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Reconstruction Algorithms and Scanning Geometries in Tomographic Imaging

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Imaging in Nuclear Medicine

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Abstract

In general tomographic imaging consists of two steps: the acquisition of data and the reconstruction. Thereby the following triple problem has to be solved: choose an efficient reconstruction algorithm, identify the optimal sampling conditions imposed on measured data as required by this reconstruction algorithm, and find an efficient way of collecting such data in the practice.

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Notes

  1. 1.

    That is, \( \hat{\varphi}(\omega )\approx 0 \) if \( \omega \in [-w,w] \) .

  2. 2.

    Polynomial \( P \) of degree \( n \) is orthogonal on \( \mathrm{ D} \) with respect to the weight function \( \mu \) if \( \int_{\mathrm{ D}} {P(x,y)Q(x,y)\mu (x,y)\mathrm{ d}x\mathrm{ d}y} =0 \) for any polynomial \( Q \) of degree less than \( n \).

  3. 3.

    That is, \( ||{A_n}f-f|{|_2}=\mathop{\min}\limits_{{g\in \boldsymbol{ \Pi}_n^2}}||g-f|{|_2} \) .

  4. 4.

    In this case, the output of acquisition system is the convolution with the impulse response of the ideal filter.

  5. 5.

    Clearly, if \( z\ne 0 \), then for different \( \alpha \), there are different planes that contain this point, so that all planes of projections that contribute to the reconstruction at given \( {\rm\bf r} \) may be visualized as forming a bundle.

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Authors and Affiliations

  1. Research Unit Medical Radiation Physics and Diagnostics, Helmholtz Zentrum München - German Research Center for Environmental Health, Neuherberg, Germany

    Oleg Tischenko & Christoph Hoeschen

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  1. Oleg Tischenko
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  2. Christoph Hoeschen
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Correspondence to Oleg Tischenko .

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Editors and Affiliations

  1. , Department of Radiation Protection and H, BfS - Federal Office for Radiation Prote, Ingolstädter Landstr. 1, Oberschleißheim, 85764, Germany

    Augusto Giussani

  2. , Research Unit Medical Radiation Physics, Helmholtz Zentrum München - German Resea, Ingolstädter Landstr. 1, Neuherberg, 85764, Germany

    Christoph Hoeschen

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Tischenko, O., Hoeschen, C. (2013). Reconstruction Algorithms and Scanning Geometries in Tomographic Imaging. In: Giussani, A., Hoeschen, C. (eds) Imaging in Nuclear Medicine. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31415-5_6

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  • DOI: https://doi.org/10.1007/978-3-642-31415-5_6

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