CT and MRI Data Processing for Rapid Prototyping

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1 CT and MRI Data Processing for Rapid Prototyping VACLAV SKALA Department of Computer Science and Engineering University of West Bohemia Univerzitni 8, CZ Plzen CZECH REPUBLIC Abstract: - Volumetric data are very often used in engineering and medical applications. Surface extraction methods for rapid prototyping differ from methods for visualization. This paper presents a pragmatic approach for generating data for rapid prototyping systems enabling production of physical models. In visualization systems only a surface is usually generated regardless to the restrictions of the actual rapid prototyping systems, while in visualization it is necessary to accept physical properties of the physical rapid prototyping systems. CAD/CAM systems mostly rely on general tetrahedral meshes, while medical or industrial CT and MRI data are organized in structured rectangular regular meshes. This type data structures enables fast and robust data processing for visualization, while for the rapid prototyping more interaction of a user is needed. Identification of duplicities, finding of identical points is made by using hash data structure and new hash function for geometrical purposes is described. Experimental results proved the correctness of the approach. Key-Words: - Rapid prototyping, surface extraction, CT, MRI, volumetric data, hash function, CAD/CAM 1 Introduction Rapid prototyping is a method for producing 3D physical models of objects, mostly for used for verification of designed mechanical parts. There are wide applications in fields of medical data processing, especially in production of specific parts for replacing missing body parts etc. There are two major data structures representations used for representing objects, i.e. surface description mostly based on triangular meshes and 3D tessellations techniques, usually tetrahedral meshes or structured 3D orthogonal meshes, like CT and MRI images. Orthogonal structured meshes have a significant advantage over the unstructured tetrahedral meshes as they are easy to store, manipulate and process. Let us assume a typical example of 3D structured orthogonal mesh, i.e. a 3D mesh with values representing physical phenomena in the corner points. This is a very representative case for medical CT and MRI images and also for industrial CT scanners. The structured orthogonal meshes with associated values in the corners are called volumetric data. Let us imagine that we have some physical phenomena in 3D, e.g. a tissue absorption in medical applications etc. Our task is to extract boundary of a specific tissue, e.g. a skull, from the given data set and produce a physical model of it. This seems to be a very simple task. Nevertheless there are several critical points in 3D printing as: we need to extract not only a surface, but a volume actually as if the model should be physically produced we need to generate a volume, not just infinitely thin surface, the generated volume must respect physical restriction of a device intended for the production of the physical 3D model, some parts must be supported by an additional volume structure due to fixing, as non connected parts need to be positionally fixed as they are fixed by softer or harder tissues in the case of medical data or by different material in the case of industrial data. 2 Iso-surface Extraction Let us assume a simple case, when a cell of an orthogonal structured mesh is used and values are stored in the corners of the mesh, see Fig.1. Fig.1 Values are stored in vertices of the 3D cubical mesh ISBN:

2 In the case of visualization of volumetric data we are extracting a surface as an iso-surface, i.e. a surface with a constant value, i.e. by the given threshold. The majority of techniques used are targeted to visualization, i.e. rendering of the iso-surface extracted for a specific value, e.g. a tissue, bone etc. Our goal is not primarily visualization of a surface but a production of physical 3D model made of some hard material. Such model is usually used for verification before the mechanical part is produced from steel or from a special material in medical applications, e.g. in replacing missing or damaged parts of a bone or a skull. 