Mojette Transform on Densest Lattices in 2D and 3D

  • Vincent RicordelEmail author
  • Nicolas NormandEmail author
  • Jeanpierre GuédonEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10502)


The Mojette Transform (MT) is an exact discrete form of the Radon transform. It has been originally defined on the lattice \(Z^n\) (where n is the dimension). We propose to study this transform when using the densest lattices for the dimensions 2 and 3, namely the lattice \(A^2\) and the face-centered cubic lattice \(A^3\). In order to compare the legacy MT using \(Z^n\), versus the new MT using \(A^n\), we define a fair comparison methodology between the two MT schemes. In particular we detail how to generate the projection angles by exploiting the lattice symmetries and by reordering the Haros-Farey series. Statistic criteria have been also defined to analyse the information distribution on the projections. The experimental results study shows the specific nature of the information distribution on the MT projections due to the high compacity of the \(A^n\) lattices.


Mojette Transform Discrete tomography Lattices Densest lattices Haros-Farey series 


  1. 1.
    Guedon, J.-P.: The Mojette transform: theory and applications. Wiley-ISTE (2009). ISBN 978-1-84821-080-6.
  2. 2.
    Katz, M.B.: Questions of Uniqueness and Resolution in Reconstruction from Projections. Lecture Notes in Biomathematics, vol. 26. Springer, Heidelberg (1978). doi: 10.1007/978-3-642-45507-0 zbMATHGoogle Scholar
  3. 3.
    Guedon, J.-P., Normand, N., Lecoq, S.: Transformation Mojette en 3D: Mise en oeuvre et application en synthéses d image. GRETSI, Groupe d’Etudes du Traitement du Signal et des Images, September 1999.
  4. 4.
    Normand, N., Servières, M., Guédon, J.P.: How to obtain a lattice basis from a discrete projected space. In: Andres, E., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 153–160. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-31965-8_15 CrossRefGoogle Scholar
  5. 5.
    Servieres, M.: Reconstruction Tomographique Mojette. Theses, Université de Nantes; Ecole Centrale de Nantes (ECN), December 2005.
  6. 6.
    Normand, N., Guedon, J.-P.: La transformee mojette: Une representation redondante pour l’image. In: Comptes Rendus de l’Academie des Sciences - Series I - Mathematics, vol. 326, no. 1, pp. 123–126 (1998)., ISSN 0764–4442
  7. 7.
    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. Grundlehren der mathematischen Wissenschaften, vol. 290. Springer, New York (1993). doi: 10.1007/978-1-4757-6568-7 zbMATHGoogle Scholar
  8. 8.
    Rashid, M.A., Iqbal, S., Khatib, F., Hoque, M.T., Sattar, A.: Guided macro-mutation in a graded energy based genetic algorithm for protein structure prediction. Comput. Biol. Chem. 61, 162–177 (2016). doi: 10.1016/j.compbiolchem.2016.01.008 CrossRefGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université de Nantes, LS2N UMR CNRS 6004 Polytech NantesNantes Cedex 3France

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