Abstract:
One embodiment is a unique method employing an improved image resolution recovery technique. Another embodiment is a unique system implementing an improved image resolution recovery technique. A further embodiment is a method including obtaining a sinogram based upon a measurement of an imaging system, processing the sinogram using a smoothing or fitting technique, deconvolving a system response function of the imaging system from the sinogram to provide a deconvolved sinogram, and constructing an image based upon the deconvolved sinogram. Further embodiments, forms, objects, features, advantages, aspects, and benefits shall become apparent from the following description and drawings.

Description:
CROSS REFERENCE 
     The present application is a continuation of PCT/US2008/002339 filed on Feb. 22, 2008, which claims the benefit of U.S. Provisional Patent Application No. 60/902,678 filed Feb. 22, 2007, each of which is incorporated herein by reference. The present application claims the benefits and the rights of priority of both aforementioned applications. 
    
    
     TECHNICAL FIELD 
     The technical field relates to imaging resolution recovery techniques operable in connection with positron emission tomography (“PET”), other tomographic imaging systems and modalities, and/or planar imaging systems and modalities such as projection x-ray systems and others. 
     BACKGROUND 
     Present approaches to PET and other imaging systems and modalities suffer from a variety of drawbacks, limitations, disadvantages and problems including those respecting image resolution, computational cost, and others. There is a need for the unique and inventive techniques, methods, and systems disclosed herein. 
     SUMMARY 
     One embodiment is a unique method employing an improved image resolution recovery technique. Another embodiment is a unique system implementing an improved image resolution recovery technique. A further embodiment is a method including obtaining a sinogram based upon a measurement of an imaging system, processing the sinogram using a smoothing or fitting technique, deconvolving a system response function of the imaging system from the sinogram to provide a deconvolved sinogram, and constructing an image based upon the deconvolved sinogram. Further embodiments, forms, objects, features, advantages, aspects, and benefits shall become apparent from the following description and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  is an illustration of an exemplary imaging system. 
         FIG. 2  is an illustration of an exemplary measurement by a imaging system and a sinogram based upon the measurement. 
         FIG. 3  is a flow diagram of an exemplary resolution recovery technique. 
         FIG. 4  is a PET image. 
         FIG. 5  is the PET image of  FIG. 4  to which a two dimensional local regression technique has been applied. 
         FIG. 6  is the PET image of  FIG. 4  to which a two dimensional local regression and a deconvolution technique have been applied. 
     
    
    