3D printers have been extremely expensive and for small companies, university laboratory and home use were totally out of the question. Nevertheless, nowadays simple 3D printers are affordable even for a domestic use, see Fig.2. Such simple 3D printers have several restrictions, but enable to print quite complicated 3D models. Of course there are more expensive 3D printers based on different principles. algorithms in general with some specific modifications. There are standard techniques, like Marching Cubes. Marching Tetrahedral, Cubic Lattice etc. Those techniques rely on a tessellation of the given structured mesh to tetrahedral meshes followed by a triangular mesh generation. Actually a set of triangles is actually generated due to computational scheme of such iso-surface extraction scheme. There are several schemes for a cube tessellation, like Marching Cubes (MC), Marching Tetrahedra (MT) with different tessellation details, i.e. MT5, MT6, MT12, MT24, MT48, or a Lattice Cells very often used in visualization of technical and medical data and standard datasets are used in computer visualization courses for demonstration of different techniques. The MC technique is actually based on a sliceby-slice processing of a single cube, mostly in the x- y with successive z slicing. It is important to note that in this case only two slices are needed to be stored in the memory, i.e. the recent and the actual ones. It reduces the memory requirements significantly and enables better caching those results into processing speed up. Also from the implementation point of view the 3D data structure is to be mapped into 1D data structure to eliminate mapping function T[i,j,k] Q[kk]. The standard data sets are usually of the size [B], i.e. approx of processed values is required with the request of real-time processing for visualization. Fig.2 Simple 3D printer Algorithms for application with 3D printers have some very specific requirements - mostly due to physical properties, restrictions etc. Such algorithms are generally based on iso-surface extraction Fig.3 Marching Cube The simplest technique Marching Cube rely on evaluation of a threshold value given by a user or application for the iso-surface extraction. Such evaluation splits corners to two sets and ISBN:

3 determines the intersection of the iso-surface with the given cube, see Fig.3. It can be seen that there are 256 possible situations, but if operations like rotation, mirroring etc. are used, only 15 fundamental cases have to be implemented. However, there are several tricky points in the actual implementation. There are other tessellation schemes and tessellation to tetrahedronal meshes. One possibility is to use tessellation of a cube to 5 (MT5) or 6 (MT6) tetrahedra. Application of this tessellation requires a careful implementation as two edges of neighboring tetrahedrals do not share a common edge in general, so rotation operation is required. The advantage of the MT5 or MT6 is that iso-surface intersection with a tetrahedron produces a triangle or a four sided non-planar polygon easy to split into two triangles. Other methods like MT12, MT24, MT48 are primarily based on inserting new points to the centre of a cube, to a center of a face or to the centre of an edge. Those scheme results into a surface with more triangles in general. The above mentioned techniques actually tessellate one cube only. There is an alternative approach based on tessellation of two neighboring cubes called Cubic Lattice, see Fig.4.b. This scheme has a significant advantage over the Marching Cubes or Marching Tetrahedra techniques as all the tetrahedra generated have the same shape, while in MC or MT tetrahedra generated do have different shapes. a) 5/6 tetrahedra scheme b) Lattice cubic scheme Fig.4 Different tessellations Fig.5a Typical example of the rendered data D E F B A C Fig.5b MT polygons There are several disadvantages of MT5-MT48 techniques. The most important are: if MTx scheme is used, more triangles are generated for higher x. for higher x rendered images are more pleasant for an observer, but the precision is decreased as far as the volume and surface is concerned. It can be seen that the MT24 or MT48 gives more pleasant rendered image, but the surface generated consists of much more triangles that result into higher surface generated at the end. In the case of the rapid prototyping a selection of methods for isosursurface is to be selected according to a resolution of the actual 3D printer and physical properties. 