reconstruction of voxels from sensor data

University of Coimbra

Reconstruction of Voxels from Sensor Data

Ricardo Martins

Coimbra, 19th January 2010

Doctoral Programme in Electrical Engineering and Computer Science

Computer Graphics and 3D Modeling


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




3D Object Representation

Points -Range images -Point cloud



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



Solid Modeling Representation

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 representationsRendering 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.




Recursive partition of space by planes. -Mark leaf cells as inside or outside or outside object.




Solids -Voxels -Octrees -Binary Space Partitions - BSP Trees -Construtive Solid Geometry - CSG


Represent a solid object as hierarchy of Boolean operations -Union-Interception-Difference






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 representationsRendering 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.


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

colordensityheat

pressureoccupancy


Voxels



Boolean Operations -simple and intuitive

Union

Interception

Top view of one slice of the grid

Union

Interception


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.



Octrees



Information representation -tree data structure



Octrees



Boolean Operations -simple and intuitive


Union Interception


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






CT/PET

Range Data Voxels/Octrees Mesh

Surfaces

Reprojection

Voxelization

Space Carving






CT


Surfaces

Reprojection

Voxelization

Space Carving





CT/PET

Computer Tomography (CT) Positron Emission Tomography (PET)





CT

270º

180º0º

90º





CT/PET






CT/PET


Surfaces

Reprojection

Voxelization

Space Carving





Reconstruction of octrees from range data

Pulli et al.’ 97

-Volumetric reconstruction from range data involves four main steps:

1.Data AcquisitionRange data sets covering the object to be modeled are obtained. Usually implies range data acquisition from multiple views.

2.RegistrationEach range view has its own coordinate system. The collection of views should be registered in a common object-centric coordinate system.

3.IntegrationThe separated registered range maps are integrated into a single data points representation.

4.Creation of the volumetric representation





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.





Reconstruction of octrees from range data

Pulli et al.’ 97

2 and 3 – Registration and Integration

The registered point set





Reconstruction of octrees from range dataPulli 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






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






4 – Creation of the volumetric representation – Generalization to multiple views

The chair octree after 4,5,6, and 7 subdivisions






CT/PET


Surfaces

Reprojection

Voxelization

Space Carving





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





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






CT/PET


Surfaces

Reprojection

Voxelization

Space Carving





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





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)







-Elementary configurations







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






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.


Volume graphics vs Surface Grafics




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/

reconstruction of voxels from sensor data

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