     DETAILED DESCRIPTION 
     For purposes of promoting an understanding of the principles of the invention, reference will now be made to the embodiments illustrated in the figures and specific language will be used to describe the same. It will nevertheless be understood that no limitation of the scope of the invention is thereby intended, such alterations and further modifications in the illustrated embodiments, and such further applications of the principles of the invention as illustrated therein being contemplated as would occur to one skilled in the art to which the invention relates. 
     With reference to  FIG. 1 , there is illustrated an exemplary imaging system  100 . In the illustrated embodiment, system  100  is a PET system including a detector ring  105  which includes detectors, such as detectors  110  and  112 , which are operable to detect photons emitted during the annihilation of a positron and an electron. The detectors may be a scintillating crystal coupled with one or more photomultiplier tubes or silicon avalanche photodiodes, or any other detector operable to detect annihilation photons. While only detectors  110  and  112  are illustrated for purposes of clarity and simplicity, detector ring  105  includes a plurality of additional detectors which could be arranged in a variety of configurations about detector ring  105 . 
     A patient  103  (or another object to be scanned) is positioned in detector ring  105 . A compound labeled with positron emitting isotopes (“labeled compound”) has been previously introduced into the patient  103 . The labeled compound will emit a positron which will be annihilated when it encounters an electron. The annihilation event will emit a pair of annihilation photons which travel in opposite directions along line of response  120  and will be detected by detectors  110  and  112 , respectively. Since positron annihilation produces two photons which travel in opposite directions, only coincident detection events (detection of photons by detectors at opposite ends of a line of response at very nearly the same time) are meaningful and non-coincident detection events can be ignored. 
       FIG. 1  further illustrates a more detailed portion  101  of PET system  100 . As illustrated in portion  101 , line of response  120  has a volume since detectors  110  and  112  have an area. The number of coincident detection events is proportional to the quantity of labeled compound  121  in patient  103  which is in the volume of response located between detectors  110  and  112 . While system  100  is illustrated and described as a PET system, additional embodiments involve the use of other imaging systems, techniques and instrumentation including, for example, other tomographic imaging systems and modalities, and/or planar imaging systems and modalities such as projection x-ray systems and others. 
     With reference to  FIG. 2 , there is illustrated a measurement  200  being taken by an exemplary PET system, such as PET system  100 . Measurement  200  acquires data for a set of lines of response  220  for a given angle θ. Lines of response  220  may be organized into a projection which includes the line integrals for all s for an angle θ. Projections over the range 0≦θ≦2π may be organized into sinogram  201  which is a two dimensional function of s and θ. For purposes of clarity the measurement illustrated in  FIG. 2  has been greatly simplified so that only a single positron annihilation event in patient  103  (indicated by point f(x,y)) is being measured. This single annihilation event produces two annihilation photons which travel along the line of response which intersects point f(x,y) and are detected by detectors in detector ring  105 . The projection of this detection is given by p(s,θ) and point f(x,y) appears as the sinusoid on sinogram  201 . As stated above, this example is greatly simplified. In a typical measurement, sinogram  201  will include a superposition of all sinusoids corresponding to each point of an annihilation event in patient  103 . A measured sinogram may be obtained in the foregoing manner, or in other manners of detection used in connection with other tomographic imaging systems and modalities, and/or planar imaging systems and modalities such as projection x-ray systems and others. 
     With reference to  FIG. 3 , there is illustrated a flow diagram  300  describing operations relating to a preferred resolution recovery technique. At operation  310  photon detection occurs, for example, as described above in connection with  FIGS. 1 and 2 . Photon detection may be accomplished using a variety of techniques, systems and instrumentation including, for example, a two dimensional PET system, a three-dimensional PET system, or a combined PET/CT system. It is also contemplated that detection other than photon detection could be accomplished using other systems, techniques and/or instrumentation such as, for example, other tomographic imaging systems and modalities, and/or planar imaging systems and modalities such as projection x-ray systems and others. 
     From operation  310  flow diagram  300  proceeds to operation  320  where coincidence detection occurs, for example, through coincidence detection as described above in connection with  FIGS. 1 and 2 . Coincidence detection may be accomplished, for example, using a coincidence processing unit which receives detection information from a detector ring and identifies coincident detection events. The coincidence processing unit and other components of the systems disclosed herein may include one or more processors, ASICs, computers, computer readable memories or combinations thereof. The computer readable memories may be configured to store executable instructions which may be created or programmed using a variety of programming languages, compilers, and tools. It is also contemplated that detection other than coincidence detection could be accomplished using other systems, techniques and/or instrumentation such as, for example, other tomographic imaging systems and modalities, and/or planar imaging systems and modalities such as projection x-ray systems and others. 
     From operation  320  flow diagram  300  proceeds to operation  330  where a measured sinogram is generated, constructed, and/or stored in a computer readable memory. Generation of a measured sinogram may be accomplished, for example, using one or more processors, ASICs, computers or combinations thereof. In a preferred embodiment, a measured sinogram p(s,θ,z) is obtained, where p(s,θ,z) is a set of measured projections, for example, a set of coincident detector responses, s is the projection position, θ is the projection angle, and z is the axial (slice) distance. The measured sinogram includes the transverse component of the system response function is l(s) where l(s) is independent of θ and is an even function of s. There is also an ideal sinogram q(s,θ,z) whose estimate may be calculated and used to construct an enhanced resolution image. The measured sinogram is a function of the convolution of the transverse component of the system response function and the ideal sinogram which can be expressed according to Equation 1: 
     
       
         
           
             
