Elsevier

Medical Image Analysis

Volume 39, July 2017, Pages 1-17
Medical Image Analysis

Geodesic shape regression with multiple geometries and sparse parameters

https://doi.org/10.1016/j.media.2017.03.008Get rights and content

Highlights

  • Geodesic shape regression with sparse parameters is developed.

  • Allows for the inclusion of multiple shapes in any combination.

  • Model estimation is robust across a variety of parameter settings.

  • Freely available software at www.deformetrica.org.

Abstract

Many problems in medicine are inherently dynamic processes which include the aspect of change over time, such as childhood development, aging, and disease progression. From medical images, numerous geometric structures can be extracted with various representations, such as landmarks, point clouds, curves, and surfaces. Different sources of geometry may characterize different aspects of the anatomy, such as fiber tracts from DTI and subcortical shapes from structural MRI, and therefore require a modeling scheme which can include various shape representations in any combination. In this paper, we present a geodesic regression model in the large deformation (LDDMM) framework applicable to multi-object complexes in a variety of shape representations. Our model decouples the deformation parameters from the specific shape representations, allowing the complexity of the model to reflect the nature of the shape changes, rather than the sampling of the data. As a consequence, the sparse representation of diffeomorphic flow allows for the straightforward embedding of a variety of geometry in different combinations, which all contribute towards the estimation of a single deformation of the ambient space. Additionally, the sparse representation along with the geodesic constraint results in a compact statistical model of shape change by a small number of parameters defined by the user. Experimental validation on multi-object complexes demonstrate robust model estimation across a variety of parameter settings. We further demonstrate the utility of our method to support the analysis of derived shape features, such as volume, and explore shape model extrapolation. Our method is freely available in the software package deformetrica which can be downloaded at www.deformetrica.org.

Introduction

The analysis and monitoring of change over time is fundamental to many problems in medicine, where anatomical change is often driven by a continuous dynamic process, such as in early childhood development, aging, or disease progression. Measuring and understanding change over time is required to assess development. For example, head circumference is measured during pediatric examination and compared to a standardized model to determine if a child is developing along a normative trajectory. In addition to assessment, measuring change over time is essential to monitor treatment, or the effectiveness of drug therapy. Furthermore, the analysis of change can improve our understanding of the time-course of psychiatric disorders or pathologies, which may provide additional information to steer treatment efforts. For example, neuroscientists discovered a decrease of striatum volume (Paulsen, Magnotta, Mikos, Paulson, Penziner, Andreasen, Nopoulos, 2006, Paulsen, Nopoulos, Aylward, Ross, Johnson, Magnotta, Juhl, Pierson, Mills, Langbehn, et al., 2010) in Huntington’s disease, a finding which has led to research efforts into gene therapy targeted at the putamen (Benraiss and Goldman, 2011).

Magnetic resonance imaging (MRI) is now ubiquitous in clinical practice, and represents a powerful tool to monitor and measure change in anatomical tissue in vivo. Medical imaging technology is improving in fidelity, as well as becoming more readily available, which has led to the proliferation of imaging studies. Such studies may be cross-sectional, where imaging data belongs to a representative population distributed in time, but each subject is scanned only once. More recently, the focus has shifted towards subject-specific and individualized analysis, where data comes from a single subject, or large scale longitudinal studies where serial scans are acquired from the same subjects over time. In any case, such time-indexed imaging databases provide a rich environment for research activity, and are essential for improving our understanding of various disorders and pathologies.

Medical imaging studies, either cross-sectional or longitudinal, rely on medical images which represent measurements sparsely distributed in time. Such images can be thought of as snapshots or frames of the underlying continuous sequence. Further, MR images represent imperfect observations of anatomy, with noise introduced by the scanner, image reconstruction, patient movement, among other possible sources. What is needed are statistical models to capture the trend in the data, which also characterize the underlying continuous anatomical change. Such statistical models can be used to age match subjects, or alternatively, match subjects along disease progression, to temporally align imaging data with clinical scores not taken at the same time as the image acquisition, for interpolation or extrapolation to generate new unobserved shapes, or as a mathematical representation for statistical hypothesis testing. The most ambitious is the construction of normative models of development and aging for comparison and monitoring patient progress.

In many clinical studies, measurements such as volume are extracted and regression models are estimated absent any imaging or geometric information. Typically, linear models are chosen for convenience and simplicity, rather than motivated by the changes in the anatomy. We advocate for modeling at a higher level, where our understanding of anatomical change can be introduced. For example, a shape regression model based on the flow of diffeomorphisms guarantees structures cannot be created, destroyed, holes introduced, or folded over on themselves, which are desirable and required properties for many clinical applications. Furthermore, shape models support traditional analysis of scalar measurements derived from morphometric features, as any value of interest can be extracted continuously from the shape trajectory.

