Multimodal Medical Data Registration

Transcription

1 INF358 Seminar in Visualization (2008) Ivan Viola and Helwig Hauser (Editors) Multimodal Medical Data Registration S.Eikeland 1 1 Institute of Informatics, University of Bergen, Norway, stian.eikeland@student.uib.no Abstract Image registration is the task of aligning two images by providing a spatial mapping from one image to the other. This state-of-the-art report aims to give an overview of medical image registration by presenting the registration pipeline and it s different components. Several transformation models (rigid, non-rigid), similarity metrics (sum of squared differences, mutual information, normalized gradient fields) and optimizer algorithms (gradient descent, bfgs) will be presented. This report also takes a look at what is used in current applications and current research, focusing on registration toolkits such as ITK, DROP and the case of Ultrasound CT registration. 1. Introduction The field of medical imaging has been accelerating rapidly since the invention of ultrasound (US) in 1953 and the first computer tomography (CT) scanning technology back in the early 1970s. Both magnetic resonance imaging (MRI), positron emission tomography (PET) and single-photon emission computer tomography (SPECT) followed quickly before the decade was over and has since then been improved, making better and better images. Medical imaging has become a fundamental tool for doctors, providing large amounts of information about their patients in a variety of different modalities, each with it s own strengths and weaknesses. Patients are often scanned multiple times, for example before an operation and after an operation or regularly over time to track the development of a condition. For the doctor it s preferably if the different images are aligned, making it much easier to spot differences in the datasets. The same goes for images from different modalities, which allows us to build on each modality s strength that can be presented in one combined fused view. One example is to combine the functional information from a PET image (showing metabolic activity, but giving few anatomical hints) with CT images (clearly showing the anatomy), the result is an image showing both the complementary functional and anatomical information at the same time, see Figure 1. Combining such images mentally is challenging, so we need a method to bring these datasets into one spatial representation that can later be fused and interpreted by the medical personnel. This method is called image registration and it s the task of finding the spatial mapping that will bring one image into alignment with another fixed image. Image registration is used in a large number of applications and research fields [FM08], such as robotics (tracking of objects) to security (comparing images with a database), remote sensing and computer vision. Even though it s a multidisciplinary problem, according to [FM08] - no general theory for image registration has been made. Each application area has basically researched it s own solutions for solving the registration problem. This report will start by introducing the taxonomy of med- Figure 1: (a) CT scan clearly showing the anatomy. (b) PET scan showing metabolic activity (c) a fused view showing the metabolic activity from a PET scan with the contextual anatomical information from a CT scan.

2 ical image registration, move on to explain the image registration pipeline and it s different components. It will then continue by presenting a few real life examples before ending with a discussion and concluding remarks about the state of the field. 2. Taxonomy of Image Registration 2.1. Basis for registration Medical image registration techniques can be classified in several different ways [MV97], one is the basis used for registration. Methods that rely on artificial objects introduced into the image are called extrinsic methods. These objects can be both invasive and non-invasive and are designed to be well visible and accurately detectable in all modalities. The most effective (and also most extreme) example of an invasive object used for registration is the stereotactic frame, a large device (usually) mounted around the head, attached with screws to the skull as in Figure 2c. Other types of invasive markers can be screwed or implanted at proper locations in the patient s body. Noninvasive markers that can be glued to the skin are also used (see Figure 2b), such a marker is often called a fiducial or a fiduciary marker, these can provide good accuracy when the object of interest is tissue or skin. Algorithms that work on image content only are called in- Figure 2: (a) MR scan of a human head with manually set landmarks. (b) a fiducial marker attached with glue to a model of a skull. (c) stereotactic frame. trinsic methods and can further be categorized into landmark based, segmentation based and voxel property based. Landmark based methods works by identifying