3 Rapid Prototyping So far we have dealt mostly with methods aimed for visualization purposes. Those methods: are targeted to speed as a user wants to see the rendered image in a real time do not necessarily produce a triangular mesh - mostly single triangles are generated and normal vectors in vertices for smooth rendering are computed as an interpolation of estimated gradient in the orthogonal mesh do not care too much about consistency of the iso-surface generated as this is not important for the visual perception. On the other hand methods for rapid prototyping have some specific requirements, e.g.: the surface generated must be closed, as it actually defines a volume to be printedgenerated triangles should be of a reasonable size with regard to the printer resolution and the object size all non-connected parts have to be connected somehow, otherwise the object will be fragmented, e.g. the spine would appeared as separated from a skull and also all verbata must be somehow connected together, see Fig.5.a. in the case of hollow object, there should be made some holes, for unused powder removal (it is also related to the final cost of the object generated) if the actual 3D printer is based on a powder technology, an additional base block is usually added to fix a position if supporting is needed, see Fig.12. There is one more factor that is not so simple, as 3D printers do have limited resolution in print. The thickness of the generated volume must be higher than a minimal value. This can be made semiautomatically only as the user must decide which ISBN:

4 part(s) of a surface can be moved to fit the printer s properties. The orthogonal mesh structure enables fast and simple solution. The problem is hidden in the fact, that we need to find all triangles sharing the given vertex, i.e. for every vertex, we have to process all triangles and as we have N vertices and vertex is shared approx. by 5-6 triangles, the final computational complexity of O(N 2 ) complexity. vertex coordinates hash table x Index = f(x) in dex cluster pointers to the vertex_array free vertex (x,y,z) vertex_array Fig.6 Detail of the generated images for MT5, MT6, MT12, MT24 schemes j-th triangle i-th triangle triangle_array free triangle As the generated surface consists of triangles of different sizes, the following two steps are usually needed: reconstruction of a triangular mesh from a set of triangles, i.e. we need to reconstruct information which triangles share the vertex and information about triangle neighbors, reduction of the mesh generated as many small triangles should be removed due to processing time and limited memory in some cases as well. Fig.8 Data structure for triangular mesh reconstruction One possibility how to decrease the computational complexity is to use hash function modified for geometrical purposes. Well designed hash function enables to find a vertex stored with the hash data structure with O(1) complexity, i.e. all points can be found with O(N) complexity. It can be seen that the data structure is a little bit more complicated in comparison with the pure hash data structure, see Fig.7. It is due to necessity to store information on triangles sharing the vertex and also storing x,y,z coordinates of the vertex. The design of the hash function for geometrical purposes is described in [5]. The hash data structure enables very fast processing as far as the mesh reconstruction is concerned. After this step, triangular mesh consistency check must be used and any inconsistency of the mesh must be corrected. Fig.7 Hash data structure As we are processing many triangles, usually of triangles the process for reconstruction of a triangular mesh from the set of triangles is quite complicated and takes very long time. Naive implementation is of O(N 2 ) complexity and processing time can be estimated in hours. Fig.9 Car chassis approx of vertices ISBN:

5 Applications of standard hash function design leads to long clusters of different points having the same index to the hash table. According to the experiments made, the best hash function can be described as index = C *( α x + β y + γ z) mod2 n where α, β, γ are irrational numbers, C is a constant so that (α+β+γ)*c = , if index is implemented as 64 bit unsigned integer, and 2 n is the length of the hash table. It is expected that x,y,z values are transformed to <0,1> interval. This can be done easily if the x min and x max values are known or by transformed values for addressing purposes x x ' = (1 + )*0.5 x + k Number of the clusters cluster length Fig.10 Car chassis, Q = 10-3, α = 3, β = 5, γ = 7 (Q insensitivity factor- see Glassner) Coefficients α = 2, β = 3, γ = ek, = 10 were used in experiments. As it can be seen, the properly developed geometric hash function can speed up the processing time significantly, see Fig. 10 and Fig.11. When the mesh is reconstructed and consistent the mesh reduction is applied. This step removes small triangles and also nearly flat parts replaces by smaller number of triangles. Principles of a mesh reduction are described in [3] in detail. Number of buckets Fig.12 Example of a skull generated 4 Special Operations When the data set is finally consistent and