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     From operation  330  flow diagram  300  proceeds to operation  340  where a smoothing or fitting technique is applied to the measured sinogram. The smoothing or fitting technique is preferably operable to reduce noise associated with the sinogram and/or to improve resolution of the sinogram. Operation  340  and subsequent operations may be performed in an image processing or reconstruction system which may include one or more processors, ASICs, computers, or combinations thereof. Preferably the technique applied at operation  340  is a non-parametric fitting technique. More preferably the technique applied at operation  340  is a two dimensional local regression. In a most preferred embodiment, a two dimensional local regression technique is applied to the projection of the measured sinogram along its projection s axis and the slice distance z axis. This is done for all projection angles and yields a sinogram estimate {circumflex over (p)}(s,θ,z) with reduced noise and improved data consistency. This technique can be performed by selecting coefficient estimates ĉ,â 1 ,â 2 ,{circumflex over (b)} 1 ,{circumflex over (b)} 2 ,{circumflex over (d)} to minimize the summation given by Equation 2: 
               ∑     j   =   1     M     ⁢       ∑     i   =   1     N     ⁢       W   ⁡     (           s   i     -   s     h     ,         z   j     -   z     h       )       ⁢       (             p   ⁡     (       s   i     ,   θ   ,     z   j       )       -   c   -       a   1     ⁡     (       s   i     -   s     )       -                     a   2     ⁡     (       s   i     -   s     )       2     -       b   1     ⁡     (       z   j     -   z     )       -                     b   2     ⁡     (       z   j     -   z     )       2     -       d   ⁡     (       s   i     -   s     )       ⁢     (       z   j     -   z     )               )     2               
for s−h≦s i ≦s+h and z−h≦z j ≦z+h. W(α,β) is a 2 dimensional weighting function which is non-zero for |α|≦1.0 and |β|≦1.0, and zero elsewhere. h can be viewed as the local neighborhood-size parameter. The regression estimate {circumflex over (p)}(s I ,θ,z J ) at the point (s I ,θ,z J ) is equal to ĉ which minimizes Equation 2 at the point s=s I ,z=z J . Thus, a fit is most preferably performed in two dimensions. In other embodiments, other techniques may be applied to a measured sinogram, for example, a two dimensional Hanning filter, a two dimensional Butterworth filter, a two dimensional low pass cosine filter, a two dimensional Weiner filter, or another two dimensional filter.
 
     From operation  340  flow diagram  300  proceeds to operation  350 . At operation  350  a deconvolving operation is performed. At operation  350  an estimate of the ideal sinogram q(s,θ,z) can be obtained, for example, by using Equation 4 or Equation 5 to calculate an approximation or estimate for the ideal sinogram q(s,θ,z). 
     Within a local range, a sinogram may be approximated by its Taylor series expansion. In a preferred embodiment a second order Taylor series expansion is performed. In additional embodiments, higher order Taylor series expansions may be utilized. The sinogram can be expressed as a Taylor series expansion expanded around s according to Equation 3: 
     
       
         
           
             
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     Since the transverse component of the system response function is assumed to be even, the odd higher order Taylor series expansion terms are all zero. Noting that the sinogram estimate is a good approximation of the ideal sinogram, a substitution of the derivative of p for the derivative of q can be made and an estimate of the ideal sinogram can be expressed according to Equation 4: 
               q   ⁡     (     s   ,   θ   ,   z     )       ≅       p   ⁡     (     s   ,   θ   ,   z     )       -         m   2     2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   2                   
where m 2  is the second order moment of the transverse component of the system response function. In another embodiment, an estimate of the ideal sinogram can be expressed according to Equation 5:
 
               q   ⁡     (     s   ,   θ   ,   z     )       ≅       p   ⁡     (     s   ,   θ   ,   z     )       -         m   2     2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   2           +       (         m   2   2     4     -       m   4     24       )     ⁢         ⅆ   4     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   4                   
where m 2  is the second order moment of the transverse component of the system response function and m 4  is the fourth order moment of the transverse component of the system response function. An estimate for the ideal sinogram can be obtained by using the second order derivative of the measured sinogram projections and the second order moment of the transverse component of the system response function which can be measured and/or modeled, for example, as a Gaussian with a certain full width at half maximum (“FWHM”), or a Laplacian with a certain FWHM that can be selected according to the FWHM of the particular imaging system in use. If desired, the fourth order derivative of the measured sinogram projections and the fourth order moment of the transverse component of the system response function can be used to provide correction up to the fourth order. Additional embodiments contemplate an estimate for the ideal sinogram can be obtained by using the higher order derivatives and moments.
 