corresponding prominent points in the two images, these points can often be easily recognizable anatomical features as in Figure 2a. The landmark locations are usually identified interactively by a user, since detecting corresponding landmarks is in general a very challenging task for fully automatic algorithms [FM08]. Segmentation based approaches utilizes segmentation techniques to extract anatomical or pathological structures from the image data. Once this knowledge is acquired and the content of both images have been categorized, one can elastically deform the structures in one image to match the other. The accuracy of such algorithms are usually very high given a sufficiently well segmented image set, with the flipside that the registration step can only be as accurate as the segmentation step [MV97]. Voxel property based algorithm work directly on the image intensities (image grey values) and are usually very automatic, requiring little to no preparation of the data (such as marking features or segmentation). These kinds of techniques seems to have been the main focus of research the last decade, and this report will focus mostly on this category Modalities Another way to categorize registration methods is by the modalities involved [MV97]. We can do registration on monomodal data, multimodal data, modality to atlas or patient to modality. Algorithms for one purpose doesn t necessarily work for the others. Monomodal registration means to register images coming from the same scanning technology, this usually means scans taken at different time intervals of the same patient. Often scans are taken regularly of a patient to monitor a condition over time, but it can also be over very short time intervals to make a time-lapse scan where the goal is to observe something functional (like wash-in/washout/diffusion of a contrast agent in the region of interest). This requires monomodal registration techniques to compensate for movement or other changes that have happened between the scans. Multimodal registration is used for registering data from different scanning technologies, for example CT MRI. The output of such scanning technologies can often be quite different, so much more advanced algorithms are usually required for multimodal registration compared to some of the simpler monomodal techniques. This is especially true for the more special cases of multimodal registration, such as aligning ultrasound images with CT for contextual anatomical information. Modality to atlas means to register the image against a map, it can for example be to align a scan of a patient s

3 brain with a mathematical model. Patient to modality on the other hand means to map the patient against an image, for example a pre-op scan that is displayed on the patient using augmented reality technology during an operation. This report will however only focus on multimodal and monomodal techniques. Image 1 Image 2 Initial transformation parameters 2.3. Interactivity The registration algorithms can also be categorized by their degree of interactivity. Interactive algorithms means that the system only helps the user visually, for example by showing transformation guesses as hints. The user makes all the decisions himself. In semi-automatic algorithms the user helps the algorithm, for example by making an initial transformation before the algorithms fine tunes it, or by marking corresponding landmarks in the input images. Accurate automatic locating of corresponding landmark markings is a very challenging task, no good system exist as of yet [FM08]. Because of this, most landmark based schemes are semi-automatic. Automatic algorithms means that the system does all the work, most image intensity and extrinsic based systems belong to this class. Transformation Model Similarity Metric Optimizer Figure 3: The registration pipeline and it s components - transformation model, similarity metric and optimizer. 3. Registration Pipeline There are many different ways of doing image registration, but for the medical domain - image similarity-based approaches are most used. In it s most general case registration is an optimization problem based around the pipeline in Figure 3. The system is given two images, one which will (usually) remain fixed throughout the procedure and one which will be spatially mapped to align up with the first image. The pipeline stays the same both for 2D and 3D datasets. Often the system is interactively provided with an initial positioning or transformation of the images by an user, however this can also be estimated by the system. Many types of transformations can be used, ranging from simple linear/affine transformations (assuming a rigid body model) to advanced non-rigid transformation models. After the initial transformation the images are run through a similarity measurement - a metric that in some way measure how well the two images correspond to each other. The result of this metric is used as an input for an optimizer, the optimizer s job is to efficiently search through the parameter space of the transformation to find a set of values that maximizes the image similarity. This is an iterative process that runs many times before the final transformation is derived. 