A New Combined Surface and Volume Registration
Natasha Lepore´1,2 ⋆ , Anand A. Joshi1⋆, Richard M. Leahy3, Caroline Brun1 Yi-Yu Chou1,
Xavier Pennec4, Agatha D. Lee1, Marina Barysheva1, Greig I. de Zubicaray5, Margaret J.
Wright6, Katie L. McMahon5, Arthur W. Toga1, Paul M. Thompson1
1 Laboratory of Neuro Imaging, UCLA School of Medicine, Los Angeles, CA, USA
2 Department of Radiology, University of Southern California, Children’s Hospital Los
Angeles, CA, USA
3 Department of Engineering, University of Southern California, Los Angeles, CA, USA
4 Asclepios Research Project, INRIA Sophia, Sophia-Antipolis Cedex, France
5 Centre for Magnetic Resonance, University of Queensland, Brisbane, Australia
6 Genetic Epidemiology Lab, Queensland Institute of Medical Research, Brisbane, Australia
ABSTRACT
3D registration of brain MRI data is vital for many medical imaging applications. However, purely intensity-
based approaches for inter-subject matching of brain structure are generally inaccurate in cortical regions, due
to the highly complex network of sulci and gyri, which vary widely across subjects. Here we combine a surface-
based cortical registration with a 3D fluid one for the first time, enabling precise matching of cortical folds,
but allowing large deformations in the enclosed brain volume, which guarantee diffeomorphisms. This greatly
improves the matching of anatomy in cortical areas. The cortices are segmented and registered with the software
Freesurfer. The deformation field is initially extended to the full 3D brain volume using a 3D harmonic mapping
that preserves the matching between cortical surfaces. Finally, these deformation fields are used to initialize a 3D
Riemannian fluid registration algorithm, that improves the alignment of subcortical brain regions. We validate
this method on an MRI dataset from 92 healthy adult twins. Results are compared to those based on volumetric
registration without surface constraints; the resulting mean templates resolve consistent anatomical features
both subcortically and at the cortex, suggesting that the approach is well-suited for cross-subject integration of
functional and anatomic data.
Keywords: brain, image analysis, Magnetic Resonance Imaging, image registration
1. INTRODUCTION
Registration of brain MRI scans is a key step in many medical imaging studies, for multimodality integration,
computational anatomy studies, and longitudinal tracking of disease. In non-linear image registration, a template
image T is transformed into a study image S, by aligning the geometrical features of the images. The anatomical
correspondences between images can be measured and optimized in two ways, either based on intensity-derived
information at each voxel such as the squared difference in intensities or mutual information between the images,
or by using anatomical landmarks, which may include 3D parametric curves and surfaces extracted from the
images. These feature correspondences can be used to drive the registration. The image transformation is
also typically constrained via a regularizer that is chosen to ensure that the transformation is spatially smooth,
invertible, and so that it satisfies other desirable properties such as inverse-consistency or transitivity.15
Registration algorithms for brain MR images may also be divided into two major categories based on whether
they attempt to register the entire 3D brain volume, or are restricted to the cortical surfaces. Volume-based
algorithms perform quite well for subcortical structures, but have difficulty in matching cortical surfaces, due
to the high complexity and variability in cortical patterns. Cortical registration is therefore used when the
subsequent analysis focuses more specifically on the cortex, e.g. for population-based studies of cortical thickness.
In this case, the registration is restricted to the cortical surface, and does not take into account the rest of the
brain volume.
⋆: equal contribution

Cortical registrations are usually performed by first transforming the cortices to more regular surfaces such
as the plane or the sphere, though some recent algorithms have been developed that directly register the cortices
using 3D harmonic maps.25 Typically the surface-based registrations either use sulci or gyri as anchors to
drive the transformation,30 or alternatively, make use of quantities defined over the whole brain such as the
curvature7 or the local conformal factor.31 To compute the flows in surface coordinates, specialized approaches
using covariant PDEs, currents, or holomorphic 1-forms have been developed.9
There are many ways to perform the volume-based registrations. In this work, we focus on regularizers
that assign continuum mechanical laws to the deforming image medium. The most popular model for these
regularizers is the elastic one. Here a similarity function is selected, and a force is derived from it and applied
to the system, while the elastic forces attempt to restore the original shape of the medium. Elastic registration
is a good choice for studies where small deformations are expected, such as longitudinal studies where the same
subject is imaged at two time points, and small changes are expected. However, in cases where large differences
exist between the two images, the deformation can induce tearing or shearing of the elastic medium. In this case,
other methods are needed. Among these, the fluid registration method was first introduced in.6 In this case, the
image is treated as a viscous fluid that obeys a linearized version of the Navier-Stokes equation. The velocity ~v
of the fluid is the time derivative of the deformation field ~v(~r, t) = d~u(~r,t)dt . The driving force for the flow is again
generated by a similarity function. The fluid paradigm allows large deformations without shearing or tearing of
the image.
A new elastic regularizer was recently introduced in.21 In this case, the regularization is based on the
deformation tensors Σ = (∇~(u) + I)T (∇~u + I), with I the identity matrix. The Σ’s characterize the voxelwise
changes in shape and volume caused by the registration. As the Σ’s are positive-definite symmetric matrices,
the computations are performed in the Log-Euclidean framework,1 which allows analytical computations on the
manifold of Σ’s. A fluid version of this method was developped in,5 where the rate of change of Σ was regularized
instead of Σ.
In this work, we combined surface and volume based algorithms, to overcome serious weaknesses of each
component code that severely limit their usefulness (one is restricted to the surface, the other cannot handle the
cortex accurately). Although combination of the two is likely to be much more useful, as it allows good matching
everywhere, no hybrid registration approach is yet widely used,27.29
There are several ways to achieve a good registration over the whole brain volume, including the cortex. The
simplest one, which we investigate here, is to use a cortical surface registration to obtain an initialization for the
volume registration. This allows a slightly simpler implementation than one that allows the surface deformation
to evolve while the volumes are matched. A similar method was designed in,13 where a new algorithm was
introduced that performs both surface and volume registrations (see also22). A landmark-based registration is
first performed on the cortical surfaces, by constraining the matching between sulci. The correspondence was
then extended to the whole brain using a constrained 3D harmonic mapping. The third step consists of refining
the volume alignment via elastic registration.
Here we improve on the method from13 in two ways. The algorithm in13 gives excellent matching in cortical
areas. However, an elastic registration is used to align the subcortical structures. This method gives poor
results when large deformations are required, and is not provably diffeomorphic, i.e., there is no mathematical
guarantee that the interior mapping is invertible with a smooth inverse. This latter property is vital for tensor-
based morphometry, a powerful approach for large-scale analysis of regional brain volumes and shape using
statistics of the local deformation,17.11
Here we combine a cortical registration with a fluid one for the first time, enabling precise matching of cortical
folds, but allowing large deformations in the enclosed brain volume, which guarantee diffeomorphisms. For this,
we will use the Riemannian fluid algorithm developed in.5 The Riemannian fluid is regularized to minimize the
distortion at each voxel from the registration. Furthermore, the surface matching algorithm designed in13 uses
sulci as landmarks to anchor the cortical registration. This method is prohibitive for large datasets, as the sulci
need to be traced by hand. Here instead we use the Freesurfer software7 (which makes use of a curvature-based
registration algorithm) to align the cortical surfaces. As a result, the entire registration process is automated.