     From operation  350  flow diagram  300  proceeds to operation  360 . At operation  360  image construction is performed using the estimate of the ideal sinogram q(s,θ,z). In a preferred embodiment image construction is accomplished using a back projection technique, preferably a filtered back projection technique. Further embodiments contemplate the use of alternate or additional techniques of image reconstruction. 
     Equations 4 and 5 can be generalized for the case where the axial component of the system response function h(z) is also included along with the transverse component of the system response function l(s) such that the measured sinogram is expressed in Equation 6: 
     
       
         
           
             
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     The generalization can be expanded using the Taylor series expansion (extended to 2 dimensions) around s and z and the result can be expressed according to Equation 7: 
               q   ⁡     (     s   ,   θ   ,   z     )       ≅       p   ⁡     (     s   ,   θ   ,   z     )       -         m     L   ⁢           ⁢   2       2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   2           -         m     H   ⁢           ⁢   2       2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     z   2                   
where m L2  is the second order moment of the transverse component of the system response function, l(s) and m H2  is the second order moment of the axial component of the system response function, h(z). An estimate for the ideal sinogram can be obtained by using the second order derivatives of the measured sinogram along the axial and transverse axes, and the second order moments of the transverse and axial components of the system response function.
 
     Additional embodiments contemplate variations on the operations described in connection with flow diagram  300 . In certain embodiments, operation  310  is not performed and input for operation  320  is provided or received form another source. In certain embodiments, operation  320  is not performed and input for operation  330  is provided or received form another source. In certain embodiments, operation  330  is not performed and input for operation  340  is provided or received form another source. In certain embodiments, operation  340  is not performed and deconvolution at operation  350  may be performed on the output of operation  330 . In certain embodiments, operation  350  is not performed and an image is constructed at operation  360  from the output of operation  340 . In certain embodiments, operation  360  is not performed and the output of operation  350  is stored or passed or transmitted to another system or location for image reconstruction or other operations. It should also be understood that techniques and operations described in connection with a measured sinogram, p, could be performed in connection with a smoothed or fitted sinogram estimate, {circumflex over (p)}, and that techniques and operations described in connection with a smoothed or fitted sinogram estimate {circumflex over (p)}, could be performed in connection with a measured sinogram, p. 
     With reference to  FIGS. 4-6 , there are illustrated transaxial image slices of a mouse brain scan reconstructed from an F-18 mouse study conducted using a small animal PET scanner. The two dimensional local regression neighborhood was a 5*0.87/2 mm by 5*0.87/2 mm square. The transverse component of the system response function was modeled as a Gaussian with a FWHM of 1.5 mm. For this example, the two dimensional weighting function W is taken to be all 1.0 with h=5. With reference to  FIG. 4 , there is illustrated a transaxial image slice generated using filtered back projection (“FBP”) using a Hanning 70% cutoff window with no two dimensional local regression post-processing. With reference to  FIG. 5 , there is illustrated the image slice of  FIG. 4  after being fit using two dimensional local regression post-processing and a FBP ramp. With reference to  FIG. 6 , there is illustrated the image slice of  FIG. 4  after being fit using two dimensional local regression post-processing and a FBP ramp and de-convolved from the system response. 
     One embodiment is a method including obtaining a sinogram based upon a measurement of an imaging system, processing the sinogram using a smoothing or fitting technique, and constructing an image based upon the sinogram. 
     Another embodiment is a method including obtaining a sinogram based upon a measurement of an imaging system, deconvolving a system response function of the imaging system from the sinogram to provide a deconvolved sinogram, and constructing an image based upon the deconvolved sinogram. 
     Another embodiment is a method including obtaining a sinogram based upon a measurement of an imaging system, processing the sinogram using a smoothing or fitting technique, deconvolving a system response function of the imaging system from the sinogram to provide a deconvolved sinogram, and constructing an image based upon the deconvolved sinogram. 
     In additional method embodiments the imaging system is a positron emission tomography system. 
     In additional method embodiments the smoothing or fitting technique is a non-parametric fitting technique. 
     In additional method embodiments the smoothing or fitting technique is a two dimensional local regression using weighted least squares. 
     In additional method embodiments the smoothing or fitting technique includes selecting coefficient estimates ĉ,â 1 ,â 2 ,{circumflex over (b)} 1 ,{circumflex over (b)} 2 ,{circumflex over (d)} to minimize the summation given by: 
     
       
         
           
             
