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Reconstruction of Voxels from Sensor Data

Reconstruction of Voxels from Sensor Data. Ricardo Martins. Coimbra, 19 th January 2010. Computer Graphics and 3D Modeling. Doctoral Programme in Electrical Engineering and Computer Science. Contents. 3D object representation Solid modeling representation *Voxel *Octree

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Reconstruction of Voxels from Sensor Data

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  1. Reconstruction of Voxels from Sensor Data Ricardo Martins Coimbra, 19th January 2010 Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  2. Contents • 3D object representation • Solid modeling representation *Voxel *Octree • Data Acquisition/Conversion *Computer Tomography *Reconstruction of octrees from range data *Voxelization *Surface reconstruction from volumetric data • Volume Graphics vs Surface Graphics • References Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  3. 3D Object Representation • Points -Range images -Point cloud  Solids-Voxels -Octrees -Binary Space Partitions - BSP Trees -Construtive Solid geometry - CSG  High Level Structures -Scene Graph -Application specific • Surfaces-Polygonal mesh -Subdivision surfaces -Parametric surfaces: -Implicit surfaces Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  4. Solid Modeling Representation  Solids-Voxels -Octrees -Binary Space Partitions - BSP Trees -Construtive Solid geometry - CSG • Representation of solid interior of objects -Surface may not describe explicitly the physical characteristics of the object • Data acquisition devices generate solid type data representations • Applications require solid object representations • Rendering algorithms which require solid object representations -Ray tracing with refraction. The considered path of the rays depends on the internal physical characteristics of the object representation. Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  5. Solid Modeling Representation  Solids-Voxels -Octrees -Binary Space Partitions - BSP Trees -Construtive Solid geometry - CSG • Recursive partition of space by planes. -Mark leaf cells as inside or outside or outside object. Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  6. Solid Modeling Representation  Solids-Voxels -Octrees -Binary Space Partitions - BSP Trees -Construtive Solid Geometry - CSG • Represent a solid object as hierarchy of Boolean operations -Union -Interception -Difference Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  7. Solid Modeling Representation  Solids-Voxels -Octrees -Binary Space Partitions - BSP Trees -Construtive Solid geometry - CSG • Representation of solid interior of objects -Surface may not describe explicitly the physical characteristics of the object • Data acquisition devices generate solid type data representations • Applications require solid object representations • Rendering algorithms which require solid object representations -Ray tracing with refraction. The considered path of the rays depends on the internal physical characteristics of the object representation. Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  8. Voxels • Partition of the space in a uniform, orthogonal grid-Grid cells are called voxel – “volume pixel” • Data type:-Binary data: {1,0}, full/empty, object/background; -Multivalued data: value representing some measurable property of the data color density heat pressure occupancy Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  9. Voxels • Boolean Operations-simple and intuitive Union Union Interception Interception Top view of one slice of the grid Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  10. Octrees • Refine resolution of voxels hierarchically-Octrees are almost often used to partition a 3D space by recursively subdividing it in eight octants. -Cube nodes: black/white/gray -More concise and efficient for non-uniform objects. -Adaptive definition of elementary size of grid cells. Top view of one slice of the grid Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  11. Octrees • Information representation-tree data structure Top view of one slice of the grid Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  12. Octrees • Boolean Operations-simple and intuitive Union Interception Top view of one slice of the grid Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  13. Data Acquisition/ Conversion • Data flow of volume visualization and volume graphics -Major sources of volumetric data: *Sampled/computed data *Geometrical models -Reconstructed sampled/computed 3D data is stored is a volume buffer -A geometrical model in 3D continuous space can be scan converted into a set of voxels and stored in the volume buffer -Volume buffer data visualization *Conversion to a geometric model *Direct projection on a 2D píxel buffer Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  14. Data Acquisition/ Conversion • Data flow of volume visualization and volume graphics CT/PET Reprojection Voxels/Octrees Space Carving Range Data Mesh Surfaces Voxelization Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  15. Data Acquisition/ Conversion • Data flow of volume visualization and volume graphics CT Reprojection Voxels/Octrees Space Carving Range Data Mesh Surfaces Voxelization Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  16. Data Acquisition/ Conversion • CT/PET Computer Tomography (CT) Positron Emission Tomography (PET) Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  17. Data Acquisition/ Conversion 90º 0º 180º 270º • CT Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  18. Data Acquisition/ Conversion • CT/PET Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  19. Data Acquisition/ Conversion • Data flow of volume visualization and volume graphics CT/PET Reprojection Space Carving Voxels/Octrees Range Data Mesh Surfaces Voxelization Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  20. Data Acquisition/ Conversion • Reconstruction of octrees from range data -Volumetric reconstruction from range data involves four main steps: 1.Data Acquisition Range data sets covering the object to be modeled are obtained. Usually implies range data acquisition from multiple views. 2.Registration Each range view has its own coordinate system. The collection of views should be registered in a common object-centric coordinate system. 3.Integration The separated registered range maps are integrated into a single data points representation. 