1. Reconstruction from Voxels (GATE-540)
Dr.Çağatay ÜNDEĞER
Instructor
Middle East Technical University, GameTechnologies
&
General Manager
SimBT Inc.
Reference: William E. Lorensen and Harvey E. Cline
Game Technologies Program – Middle East Technical University – Spring 2010

2. Outline
Reconstruction

3. Goals Develop 3D Analysis Algorithms: Reconstruction Segmentation
Feature Detection
Labeling
Matching
Classification
Retrielval
Recognition
Clustering

4. Voxel A voxel (volumetric pixel)
A volume element, representing a value on a regular grid in three dimensional space.
Analogous to a pixel, which represents 2D image data in a bitmap

5. Voxel Data
Do not typically have their position (their coordinates) explicitly encoded along with their values.
Stores:
Binary data:
empty / full
Float data:
density / color / distance to surface

6. Voxel vs Polygon Polygons:
Often explicitly represented by the coordinates of their vertices.
Able to efficiently represent simple 3D structures with lots of empty or homogeneously-filled space.
Voxels:
Good at representing regularly-sampled spaces that are non-homogeneously filled.
Have a limited resolution, as precise data is only available at the center of each cell.

7. Usage of Voxel
Frequently used in the visualization and analysis of medical and scientific data.
Some volumetric displays use voxels to visualize models and describe their resolution in 3D dimension (512×512×256 voxels).

8. Usage of Voxel In games and simulations,
Used for representation of terrain containing overhangs and caves.
Concave features cannot be represented by heightmaps.

9. Visualizing Voxels Visualization: Direct volume rendering
Extraction of polygon iso-surfaces which follow the contours of given threshold values.
The marching cubes algorithm is often used for iso-surface extraction.

10. Iso-Surface A three-dimensional analog of an iso-contour.
A surface that represents points of a constant value (e.g. pressure, temperature, velocity, density) within a volume of space.
İso-surface

11. Marching Cubes
Marching Cubes is an algorithm which “creates triangle models of constant density surfaces (iso-surfaces) from 3D medical data.”

12. Medical Data Acquisition
Computed Tomography (CT)
Magnetic Resonance (MR) Imagining (MRI)
Single-Photon Emission Computed Tomography (SPECT)
Each scanning process results in two dimensional “slices” of data.

13. Data Slices

14. Marching Cubes Extraction
Extracts surfaces from adjacent pairs of data slices using cubes.
Cubes “march” through the pair of slices until the entire surface of both slices has been examined.

15. Marching Cubes Overview
Load slices.
Create a cube from pixels on adjacent slices.
Find vertices on the surfaces.
Determine the intersection edges.
Interpolate the edge intersections.
Calculate vertex normals.
Output triangles and normals.

16. How Are Cubes Constructed
Uses identical squares of four pixels connected between adjacent slices.
Each cube vertex is examined to see if it lies “on” or “off” the surface.

17. How Are The Cubes Used
Pixels on the slice surfaces determine 3D surfaces.
256 surface permutations, but only 15 unique patterns.
A normal is calculated for each triangle vertex for rendering.

18. 15 Unique Patterns

19. Triangle Creation Determine triangles contained by a cube.
Determine which cube edges are intersected.
Interpolate intersection point using pixel density.
Calculate unit normals for each triangle vertex using the gradient vector.

20. Determining Triangles
An index to a pre-calculated array of 256 possible polygon configurations (28 = 256) within the cube
Treat each of the 8 scalar values (cube corners) as a bit in an 8-bit integer.
8 scalars (8 bits) determine the actual index to the polygon configuration array.

21. Determining Inside/Outside
Select a iso-value that the surface will pass through.
If the scalar's value is higher than the iso-value then
The appropriate bit is set to one (inside)
If it is lower then
The appropriate bit is set to zero (outside)
0.1
0.8
0.5
0.0
iso-value = 0.3

22. Determining Intersections
Determine intersection points to iso-suface by interpolation.
0.1
0.8
0.5
0.3
0.0
iso-value = 0.3

23. Determining Intersections
Gradient of the scalar field at each grid point is also the normal vector of a hypothetical iso-surface passing from that point.

24. Determining Intersections
Interpolate these normals along the edges of each cube to find the normals of the generated vertices. N N

25. Grid Resolution
Variations can increase/decrease surface density.

26. Examples

27. Marching Squares 2D version of Marching Cubes
Aims at drawing lines between interpolated values along the edges of a square considering given weights of the corners and a reference value.

28. Conclusion
Marching Squires / Marching Cubes provides a simple algorithm to translate a series of 2D medical scans into 2D / 3D objects.