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     for s−h≦s i ≦s+h and z−h≦z j ≦z+h, where W(α,β) is a 2 dimensional weighting function which is non-zero for |α|≦1.0 and |β|≦1.0, and zero elsewhere, h is the local neighborhood-size parameter, and he regression estimate {circumflex over (p)}(s I ,θ,z J ) at the point (s I ,θ,z J ) is equal to ĉ which produced a minimization at the point s=s I ,z=z J . 
     In additional method embodiments the deconvolving includes subtracting a term including the second order derivative of the sinogram from the sinogram. 
     In additional method embodiments the deconvolving includes subtracting a term including the second order moment of the system response of the imaging system from the sinogram. 
     In additional method embodiments the deconvolving includes calculating a solution or estimation of 
               q   ⁡     (     s   ,   θ   ,   z     )       ≅       p   ⁡     (     s   ,   θ   ,   z     )       -         m   2     2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   2                   
for q(s,θ,z), where m 2  is the second order moment of the transverse component of the system response function of the imaging system, and p(s,θ,z) is the sinogram.
 
     In additional method embodiments the deconvolving includes calculating a solution or estimation of 
               q   ⁡     (     s   ,   θ   ,   z     )       ≅       p   ⁡     (     s   ,   θ   ,   z     )       -         m   2     2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   2           +       (         m   2   2     4     -       m   4     24       )     ⁢         ⅆ   4     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   4                   
for q(s,θ,z), where m 2  is the second order moment of the transverse component of the system response function of the imaging system, m 4  is the fourth order moment of the transverse component of the system response function of the imaging system, and p(s,θ,z) is the sinogram.
 
     In additional method embodiments the deconvolving includes calculating a solution or estimation of 
               q   ⁡     (     s   ,   θ   ,   z     )       ≅       p   ⁡     (     s   ,   θ   ,   z     )       -         m     L   ⁢           ⁢   2       2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   2           -         m     H   ⁢           ⁢   2       2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     z   2                   
for q(s,θ,z), where m L2  is the second order moment of the transverse component of the system response function l(s) of the imaging system, m H2  is the second order moment of the axial component of the system response function h(z) of the imaging system, and p(s,θ,z) is the sinogram.
 
     In additional method embodiments the second order moment of the system response function of the imaging system is modeled as a Gaussian. 
     In additional method embodiments the Gaussian is selected to have a FWHM based upon the FWHM of the imaging system. 
     In additional method embodiments the deconvolving occurs after said processing. 
     In additional method embodiments the constructing an image utilizes a back projection technique. 
     In additional method embodiments the constructing an image utilizes a filtered back projection technique. 
     One embodiment is a system including an imaging system operable to provide a sinogram based upon a measurement of the imaging system, at least one computer readable memory configured to store first executable instructions to perform a smoothing or fitting operation on the sinogram, second executable instructions to deconvolve a system response function of the imaging system from the sinogram to provide a deconvolved sinogram and third executable instructions to construct an image based upon the deconvolved sinogram, and at least one processor operable to execute said first executable instructions, said second executable instruction, and said third executable instructions. 
     Another embodiment is a system including an imaging system operable to provide a sinogram based upon a measurement of the imaging system, at least one computer readable memory configured to store executable instructions to perform a smoothing or fitting operation on the sinogram and further executable instructions to construct an image based upon the deconvolved sinogram, and at least one processor operable to execute said executable instructions and said further executable instructions. 
     Another embodiment is a system including an imaging system operable to provide a sinogram based upon a measurement of the imaging system, at least one computer readable memory configured to store executable instructions to deconvolve a system response function of the imaging system from the sinogram and further executable instructions to construct an image based upon the deconvolved sinogram, and at least one processor operable to execute said executable instructions and said further executable instructions. 
     In additional system embodiments said imaging system is a positron emission tomography system. 
     In additional system embodiments the executable instructions to perform a smoothing or fitting operation on the sinogram include instructions to perform a non-parametric fitting technique. 
     In additional system embodiments the executable instructions to perform a smoothing or fitting operation on the sinogram include instructions to perform a two dimensional local regression using weighted least squares. 
     In additional system embodiments the executable instructions to perform a smoothing or fitting operation on the sinogram are executable to select coefficient estimates ĉ,â 1 ,â 2 ,{circumflex over (b)} 1 ,{circumflex over (b)} 2 ,{circumflex over (d)} to minimize the summation given by: 
     
