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Non-linear Hypothesis Testing of Geometric Object Properties of Shapes Applied to Hippocampi

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Abstract

This paper presents a novel method to test mean differences of geometric object properties (GOPs). The method is designed for data whose representations include both Euclidean and non-Euclidean elements. It is based on advanced statistical analysis methods such as backward means on spheres. We develop a suitable permutation test to find global and simultaneously individual morphological differences between two populations based on the GOPs. To demonstrate the sensitivity of the method, an analysis exploring differences between hippocampi of first-episode schizophrenics and controls is presented. Each hippocampus is represented by a discrete skeletal representation (s-rep). We investigate important model properties using the statistics of populations. These properties are highlighted by the s-rep model that allows accurate capture of the object interior and boundary while, by design, being suitable for statistical analysis of populations of objects. By supporting non-Euclidean GOPs such as direction vectors, the proposed hypothesis test is novel in the study of morphological shape differences. Suitable difference measures are proposed for each GOP. Both global and simultaneous GOP analyses showed statistically significant differences between the first-episode schizophrenics and controls.

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Abbreviations

CDF:

Cumulative distribution function

CG:

Control group

CPNG:

Composite principal nested great spheres

CPNS:

Composite principal nested spheres

DiProPerm:

Direction projection permutation

DWD:

Distance-weighted discrimination

FDR:

False discovery rate

FWER:

Familywise error rate

GOP:

Geometric object property

KDE:

Kernel density estimate

MD:

Mean difference

MRI:

Magnetic resonance imaging

PCA:

Principal component analysis

PDM:

Point distribution model

PGA:

Principal geodesic analysis

PNG:

Principal nested great spheres

PNS:

Principal nested spheres

PP1:

Pre-processing step 1

PP2:

Pre-processing step 2

RFT:

Random field theory

ROC:

Receiver operating characteristic

S-rep:

Skeletal representation

SG:

Schizophrenia group

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Acknowledgments

The following researchers have also contributed to this work: Jared Vicory (UNC) gave advice on running Pablo and provided earlier fits of 62 hippocampi, Juan Carlos Prieto (CREATIS-INSA, France) provided the implementation of a crest interpolation term in Pablo and removed bugs from the program, Sungkyu Jung (University of Pittsburgh, USA) provided Fig. 2, program code and additional discussions about CPNS, Martin Styner (UNC) provided the hippocampus dataset and answered questions. The first author acknowledges support from the Norwegian Research Council through grant 176872/V30 in the eVita program and additional support from the Tromsø Telemedicine Laboratory and the Department of Electrical Engineering and Computer Science at the University of Stavanger, Norway.

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Schulz, J., Pizer, S.M., Marron, J.S. et al. Non-linear Hypothesis Testing of Geometric Object Properties of Shapes Applied to Hippocampi. J Math Imaging Vis 54, 15–34 (2016). https://doi.org/10.1007/s10851-015-0587-7

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