4.Creation of the volumetric representation Pulli et al.’ 97 Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  21. Data Acquisition/ Conversion • Reconstruction octrees from range data 1.Data Acquisition Eight intensity images corresponding to the views of the miniature chair The data of the corresponding range images is acquired to each view. Pulli et al.’ 97 Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  22. Data Acquisition/ Conversion • Reconstruction of octrees from range data 2 and 3 – Registration and Integration The registered point set Pulli et al.’ 97 Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  23. Data Acquisition/ Conversion Pulli et al.’ 97 • Reconstruction of octrees from range data 4 – Creation of the volumetric representation - Processing a single range view -Initial volume that surrounds all the range data. -For each of the cubes, the 8 vertex are project in the image plane – hexagonal convex hull projection -The hexagonal cone is truncated so it just encloses the cube -If all the data points projecting on the hexagon are behind the truncated cone  Outside -If those points are closer than the closest corner of the cube  Inside -Otherwise  Boundary  Subdivision of the cube in 8 children cubes and apply the algorithm Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  24. Data Acquisition/ Conversion Pulli et al.’ 97 • Reconstruction of octrees from range data 4 – Creation of the volumetric representation – Generalization to multiple views Two possible processing orders: - Simultaneous processing: At each level, each cube is labeled only after conjugating the labels from all available views. - Sequential processing One view is processed at a time. Final conjugation of individual view results Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  25. Data Acquisition/ Conversion Pulli et al.’ 97 • Reconstruction of octrees from range data 4 – Creation of the volumetric representation – Generalization to multiple views The chair octree after 4,5,6, and 7 subdivisions Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  26. Data Acquisition/ Conversion • Data flow of volume visualization and volume graphics CT/PET Reprojection Voxels/Octrees Space Carving Range Data Mesh Surfaces Voxelization Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  27. Data Acquisition/ Conversion • Voxelization -Motivation: -Conversion of a geometric object from their continuous geometric representation into a set of voxels that best approximate the continuous object; -Discrete digitalization of a continuous object -Approaches - Straight forward and intuitive method  point sampling *The continuous object is evaluated at voxel center: 0 or 1 is assigned to each voxel *Binary classification of the voxel: the resolution of the grid determine theprecision of the discrete model. *Jagged surfaces Object space aliasing Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  28. Data Acquisition/ Conversion • Voxelization -Approaches - 3D object shape anti-aliasing technique  Volume Sampling -For each voxel visited by the binary voxelization algorithm, it is estimated the density contribution of the geometric object to the voxel. -Multi-valued volumetric representation – Smoother Representation Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  29. Data Acquisition/ Conversion • Data flow of volume visualization and volume graphics CT/PET Reprojection Voxels/Octrees Space Carving Range Data Mesh Surfaces Voxelization Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  30. Data Acquisition/ Conversion • Surface reconstruction from volumetric data -Motivation: Extraction and visualization of Isosurfaces from the volumetric data sets (multivalued data sets) -Isosurfaces display is usually fast since most isosurfacing methods output a mesh composed of triangular polygons  fast on typical graphics harware -Marching Cubes - Popular methods was developed by Lorensen and Cline (1987) *Creation of a polygonal representation of constant value surface for a 3D array of data 1. Location of the surface corresponding to a user specific value and triangle creation 2.Surface normal calculation at each vertex of each triangle Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  31. Data Acquisition/ Conversion • Surface reconstruction from volumetric data - Marching cube 1. Location of the surface and triangle creation -Cube-by-cube determination of the surface configuration inside the cube Comparison of the data value for the isosurface and the data value of each vertex 1- Data value of the vertex exceeds or equals the surface value – Inside surface 0- Data value of the vertex is below than the surface value – Outside surface 28 – 256 different topological configurations  Look-up table which contains the edges intercepted for each case Simplification: Reflective Symmetry ( 256 128) Rotational Symmetry (128 14) Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  32. Data Acquisition/ Conversion • Surface reconstruction from volumetric data - Marching cube 1. Location of the surface and triangle creation -Elementary configurations Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  33. Data Acquisition/ Conversion • Surface reconstruction from volumetric data - Marching cube 1. Location of the surface and triangle creation Index-pointer to an edge table that stores all edges interception given a cube configuration. Identification of intercepted edges  Interpolation to determine the precise location interception point triangle(s) definition Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  34. Data Acquisition/ Conversion • Surface reconstruction from volumetric data - Marching cube 2. Unit Normal determination for each triangle vertex -The normal will be used by the rendering algorithms to produce shaded images. -Normal determination based on the gradient vector on each vertex (i,j,k) -D(i,j,k) is the density at pixel (i,j) in slice k. -x, y, z are the lengths of the cube edges -The normal is linearly interpolated to the point of interception. Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  35. Volume graphics vs Surface Grafics Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

  36. References -Kaufman, A.; Cohen, D.; Yagel, R., Volume Graphics, IEEE Computer, Volume: 26 7 , July 1993 , Page(s): 51 -64. -S. Wang and A. Kaufman, Volume-Sampled 3D Modeling, IEEE Computer Graphics & Appplications14(5), September 1994, pp.26-32. -Oomes, S.[Stijn], Snoeren, P., Dijkstra, Tj.,3D Shape Representation: Transforming Polygons into Voxels, ScaleSpace97 (xx)-K Pulli, T. Duchamp, H. Hoppe, J. McDonald, L. Shapiro, W. Stuetzle, Robust Meshes from Multiple Range Maps, -W.E. Lorensen and H.E. Cline, Marching Cubes: A High Resolution 3D Surface Reconstruction Algorithm, SIGGRAPH 87, 163-169. -http://www.cs.princeton.edu/courses/archive/fall00/cs426/-http://www.cs.princeton.edu/courses/archive/spring00/cs598b/ Computer Graphics and 3D Modeling Doctoral Programme in Electrical Engineering and Computer Science

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