Kluwer - Handbook of Biomedical Image Analysis Vol

which a user can modify the result of a segmentation in any way desired. There are no limitations in shape, size, or complexity of a region under consideration. The only requirement is that the given region have a spherical topology.

7.4 Conclusions

Image segmentation is an important component of any image analysis system. In medical imaging, it is essential that an image is accurately segmented so that different measurements about the region are accurately determined. In this paper, the idea of using a computer-aided design system to effectively revise the result of an automatically determined segmentation was introduced. In the proposed system, a RaG surface is fitted to voxels representing a 3D region by the least-squares method. The surface and the original volumetric image are then overlaid and the surface is interactively revised until the desired segmentation is achieved.

The system provides the option of using the output of an automatically obtained segmentation as the input or manually creating an initial segmentation by selecting a number of 3D points in the given image volume. In the latter case, an initial surface is created from the points and overlaid with the image. The user can then observe the image data and revise the surface to a desired shape. Because a region of interest is represented by a parametric surface, the surface may be sent to a computer-aided manufacturing system for construction of an actual 3D model of the region.

7.5 Acknowledgements

The image used in Fig. 7.10 is from Georg Gotschuli and Erich Sorantin, University of Garz, Austria; the cardiac MR image in Fig. 7.9 is from David Turner, Presbyterian-St. Luke’s Hospital, Chicago, IL; and the MR brain image containing only the brain in Fig. 7.9 is from Terry Oaks, University of Wisconsin, Milwaukee. We appreciate all these contributions.

Bibliography

[1]Barrett, W. and Mortensen, E., Fast, accurate, and reproducible live-wire boundary extraction, In: Proc. Visualization in Biomedical Computing, Hamburg, 1996.

[2]Brechbuhler,¨ Ch., Gerig, G., and Kubler,¨ O., Surface parametrization and shape description, In: SPIE Workshop on Visualization in Biomedical Computing, Vol. 1808, pp. 80–89, 1992.

[3]Brechbuhler,¨ Ch., Gerig, G., and Kubler,¨ O., Parametrization of closed surfaces for 3D shape description, Computer Vision and Image Understanding, Vol. 61, No. 2, pp. 154–170, 1995.

[4]Cabral, Jr., J. E., White, K. S., Kim, Y., and Effmann, E. L., Interactive segmentation of brain tumors in MR images using 3D region growing, SPIE Medical Imaging Conference, Vol. 1989, pp. 171–179, 1993.

[5]Cagnoni, S., Dobrzeniecki, A. B., Ploi, R., and Yanch, J. C., Genetic algorithm-based interactive segmentation of 3D medical images, Image and Vision Computing, Vol. 17, pp. 881–895, 1999.

[6]Canny, J., A Computational Approach to Edge Detection, IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 8, No. 6, pp. 679–698, 1986.

[7]Falcao,˜ A. X. and Udupa, J. K, A 3D generalization of user-steered live-wire segmentation, Medical Image Analysis, Vol. 4, pp. 389–402, 2000.

[8]Falcao,˜ A. X., Udupa, J. K., Samarasekera, S., Sharma, S., Hirsch, B. E., and Lotufo, R. A., User-steered image segmentation paradigms: livewire and live-lane, Graphical Models and Image Processing, Vol. 60, No. 4, pp. 233–260, 1998.

[9]Falcao,˜ A. X., Udupa, J. K., and Miyazawa, F. K., An ultra-fast usersteered image segmentation paradigm: live-wire-on-the-fly, In: SPIE Medical Imaging Conference, San Diego, CA, Vol. 3661, pp. 184–191, 1999.

[10]Galvez, J. M. and Canton, M., Normalization and shape recognition of three-dimensional objects by 3D moments, Pattern Recognition, Vol. 26, No. 5, pp. 667–682, 1993.

[11]Goshtasby, A., Design and recovery of 2D and 3D shapes using rational Gaussian curves and surfaces, International Journal of Computer Vision, Vol. 10, No. 3, pp. 233–256, 1993.

[12]Goshtasby, A., Geometric modeling using rational Gaussian curves and surfaces, Computer-Aided Design, pp. 363–375, May 1995.