4. Transformation models The most simplest of all possible transformation models used for medical image registration [MME 93] is to assume a fully rigid body model, in such a transformation model Figure 4: a) Affine transformation (12 parameters), b) Non- Rigid (about 2000 parameters) only translations and rotations are allowed. This can be taken one step further by including scaling and shear to end up with an affine transformation model [RTL 93]. Both of these are very simple as they can be represented by a small transformation matrix (3x4 variables in 3D) and works pretty good because a rigid model is close enough for many medical imaging purposes where the structures of interest are either bone or enclosed in bone. The model is global since it applies to the entire image. It also makes the job very easy for the optimizer, since there are only 12 variables to consider (Figure 4a). Because of lacking computing power, many older registration approaches preferred this for that very reason. A rigid body model is however not good enough when you want to deal with tissue and internal organs that move around or deform. Several transformation models have been tried

4 Figure 5: Examples showing the interpolation artifacts that can cause local optima [PMV99]. (a) linear interpolation, (b) B-Spline interpolation. from standard polynomial [WGW 98] and the most popular one, B-Spline surfaces [RSH 99] to physics based elasticity and fluid flow models [HBHH01]. This increases the complexity of the parameter space the optimizer has to search dramatically, since it now has to search for a grid of control points for B-Splines or in the more extreme cases, find a vector for every single voxel position (Figure 4b). Because of computational complexity a combination of rigid and non-rigid models are often used, the first adjustment is usually done using affine transformations before spline-based or similar non-rigid models are used. In later work one also tries to incorporate local rigidity into nonrigid models [LMVS04], by the use of a penalty system one can modify the transformation model and metric in such a way that it will try to keep certain structures rigid. When dealing with CT-scans one can for example penalize nonrigid transformation of bones (which have a high density value). Some systems even tries to create a personal motion model for each patient based on inputs from respiratory movement measurements [MRN 02]. While these non-rigid models provide a great deal of freedom, are they inherently hard to verify for correctness. Validation is mostly based on eye-balling by domain experts or tests on artificial phantoms [FM08]. Recent work tries to provide a validation scheme for registration algorithms [CJ07] Interpolation Often an interpolation scheme is used to obtain data outside of grid positions, since in many cases the transformed sample locations won t perfectly overlap. The most used interpolation scheme is d-linear interpolation (linear, bi-linear, tri-linear, etc). D-linear interpolation works by finding a weighted average based on the surrounding voxels, it s computational fast, but is known to introduce artifacts that can end up as local minima (Section 6) [PMV99]. More accurate interpolation schemes such as higher order spline interpolation can be used at the cost of computation time, but they also have their problems [UAE93], see Figure5 for an example. 5. Similarity metric The similarity metric is an important part of the registration pipeline, taking two images as an input it tries to quantify the similarity of the two pictures. Similarity metrics can work on both image content and markers (fiducials or landmarks deduced after imaging). Perhaps the simplest approach is to only consider landmarks [FM08], given a number of points r 1,...,r n R d from the reference image and a number of corresponding points in the transformed image t 1,...,t n T d one can then calculate the Sum of Squared Differences (SSD) of the positions using the following formula: D LMSSD n [y] = y(r j ) t j ) 2 (1) i=1 This only considers a few points and is therefore very computational fast, but since it only sees parts of the image it s very likely to produce bad results outside of these markers. The same technique also works on image intensities, by applying SSD on the intensity value for every voxel in the images you get a very good similarity measure for images of the same modality [HBHH01]. This is the optimal measure when the difference between two images is only Gaussian noise [Vio95], this of course makes the method very bad for multi-modal registration. When performing multi-modality registration on image content, a possible technique to use is (Cross) Correlation [HHH01]. It s a method borrowed from signal processing, and can therefore be used in both the spatial domain or the frequency domain. It assumes that there is a linear relationship between the intensity values in the images and the optimum similarity measure is described as C = 1 N A(x A ) B T (x A ) (2) x A Ω T A,B where A is the fixed image, B is a transformed image and Ω T A,B is the overlap domain. This method can also easily be normalized by dividing by the standard deviation in each summation step, to make it more accurate in cases with different intensity exposures in the images. The most used approach for multi-modality registration is based on information theory and is called Mutual Information (MI) [WVA 96] [VW97]. MI is defined as I(X,Y) = H(X) H(X Y) (3a) = H(Y) H(Y X) (3b) n m n = p i logp i + p i j logp i j i=1 j=1 i=1 n m p i j = p i j i=1 j=1 p i q j (3c) (3d) where H(X) is the Shannon entropy of image X s probability distribution of the intensity values, while H(X Y) is the

5 is used. n(i,x) := { I(x) when I(x) 0,otherwise 0 (4) I(x) By looking at two related points in image A and the transformed image B, one uses the vectors n(a, x) and n(trans(b), x). These two vectors form an angle, by using the inner product (related to the cosine) one can maximize the square of the cosine to align the images. Most of these image intensity based metrics can also be used in correlation with markers/landmarks by the use of penalty systems [FM08]. By penalizing solutions where the landmarks don t correspond with each other one can force the optimizer to transform the image in such a way that positioning the markers are prioritized over the other image content. By using the corresponding landmarks as a starting guess this also helps to avoid the problem of local optimums (see next section), since it pushes the optimizer towards a specific solution-area. 6. Optimizer Figure 6: Showing Mutual Information and Normalized Gradient Fields utilized on the same dataset, in this case MI needed a good starting guess to converge to the correct minima [HM06]. (a) Reference CT image, (b) Template PET image, (c) MI registered T and it s deformed grid, (d) Normalized Gradient Fields registered T and it s deformed grid. conditional entropy based on p(b a) - the chance of value b in B when A has value a [SFR 07]. MI is a measure of how much information one variable tell about another. A high mutual information means low entropy, if the conditional entropy is 0 and we know the intensity value A(x) we could exactly predict the intensity in B(x) [HBHH01]. In the imaging domain it can be thought of how well one image explains the other, and when MI is maximized you arrive at the optimal alignment of the two images. MI has been proved to be a very good metric both for intra- and multimodality registration [PMV03]. MI has been refined several times since it was introduced, both for accuracy (such as the Normalized Mutual Information [SHH99]) and speed (Mattes MI [MHV 01]). Mutual Information approaches have certain weaknesses, such as having many local maxima that can confuse the optimizer, work has been done to try to avoid this. Recently work has also been done in other directions, such as new methods based on Normalized Gradient Fields (NGF) [HM06] (Figure 6). NGF is based on a very simple idea - two images are considered similar if intensity changes occur at the same locations. The image s intensity gradient is used as a measure of intensity change, but since the magnitude of this gradient is dependent of the modality of the image a normalized value There are two ways for an optimizer to solve the registration problem, one is to directly calculate the optimal transformation parameters, the other is to use an iterative approach where an initial estimate is gradually refined by trial and error. The first is however only feasible in the most simplest of cases, such as calculating the optimal transformation parameters for a rigid or affine transformation model by solving a set of linear equations. This section will therefore only cover iterative methods. In every iterative step the fixed and the transformed image are run through a metric which provides the optimizer with a quantified similarity measure, the optimizer then makes a new guess of the transformation and uses the metric again to find out whether it s transformed in the right or wrong direction. One of the big challenges with these techniques is that the optimizer can stumble upon local optima - a transformation that leads to a semi-good measurement and moving away from it seemingly provides worse results, but globally there is a better value - see Figure 7. Such local optima can exist inherently in the images, but can also be caused by, for example, the interpolation scheme or the metric used. Many techniques can be used to avoid getting stuck in a local optima, one that can also provides a decent speed-up is to re-sample the images at different resolutions [SHH97], see Figure 8. The optimizer is first run on a very low resolution which enables it to try more combination to cover larger areas and also blurring out many of the smaller local optima, thereby avoiding or escaping them. It then moves on and uses the derived transformation on a higher resolution image. This use of multi-resolution images will help the optimizer converge faster on the solution, since it can take coarse steps in the beginning which are then gradually refined. Different optimization techniques can also be combined, using