Just as it is important to have an anatomically realistic model of change, it is desirable for a model to include multiple sources of geometric information as a multi-object complex. There is a large variety of geometric information which can be extracted from medical images such as: surfaces, curves, point clouds, and landmarks. As we will see later in the paper, shapes may derive from different modalities, such as subcortical shapes from structural imaging and white matter connections from diffusion tensor imaging. The different sources of geometry may complement each other, giving a more complete description of change over time. Due to the wide variety of shape representations derived from medical imaging data, we desire a model which is independent of a given shape parameterization, instead it must be able to handle numerous shape representations arranged in various combinations.

In the field of medical image analysis, the problem of regression has received considerable attention over the last 10 years, as regression is a necessary tool in many longitudinal statistical analysis pipelines (Datar, Muralidharan, Kumar, Gouttard, Piven, Gerig, Whitaker, Fletcher, 2012, Durrleman, Pennec, Trouvé, Braga, Gerig, Ayache, 2013, Fishbaugh, Prastawa, Durrleman, Piven, Gerig, 2012, Hart, Shi, Zhu, Sanchez, Styner, Niethammer, 2010, Singh, Hinkle, Joshi, Fletcher, 2016). There are a variety of methods introduced in the general Riemannian setting, such as geodesic regression (Fletcher, 2011, Fletcher, 2013). This idea was extended to polynomial regression (Hinkle et al., 2014), with geodesics being a special case. In addition to linear models, other work includes non-linear regression in the general Riemannian setting (Banerjee et al., 2015). The application of these methods is typically finite dimensional manifolds, most commonly shapes represented in Kendall shape space (Kendall, 1984). There has also been attention towards methods focused on a specific manifold, such as the Grassmannian (Hong, Kwitt, Singh, Davis, Vasconcelos, Niethammer, 2014, Hong, Singh, Kwitt, Vasconcelos, Niethammer, 2016).

Regression on medical images has also been explored, including the extension of kernel regression to images (Davis et al., 2007) and splines for diffeomorphic image regression (Singh et al., 2015). Geodesic regression has been developed for imaging data in Niethammer et al. (2011) which leverages the initial momenta formulation of the EPDiff equation (Vialard et al., 2012). In Niethammer et al. (2011), the momenta are a scalar field of the same dimension of the image, as such, they can be thought of as attached to each voxel. The direction of the initial momenta is orthogonal to the gradient of the deforming baseline image (Miller et al., 2006). Rather than use scalar initial momenta, the work of Singh et al. (2013) introduces a vector formulation to ease the estimation of the baseline image. The optimization procedure need not jointly compute both baseline image and initial momenta, rather only momenta are estimated, and a new baseline image is computed in turn. This leads to faster convergence in terms of the number of iterations of gradient descent.

Regression on geometric structures extracted from imaging has also been explored. In Vialard and Trouvé (2012), a nonparametric spline model is proposed as perturbations of a geodesic path. In Datar et al. (2009), each landmark point in correspondence across the population are assumed to follow a linear trajectory. As with the Riemannian methods, these methods are applicable to shapes represented as landmarks.

Several regression methods have been proposed for multi-object complexes containing a variety of shape representations, such as piecewise-geodesic regression (Durrleman et al., 2009) and regression based on controlled acceleration (Fishbaugh et al., 2011). However, the methods do not provide a solution for estimating a baseline shape, rather the regression is constrained to start from the observation earliest in time. The dimensionality of the models is directly related to the sampling of the data, as the model parameters in Durrleman et al. (2009) and Fishbaugh et al. (2011) are located on the vertices of the shapes. Furthermore, the models are not based on a shooting formulation from initial conditions, instead requiring model parameters at every shape point and every time-point in the discretization. As a consequence, many thousands of parameters are needed to describe shape evolution.

To summarize, currently available methods are either limited to specific data types, such as landmarks in correspondence or images, or require a huge number of deformation parameters. The high dimensionality can be due to a shared parameterization between deformation and shape, as is the case of momenta attached to image voxels or shape vertices, or to nonparametric models whose parameters are functions of time. What is lacking is a model of shape change which is flexible to the data representation, which is also a compact generative model which describes shape evolution with a small number of parameters.

In this paper, we present a geodesic shape regression model in the large deformation (LDDMM) framework that incorporates multiple sources of geometry in different combinations as multi-object complexes which drive the estimation of a single continuous deformation of the ambient space. The proposed generative model uses a sparse representation of diffeomorphisms, which describe complex nonlinear changes over time with a small number of model parameters defined by the user. By analogy with simple linear regression, we estimate an intercept as the initial baseline shape configuration, as well as a slope, which in our model is the initial momenta vectors as well as their location. We derive the Euler-Lagrange equations and propose a gradient descent algorithm for model estimation as well as systematic experimentation to expand on our previous published conference work (Fishbaugh et al., 2013b).

