Skip to main page content
U.S. flag

An official website of the United States government

Dot gov

The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

Https

The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
. 2009 Mar 12:7263:813799.
doi: 10.1117/12.813799.

Singular Value Decomposition of Pinhole SPECT Systems

Robin Palit  1 , Matthew A Kupinski, Harrison H Barrett, Eric W Clarkson, John N Aarsvold, Lana Volokh, Yariv Grobshtein
Affiliations

Singular Value Decomposition of Pinhole SPECT Systems

Robin Palit et al. Proc SPIE Int Soc Opt Eng. .

Abstract

A single photon emission computed tomography (SPECT) imaging system can be modeled by a linear operator H that maps from object space to detector pixels in image space. The singular vectors and singular-value spectra of H provide useful tools for assessing system performance. The number of voxels used to discretize object space and the number of collection angles and pixels used to measure image space make the matrix dimensions H large. As a result, H must be stored sparsely which renders several conventional singular value decomposition (SVD) methods impractical. We used an iterative power methods SVD algorithm (Lanczos) designed to operate on very large sparsely stored matrices to calculate the singular vectors and singular-value spectra for two small animal pinhole SPECT imaging systems: FastSPECT II and M(3)R. The FastSPECT II system consisted of two rings of eight scintillation cameras each. The resulting dimensions of H were 68921 voxels by 97344 detector pixels. The M(3)R system is a four camera system that was reconfigured to measure image space using a single scintillation camera. The resulting dimensions of H were 50864 voxels by 6241 detector pixels. In this paper we present results of the SVD of each system and discuss calculation of the measurement and null space for each system.

PubMed Disclaimer

Figures

Figure 1
Figure 1
(a) The largest 5000 singular values for the FastSPECT II system. Our current computing resources were unable to allocate enough memory to compute more than this number of singular values for the FastSPECT II system. However, this number of elements may be sufficient for system assessment tasks since the spectrum exhibits a significant drop in magnitude within the 5000 computed singular values. (b) The largest 6082 singular values for the M3R system. The Lanczos method is subject to numerical instability for small singular values. The 6082 computed values represents the number of eigenvalues that could be computed for this system before arriving at this stability threshold.
Figure 2
Figure 2
A 2D slice through object space singular vectors 1–16 for the FastSPECT II system.
Figure 3
Figure 3
2D slice through object space singular vectors 17–32 for the FastSPECT II system.
Figure 4
Figure 4
Image space singular vector 1 for the FastSPECT II system.
Figure 5
Figure 5
2D slice through object space singular vectors 1–16 for the M3R system.
Figure 6
Figure 6
2D slice through object space singular vectors 17–32 for the M3R system.
Figure 7
Figure 7
Image space singular vectors 1–16 for the M3R system.
See this image and copyright information in PMC

References

    1. Cullum J, Willoughby R. Theory. I. Birkhauser; Boston: 1985. Lanczos Algorithms for Large Symmetric Eigenvalue Computations.
    1. Sehmi N. Large Order Structural Eigenanalysis Techniques: Algorithms for Finite Element Systems. Ellis Horwood; 1989.
    1. Berry M, Do T, OBrien G, Krishna V, Varadhan S. SVDPACKC: Version 1.0: User’s Guide. University of Tennessee, Computer Science Dept; 1993.
    1. Barrett H, Myers K. Foundations of Image Science. John Wiley and Sons; 2004.
    1. Chen Y. PhD thesis. University of Arizona; 2006. System calibration and image reconstruction for a new small-animal SPECT system.

LinkOut - more resources