6 Figure 9: ITK used to register a 3D CT dataset to a MRI T1 dataset using 4 resolution levels, mutual information metric, a gradient descent optimizer and a quaternion rigid transformation model. Figure 7: An optimizer searching for a solution, managing to avoid the local optima. A not so good optimization algorithm might get stuck at point 2. a fast and coarse algorithm for the first few iterations, then moving on with a more accurate but slow one. A popular search technique used is a stochastic gradient descent method by Robbins-Monro (RM) [RM51]. It can (simplified) be described as µ k+1 = µ k a k g k (5a) g k = g(µ k )+ε k (5b) where a k is a decaying function (larger than 0), g k is an approximation of the gradient in the current position influenced by ε. This method will make smaller and smaller steps (following the a k function) and mostly follow the local gradient. Some techniques use Robbins-Monro on small subsets of the image, making the calculation very fast. Methods based on the old Newton-Raphson algorithm are also used, usually called Quasi-Newton methods [KSP07]. µ k+1 = µ k a k L k g(µ k ) (6) where L k is an approximation of the inverse Hessian used in the original Newton-Raphson. A popular way to calculate this approximation is a method from Broyden et al. called BFGS [Bro70]. This matrix can get very large, so a limited memory version (LBFGS) has been developed, which do not need to store as much data [Meh99]. Figure 8: An example of multi-resolution images used to help the optimizer. According to a recent paper by Klein et al. [KSP07] who benchmarked eight different optimization techniques, Robbins-Monro is the best choice for most application, providing very good accuracy while still staying fast. Newer optimizers such as FastPD (described in Section 7.2) was not benchmarked in this paper. 7. Examples from current applications This section will present current registration applications and their use of many of the techniques that has been described above Insight Segmentation and Registration Toolkit The Insight Segmentation and Registration Toolkit (ITK) is a registration and segmentation framework orginally developed by National Library of Medicine (U.S.) for the use of analyzing data from the Visible Human Project [YM05]. Development started in 1999 and the source code is released under the BSD license, since then ITK has grown outside the scope of the original Visible Human Project. Today ITK has been designed to be plugged into existing visualization tools and it s is currently used in many real-world medical applications and projects such as VolView from Kitware, SCIRun/BioPSE and ITK-SNAP. From a registration point of view, ITK is basically a highly modular implementation of the registration pipeline allowing users to mix and match from a large array of registration algorithms and methods, including most of the methods that have been presented in this report. The ITK registration frameworks supports deformable transformations such as B-splines, advanced Mutual Information based metrics and modern optimizers such as LBFGS. Because of it s highly modular design it s often used for easy prototyping of new techniques, for example a new deformable registration approach based on polynomial expansion by Farneback et al. [FW06].

7 7.2. Deformable Registration using Discrete Optimization S.Eikeland / Multimodal Medical Data Registration DROP is a relatively new (2008) application for 2D-2D/3D- 3D image registration, developed at the Technische Universitat Munchen, Germany. DROP works on image content only and is based around a new Markov Random Field (or Markov network) optimization technique called FastPD [KTP07] [GKP 07], which is able to find almost optimal solutions to many of the NP-hard MRF problems very fast. DROP supports deformable grids and a wide array of different similarity metrics including Mutual Information, SSD and correlation approaches. It also supports a series of penalty systems to control the behavior of the deformation field. See Figure 10 for an example. Figure 11: Registration results from Wein et al Ultrasound - CT [WRN05]. (a) A series of ultrasound images registered to a CT volume rendering. (b) Volume rendering of compounded 3d ultrasound Ultrasound CT registration Ultrasound-to-CT registration techniques have been a focus of research for the last couple of years [BPC 05] [WRN05] [LMPT07]. Matching US and CT data is a difficult