Brigitte Gundlich

Multiple Models — Fixed, Switching, Interacting

In dynamic models the dynamic and the observation equations are based on a known system model. The multiple model approach introduces uncertainties about the system model by a set of possible system models. In the multiple model approach for fixed models the true system does not change during the whole observation process, wheareas in the approach for switching models a jump from one model to another is allowed. In the later case the state estimation usually has to be approximated, e.g. by so-called interacting multiple models. The multiple model approach for fixed, switching and interacting models are presented and their application for GNSS ambiguity resolution is discussed, but open questions still remain.

From 2D PET to 3D PET: issues of data representation and image reconstruction.

Positron emission tomography (PET), intrinsically a 3D imaging technique, was for a long time exclusively operated in 2D mode, using septa to shield the detectors from photons emitted obliquely to the detector planes. However, the use of septa results in a considerable loss of sensitivity. From the late 1980s, significant efforts have been made to develop a methodology for the acquisition and reconstruction of 3D PET data. This paper focuses on the differences between data acquisition in 2D and 3D mode, especially in terms of data set sizes and representation. Although the real time data acquisition aspect in 3D has been mostly solved in modern PET scanner systems, there still remain questions on how to represent and how to make best use of the information contained in the acquired data sets. Data representation methods, such as list-mode and matrix-based methods, possibly with additional compression, will be discussed. Moving from 2D to 3D PET has major implications on the way these data are reconstructed to images. Two fundamentally different approaches exist, the analytical one and the iterative one. Both, at different expenses, can be extended to directly handle 3D data sets. Either way the computational burden increases heavily compared to 2D reconstruction. One possibility to benefit from the increased sensitivity in 3D PET while sticking to high-performance 2D reconstruction algorithms is to rebin 3D into 2D data sets. The value of data rebinning will be explored. An ever increasing computing power and the concept of distributed or parallel computing have made direct 3D reconstruction feasible. Following a short review of reconstruction methods and their extensions to 3D, we focus on numerical aspects that improve reconstruction performance, which is especially important in solving large equation systems in 3D iterative reconstruction. Finally exemplary results are shown to review the properties of the discussed algorithms. This paper concludes with an overview on future trends in data representation and reconstruction.

Compensation strategies for PET scanners with unconventional scanner geometry

The small animal PET scanner ClearPETtradeNeuro, developed at the Forschungszentrum Julich GmbH in cooperation with the Crystal Clear Collaboration (CERN), represents scanners with an unconventional geometry: due to axial and transaxial detector gaps ClearPettradeNeuro delivers inhomogeneous sinograms with missing data. When filtered backprojection (FBP) or Fourier rebinning (FORE) are applied, strong geometrical artifacts appear in the images. In this contribution we present a method that takes the geometrical sensitivity into account and converts the measured sinograms into homogeneous and complete data. By this means artifactfree images are achieved using FBP or FORE. Besides an advantageous measurement setup that reduces inhomogeneities and data gaps in the sinograms, a modification of the measured sinograms is necessary. This modification includes two steps: a geometrical normalization and corrections for missing data. To normalize the measured sinograms, computed sinograms are used that describe the geometrical sensitivity for a given measurement setup. Corrections for the data gaps are achieved by a provisional reconstruction followed by a forward projection of the image. The modified sinograms are homogeneous and complete. Modification of the sinograms and reconstruction with FBP or FORE lead to images without geometrical artifacts and still cost less computation time than using iterative reconstruction algorithms

Dynamic List-Mode Reconstruction of PET Data based on the ML-EM Algorithm

In dynamic reconstruction of positron emission tomography (PET) data a sequence of measured data sets is usually reconstructed independently from each other. Using this timeframe reconstruction, an appropriate trade-off between time resolution and noise has to be found. To overcome these drawbacks smoothing techniques and advanced dynamic reconstruction algorithms are more and more applied. Especially for the last, list-mode reconstruction is the predestinated approach, as the data are acquired in the highest possible spatial and temporal resolution. In this contribution we study dynamic reconstruction algorithms that base on the ML-EM algorithm for the small animal PET scanner ClearPETtradeNeuro. In a simulated example we generate list-mode data and compute time activity curves from the reconstructed images. We compare dynamic reconstruction methods, like time-frame reconstruction - with and without temporal smoothing - and reconstruction with B-splines as temporal basis functions.

Monte Carlo Integration for Quasi–linear Models

In this contribution we consider the inversion of quasi-linear models by means of Monte-Carlo methods. Quasi-linear models are a special class of non-linear models, which can be formally written in matrix-vector formulation but whose design matrix depends on a subset of the unknown parameters. A large class of geodetic problems can be recast as quasi-linear models. As there exist no general analytical solutions for the quasi-linear model, Monte Carlo optimization techniques in the context of a Bayesian approach are investigated here. In order to accelerate the Monte Carlo method we utilize the analytical solution of the linear model under the condition that the unknown parameters in the design matrix are considered as constant. Thereby the sampling dimension in the Monte Carlo approach can be reduced. The estimators for expectation and covariance of the parameters that we derive turn out as weighted means of the individual sample least-squares solutions. We develop an efficient set of algorithms for the solution of quasi-linear models using Monte Carlo techniques and demonstrate the efficiency of the method in a numerical example which is taken from satellite geodesy and gravity field recovery. Two groups of unknown parameters are relevant in this example: the spherical harmonic coefficients of a gravity field model, and the state vectors of the satellite(s) which affect the observation model through the design matrix.

Scatter Analysis of the ClearPET™ Neuro Using Monte Carlo Simulations

Scatter reduces the image quality in Positron Emission Tomography (PET). In this paper we discuss a) methods to estimate the scatter fraction in the raw data set as well as b) analysis of the different scatter components (e.g. phantom scatter, gantry scatter) arising from activity outside the field of view (OFOV). The PET detection system does not allow a discrimination of scattered events, thus Monte Carlo Simulations were used. The accuracy of the different scatter estimation methods was analyzed. Simulations with OFOV activity showed that small animal PET systems are indeed sensitive to random and scattered events from OFOV.

Influence of Corrections During Image Reconstruction on the Spatial Resolution of ClearPET Neuro

The small-animal PET scanner ClearPET Neuro developed at the Research Centre in Julich is based on an unconventional scanner geometry. It represents axial and transaxial gaps that lead to sinograms with missing data. Images reconstructed from uncorrected data include artefacts and a high variation of spatial resolution between different slices. Methods to compensate these artefacts are applied by taking the geometrical sensitivity into account. In this work the effects of compensation strategies with regard to the slice by slice variation of spatial resolution are examined.

Dynamic Reconstruction for the ClearPET™ Neuro Using Temporal B-Splines

Dynamic reconstruction methods are studied for the small animal PET (positron emission tomography) scanner ClearPET™ Neuro. In dynamic reconstruction the data are usually sorted into timeframes and reconstructed independently of each other. Using this timeframe approach, an appropriate trade-off between time resolution and noise has to be found. A more advanced method is dynamic reconstruction with temporal basis functions, where voxel values are time dependent modeled as weighted sum of basis functions. In a simulated example list-mode data are generated for the ClearPET™ Neuro and reconstructed with timeframe reconstruction and with dynamic reconstruction using B-Splines as temporal basis functions. Time activity curves are computed for various reconstructions with different timeframes and B-Splines. The example demonstrates the potential of dynamic reconstruction with temporal B-Splines.