page_title: Computational anatomy
section_title: Riemannian exponential (geodesic positioning) and Riemannian logarithm (geodesic coordinates)
hierarchical_section_title: Computational anatomy / Riemannian exponential (geodesic positioning) and Riemannian logarithm (geodesic coordinates)
context_page_description: Computational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical methods for modelling and simulation of biological structures.
  The field is broadly defined and includes foundations in anatomy, applied mathematics and pure mathematics, machine learning, computational mechanics, computational science, biological imaging, neuroscience, physics, probability, and statistics; it also has strong connections with fluid mechanics and geometric mechanics. Additionally, it complements newer, interdisciplinary fields like bioinformatics and neuroinformatics in the sense that its interpretation uses metadata derived from the original sensor imaging modalities. It focuses on the anatomical structures being imaged, rather than the medical imaging devices. It is similar in spirit to the history of Computational linguistics, a discipline that focuses on the linguistic structures rather than the sensor acting as the transmission and communication media.
context_section_description: In medical imaging and computational anatomy, positioning and coordinatizing shapes are fundamental operations; the system for positioning anatomical coordinates and shapes built on the metric and the Euler-Lagrange equation a geodesic positioning system as first explicated in Miller Trouve and Younes.
  Solving the geodesic from the initial condition  is termed the Riemannian-exponential, a mapping  at identity to the group.
  The Riemannian exponential satisfies  for initial condition , vector field dynamics ,
  for classical equation diffeomorphic shape momentum , , then
  for generalized equation, then ,,
  Computing the flow   onto coordinates Riemannian logarithm, mapping  at identity from  to vector field ;
  Extended to the entire group they become
   ;    .
  These are inverses of each other for unique solutions of Logarithm; the first is called geodesic positioning, the latter geodesic coordinates (see Exponential map, Riemannian geometry for the finite dimensional version).The geodesic metric is a local flattening of the Riemannian coordinate system (see figure).
caption: Showing metric local flattening of coordinatized manifolds of shapes and forms. The local metric is given by the norm of the vector field 
    
      
        
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      {\displaystyle \|v_{0}\|_{V}}
    
   of the geodesic mapping 
    
      
        
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      {\displaystyle Exp_{id}(v_{0})\cdot m}
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