task given how different the nature of the images are, but a range of methods that solve this problem to some degree has been proposed. Locating the area which is scanned by US can greatly be helped by tracking the US-probe using a optical or magnetic tracking system, this means that the relative position between the US images can be calculated. If one of these slices are registered manually or the first scan is of a preassigned area one can with some degree of accuracy coarsely place the slices. After this, one needs to rely on image content or markers to further accurately register the images. One such image based technique has been proposed by Wein et al [WRN05], and is really a composite technique built on several weighted similarity measures. Edge detection is used on the CT-data to derive edge structures, the idea is to match these edges against ultrasound intensity values and use this as one of the similarity measures. The next measure is a Normalized Mutual Information (NMI) approach which works on the CT and US intensity values. The algorithm also considers the fact that US-probes compress the skin and tissue of the area where it s place, so a penalty is assigned in this area to prevent some unwanted transformations. The rest of the registration is then performed using a rigid transformation system and a standard optimizer (Different hill climbers and Powell-Brent was used), the result is a system that can do US to CT registration within a few seconds with a few millimeters error margin. Another technique proposed by Barratt et al [BPC 05] uses bone structures as base for a point registration system. Bones show up as occlusions when looked at by ultrasound, intensity values behind such a structure will simply be noise, this means that surfaces of bone structures can be determined. The bone structures are used as points and matched against bone structures in a previously acquired CT-scan of the patient. The system then uses a modified least-squares similarity measure on the points accompanied by a optimizer. The system also calibrates the optical tracking system based on the registration data to further improve the accuracy. It uses a rigid transformation model, and according to tests performed by the authors, it provides accurate registration with a few millimeters error margin in most cases. Other approaches based on freehand ultrasound working on image intensities has been proposed, one presented by Leroy et al [LMPT07] is based on Correlation Ratio (CR) coupled with preprocessing steps such as shadow and speckle removal. Normalized Mutual Information was also tried, but CR gave better results. The system achieves decent accuracy, but is of course not up to pair with a tracked system. 8. Conclusions Figure 10: DROP: 3D-MRI registration using a Cubic B- Spline deformable grid, SSD metric and the FastPD optimizer. (a) Source image, (b) Target image, (c) The two images split in a quad after registration. This report has introduced the field of image registration and shown several of the techniques currently used. Problems dealing with rigid transformation models seems to have been solved successfully [MV97], and the field has focused mostly on different non-rigid models lately. A problem that

8 still remains largely unanswered is the verifiability of correctness for such models [FM08] [CJ07], this is probably a research area that will see a lot of focus over the next couple of years. The focus of the basis of registration seems to have shifted away from extrinsic to intrinsic methods over the last decade, most modern approaches are based on image content only (some can be helped by landmarks). The most popular technique is still Mutual Information which has been gradually refined since it was introduced, research on other possible directions is being done, but MI based approaches are still considered to be state-of-the-art. Many of the multi-modality registration problems have been solved, but research is still very active on some of the more special multi-modal cases, such as CT and ultrasound registration. This will most likely also be a focus of research for years to come. Even state-of-the-art optimizers are very slow when working on large resolution images, large deformation grids and computational heavy metrics - computing time is a large constraint. Some of the more heavier registration techniques can even take a long time on super computers, so there is a need for faster optimization algorithms. References [BPC 05] BARRAT D. C., PENNEY G., CHAN C. S., SLOMCZYKOWSKI M., CARTER T. J., EDWARDS P. J., HAWKES D. J.: Self-calibrating ultrasound-to-ct bone registration. Medical Image Computing and Computer- Assisted Intervention - MICCAI (2005). [Bro70] BROYDEN C. G.: The convergence of a class of double-rank minimization algorithms. Journal of the Institute of Mathematics and Its Applications (1970). [CJ07] CASTRO S., JAVIER F.: Nonrigid medical image registration: algorithms, validation and applications. PhD thesis, EPFL, [FM08] FISCHER B., MODERSITZKI J.: Ill-posed medicine - an introduction to image registration. Inverse problems (2008). [FW06] FARNEBAÌĹCK G., WESTIN C.