Diffeomorphic statistical shape models

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Abstract

We describe a method of constructing parametric statistical models of shape variation which can generate continuous diffeomorphic (non-folding) deformation fields. Traditional statistical shape models are constructed by analysis of the positions of a set of landmark points. Here, we describe an algorithm which models parameters of continuous warp fields, constructed by composing simple parametric diffeomorphic warps. The warps are composed in such a way that the deformations are always defined in a reference frame. This allows the parameters controlling the deformations to be meaningfully compared from one example to another. A linear model is learnt to represent the variations in the warp parameters across the training set. This model can then be used to generalise the deformations. Models can be built either from sets of annotated points, or from unlabelled images. In the latter case, we use techniques from non-rigid registration to construct the warp fields deforming a reference image into each example. We describe the technique in detail and give examples of the resulting models.

Introduction

This paper proposes a method of building generative statistical models of diffeomorphic (i.e. smooth, invertible) deformation fields. It is part of a larger programme aimed at exploiting the synergy between work on constructing statistical models of shape and that on non-rigid registration.

Statistical models of shape variation [1] have been shown to be powerful tools for image interpretation. Most approaches to constructing such models assume (either explicitly or implicitly) that the object shape can be represented by a set of points which exist on all examples, and essentially define the correspondences across those examples.

The act of annotating a training set with such points is essentially the same as the goal of non-rigid image registration, widely used in the medical image analysis community. In such cases the aim is to find the deformation field which maps one image into another. Such a field defines a dense correspondence between two images. Given a set of such dense correspondences, one can build a statistical model of the deformation field [2].

When registering two images one usually assumes that similar structures are present in each. Since in general the ordering of images should not be important, it is desirable that the deformation field be smooth and invertible (so that every point in one image has a corresponding point in the other). Such smooth, invertible transformations are called diffeomorphisms.1

In the non-rigid registration literature there are a variety of ways of constructing diffeomorphic maps between pairs of images. The diffeomorphism is either achieved by placing constraints on the Jacobian of more arbitrary warps [3], [2], [4] or by composing many simple diffeomorphisms [5].

There has been little attention paid to generating parameterisable diffeomorphisms suitable for representing the statistics of shape change across a set of examples.

Point-based statistical models of shape [1] give us a parameterised representation of the movement of a set of landmark points. If we could interpolate the deformation field between the points we would have a statistical model of continuous deformation. Unfortunately it is difficult to interpolate in such a way as to ensure the resulting mapping is diffeomorphic.

For instance, the simplest approach, that of piecewise linear interpolation using a triangulation of the points, is clearly unsatisfactory. Not only is the resulting deformation field not smooth, it is quite likely to fold up on itself (see for instance, Fig. 1).

Smoother interpolation schemes such as Thin-Plate Splines [6] are also prone to tearing space. However, Twining et al. describe Geodesic Interpolating Splines, which are capable of constructing a diffeomorphism between two sets of landmark points [7]. Unfortunately the method requires a relatively complex optimisation, so is not very efficient for large numbers of points.

Thus, a conceptually simple approach, that of using a standard point-based statistical model, then interpolating to estimate the diffeomorphism between the mean shape and points generated by the model, can be rather slow.

In this work we describe how we can construct statistical models of continuous deformation fields in such a way that they can only generate smooth invertible mappings. The key to the approach is to create parameterised deformations by composing simple diffeomorphic functions, then to apply statistics in the space of the parameters, rather than to the point positions directly. It is possible to generate functions such that linear interpolation in the parameter space always leads to legal deformations.

In the following we will describe the construction of parameterised diffeomorphic warps, how their parameters can be estimated and how one can model their parameters given a training set of registered data. We will give examples of the models and discuss how such models can be used.

Rueckert et al. [2] describe statistical shape models in which deformation is represented by the control points of a B-spline. Since the B-spline is linear in the control point positions, the resulting model is essentially identical to the original PDMs of Cootes et al. [1]. Such a model is prone to generating non-diffeomorphic deformations. However, one could use the constraints on the control point displacements described by Chio and Lee [4] to ensure that B-spline-based deformations are diffeomorphic.

The work by Pizer’s group on medial representations (M-Reps) [8] explores modelling compound structures and their deformation. In work by Fletcher et al. [9] they investigate statistically modelling the variation in shape in such a way as to preserve correspondence.

Marsland and Twining [10] describe a pairwise non-rigid registration scheme in which the deformation field is defined by interpolation of a sparse set of correspondences. These correspondences are defined in the frame of the reference image. The approach could be used to generate parameterisations capable of statistical analysis across a set.

Vaillant et al. [11] describe a method of representing a diffeomorphism relative to a reference frame in terms of the momentum of a set of control points. Given a momentum the position of the control points at a given time can be uniquely determined, leading to a relationship between the initial momentum and a final diffeomorphic deformation. The momentum is defined in a linear tangent space, allowing statistics to be applied. This is a very elegant approach, but is fairly computationally expensive.

Section snippets

Representing diffeomorphisms

We can construct complex diffeomorphic functions by composing simple diffeomorphisms.

Let f(x|ϕi) be a diffeomorphic mapping controlled by parameters ϕi. We define f1f2(x) = f1(f2(x)).

We can define the composition of a set of diffeomorphisms asF(x|Φ)=f1f2fn-1fn(x)where the parameter vector Φ is simply a concatenation of the parameter vectors for each individual function, {ϕi}. As long as the component functions are all diffeomorphic, so is F.

In the following we will consider diffeomorphisms of

Estimating diffeomorphisms

We will consider estimating the parameters for the compositional warps described above for two cases; one in which we have a set of landmark positions on a reference shape and a target shape, and one in which we have only un-annotated images (one reference, one target). In both cases we wish to find the parameters of the compositional warp, ϕA (the affine component) and Φ (the non-linear components), which best match the reference to the target data.

Statistical models of diffeomorphisms

Suppose we have a set of training images. If we choose one as a reference, and choose a suitable class of compositional warps (such as the grid-based deformations), we can use the method described above to find the diffeomorphic deformation from the reference image into every other image. Each such deformation is summarised by the affine parameters, ϕA,i and the parameters controlling the non-rigid warps, concatenated into vectors Φi.

Shape is usually defined as the geometric properties of an

Discussion

The key to the models is the method of generating parameterised classes of diffeomorphisms by composing relatively simple basis functions. Though we have presented results with a particular type of grid-based interpolating warp, a wide range of alternatives could be used. One obvious choice would be the B-splines used to such success by Rueckert et al. [2], as long as suitable constraints are used to ensure diffeomorphisms [4].

Of course, a different choice of basis functions will lead to a

Conclusions

We have described a method of modelling diffeomorphic shape deformation statistically, leading to a parameterised model capable of synthesising diffeomorphisms efficiently. The resulting models have modes of variation similar to those of linear models of point position, but unlike the latter, explicitly define a full diffeomorphic deformation field. We anticipate that the models will find wide application in the fields of modelling and interpretting deformable objects, particularly in the

Acknowledgements

The authors thank their collaborators in the EPSRC/MRC funded MIAS-IRC Project for many fruitful discussions.

References (15)

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