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) yields negative values for material concentrations if - due to noise - a region of interest's (ROI) CT values falls outside the boundary of CT values describing all possible pure or mixed basis materials. This may be addressed geometrically by projecting these points (the CT values of the ROI in question) onto the closest point on this boundary (the space of allowable CT values). 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 voxels to be included within. Therefore, projection onto the boundary becomes an important option. But, projection in higher than three 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 energy threshold combinations. Comparisons 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
Article number015007
JournalBiomedical Physics and Engineering Express
Issue number1
StatePublished - Jan 2018


  • Clifford
  • algebra
  • counting
  • decomposition
  • photon


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