             
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                                                 , 
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                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The equation can be solved for ε k , taking into account, that ω ρθζ,ij =0 if i=j, since the crystal is never in coincidence with itself. Therefore, the second multiplication factor in (6) does not contain ε k . Then the equation is linear with regards to ε k . This can be written explicitly, extracting ε k  from the first multiplicative factor in (6): 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           ρ 
                           , 
                           θ 
                           , 
                           ζ 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               r 
                               ρθζ 
                             
                             - 
                             
                               
                                 1 
                                 2 
                               
                               ⁢ 
                               
                                 
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                                       ≠ 
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                                     , 
                                     
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                                       ≠ 
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                                 ⁢ 
                                 
                                   
                                     ω 
                                     
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                                       , 
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                                   ⁢ 
                                   
                                     s 
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                                   ⁢ 
                                   
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                         ⁢ 
                         
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                         s 
                         k 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
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                         ⁢ 
                         
                           
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                               ⁢ 
                               
                                 
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                                     , 
                                     kj 
                                   
                                 
                                 ⁢ 
                                 
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                   = 
                   0. 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     It can be rewritten in additive iterative form to update variable ε k  with respect to a previous value. Such expression will have term  r   ρθζ , estimated using the previous value of ε k : 
     
       
         
           
             
               
                 
                   
                     s 
                     k 
                     
                       ( 
                       new 
                       ) 
                     
                   
                   = 
                   
                     
                       s 
                       k 
                       
                         ( 
                         old 
                         ) 
                       
                     
                     + 
                     
                       
                         
                           
                             ∑ 
                             
                               ρ 
                               , 
                               θ 
                               , 
                               ζ 
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   r 
                                   ρθζ 
                                 
                                 - 
                                 
                                   
                                     
                                       r 
                                       _ 
                                     
                                     ρθζ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       ɛ 
                                       
                                         ( 
                                         old 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     ≠ 
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                                 ⁢ 
                                 
                                   
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                                       , 
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                                   ⁢ 
                                   
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                                     j 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             ∑ 
                             
                               ρ 
                               , 
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                               , 
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                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     ≠ 
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                                 ⁢ 
                                 
                                   
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                                       , 
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                                   ⁢ 
                                   
                                     s 
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                               ) 
                             
                             2 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     One convenient implementation is crystal-driven. For each crystal k, one needs to consider all corresponding opposite crystals in coincidence. From such knowledge of two particular crystals, a corresponding sonogram bin is accessed. An exemplary crystal-driven equation is: 
     
       
         
           
             
               
                 
                   
                     
                       s 
                       k 
                       
                         ( 
                         n 
                         ) 
                       
                     
                     = 
                     
                       
                         s 
                         k 
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             ∑ 
                             
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                               ≠ 
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                           ⁢ 
                           
                             
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                                 ( 
                                 
                                   n 
                                   , 
                                   k 
                                 
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                             ⁢ 
                             
                               
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                                   , 
                                   kj 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     r 
                                     ρθζ 
                                   
                                   - 
                                   
                                     
                                       
                                         r 
                                         _ 
                                       
                                       ρθζ 
                                     
                                     ⁡ 
                                     
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                                             n 
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                                           ) 
                                         
                                       
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                             ⁢ 
                             
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                                 ⁢ 
                                 
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                                     n 
                                     , 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     s 
                     
                       ( 
                       
                         n 
                         , 
                         k 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           s 
                           0 
                           
                             ( 
                             n 
                             ) 
                           
                         
                         , 
                         
                           s 
                           1 
                           
                             ( 
                             n 
                             ) 
                           
                         
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         
                           
                             s 
                             
                               
                                 k 
                                 - 
                                 1 
                               
                               , 
                             
                             
                               ( 
                               n 
                               ) 
                             
                           
                           ⁢ 
                           
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                             k 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         , 
                         
                           s 
                           
                             k 
                             + 
                             1 
                           
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                         , 
                         … 
                       
                       ⁢ 
                       
                           
                       