Section snippets

Shape regression

In its most basic form, regression analysis involves exploring the relationship between a dependent variable and one or more independent variables. The most ubiquitous model is simple linear regression, where we assume a linear relationship between one dependent and one independent variable. Given the parametric form of a line y=mx+b, linear regression can be expressed as E(m,b)=i=1N((mxi+b)yi)2given measurements [y1,y2,,yn] and corresponding explanatory variables [x1,x2,,xn]. Model

Geodesic flow of diffeomorphisms

The geodesic path connecting ϕ0 to ϕ1 is the path with constant velocity, which is equivalent to the path which minimizes the total kinetic energy of the velocity field vt 1201vtV2dt=01p=1Ncq=1Ncαp(t)tK(cp(t),cq(t))αq(t)dt,which is defined entirely by the state of the system S(t). The α(t) which minimize (11) satisfy a set of differential equations defining the evolution of momenta over time (Miller et al., 2006). Combining this with the motion of the control points (10) gives {c˙i(t)=p=1

Impact of parameter selection via cross-validation

There are three main parameters which influence model estimation:

  • σV: the size of the kernel which defines the deformation. It is the distance at which points move in a correlated way. Higher values result in mostly rigid deformation, while lower values allow each point to move independently.

  • σW: the size of the kernel which defines the metric on currents. This parameter allows you to tune the metric properties of the space of currents to suit your application. Intuitively, this parameter is the

Impact of missing data

The previous section explored the ability of the geodesic model to match a cross-sectional population with considerable inter-subject variability. Here, we focus on the geodesic model applied to estimating subject-specific growth trajectories from longitudinal data. In such cases, only a few observations sparsely distributed in time are available. Further, the time between observations is on the order of months or even years. There can potentially be dramatic changes and differences in

Multimodal shape regression analysis

The control point formulation of diffeomorphic flow separates the deformation parameterization from any specific shape representation. As a consequence, we can embed several shapes with different representations (i.e. points, curves, meshes, etc.) into the same ambient space without impacting the dimensionality or parameterization of the geodesic model. By including multiple sources of geometric information in the analysis, we get a more complete picture than is possible from any single source.

Conclusions

In this paper, we detailed a sparse representation of diffeomorphisms, where momenta are located at discrete control points. From the discrete momenta, dense deformations of the whole space can be computed. Then, through geodesic shooting, a geodesic flow of diffeomorphisms can be constructed and used to deform various shapes embedded in the ambient space. This machinery became the foundation around which we developed a dedicated algorithm for geodesic shape regression. Indeed, the control

Acknowledgements

Supported by the European Research Council (ERC) under grant agreement No 678304, European Union’s Horizon 2020 research and innovation program under grant agreement No 666992, the program “Investissements d’ avenir” ANR-10-IAIHU-06, RO1 HD055741 (ACE, project IBIS), U54 EB005149 (NA-MIC), and U01 NS082086 (HD).

References (47)

  • P. Dupuis et al.

    Variational problems on flows of diffeomorphisms for image matching

    Q. Appl. Math.

    (1998)
  • S. Durrleman et al.

    Sparse adaptive parameterization of variability in image ensembles

    Int. J.Comput. Vision (IJCV)

    (2013)
  • S. Durrleman et al.

    Toward a comprehensive framework for the spatiotemporal statistical analysis of longitudinal shape data

    Int. J. Comput. Vision

    (2013)
  • S. Durrleman et al.

    Spatiotemporal atlas estimation for developmental delay detection in longitudinal datasets

  • S. Durrleman et al.

    Morphometry of anatomical shape complexes with dense deformations and sparse parameters

    Neuroimage

    (2014)
  • S. Durrleman et al.

    Optimal data-driven sparse parameterization of diffeomorphisms for population analysis

    Information Processing in Medical Imaging (IPMI)

    (2011)
  • S. Durrleman et al.

    Topology preserving atlas construction from shape data without correspondence using sparse parameters

    Medical Image Computing and Computer Assisted Intervention (MICCIA)

    (2012)
  • A. Fedorov et al.

    3D slicer as an image computing platform for the quantitative imaging network

    Magn. Reson. Imaging

    (2012)
  • J. Fishbaugh et al.

    Estimation of smooth growth trajectories with controlled acceleration from time series shape data

  • J. Fishbaugh et al.

    Analysis of longitudinal shape variability via subject specific growth modeling.

  • J. Fishbaugh et al.

    Geodesic image regression with a sparse parameterization of diffeomorphisms

    Geometric Science of Information (GSI)

    (2013)
  • J. Fishbaugh et al.

    Geodesic shape regression in the framework of currents

    Information Processing in Medical Imaging (IPMI)

    (2013)
  • J. Fishbaugh et al.

    Geodesic regression of image and shape data for improved modeling of 4D trajectories

    International Symposium on Biomedical Imaging (ISBI)

    (2014)
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