Purpose: Multi-energy CT (e.g., dual energy or photon counting) facilitates the identification of certain compounds via data decomposition. However, the standard approach to decomposition (i.e., solving a system of linear equations) fails if - due to noise - a pixel's vector of HU values falls outside the boundary of values describing possible pure or mixed basis materials. Typically, this is addressed by either throwing away those pixels or projecting them onto the closest point on this boundary. However, when acquiring four (or more) energy volumes, the space bounded by three (or more) materials that may be found in the human body (either naturally or through injection) can be quite small. Noise may significantly limit the number of those pixels to be included within. Therefore, projection onto the boundary becomes an important option. But, projection in higher than 3 dimensional space is not possible with standard vector algebra: the cross-product is not defined. Methods: We describe a technique which employs Clifford Algebra to perform projection in an arbitrary number of dimensions. Clifford Algebra describes a manipulation of vectors that incorporates the concepts of addition, subtraction, multiplication, and division. Thereby, vectors may be operated on like scalars forming a true algebra. Results: We tested our approach on a phantom containing inserts of calcium, gadolinium, iodine, gold nanoparticles and mixtures of pairs thereof. Images were acquired on a prototype photon counting CT scanner under a range of threshold combinations. Comparison of the accuracy of different threshold combinations versus ground truth are presented. Conclusions: Material decomposition is possible with three or more materials and four or more energy thresholds using Clifford Algebra projection to mitigate noise.

Original languageEnglish
Title of host publicationMedical Imaging 2017
Subtitle of host publicationPhysics of Medical Imaging
EditorsTaly Gilat Schmidt, Joseph Y. Lo, Thomas G. Flohr
ISBN (Electronic)9781510607095
StatePublished - 2017
EventMedical Imaging 2017: Physics of Medical Imaging - Orlando, United States
Duration: 13 Feb 201716 Feb 2017

Publication series

NameProgress in Biomedical Optics and Imaging - Proceedings of SPIE
ISSN (Print)1605-7422


ConferenceMedical Imaging 2017: Physics of Medical Imaging
Country/TerritoryUnited States


  • Algebra
  • Clifford
  • Counting
  • Decomposition
  • Photon


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