                       } 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Here n is iteration number. A single iteration may consist of updating all crystal single values. The summation over sonogram indices was explicitly removed: ρθζ is uniquely defined by the k-j pair. In the denominator expression we also assumed that 
                   ∑     ρ   ,   θ   ,   ζ       ⁢       ω       ρ   ⁢           ⁢   θ   ⁢           ⁢   ζ     ,   kj       ⁢     ω       ρ   ⁢           ⁢   θ   ⁢           ⁢   ζ     ,     kj   ′             ≠   0     ,       if   ⁢           ⁢   j     =     j   ′       ,         
since it was true for our mashing/rebinning scheme. Definition (10) represents an exemplary idea of sequential CD algorithm: the very last update of crystal efficiencies should be used in estimation of current crystal single. This definition implies that updates are performed according to crystal index. The crystal with index zero updated first, then the crystal with index 1 is updated and so on.
 
     The CD algorithm is difficult to parallelize in the case of compressed data, comparing with list mode data acquisition. The value  r   ρθζ  depends on the cluster of crystals of the same blocks due to mashing and rebinning. Therefore, this term should be updated immediately after the update of every crystal single. When k crystal is updated, a loop over corresponding opposite crystals in coincidence is performed, subtracting an old value of product of k crystal and crystal in coincidence singles and adding multiplicative term with a new crystal single in corresponding  r   ρθζ . Such updates are prone to round off error when many iterations are performed. To exclude this effect  r   ρθζ  may be recomputed at every iteration, when the value of the objective function is estimated. 
     In the following we will reference update equation (9) as the CD-LS algorithm. 
     Simultaneous Update (SU) Algorithms 
     In this section, exemplary algorithms of gradient type are derived. In case of the LS objective function, the algorithm has a simple update equation at iteration n: 
                           s   k     (   n   )       =       s   k     (     n   -   1     )       -       α   k     (     n   -   1     )       ⁢       ∂   LS       ∂     s   k                      s   k     (     n   -   1     )         .           (   11   )               
A surrogate function is used to derive positive value step size α. First, it is assumed that the objective function can be well approximated by a second order Taylor&#39;s expression with regards to a current singles estimate of iteration n−1
 
                                     LS   ⁡     (   s   )       ≈       T   1     ⁡     (     s   ,     s     (     n   -   1     )         )         =       LS   ⁡     (     s     (     n   -   1     )       )       +       ∂   LS       ∂     s   l                    s     (     n   -   1     )         ⁢     (       s   l     -     s   l     (     n   -   1     )         )       +       1   2     ⁢         ∂   2     ⁢   LS         ∂     s   l       ⁢     ∂     s   m                      s     (     n   -   1     )         ⁢           ⁢     (       s   l     -     s   l     (     n   -   1     )         )     ⁢       (       s   m     -     s   m     (     n   -   1     )         )     .             (   12   )               
Δs l  may be defined as, for example, Δs l =(s l −s l   (n−1) ). Using non equalities
 
                       Δ   ⁢           ⁢     s   l     ⁢   Δ   ⁢           ⁢     s   m       ≤         1   2     ⁢     (       Δ   ⁢           ⁢     s   l   2       +     Δ   ⁢           ⁢     s   m   2         )       -     Δ   ⁢           ⁢     s   l     ⁢   Δ   ⁢           ⁢     s   m         ≤       1   2     ⁢     (       Δ   ⁢           ⁢     s   l   2       +     Δ   ⁢           ⁢     s   m   2         )         ,           (   13   )               
a separable surrogate function can be constructed
 
                                 T   1     ⁡     (     s   ,     s     (     n   -   1     )         )       ≤       T   2     ⁡     (     s   ,     s     (     n   -   1     )         )         =       LS   ⁡     (     s     (     n   -   1     )       )       +       ∂   LS       ∂     s   l                    s     (     n   -   1     )         ⁢     (       s   l     -     s   l     (     n   -   1     )         )       +       1   2     ⁢                ∂   2     ⁢   LS         ∂     s   l       ⁢     ∂     s   m                  s     (     n   -   1     )         ⁢                  (       s   l     -     s   l     (     n   -   1     )         )     2     +       (       s   m     -     s   m     (     n   -   1     )         )     2       2     .                 (   14   )               
Here absolute value of
 
