Abstract The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong–Reeb line. Two of the new models, contrary to the Harthong–Reeb line, provide accurate integer “views” on real numbers at a sequence of growing scales $B^n$ ( $B\ge2$ ).

A cryo Electron Microscopy dataset is composed of tomographic projections of an object (e.g. a macromolecule). The projection orientation information is unknown. The scope of this paper is the tomographic reconstruction of the observed object in the ab-initio case where the volume has to be estimated only from a raw projection dataset. A new approach based on a parametric model of the volume is presented. The description of the model and the search of the parameters are detailed. The accuracy and robustness of the proposed reconstruction method is shown on synthetic and real databases.

This paper deals with the estimation of angular difference between two tomographic projections belonging to a set of projections taken at unknown directions. The proposed method extends our former work from 2D to 3D. The method is potential for many applications such as projection refinement or projection classification, which are important in the process of tomographic reconstruction. Unlike to common line based angular estimation, the proposed method does not need reference projections. Our method relies on the selection of projection neighbors with local adaptive thresholds, the calculus of the angular difference for neighboring projections by using properties of moments. The accuracy and the robustness of our method are shown on a test database including 50 3D gray-level images at different resolutions and with different levels of noise.

This paper introduces a new measure for estimating the angular difference between two tomographic projections belonging to a set of projections taken at unknown directions. The measure is potential for many applications such as projection refinement or projection classification, which are important in the process of tomographic reconstruction. Our measure relies on the construction of a neighborhood graph for projection moments, the calculus of the angular difference for neighboring projections and the computation of geodesics on this graph. The accuracy and the robustness of our measure is shown on a test database including 50 2D gray-level images at different resolutions and with different levels of noise.