Pattern Recognition 32 (1999) 71—86An overlap invariant entropy measure of 3D medical imagealignmentC. Studholme *, D.L.G. Hill, D.J. HawkesDepartment Diagnostic Radiology and Electrical Engineering, Yale University, BML 332, 333 Cedar Street, New Haven 06511, USAComputational Imaging Science Group, Radiological Sciences, UMDS, Guys Hospital, London SE1 9RT, UKReceived 31 October 1997; in revised form 29 April 1998AbstractThis paper is concerned with the development of entropy-based registration criteria for automated 3D multi-modalitymedical image alignment. In this application where misalignment can be large with respect to the imaged field ofview, invariance to overlap statistics is an important consideration. Current entropy measures are reviewedand a normalised measure is proposed which is simply the ratio of the sum of the marginal entropies and thejoint entropy. The effect of changing overlap on current entropy measures and this normalised measure are comparedusing a simple image model and experiments on clinical image data. Results indicate that the normalised entropymeasure provides significantly improved behaviour over a range of imaged fields of view.  1999 Pattern RecognitionSociety. Published by Elsevier Science Ltd. All rights reserved.Keywords: Multi-modality; 3D medical images; Registration criteria; Information theory; Entropy; Mutual information;Normalisation1. IntroductionRecently there has been active research into the use ofvoxel-similarity-based measures of multi-modality medi-cal image alignment [1,2]. Such an approach avoids thetask of identifying corresponding anatomical structuresin two different modalities, which is difficult to automatefor a broad range of clinical data. The use of imagesimilarity measures, especially those derived from in-formation theory, have been shown to allow fully auto-mated alignment in a number of important clinicalapplications [3,4]. In this paper we are concerned*Corresponding author.specifically with entropy based measures of alignmentbetween different 3D medical modalities [5—9]. In par-ticular, we examine one limitation of the measures thathave so far been proposed, which is their sensitivity toimage overlap statistics. Invariance to image overlap isan important property for this application, where mis-alignment may be significant with respect to the volumeof overlap of the images. Such misalignment is illustratedby the image pair in Fig. 1.The paper begins by introducing some of the ideasbehind the measurement of information content in animage and how this can be used to quantify image align-ment. Joint entropy derived from the joint probabilitydistribution of image values provides the starting pointto relate the information content of a pair of images. Itsproperties are examined as a measure of image alignment0031-3203/99/$19.00#0.00  1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.PII: S 0 0 3 1 - 3 2 0 3 ( 9 8 ) 0 0 0 9 1 - 0Fig. 1. Transaxial (top) and Sagittal (bottom) slices through clinical MR (left) and CT (right) image volumes of the brain (here manuallyaligned) illustrating the limited volume of overlap of two clinical images. CT in particular can have a limited extent to reduce X-Ray doseto the patient.and from this a set of related measures including mutualinformation and normalised mutual information are de-veloped and compared. The following sections then com-pare the behaviour of joint entropy, mutual informationand the normalised measure to varying image statisticsin their region of overlap. This is achieved using a simpleimage model and also by examining the behaviour of anautomated registration algorithm using these measuresas the initial misalignment and field of view of the imagesis varied.2. Information measures and image alignmentIn imaging the head using a 3D medical imagingdevice we form an image of measurements of a physicalproperty of material within the patient. Such propertiesdelineate regions of material, for example grey matter,white matter, bone or tumor tissue, within the patient.Different modalities are commonly used in clinical diag-nosis and therapy planning which record different ana-tomical or physiological properties within the patient,and can therefore delineate different structures of clinicalinterest (for example to distinguish between tumor andnon-tumor tissues).In order to accurately relate different structures ofinterest delineated by the two modalities we need to alignthe images into a common coordinate system. In practicethese modalities will delineate both common and com-plementary structure within the images. To definealignment we wish to make use of any shared regionsdelineated by the two modalities.2.1. Information measuresThe concept of uncertainty can provide a useful basisfor developing ideas about information theory [10]. Ifwe examine the probabilities of values occurring in animage, some values are rare and some are common. In72 C. Studholme et al. / Pattern Recognition 32 (1999) 71—86Fig. 2. The information content of a binary symmetric channel: This can be equated to the information content of a binary imageconsisting of foreground (m) and background (m) where p(m)"1!p(m). The horizontal axis in this graph therefore corresponds tovarying the ratio of background to foreground area in the images of the white circle shown below.predicting what value a voxel has we can form an esti-mate of the uncertainty of our guess at the value, giventhe observed distribution of probabilities. If all probabil-ities are equal, uncertainty in guessing what value a givenvoxel might have is largest. If there is only one value inthe scene then the uncertainty in guessing the value ofa given voxel would be zero.If we learn the value of a measurement (pixel or voxel)that we were very uncertain about (i.e. it would be diffi-cult to guess) then we gain a large amount of information.Conversely, if we learn a value that would be easy toguess because it has a high probability of occurrence,then we only gain a small amount of information. If wehave a set of values (for example the set of voxel valuesoccurring in an image) we can then look at the averageamount of information provided by the whole set. Essen-tially, we can say an image with similar numbers of eachvoxel value contains more information than an imagewhere the majority of voxels have the same value.There have been many plausible functions proposed