C.J. Taylor and J.A. Noble (Eds.): IPMI 2003, LNCS 2732, pp. 378–387, 2003.
© Springer-Verlag Berlin Heidelberg 2003
Information Theoretic Similarity Measures in
Non-rigid Registration
William R. Crum, Derek L.G. Hill, and David J. Hawkes
Division of Imaging Sciences,
The Guy’s, King’s and St Thomas’ School of Medicine,
Guy’s Hospital, London SE1 9RT,UK
bill.crum@kcl.ac.uk
Abstract. Mutual Information (MI) and Normalised Mutual Information (NMI)
have enjoyed success as image similarity measures in medical image registra-
tion. More recently, they have been used for non-rigid registration, most often
evaluated empirically as functions of changing registration parameter. In this
paper we present expressions derived from intensity histogram representations
of these measures, for their change in response to a local perturbation of a de-
formation field linking two images. These expressions give some insight into
the operation of NMI in registration and are implemented as driving forces
within a fluid registration framework. The performance of the measures is
tested on publicly available simulated multi-spectral MR brain images.
1 Introduction
Mutual Information (MI) was proposed as an image similarity measure useful for im-
age registration independently by Collignon et al [1] and Wells [2] and Viola [3]. It is
now widely used in rigid and affine registration. Some modifications to MI have been
suggested which make it more robust to large mis-alignment of images; the Normal-
ised Mutual Information (NMI) [4] and the Entropy Correlation Coefficient [5] are
the best known of these. The evolution of all of these measures has been rather ad
hoc; they are used because they appear to work. Non-rigid registration is now being
applied to more challenging tasks for which MI and NMI are currently the best avail-
able similarity measures. Although, under certain circumstances, a function of MI can
be derived as the optimal similarity measure linking two images [6], the use of NMI
in particular has little theoretical support.
In registration, the response of a similarity measure to a change in registra-
tion parameter must be evaluated. This can be done in several ways but an attractive
approach is to compute derivatives of the similarity measure with respect to the trans-
formation parameters. These derivatives can be used in an optimisation scheme to
find the registration parameters, which maximise the image-similarity measure. This
maximum is assumed to correspond with the optimal image alignment. For similarity
measures such as mean-square intensity difference or intensity cross-correlation,
analytic expressions can be derived for these derivatives. However for MI and NMI,
an empirical calculation is often performed to obtain numerical values for the deriva-

Information Theoretic Similarity Measures in Non-rigid Registration 379
tives, perhaps because the most common way to compute these measures is from dis-
crete intensity histograms.
In this paper we derive analytic expressions for derivatives of Joint Entropy
(JE), MI and NMI by considering a point perturbation of a displacement vector field.
The expressions give a gradient vector at each voxel which indicates the direction in
which perturbing the associated displacement vector results in the largest increase in
the image similarity measure. We have used these gradient vectors as driving forces
in a fluid registration framework [7] and present some initial experiments inspired by
[8] to establish the behaviour of the technique. These experiments involve a compari-
son between fluid registration driven by JE, MI and NMI and fluid registration driven
by intensity cross-correlation using simulated image data where the ground truth
transformation is known.
2 Methods
First, the analysis, which leads to the derivatives of JE, MI and NMI, is presented to-
gether with a brief discussion of the operation of these measures as driving forces. In
the second part of the methods, the experiments using these derivatives to drive non-
rigid registration are described.
2.1 Theory : The Force at a Voxel in Non-rigid Registration
We consider a non-rigid registration scenario where a source image, (B), is being
warped into the space of a target image, (A). The registration is an iterative process
and a pseudo-time variable, t, defines the current state of the registration. The source
image, (B), is assumed to be a function of t as it is updated to reflect the most up-to-
date estimate of the transformation linking the two images. We assume the target im-
age, (A), is static and defines a fixed region over which the similarity measures are
computed. Writing HAB(t) for the joint entropy and HA and HB(t) for the marginal entro-
pies of the two images the standard definitions of the Mutual Information and Nor-
malised Mutual Information are:
( ) ( ) ( )tABtBAtAB HHHM += (1)
( ) ( )( ) ( )tABtBAtAB HHHN += (2)
The marginal entropies of the target and source image and the joint entropy are de-
fined in the usual way, as functions of the marginal (Qj and Pi) and joint (pij) intensity
probabilities.
[ ]=
j
jjA QQH log and [ ]=
i
iiB PPH log (3)