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

University of Coimbra 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 Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling

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

University of Coimbra  Solids -Voxels -Octrees -Binary Space Partitions - BSP Trees -Construtive Solid geometry - CSG Solid Modeling Representation  Representation of solid interior of objects -Surface may not describe explicitly the physical characteristics of the object Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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.

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

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

University of Coimbra  Solids -Voxels -Octrees -Binary Space Partitions - BSP Trees -Construtive Solid geometry - CSG Solid Modeling Representation  Representation of solid interior of objects -Surface may not describe explicitly the physical characteristics of the object Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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.

University of Coimbra Voxels Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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

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

University of Coimbra Octrees Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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

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

University of Coimbra Octrees Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  Boolean Operations -simple and intuitive Top view of one slice of the grid UnionInterception

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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

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

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

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  CT/PET Computer Tomography (CT) Positron Emission Tomography (PET)

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  CT 270º 180º 0º 90º

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  CT/PET

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

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  Reconstruction of octrees from range data Pulli et al.’ 97 -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

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  Reconstruction octrees from range data Pulli et al.’ 97 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.

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

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  Reconstruction of octrees from range data Pulli et al.’ 97 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

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  Reconstruction of octrees from range data Pulli et al.’ 97 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

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

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

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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 the precision of the discrete model. *Jagged surfaces  Object space aliasing

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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

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

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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 2 8 – 256 different topological configurations  Look-up table which contains the edges intercepted for each case Simplification: Reflective Symmetry ( 256  128) Rotational Symmetry (128  14)

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

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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

University of Coimbra Data Acquisition/ Conversion Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling  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.

University of Coimbra Volume graphics vs Surface Grafics Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling

University of Coimbra References Doctoral Programme in Electrical Engineering and Computer Science Computer Graphics and 3D Modeling -Kaufman, A.; Cohen, D.; Yagel, R., Volume Graphics, IEEE Computer, Volume: 26 7, July 1993, Page(s): S. Wang and A. Kaufman, Volume-Sampled 3D Modeling, IEEE Computer Graphics & Appplications 14(5), September 1994, pp 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,