-F.: Affine and deformable registration based on polynomial expansion. MICCAI (2006). [GKP 07] GLOCKER B., KOMODAKIS N., PARAGIOS N., TZIRITAS G., NAVAB N.: Inter and intra-modal deformable registration: Continuous deformations meet efficient optimal linear programming. In Information Processing in Medical Imaging (2007). [HBHH01] HILL D. L. G., BATCHELOR P. G., HOLDEN M., HAWKES D. J.: Medical image registration. Physics in medicine and biology (IOP) (2001). [HHH01] HAJNAL J. V., HAWKES D. J., HILL D. L. G.: Medical Image Registration. CRC Press, [HM06] HABER E., MODERSITZKI J.: Intensity gradient based registration and fusion of multi-modal images. Medical Image Computing and Computer-Assisted Intervention - MICCAI (2006). [KSP07] KLEIN S., STARING M., PLUIM J. P. W.: Evaluation of optimization methods for nonrigid medical image registration using mutual information and b-splines. IEEE Transactions on Image Processing (2007). [KTP07] KOMODAKIS N., TZIRITAS G., PARAGIOS N.: Fast, approximately optimal solutions for single and dynamic MRFs. In IEEE Computer Vision and Pattern Recognition (2007). [LMPT07] LEROY A., MOZER P., PAYAN Y., TROCCAZ J.: Intensity-based registration of freehand 3d ultrasound and ct-scan images of the kidney. International Journal of Computer Assisted Radiology and Surgery (2007). [LMVS04] LOECKX D., MAES F., VANDERMEULEN D., SUETENS P.: Nonrigid image registration using free-form deformations with a local rigidity constraint. Medical Image Computing and Computer-Assisted Intervention - MICCAI 2004 (2004). [Meh99] MEHIDDIN A.-B.: Improved hessian approximations for the limited memory bfgs method. Numerical Algorithms (1999). [MHV 01] MATTES D., HAYNOR D. R., VESSELLE H., LEWELLEN T. K., EUBANK W.: Nonrigid multimodality image registration. Medical Imaging: Image Processing, SPIE (2001). [MME 93] MORRIS E., MUSWICK G., ELLERT E., STEAGALL R., GOYER P., SEMPLE W.: Computer-aided techniques for aligning interleaved sets of nonidentical medical images. Proc. SPIE (1993). [MRN 02] MANKE D., ROSCH P., NEHRKE K., BORNERT P., DOSSEL O.: Model evaluation and calibration for prospective respiratory motion correction in coronary mr angiography based on 3-d image registration. Medical Imaging, IEEE Transactions on (2002). [MV97] MAINTZ J. B. A., VIERGEVER M. A.: A Survey of Medical Image Registration. Tech. rep., Image Sciences Institue, Utrecht University Hospital, [PMV99] PLUIM J. P. W., MAINTZ J. B. A., VIERGEVER M. A.: Mutual information matching and interpolation artefacts. SPIE Medical Imaging (1999). [PMV03] PLUIM J. P. W., MAINTZ J. B. A., VIERGEVER M. A.: Mutual information based registration of medical images: a survey. IEEE transactions on medical imaging (2003). [RM51] ROBBINS H., MONRO S.: A stochastic approximation method. Ann. Math. Statist. (1951). [RSH 99] RUECKERT D., SONODA L., HAYES C., HILL

9 D., LEACH M., HAWKES D.: Non-rigid registration using free-form deformations: application to breast mr images. IEEE Trans. Med. Imaging (1999). [RTL 93] RUSINEK H., TSUI W., LEVY A. V., NOZ M. E., DE LEON M. J.: Principal axes and surface fitting methods for three- dimensional image registration. Journal of nuclear medicine, 34 (1993). [SFR 07] SBERT M., FEIXAS M., RIGAU J., VIOLA I., CHOVER M.: Applications of information theory to computer graphics - part i: Introduction. Eurographics (2007). [SHH97] STUDHOLME C., HILL D. L. G., HAWKES D. J.: Automated three-dimensional registration of magnetic resonance and positron emission tomography brain images by multiresolution optimization of voxel similarity measures. Medical Physics (1997). [SHH99] STUDHOLME C., HILL D. L. G., HAWKES D. J.: An overlap invariant entropy measure of 3d medical image alignment. Pattern Recognition (1999). [UAE93] UNSER M., ALDROUBI A., EDEN M.: B-spline signal processing. IEEE Trans. Signal process (1993). [Vio95] VIOLA P. A.: Alignment by Maximization of Mutual Information. PhD thesis, MIT, [VW97] VIOLA P., WELLS W.: Alignment by maximization of mutual information. International Journal of Computer Vision (1997). [WGW 98] WOODS R. P., GRAFTON S. T., WATSON J. D. G., SICOTTE N. L., MAZZIOTTA J. C.: Automated image registration. J. Comput. Assist. Tomogr. (1998). [WRN05] WEIN W., ROPER B., NAVAB N.: Automatic registration and fusion of ultrasound with ct for radiotherapy. Medical Image Computing and Computer-Assisted Intervention - MICCAI (2005). [WVA 96] WELLS W. M., VIOLA P., ATSUMI H., NAKAJIMA S., KIKINIS R.: Multi-modal volume registration by maximization of mutual information. Medical Image Analysis (1996). [YM05] YOO T. S., METAXAS D. N.: Open science - combining open data and open source software: Medical image analysis with the insight toolkit. Medical Image Analysis (2005). S.Eikeland / Multimodal Medical Data Registration