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Interpolation-based reconstruction methods for tomographic imaging in 3D positron emission tomography

Li Y., Kummert A., Boschen F., Herzog H.

International Journal of Applied Mathematics and Computer Science

2008

Vol. 18, no 1

63-73

Positron Emission Tomography (PET) is considered a key diagnostic tool in neuroscience, by means of which valuable insight into the metabolism function in vivo may be gained. Due to the underlying physical nature of PET, 3D imaging techniques in terms of a 3D measuring mode are intrinsically demanded to assure satisfying resolutions of the reconstructed images. However, incorporating additional cross-plane measurements, which are specific for the 3D measuring mode, usually imposes an excessive amount of projection data and significantly complicates the reconstruction procedure. For this reason, interpolation-based reconstruction methods deserve a thorough investigation, whose crucial parts are the interpolating processes in the 3D frequency domain. The benefit of such approaches is apparently short reconstruction duration, which can, however, only be achieved at the expense of accepting the inaccuracies associated with the interpolating process. In the present paper, two distinct approaches to the realization of the interpolating procedure are proposed and analyzed. The first one refers to a direct approach based on linear averaging (inverse distance weighting), and the second one refers to an indirect approach based on two-dimensional convolution (gridding method). In particular, attention is paid to two aspects of the gridding method. The first aspect is the choice of the two-dimensional convolution function applied, and the second one is the correct discretization of the underlying continuous convolution. In this respect, the geometrical structure named the Voronoi diagram and its computational construction are considered. At the end, results of performed simulation studies are presented and discussed.

Cyfrowy model wysokości powierzchni podczwartorzędowej na Wyżynie Woźnicko-Wieluńskiej

Szubert M.

Archiwum Fotogrametrii, Kartografii i Teledetekcji

2003

Vol. 13a

233--242

W publikacji przedstawiono doświadczenia w zastosowaniu metod komputerowej interpolacji do opracowania cyfrowego modelu wysokości powierzchni podczwartorzędowej. Staranny dobór metody interpolacji oraz jej parametrów ma w tym przypadku szczególne znaczenie, gdyż przedmiot badań nie jest dostępny do bezpośredniej obserwacji. Metoda interpolacji musi być tak dobrana, aby DEM był jak najbardziej zbliżony do rzeczywistości. Powierzchnię podczwartorzędową zrekonstruowano w oparciu o 6553 punkty pomiarowe (dane z odwiertów oraz mapy geologicznej) na obszarze ok. 800 km2. Zastosowano cztery metody interpolacji: odwrotnych odległości, najmniejszej krzywizny, krigingu oraz radialnych funkcji bazowych w programie „Surfer”. Najlepszą okazała się metoda krigingu standardowego z odpowiednio dobranym modelem semiwariogramu. O wyborze tej metody zdecydowały: poprawność kartograficzna rysunku poziomicowego oraz najmniejsze odchylenia wyinterpolowanej powierzchni wzdłuż kilku profili wykonanych na podstawie odwiertów.

The paper presents the results of experiments on the use of computer interpolation to the construction of digital elevation model (DEM) of a base Quaternary surface. A careful selection of the interpolation method and parameters is here especially important as the studied surface is not unavailable for direct observation. The method of interpolation must be chosen so that the DEM is as close to reality as possible. The shape of the base Quaternary surface was interpolated from 6,553 data points (data from boreholes and a geological map) over an area of ca. 800 km2. Four gridding methods have been used in the Surfer program: Inverse Distance to a Power, Kriging, Minimum Curvature and Radial Basis Function. The models of base Quaternary relief thus obtained were then verified by comparison of the obtained contour patterns with several tens of cross-sections based on borehole data. The best results were obtained using standard kriging with adequately chosen variogram model (fig. 3). Its parameters were: linear (Slope = 2,18E-005; Aniso = 1,0); nugget effect (Error = 0,686, Micro = 0). This method was chosen because of the cartographic correctness of the contour pattern and the smallest departures of the interpolated surface along several cross-sections constructed basing on the borehole data. The drawbacks of the contour pattern obtained by the “Inverse Distance to a Power” method (with the power parameter = 2) was the bull’s-eye effect, that is a distinct concentric arrangement of contours around data points (fig. 2). The effect disappeared with reduction in the power parameter and increased with its increase. A similar pattern of contours was obtained using the Minimum Curvature method (fig. 4), at the minimum value of error (maximum residual = 0,2). The effect disappeared with reduction of the value. The Radial Basis method resulted in the contour pattern similar to that obtained with the kriging method, but with other cartographic errors which could not be eliminated by altering the algorithm. The resultant DEM of the base Quaternary relief was used for visualisation of the fossil surface and to the study of the pre-Quaternary relief.