[13]Goshtasby, A., Parametric circles and spheres, Computer-Aided Design, Vol. 35, May 2003, pp. 487–494.

[14]Hinshaw, K. P. and Brinkley, J. F., Shape-based interactive threedimensional medical image segmentation, SPIE Medical Imaging Conference, Vol. 3034, pp. 236–242, 1997.

[15]Hohne,¨ K. H. and Hanson, W. A., Interactive 3D segmentation of MRI and CT volumes using morphological operations, Journal of Computer Assisted Tomography, Vol. 16, No. 2, pp. 258–294, 1992.

[16]Hoschek, J. and Lasser, D. Computer Aided Geometric Design, A. K. Peters, pp. 289–291, 1989.

[17]Jackowski, M., Satter, M., and Goshtasby, A., Approximating digital 3D shapes by rational Gaussian surfaces, IEEE Trans. Visualization and Computer Graphics, Vol. 9, No. 1, Jan–March 2003, pp. 56–69.

[18]Kass, A., Witkin, A., and Terzopoulos, D., Snakes: Active contour models, International Journal of Computer Vision, Vol. 1, pp. 321–331, 1987.

[19]Krivanek, A. and Sonka, M., Ovarian ultrasound image analysis: Follicle segmentation, IEEE Trans. Medical Imaging, Vol. 17, No. 6, pp. 935–944, 1998.

[20]McInerney, T. and Terzopoulos, D., A dynamic finite element surface model for segmentation and tracking in multidimensional medical images with application to cardiac 4D image analysis, Computerized Medical Imaging and Graphics, Vol. 19, No. 1, pp. 69–83, 1995.

[21]Mortensen, E. N. and Barrett, W. A., Interactive segmentation with intelligent scissors, Graphical Models and Image Processing, Vol. 60, No. 5,

pp.349–384, 1998.

[22]Mortensen, E. N., Morse, B. S., Barrett, W. A., and Udupa, J. K., Adaptive boundary detection using live-wire two-dimensional dynamic programming, In: IEEE Proc. Computer in Cardiology, pp. 635–638, 1992.

[23]Olabarriaga, S. D. and Smeulders, A. W. M., Interaction in the segmentation of medical images: A survey, Medical Image Analyis, Vol. 5,

pp.127–142, 2001.

[24]Pizer, S. M., Cullip, T. J., and Fredericksen, R. E., Toward interactive object definition in 3D scalar images. In: 3D Imaging in Medicine, K. H. Hohne¨ et al., eds., Springer, Berlin: pp. 83–105, 1990.

[25]Staib, L. H. and Duncan, J. D., Model-based deformable surface finding for medical images, IEEE Trans. Medical Imaging, Vol. 15, No. 5,

pp.720–731, 1996.

[26]Udupa, J. K., Samarasekera, S., and Berrett, W. A., Boundary detection via dynamic programming, Proc. SPIE, Vol. 1808, pp. 33–39, 1992.

[27]Welte, D., Grunert, T., Klose, U., Petersen, D., and Becker, E., Interactive 3D segmentation and visualization of vessels, Proceedings of Computer Assisted Radiology, pp. 329–335, 1996.

list of methods but to present generic and up-to-date methods. The interested reader will refer to [19, 60, 61, 87, 90, 91, 140, 143, 151] for a complete survey on this subject. This section will therefore be restricted to an overview with classification of non-rigid registration methods, more particularly applied to non-rigid registration of brains of different subjects.

Methods can generally be classified according to the following criteria:

Features that will be matched. This includes both the dimension of the data (classically from 2D to 4D) as well as the homologous structures that are chosen for matching.

Transformation type. This includes the transformation domain: local or global. A transformation is called “global” when the modification of one parameter affects the entire image. This also includes the transformation type (rigid, affine, projective and so on).

The similarity measure. The similarity models the interaction between the data (features used for matching defined above) and the variables to be estimated (parameters of the transformation for instance).

The regularization. The regularization can be implicit (regularized transformation model for instance) or explicit (first-order regularization for instance).

