A global optimisation method for robust affine registration of brain images
Introduction
Registration is an important component in many medical image analysis applications. It is used in motion correction, multi-modal fusion, mapping to Talairach space and many other tasks. When analysing large quantities of data, such as in a clinical study or within a busy imaging unit, it is desirable to have fully automatic registration methods. Such methods aim to offer reliability and repeatability as well as minimising user interaction.
A standard method of solving the registration problem is to treat it as a mathematical optimisation, using a cost (or similarity) function to quantify the quality of the alignment of the two images for any given transformation. In practice, this formulation relies on the use of a global optimisation method. This optimisation method is crucial for obtaining good registrations. However, to date most attention has been focused on other aspects of the problem, such as defining cost functions, rather than on the optimisation method.
Most of the mathematical optimisation methods that exist are only suitable for local optimisation and therefore will not find the global solution in general. These methods include gradient descent, Powell’s method, simplex methods and so on (Press et al., 1995). Furthermore, although some global optimisation methods exist (such as Genetic Algorithms and Simulated Annealing), many of the methods require a very large number of iterations to satisfy statistical convergence criteria Geman and Geman, 1984, Ingber, 1989. It is therefore the speed of the global optimisation methods that is the limiting factor for their successful application to this problem, since evaluation of the cost function is particularly time consuming for volumetric registration using voxel similarity measures. Consequently, although some existing global optimisation techniques may be viable (such as Genetic Algorithms) the majority of existing registration methods opt for local optimisation in order to increase the speed.
In an attempt to solve the global optimisation problem in a reasonable amount of time, many registration methods rely on a multi-resolution approach in the hope that local optimisation will then find the global minimum. The assumption is that it is easier to find the global minimum at a coarse resolution (when using large sub-sampling), since larger transformation steps can be taken, which should reduce the chance of there being a local minimum between the initial starting position and the global minimum. This global minimum is then refined by applying successive local optimisations at a series of different resolutions.
This paper examines affine registration of brain images using inter-modal voxel similarity measures such as the Correlation Ratio (Roche et al., 1998) and Mutual Information Viola and Wells, 1997, Maes et al., 1997. It initially considers the assumptions underlying the registration problem (Sections 2 and 3) and then proposes a fast global optimisation method (Section 4) that is specifically tailored to this registration problem. A full discussion of all the necessary implementation details (Section 5) is given next, and then results are presented (Section 6) that compare the method with several currently available registration packages in common usage. These results show that the proposed method is more reliable at finding the global minimum than the other packages. Furthermore, using the above results, together with different tests which compare various optimisation schemes and cost functions, it is demonstrated that the use of local optimisation methods together with the standard multi-resolution approach is not sufficient to reliably find the global minimum.
Section snippets
Mathematical formulation
The standard registration problem is to find a transformation that best aligns a reference image Ir to another (floating) image If. This is formulated as a mathematical problem by taking a cost function, C(I1,I2), that quantifies the quality of the registration and then finding the transformation which gives the minimum cost: where ST is the space of allowable transformations, and If∘T represents the image If after it has been transformed by the transformation T.
It is
Characterisation of cost functions
In this section, the behaviour of five different inter-modal cost functions will be examined. The cost functions that will be compared here are: the Woods function (Woods et al., 1993), Correlation Ratio (Roche et al., 1998), Joint Entropy Studholme et al., 1995, Collignon et al., 1995, Mutual Information Viola and Wells, 1997, Maes et al., 1997, and Normalised Mutual Information (Studholme et al., 1999, Maes, 1998). Definitions of these functions are given in Table 1. Note that to form the cost
Global optimisation method
This section presents a global optimisation method that is specifically tailored for volumetric registration of brain images. It uses trilinear interpolation with either the Correlation Ratio or Mutual Information as a cost function. However, the optimisation method is quite general and can be used for most cost functions, provided that they have the correct asymptotic behaviour.
Implementation
In almost all methods there are several additional choices which need to be made at the implementation stage. These choices are important and a re-implementation of a method will be unlikely to give similar results (and may even fail completely) unless these details are treated the same way.
Results
The registration method described above in Sections 4 and 5 has been implemented in C++ and is called FLIRT – FMRIB’s Linear Image Registration Tool. This program has undergone extensive trials over several months, being used by various researchers including trained neurologists, psychologists and physiologists. During this time it has been used to perform many thousands of registrations in the context of FMRI analysis and structural studies (Smith et al., 2001).
Feedback from the users has been
Discussion
In this paper the problem of global optimisation for fully automatic registration of brain images was examined. Only affine registration was examined as, although this is a much easier problem than general, non-linear transformations, finding the global minimum is still difficult. Furthermore, many non-linear methods rely on an initial affine registration to find a good starting position, and so having a good method of affine registration is important.
The standard mathematical formulation of
Acknowledgements
The authors wish to thank the Medical Research Council and the European MICRODAB project for supporting this work.
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