A Log-Euclidean Framework for Statistics on Diffeomorphisms

Abstract

In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is well-defined for diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. We also present here two efficient algorithms to compute logarithms of diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Finally, we apply these tools to compute the mean of a set of diffeomorphisms, in the context of a registration experiment between an atlas an a database of 9 T1 MR images of the human brain.

Keywords

LogarithmEuclidean geometryGeneralizationInvertible matrixComputationComputer scienceVector fieldMathematicsAlgorithmPure mathematicsMathematical analysisGeometry

MeSH Terms

AlgorithmsArtificial IntelligenceBrainComputer SimulationData InterpretationStatisticalHumansImage EnhancementImage InterpretationComputer-AssistedImagingThree-DimensionalMagnetic Resonance ImagingModelsNeurologicalModelsStatisticalPattern RecognitionAutomatedReproducibility of ResultsSensitivity and SpecificitySubtraction Technique

Affiliated Institutions

Institut national de recherche en sciences et technologies du numérique FR

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Publication Info

Year: 2006
Type: article
Volume: 9
Issue: Pt 1
Pages: 924-931
Citations: 420
Access: Closed

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APA Style

                            
                                
                                    Vincent Arsigny, 
                                
                                    Olivier Commowick, 
                                
                                    Xavier Pennec
                                
                                et al.
                            
                            (2006). 
                            A Log-Euclidean Framework for Statistics on Diffeomorphisms. 
                            Lecture notes in computer science
                            , 9
                            (Pt 1)
                            , 924-931.
                            https://doi.org/10.1007/11866565_113
                        

Identifiers

DOI: 10.1007/11866565_113
PMID: 17354979

Data Quality

Data completeness: 86%