reduced the last step has to be performed, i.e. connection of non-connected parts, creation of the supporting box and making holes for the removal of powder or similar mass which is not part of the produced object. There is well known Computer Solid Geometry (CSG) technique which enables to perform quite complicated operations on complex objects using operations like union, intersection, subtraction etc. The CSG operations are defined on implicit functions F(x) = 0. In our case of the rapid prototyping applications the same principles were used. Of course, the operations are defined in the context of the rectangular mesh and for discrete data sets. The connection of non-connected parts is made as a union of those parts with a simple object, like cylinder or similar, which connects those nonconnected parts together. This operation in the orthogonal mesh directly is quite simple and fast. Making holes into the object for the unused powder removal is simple as well as it is made as an subtraction of the object with a properly positioned cylinder which actually creates a hole into the generated object. All those operations were performed on CPU as the processing is fast enough for the currently processed data sets. Nevertheless use of GPU with a significant speed up can be made especially for the final operations with the orthogonal mesh and significant speed-up can be achieved. Bucket length Fig.11 Histogram of bucket lengths ISBN:

6 5 Experimental Results The presented approach has been used for making objects using rapid prototyping system, i.e. 3D printers. Algorithms and techniques used are quite easy to implement. Software design implementation and functionality verification was made as a part of student coursework. 6 Conclusion Basic principles for generating data from volumetric data, namely from CT and MRI images, for rapid prototyping systems have been presented in this paper. As the volumetric data are based on discrete rectangular mesh the processing is robust and algorithms are simple. For a fast processing geometrical hash function was used. It enables fast triangular mesh reconstruction from the set of extracted triangles by Marching Cube, Marching Tetrahedra or Cubic Lattice techniques. Acknowledgment The author would like to thank to colleagues at the University of West Bohemia for their comments, to students of computer graphics for their effort in implementation, especially to Marek Krejza and Martin Kuchar, and for production of a physical model, especially to Jan Rus, who handled data and their correctness, to Petr Pelikan from the Institute of Art for the physical 3D printing. Thanks belong also to Dr.Rongjiang Pan and colleagues from the Shandong University and Zhejiang University, China, who stimulated some thoughts during their research stay in Plzen. Research was supported by the Ministry of Education of the Czech Republic, projects No.ME10060, LH12181, LA10035 J.F., Soni, B.K., Weatherill, N.P.(Eds.), Handbook of Grid Generation. CRC Press, Boca Raton, FL, pp. 14 1: [5] Hradek,J., Skala,V.: Hash Function and Triangular Mesh Reconstruction, Vol.29, No.6., pp , Computers&Geosciences, Pergamon Press, ISSN , 2003 [6] Korfhage, R.R., Gibbs, N.E., Principles of Data Structures and Algorithms with Pascal. Wm. C. Brown Publishers, McGraw-Hill Science/Engineering/Math. Dubuque, Iowa. 480pp. [7] Knuth, D., The Art of Computer Programming. Vol. 3, Sorting and Searching, 2nd Edition. Addison-Wesley, Reading, MA, pp.736. [8] Skala,V., Kuchar,M.: The Hash Function and the Principle of Duality, Computer Graphics International 2001, Honk Kong, China, IEEE Proceedings, ISSN , pp , [9] Skala,V., Hradek,J., Kuchar,M.: Hash Function for Triangular Mesh Reconstruction, 13th International Conference on COMPUTERS, WSEAS, pp , ISBN: , 2009 [10] Skala,V., Hradek,J., Kuchar,M.: New Hash Fuction Construction for Textual and Geometrical Data Retrieval, Latest Trends on Computers, Vol.2, pp , ISBN , ISSN , CSCC conference, Corfu, Greece, 2010 [11] Wolfson, H.J., Rigoutsos, I., Geometric hashing: an overview. IEEE Computational Science and Engineering 4 (4), [12] Yao, A.C., Uniform hashing is optimal. Journal of the Association for Computing Machinery 32 (3), References: [1] Foley,J.D., van Dam,A., Feiner,S.K., Huges,J.F.: Computer Graphics, Principle and Practice. Addison-Wesley, [2] Glassner, A.: Building Vertex Normals from an Unstructured Polygon List, Graphic Gems IV, Academic Press, Inc., [3] Franc,M., Skala,V.: Triangular Mesh Decimation in Parallel Environment, EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, [4] Formaggia, L., Data structures for unstructured mesh generation. In: Thomson, ISBN: