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Noise Propagation from Attenuation Correction into PET Reconstructions Ing-Tsung Hsiao and Gene Gindi Departments of Radiology and Electrical

Abstract It is useful to have the ability to analyze the propagation of errors from transmission data into the resulting attenuationcorrected emission reconstruction PET. We develop theoretical expressions for the mean and covariance of the emission reconstruction when the only noise source is that in the transmission data. There are several ways to impose an attenuation correction onto the PET reconstructions, but here we theoretically analyze two cases: (1) A linear (on log data) transmission reconstruction is reprojected to get an ACF (Attenuation Correction Factor) estimate which is then used in a linear emission estimate. (2) PET reconstruction is a nonlinear estimate based on maximizing a regularized likelihood objective, and attenuation is modeled directly in the objective function. A validation study is presented for the mean and covariance expressions for case (1).

I. I NTRODUCTION Reconstruction of transmission tomographic (TT) data is needed for attenuation correction (AC) in emission tomographic (ET) reconstruction in PET and SPECT. There are a variety of ways to perform AC in PET, but a central feature of all the correction schemes is that the high noise in the TT projection data is transferred to the PET reconstruction, thus, potentially lowering image quality. Hence there is an image quality tradeoff between the benefits of AC and the attendant “extra” noise due to AC itself. Given the magnitude of errors in AC, an analysis of the propagation of noise from TT reconstruction into ET reconstruction is very important and useful, for example, in designing better TT reconstruction methods [1, 2] or optimizing the scan time between TT and ET in post-injection PET [3]. An AC-corrected PET scan has a variety of sources of noise that gets propagated into the reconstruction. In this work, we consider only ET and TT photon noise in our theoretical expressions, and in some instances, assess the effects of TT noise only, implicitly assuming a limiting case of noiseless ET projection data.

A. Notation We will use vector component notation, with lowercase bold quantities indicating a vector. Given vectors
and  , then  is a vector whose components are     , the product
 has component 
   ,  has component        ,
 is  an inner product with  indicating transpose, and 
a matrixvector product. Notation diag !"
$# means a diagonal matrix with vector
denoting diagonal elements, i.e.  diag !"
$#" % &' . It turns out that
( diag !"
$#)* diag !#+
, and that the following relation holds for vector
and matrices , , - : . / . -1032 34 65- diag !"102  #7,98 (1)


Computer Engineering, SUNY Stony Brook / We use the notation : 4 to designate an operator, and : matrix, with :; its transpose.

a

B. Imaging Model Let < and = be the emission and transmission data, > ?A@ , and let here as vectors /OofN3DFPsize /CErepresented DFG G B  (@IHE8J8%HLK 4 and M  (@1HQ8%8JHRK 4 be emission and transmission object vectors of KSB ?T@ , respectively. We denote reconstruction (estimates) of and M with KU?*@ B vectors V and M V , respectively. Both < and = are independent Poisson measurements with corresponding means < W and W= as: W ZY A?i> and - is the forward >A?9K ET projection matrix (now without attenuation effects). We assume that ET and TT have the same resolution here.

C. Taxonomy of Reconstruction Methods Table 1 Taxonomy of AC methods ACF

pj l

p j nl

ACF reproject

direct k j l

ACF reproject

k j nl

[3]

[4],Sec.II [3][5],Sec.II

[6]

Sec.III

Sec.III

Model km j l in system matrix

Model ko j nel in system matrix

Sec.III

Sec.III

In table 1 we illustrate a classification of AC methods. The two rows of the table correspond to linear (q ) and non-linear (Krq ) ET estimates. Column 1 accounts for direct methods wherein the ACF estimates are computed as the ratio (perhaps smoothed) of the blank and transmission scans. Columns 2 and 3 accounts for those cases in which the ACF estimates are derived by reprojecting a TT reconstruction. Column 2 is for the somewhat special case of a linear TT estimate operating on the logarithm of the s ratio. Column 3 includes any other (e.g. penalized likelihood) method of TT reconstruction. Columns 4 and 5 account for cases in which the TT reconstruction is directly modeled in the system matrix Y of the ET scans, with col. 4 for linear , and col.5 for nonlinear TT reconstructions.

II. N OISE P ROPAGATION IN L INEAR ET R ECONSTRUCTION WITH ACF R EPROJECTION

Eqs.(8) and (9) are the main results for this section. 

A. Theory We now consider the case corresponding to row 1, cols 2, 3 of Table 1. We examine the case of TT photon noise only. Then B  the ET data are noiseless and are given by < E   032 M  02 M < , where vectors < and < denote the noiseless attenuated and unattenuated ET projections, respectively. Consider the case where AC is accomplished by multiplication of the ET
sinogram < with reprojected ACF’s, i.e.  2 M < , which is thus a random quantity by virtue of M V . We then write the linear ET reconstruction as
 E B k V ! V #o (4) !   2 M