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Conference Paper

In this paper, the authors introduce a new iterative algorithm for
reconstructing positron emission tomography (PET) images. This algorithm
seeks to minimize an objective function of weighted least squares (WLS)
type. However, unlike conventional WLS methods, the weights do not need
to be estimated from the data, but are incorporated in the objective
function and relies heavily on the Poisson nature of the data. As a
result, the objective function is not quadratic, but is convex. The
iterative algorithm is obtained in a manner similar to an analytic
derivation of the ML-EM (maximum likelihood-expectation maximization)
algorithm which employs an alternating minimization procedure between
two convex sets of matrices. However, the distance metric is quite
different in the authors' case, and much more difficult to analyze. This
algorithm is similar in form to, and shares many properties in common
with, the ML-EM algorithm. The mathematical proof of the global
convergence of the algorithm remains an open problem

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... Anderson et al. demonstrated that the WLS algorithm converges faster than the ML-EM and produces images that have significantly better resolution and contrast. Anderson's algorithm was then improved by Mair et al. [13], they applied an alternating minimization procedure to obtain a new iterative algorithm for obtaining Poisson WLS estimators of the emission in densities. ...

In this paper, we present a novel image reconstruction method based on weighted least squares (WLS) objective function for positron emission tomography (PET). Unlike a usual WLS algorithm, the proposed method, which we call it SA-WLS, combines the SAGE algorithm with WLS algorithm. It minimized the WLS objective function using single coordinate descent (SCD) method in a sequence of small “hidden” data spaces (HDS). Although SA-WLS used a strategy to update parameter sequentially just like common SCD method, the use of these small HDS makes it converge much faster and produce the reconstructed images with greater contrast and detail than the usual WLS method. In order to decrease further the actual CPU time per iteration, the adaptive variable index sets were introduced to modify SA-WLS (MSA-WLS). Instead of optimizing each pixel, this MSA-WLS method sequentially optimizes many pixels located in an index set at one time. The index sets were automatically modified during each iteration step. MSA-WLS gathers the virtue of simultaneously and sequentially updating the parameters so that it achieves a good compromise between the convergence rate and the computational cost in PET reconstruction problem. Details of these algorithms were presented and the performances were evaluated by a simulated head phantom.

The related problems of minimizing the functionals
F( x )=αKL( y ,P x )+(1-α)KL( p
, x ) and
G( x )=αKL(P x , y )+(1-α)KL( x
, p ), respectively, over the set of vectors
x ⩾0 are considered. KL( a , b ) is the
cross-entropy (or Kullback-Leibler) distance between two nonnegative
vectors a and b . Iterative algorithms for minimizing
both functionals using the method of alternating projections are
derived. A simultaneous version of the multiplicative algebraic
reconstruction technique (MART) algorithm, called SMART, is introduced,
and its convergence is proved

We present unpenalized and penalized weighted least-squares (WLS) reconstruction methods for positron emission tomography (PET), where the weights are based on the covariance of a model error and depend on the unknown parameters. The penalty function for the latter method is chosen so that certain a priori information is incorporated. The algorithms used to minimize the WLS objective functions guarantee nonnegative estimates and, experimentally, they converged faster than the maximum likelihood expectation-maximization (ML-EM) algorithm and produced images that had significantly better resolution and contrast. Although simulations suggest that the proposed algorithms are globally convergent, a proof of convergence has not yet been found. Nevertheless, we are able to show that the unpenalized method produces estimates that decrease the objective function monotonically with increasing iterations.

Previous models for emission tomography (ET) do not distinguish the physics of ET from that of transmission tomography. We give a more accurate general mathematical model for ET where an unknown emission density lambda = lambda(x, y, z) generates, and is to be reconstructed from, the number of counts n(*)(d) in each of D detector units d. Within the model, we give an algorithm for determining an estimate lambdainsertion mark of lambda which maximizes the probability p(n(*)|lambda) of observing the actual detector count data n(*) over all possible densities lambda. Let independent Poisson variables n(b) with unknown means lambda(b), b = 1, ..., B represent the number of unobserved emissions in each of B boxes (pixels) partitioning an object containing an emitter. Suppose each emission in box b is detected in detector unit d with probability p(b, d), d = 1, ..., D with p(b,d) a one-step transition matrix, assumed known. We observe the total number n(*) = n(*)(d) of emissions in each detector unit d and want to estimate the unknown lambda = lambda(b), b = 1, ..., B. For each lambda, the observed data n(*) has probability or likelihood p(n(*)|lambda). The EM algorithm of mathematical statistics starts with an initial estimate lambda(0) and gives the following simple iterative procedure for obtaining a new estimate lambdainsertion mark(new), from an old estimate lambdainsertion mark(old), to obtain lambdainsertion mark(k), k = 1, 2, ..., lambdainsertion mark(new)(b)= lambdainsertion mark(old)(b)Sum of (n(*)p(b,d) from d=1 to D/Sum of lambdainsertion mark()old(b('))p(b('),d) from b(')=1 to B), b=1,...B.

Presents an image reconstruction method for positron-emission
tomography (PET) based on a penalized, weighted least-squares (PWLS)
objective. For PET measurements that are precorrected for accidental
coincidences, the author argues statistically that a least-squares
objective function is as appropriate, if not more so, than the popular
Poisson likelihood objective. The author proposes a simple data-based
method for determining the weights that accounts for attenuation and
detector efficiency. A nonnegative successive over-relaxation (+SOR)
algorithm converges rapidly to the global minimum of the PWLS objective.
Quantitative simulation results demonstrate that the bias/variance
tradeoff of the PWLS+SOR method is comparable to the maximum-likelihood
expectation-maximization (ML-EM) method (but with fewer iterations), and
is improved relative to the conventional filtered backprojection (FBP)
method. Qualitative results suggest that the streak artifacts common to
the FBP method are nearly eliminated by the PWLS+SOR method, and
indicate that the proposed method for weighting the measurements is a
significant factor in the improvement over FBP

We have studied the properties of two iterative reconstruction algorithms, namely, the maximum likelihood with expectation maximization (ML-EM) and the weighted least squares with conjugate gradient (WLS-CG) algorithms, for use in compensation for attenuation and detector response in cardiac SPECT imaging. A realistic phantom, derived from a patient X-ray CT study to simulate ’OITP SPECT data, was used in the investigation. Both algorithms are effective in compensating for the nonuniform attenuation distribution in the thorax region and the spatially variant detector response function of the imaging system. At low iteration numbers, the addition of detector response compensation provides improvement in both spatial resolution and image noise when compared with attenuation compensation alone. However, at higher iteration numbers, there is a more rapid increase in image noise when detector response compensation is included, and the increase is more dramatic for the WLS-CG algorithm. In general, the convergence rate of the WLS-CG algorithm is about ten times that of the ML-EM algorithm. Also, the WLS-CG exhibits a faster increase in image noise at large iteration numbers than the ML-EM algorithm. This study is valuable in the search for useful and practical reconstruction methods for improved clinical cardiac SPECT imaging.

Proximity function minimization and the convex feasibility problem for jointly convex Bregman distances

- C L Byrne
- Y Censor