The optimization method. Once the registration problem has been formalized, the optimization plays a crucial role in estimating the registration variables.

We have chosen to divide non-rigid registration methods into two classes: geometrical methods that are based on the extraction and matching of sparse features; and photometric (or intensity-based) methods that exploit luminance information directly.

8.2.2 Geometric Methods

The amount of data in a 3D MR image is enormous: it contains more than 10 million voxels. The computation of a dense deformation field is a tough problem: more than 40.106 variables have to be estimated. This complexity has motivated geometric methods: sparse anatomical features reduce the dimension of the

problem. Methods that are presented here extract geometrical features from images and compute a transformation that matches these features while interpolating smoothly the deformation throughout the image.

8.2.2.1 Points

Earliest methods rely on points. The most famous one, which is still a reference in the field of neuroscience, is the Talairach stereotaxic space [130]. It has then been extended to the Talairach proportional squaring system [132]. Both methods rely on the identification of the anterior comissure AC and posterior comissure PC, as well as five brain extrema which makes it possible to specify a partition of the volume into 12 subvolumes. The transformation associated with the Talairach proportional squaring system is a piecewise linear one that makes it possible to embed the brain into a “box” centered at AC and whose anatomical axes are known. This framework is known to be quite accurate in the central region but less accurate for cortical areas.

Other authors have proposed methods based on anatomical points to register brains of different subjects [16, 26, 50, 116]. However, the number of points that can be reproducibly identified among a population of subjects is limited. It has been evaluated as 36 [38] or 26 [50]. This number of points seems limited to understand the intersubject variability; in addition to this, the extraction step might be erroneous. To limit the dependency toward extraction, some authors have proposed differential geometry operators to automate the process [135, 114, 115].

8.2.2.2 Curves

Gueziec´ [66], Subsol [126] and Declerck [41] describe methods to register two volumes thanks to curves: smoothing and curve matching in [66], application to the registration of brains in [41], building of skull atlases in [126]. Crest lines, introduced by Monga et al. [100], are defined as maximal curvature points and can be automatically extracted using the marching lines algorithm [137].

Gueziec et al. [66] approximate curves using B-splines. This enables the direct computation of features such as position, curvature and so on. Curves are then registered using an iterative approach like the Kalman filter. Subsol [126] and Declerck [41] have extended the ICP algorithm (Iterative closest points

proposed by Zhang [150] and Besl [12]). The iterative approach matches each point with the closest point on the target curve. [41] models the transformation as a B-spline while [126] estimates successively a rigid, affine, polynomial and spline transformation.

8.2.2.3 Surfaces

In 3D medical imaging, anatomical structures are likely to be closed surfaces than points or curves. Several authors [51, 56, 138, 128] have therefore proposed methods to register brain surfaces (like ventricles, central nuclei, brain surface). In a nutshell, deformable models have been extensively used for segmentation of medical images [8, 15, 94]. They can lead either to an explicit representation of contours (e.g., snakes) or to an implicit representation of contours (e.g., level sets [124]).

Thompson and Toga [138] have proposed an original method based on the extraction and matching of the cortical surface. Surfaces are first modeled by a superquadric [25] than refined by the “balloons” method [32]. Feldmar et al.

[52] have extended the curve matching methods (see Section 8.2.2 to the registration of free-form surfaces. The iterative approach matches points of similar curvature.

8.2.2.4 From “Contour” to Volume

We have previously presented registration methods based on the extraction and matching of sparse features. The next step is to extrapolate smoothly the deformation to the entire volume. To do so, two kinds of approaches can be used: thin plate spline methods and free form deformations.

The thin plate spline (TPS) approach consists in minimizing a functional under constraints. Initiated by Duchon [47] and Meinguet [95], this method is now widespread, thanks to the work of Bookstein [16]. The problem can be

formulated as the minimization of:

min |mu (x)|dx, under the constraint i E, u (ai) = αi,

u

where u is the deformation field, E is the set of contour points, ai being matched with αi. Bookstein describes the TPS method that minimizes the energy of a thin plate under constraint. The solution can be expressed as local