                 ∂   2     ⁢   LS         ∂     s   l       ⁢     ∂     s   m               
is used. The value and the gradient of the surrogate function T 2 (s,s (n−1) ) coincides with the ones of the T 1 (s,s (n−1) ) objective function at s (n−1) . The surrogate function lies above T 1  (s,s (n−1) ). In approximation
 
   r     ρθζ   ≈  r     ρθζ ,  (15)
 
which is accurate close to solution,
 
                 ∂   2     ⁢   LS         ∂     s   l       ⁢     ∂     s   m               
is positive, assuming positive s (n−1) , so the surrogate function closely approximates a truncated series. However, this term positivity is not necessary in general and the second of non equalities (13) is used to construct a surrogate function. If expansion (12) is sufficiently accurate, the algorithm will show monotonic behavior; however this property is not guaranteed by the present exemplary derivation.
 
     Finally, update equation in case of LS objective function, using approximation (15) in term 
                   ∂   2     ⁢   LS         ∂     s   l       ⁢     ∂     s   m           ,         
which defines a step size, has the form:
 
                       s   k     (     n   +   1     )       =       s   k     (   n   )       +         ∑     j   ≠   k       ⁢       s   j     (   n   )       ⁢       ω     ρθζ   ,   kj       ⁡     (       r   ρθζ     -         r   _     ρθζ     ⁡     (     s       (   n   )     ⁢               )         )               ∑     j   ≠   k       ⁢       s   j     (   n   )       ⁢     ω     ρθζ   ,   kj       ⁢     ∑   ρθζ     (   n   )                 ,     
     ⁢   where           (   16   )                 ∑   ρθζ     (   n   )       ⁢     =       1   2     ⁢       ∑   ij     ⁢       ω     ρθζ   ,   ij       ⁡     (       s   i     (   n   )       +     s   j     (   n   )         )                     (   17   )               
In the following we will reference update equation (16) as SU-LS algorithm.
 
     In the case of the L objective function, a second exemplary derivation is similar. The surrogate function is contracted to have a lower value with regards to truncated expansion. The term 
                 ∂   2     ⁢   L         ∂     s   l       ⁢     ∂     s   m               
has negative value in approximation (15) and assumption of positive s (n−1) .
 
     The crystal-driven update equation for maximizing L objective function has the following form: 
                     s   k     (     n   +   1     )       =       s   k     (   n   )       +           ∑     j   ≠   k       ⁢       s   j     (   n   )       ⁢       ω     ρθζ   ,   kj       ⁡     (         r   ρθζ           r   _     ρθζ     ⁡     (     s       (   n   )     ⁢               )         -   1     )               ∑     j   ≠   k       ⁢       s   j     (   n   )       ⁢     ω     ρθζ   ,   kj       ⁢       ∑   ρθζ     (   n   )             r   _     ρθζ     ⁡     (     s       (   n   )     ⁢               )               .               (   18   )               
In the following we will reference update equation (16) as SU-L algorithm. Once again summation over sonogram indices was explicitly removed in (16)-(18): ρθζ is uniquely defined by the k-j pair.
 
     An advantage of these exemplary algorithms is that they are straightforwardly easy to be parallelized. Non-negativity constrain is implemented in the same way as in case of CD algorithm. If single update has negative value it is replaced by some small positive value. 
     Initial Condition 
     The CD algorithm is sensitive to the initial condition. The use of truncated expansion in the SU algorithm derivation is justified close to solution. The solution also allows for a negative value of crystal singles. It is possible to find a good initial condition. As an example, an initial condition that provides a positive value of singles is obtained through the global surrogate function of the Poisson Likelihood objective function. The concavity of log function is used 
                       ln   (       1   2     ⁢       ∑   ij     ⁢       ω     ρθζ   ,   ij       ⁢     s   i     ⁢     s   j           )     ≥         1     2   ⁢     N   ρθζ         ⁢       ∑   ij     ⁢       ω     ρθζ   ,   ij       ⁢     ln   ⁡     (       s   i     ⁢     s   j       )             +     ln   ⁡     (     N   ρθζ     )           ,     
     ⁢   where           (   19   )                 N   ρθζ     =       1   2     ⁢       ∑   ij     ⁢       ω     ρθζ   ,   ij       .                 (   20   )               
is the number of LORs contribution to particular  r   ρθζ . Using
 
