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2012, 9(3): 527-537. doi: 10.3934/mbe.2012.9.527

## Fast two dimensional to three dimensional registration of fluoroscopy and CT-scans using Octrees on segmentation maps

 1 ECE Department, UCSB, Santa Barbara, CA 93106, United States 2 Department of Computer Science and Department of Mechanical Engineering, University of California at Santa Barbara, CA 93106-5070

Received  April 2011 Revised  May 2012 Published  July 2012

We introduce a computationally efficient approach to the generation of Digital Reconstructed Radiographs (DRRs) needed to perform three dimensional to two dimensional medical image registration and apply this algorithm to virtual surgery. The DRR generation process is the bottleneck of any three dimensional to two dimensional registration system, since its computational complexity scales with the number of voxels in the Computed Tomography Data, which can be of the order of tens to hundreds of millions. Our approach originates from the segmentation of the volumetric data into multiple regions, which allows a compact representation via Octree Data Structures. This, in turn, yields efficient storage and access of the attenuation indexes of the volumetric cells, required in the projection procedure that generates the DRR. A functional based on Mutual Information is then maximized to obtain the alignment of the DRR with the two dimensional X-ray fluoroscopy scans acquired during the operation. Promising experimental results on real data are presented.
Citation: Luca Bertelli, Frédéric Gibou. Fast two dimensional to three dimensional registration of fluoroscopy and CT-scans using Octrees on segmentation maps. Mathematical Biosciences & Engineering, 2012, 9 (3) : 527-537. doi: 10.3934/mbe.2012.9.527
##### References:
 [1] L. Bertelli, S. Chandrasekaran, F. Gibou and B. Manjunath, On the length and area regularization for multiphase level set segmentation, International Journal on Computer Vision, 90 (2010), 267-282. doi: 10.1007/s11263-010-0348-4.  Google Scholar [2] T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291.  Google Scholar [3] F. Gibou and R. Fedkiw, Fast hybrid k-means level set algorithm for segmentation, Technical report, Stanford, 2002, Also in proceeding of the $4^{th}$ International Conf. on Stat., Math. and Related Fields, Honolulu, 2005. Google Scholar [4] M. Levoy and P. Hanrahan, Light field rendering, Computer Graphics SIGGRAPH, 30 (1996), 31-42. Google Scholar [5] C. Min, Local level set method in high dimension and codimension, J. Comput. Phys., 200 (2004), 368-382. doi: 10.1016/j.jcp.2004.04.019.  Google Scholar [6] C. Min and F. Gibou, A second order accurate level set method on non-graded adaptive Cartesian grids, J. Comput. Phys., 225 (2007), 300-321. doi: 10.1016/j.jcp.2006.11.034.  Google Scholar [7] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.  Google Scholar [8] D. Russakoff, T. Rohlfing and C. R. Maurer, Fast intensity-based 2d-3d image registration of clinical data using light fields, IEEE International Conference on Computer Vision, 1 (2003), 416-422. Google Scholar [9] H. Samet, "The Design and Analysis of Spatial Data Structures," Addison-Wesley, New York, 1989. Google Scholar [10] H. Samet, "Applications of Spatial Data Structures: Computer Graphics, Image Processing and GIS," Addison-Wesley, New York, 1990. Google Scholar [11] J. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Natl. Acad. Sci. U.S.A., 93 (1996), 1591-1595. doi: 10.1073/pnas.93.4.1591.  Google Scholar [12] B. Smits, Efficient bounding box intersection, Ray Tracing News, 15 (2002). Google Scholar [13] J. Strain, Tree methods for moving interfaces, J. Comput. Phys., 151 (1999), 616-648. doi: 10.1006/jcph.1999.6205.  Google Scholar [14] J. Tsitsiklis, Efficient algorithms for globally optimal trajectories, IEEE Trans. on Automatic Control, 40 (1995), 1528-1538. doi: 10.1109/9.412624.  Google Scholar [15] L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the mumford and shah model, International Journal of Computer Vision, (2002), 271-293. doi: 10.1023/A:1020874308076.  Google Scholar [16] A. Williams, S. Barrus, R. K. Morley and P. Shirley, An efficient and robust ray-box intersection algorithm, International Conference on Computer Graphics and Interactive Techniques, 2005. Google Scholar [17] L. Zollei, E. Grimson, A. Norbash and W. Wells, 2D-3D rigid registration of x-ray fluoroscopy and ct images using mutual information and sparsely sampled histogram estimators, IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), 2 (2001), 696-673. Google Scholar

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##### References:
 [1] L. Bertelli, S. Chandrasekaran, F. Gibou and B. Manjunath, On the length and area regularization for multiphase level set segmentation, International Journal on Computer Vision, 90 (2010), 267-282. doi: 10.1007/s11263-010-0348-4.  Google Scholar [2] T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291.  Google Scholar [3] F. Gibou and R. Fedkiw, Fast hybrid k-means level set algorithm for segmentation, Technical report, Stanford, 2002, Also in proceeding of the $4^{th}$ International Conf. on Stat., Math. and Related Fields, Honolulu, 2005. Google Scholar [4] M. Levoy and P. Hanrahan, Light field rendering, Computer Graphics SIGGRAPH, 30 (1996), 31-42. Google Scholar [5] C. Min, Local level set method in high dimension and codimension, J. Comput. Phys., 200 (2004), 368-382. doi: 10.1016/j.jcp.2004.04.019.  Google Scholar [6] C. Min and F. Gibou, A second order accurate level set method on non-graded adaptive Cartesian grids, J. Comput. Phys., 225 (2007), 300-321. doi: 10.1016/j.jcp.2006.11.034.  Google Scholar [7] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.  Google Scholar [8] D. Russakoff, T. Rohlfing and C. R. Maurer, Fast intensity-based 2d-3d image registration of clinical data using light fields, IEEE International Conference on Computer Vision, 1 (2003), 416-422. Google Scholar [9] H. Samet, "The Design and Analysis of Spatial Data Structures," Addison-Wesley, New York, 1989. Google Scholar [10] H. Samet, "Applications of Spatial Data Structures: Computer Graphics, Image Processing and GIS," Addison-Wesley, New York, 1990. Google Scholar [11] J. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Natl. Acad. Sci. U.S.A., 93 (1996), 1591-1595. doi: 10.1073/pnas.93.4.1591.  Google Scholar [12] B. Smits, Efficient bounding box intersection, Ray Tracing News, 15 (2002). Google Scholar [13] J. Strain, Tree methods for moving interfaces, J. Comput. Phys., 151 (1999), 616-648. doi: 10.1006/jcph.1999.6205.  Google Scholar [14] J. Tsitsiklis, Efficient algorithms for globally optimal trajectories, IEEE Trans. on Automatic Control, 40 (1995), 1528-1538. doi: 10.1109/9.412624.  Google Scholar [15] L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the mumford and shah model, International Journal of Computer Vision, (2002), 271-293. doi: 10.1023/A:1020874308076.  Google Scholar [16] A. Williams, S. Barrus, R. K. Morley and P. Shirley, An efficient and robust ray-box intersection algorithm, International Conference on Computer Graphics and Interactive Techniques, 2005. Google Scholar [17] L. Zollei, E. Grimson, A. Norbash and W. Wells, 2D-3D rigid registration of x-ray fluoroscopy and ct images using mutual information and sparsely sampled histogram estimators, IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), 2 (2001), 696-673. Google Scholar
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