       
         
           
             
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     for s−h≦s i ≦s+h and z−h≦z j +h, where W(α,β) is a 2 dimensional weighting function which is non-zero for |α|≦1.0 and |β|≦1.0, and zero elsewhere, h is the local neighborhood-size parameter, and he regression estimate {circumflex over (p)}(s I ,θ,z J ) at the point (s I ,θ,z J ) is equal to ĉ which produced a minimization at the point s=s I ,z=z J . 
     In additional system embodiments the executable instructions to deconvolve a system response function of the imaging system from the sinogram include instructions to subtract a term including the second order derivative of the sinogram from the sinogram. 
     In additional system embodiments the executable instructions to deconvolve a system response function of the imaging system from the sinogram include instructions to subtract a term including the second order moment of the system response of the imaging system from the sinogram. 
     In additional system embodiments the executable instructions to deconvolve a system response function of the imaging system from the sinogram are executable to estimate a true sinogram by solving 
               q   ⁡     (     s   ,   θ   ,   z     )       ≅       p   ⁡     (     s   ,   θ   ,   z     )       -         m   2     2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   2                   
for q(s,θ,z), where m 2  is the second order moment of the transverse component of the system response function of the imaging system, and p(s,θ,z) is the sinogram.
 
     In additional system embodiments the executable instructions to deconvolve a system response function of the imaging system from the sinogram are executable to estimate a true sinogram by solving 
               q   ⁡     (     s   ,   θ   ,   z     )       ≅       p   ⁡     (     s   ,   θ   ,   z     )       -         m   2     2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   2           +       (         m   2   2     4     -       m   4     24       )     ⁢         ⅆ   4     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   4                   
for q(s,θ,z), where m 2  is the second order moment of the transverse component of the system response function of the imaging system, m 4  is the fourth order moment of the transverse component of the system response function of the imaging system, and p(s,θ,z) is the sinogram.
 
     In additional system embodiments the executable instructions to deconvolve a system response function of the imaging system from the sinogram are executable to estimate a true sinogram by solving 
               q   ⁡     (     s   ,   θ   ,   z     )       ≅       p   ⁡     (     s   ,   θ   ,   z     )       -         m     L   ⁢           ⁢   2       2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     s   2           -         m     H   ⁢           ⁢   2       2     ⁢         ⅆ   2     ⁢     p   ⁡     (     s   ,   θ   ,   z     )           ⅆ     z   2                   
for q(s,θ,z), where m L2  is the second order moment of the transverse component of the system response function l(s) of the imaging system, m H2  is the second order moment of the axial component of the system response function h(z) of the imaging system, and p(s,θ,z) is the sinogram.
 
     In additional system embodiments the second order moment of the system response function of the imaging system is modeled as a Gaussian. 
     In additional system embodiments the second order moment of the system response function of the imaging system is modeled as a Laplacian. 
     In additional system embodiments said processor is configured to execute said second instructions after said first instructions. 
     In additional system embodiments said third instructions utilize a back projection technique. 
     In additional system embodiments said third instructions utilize a filtered back projection technique. 
     Exemplary embodiments have been described herein in relation to PET. Additional embodiments relate to other tomographic imaging systems and modalities, and/or planar imaging systems and modalities. Various embodiments of the present application include detectors, measurements, sinograms and other acts, components, elements, and features respecting each of these modalities and are not limited to a particular modality unless indicated to the contrary. 
     While exemplary embodiments of the invention have been illustrated and described in detail in the drawings and foregoing description, the same is to be considered as illustrative and not restrictive in character, it being understood that only the preferred embodiments have been shown and described and that all changes and modifications that come within the spirit of the invention are desired to be protected. It should be understood that while the use of words such as preferable, preferably, preferred, more preferred or exemplary utilized herein indicate that the feature so described may be more desirable, it nonetheless may not be necessary and embodiments lacking the same may be contemplated as within the scope of the invention, the scope being defined by the claims that follow. In reading the claims, it is intended that when words such as “a,” “an,” “at least one,” or “at least one portion” are used there is no intention to limit the claim to only one item unless specifically stated to the contrary in the claim. When the language “at least a portion” and/or “a portion” is used the item can include a portion and/or the entire item unless specifically stated to the contrary.