                         s   i     ⁢     s   j       ≤       1   2     ⁢     (       s   i   2     +     s   j   2       )         ,           (   21   )               L   ≥         1   2     ⁢       ∑   ij     ⁢       ∑     ρ   ,   θ   ,   ζ       ⁢       ω     ρθζ   ,   ij       ⁢     {           r   ρθζ       N   ρθζ       ⁢     (       ln   ⁢           ⁢     s   i       +     ln   ⁢           ⁢     s   j         )       -         s   i   2     +     s   j   2       2       }             +       ∑     ρ   ,   θ   ,   ζ       ⁢       r   ρθζ     ⁢       ln   ⁡     (     N   ρθζ     )       .                   (   22   )               
The global surrogate function is separable and its maximum corresponds to
 
                     s   k     (   0   )       =             ∑     j   ≠   k       ⁢       ω     ρθζ   ,   kj       ⁢       r   ρθζ       N   ρθζ               ∑     j   ≠   k       ⁢     ω     ρθζ   ,   kj             .             (   23   )               
This may serve as the initial condition to all exemplary algorithms.
 
       FIG. 4  depicts an object used for random sonogram study. The transaxial fan of crystals in coincidence are also shown on the right-hand side. 
     The exemplary random VR algorithms were tested for data acquired in list mode format on the Siemens TruePoint scanner, schematically presented in  FIG. 5 . This scanner consists of 144 blocks of 13×13 crystals. The blocks are organized as three blocks (rings) in the axial direction and 48 blocks in the transaxial direction. The total number of crystals is 24336. The scanner consists of 39 rings; however, the list mode file contains data with a maximum ring difference of 27. Transaxially, each block is in coincidence with 26 opposite blocks, see  FIG. 4 . 
     Six hours of list mode (LM) files were acquired on one bed position with two Ge-68 cylinders spaced axially with cold arms. This phantom is depicted on  FIG. 6 . The variation in the random sonogram values was large as 35%. 
     A random mashed and rebinned sonogram was produced from list mode data with span  11  and mash factor  4 , resulting in a 336(radial)×84(azimuthal)×313(axial) sonogram bin size. Two sonogram time frames were considered. The first corresponded to the first acquisition minute. Sonogram contained ˜5M random and ˜5M true counts. The second, full six hour, random data served as a gold standard for VR random sonogram. 
     Singles were estimated directly from the list mode file by the Coordinate Ascent method [7, 8]. This estimation served as a gold standard for singles distribution, since uncompressed data provide the best possible estimation. 
       FIG. 5  shows the exemplary algorithms performance in case of high count six hours scan. All algorithms showed monotonical behavior. The SU updates provided a smoother singles distribution when compared to the CD method, when the same number of iterations was exploited. The CD algorithm eventually converged to a smaller value of the normalized LS objective function. Contrary to computer simulations, the CD method was not faster at converging at earlier iterations. 
       FIG. 6  shows the algorithm&#39;s performance in case a one minute scan. List mode estimation still provided very good singles estimation, slightly noisier with respect to a 6 hour scan. Estimation from compressed data, however, displayed noise explosion phenomena at a large iteration number. The CD-SU algorithm estimation suffers most from it due to fast high frequencies convergence. Intrinsic regularization property of SU methods allows for smoother singles estimation at early iterations. In order to optimize algorithm performance with respect to iteration number, VR random sonograms were compared with six hours raw scan by computing NRMSE difference between these two data sets. According to this figure of merit, the CD-LS algorithm should be stopped after 4 iterations. The SU-LS and SU-ML algorithms showed very similar performance and should stop after 7 or 8 iterations. The intrinsic regularization property of SU methods allows for smother useful singles estimations. 
     While the one minute raw sonogram shows nothing but noise, VR sonograms resembled six hours of raw scan. The CD-LS method produced noisy singles distribution and it in turn led to additional artificial strips in a VR sonogram. The SU methods did not produce such artifacts in the sonogram at least in the visual estimation. 
     Those of ordinary skill may vary the data collection apparatus and methods for recording and processing the images without varying from the scope of the invention as defined in the appended claims.