Marching Cubes
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Computer Graphics and Animation course TU Delft
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Marching Cubes
- 1. 1Challenge the future Marching Cubes A High Resolution 3D Surface Construction Algorithm
- 2. 2Challenge the future Introduction • Algorithm developed by William E. Lorensen and Harvey E. Cline and published in the 1987 SIGGRAPH proceedings. • Aims to create 3D models from Medical data: • X-ray computed tomography (CT) • Magnetic resonance (MR) • Single-photon emission computed tomography (SPECT)
- 3. 3Challenge the future 3D Medical Algorithms & Related Work Workflow: 1. Data acquisition: multiple 2D slices 2. Image processing to find structures or filter data 3. Surface construction 4. Display Approaches: • Contours of the surface on consecutive slices connected with triangles • Creates surfaces from cuberilles • Octree, etc.
- 4. 4Challenge the future Marching Cubes Algorithm • Locate surface to a user-specified value • Create triangles • Calculate normals to ensure the quality of the image Idea
- 5. 5Challenge the future Marching Cubes Algorithm • Divide-and-conquer to locate surface in cube • 2 adjacent slices • 4 pixels used on both slices to create vertices of cube Locate surface
- 6. 6Challenge the future Marching Cubes Algorithm • Cube vertices are assigned with binary values • One for inside (or on) the surface • Zero for outside the surface • In 2D: Create triangles
- 7. 7Challenge the future Marching Cubes Algorithm • In 3D: • 28 = 256 cases Create triangles
- 8. 8Challenge the future Marching Cubes Algorithm • Use symmetry and rotation to reduce 256 cases to 14 patterns • Index of 8 bits to number the cases • With linear interpolation the surface intersection is found Create triangles
- 9. 9Challenge the future Marching Cubes Algorithm • Final step to increase the quality of the image • With central differences an unit normal can be calculated for every cube vertex using 4 slices • Interpolation of these normals Calculate normals
- 10. 10Challenge the future Improvements • Efficiency increased by using pixel-to-pixel and line-to-line coherence. • 3 new edges are needed to interpolate • Other 9 edges are obtained from previous slices, lines or pixels • Reducing slice resolution by averaging four pixels into one • Solid modeling using the three notions “inside”, “outside”, and “on”
- 11. 11Challenge the future Implementation and Results Implementation • Number of triangles is proportional to the area of the surface = A lot! • Filtering is applied to reduce the resolution and number of triangles Results • CT • MR • SPECT
- 12. 12Challenge the future Conclusions • Realism is achieved by the calculation of the normalized gradient • Large number of triangles reduced by surface cutting and connectivity • The algorithm has some flaws: • high amount of memory needed to store resulting surface • Sign change in the 14 original patterns can lead to mistakes
Notas do Editor
- - Objective paper - Introduction to 3D medical algorithms - Related work
- http://www.byclb.com/TR/Tutorials/volume_rendering/ch1_1.htm Octree: http://http.developer.nvidia.com/GPUGems2/gpugems2_chapter37.html BUT each of these techniques throw away useful information in the original data. The marching cubes approach uses information from the original 3D data to derive inter-slice connectivity, surface location and surface gradient.
- Explain algorithm Divide and conquer Slices and cubes Intersection cubes and surface 14 patterns Normalized vector and density
- Implementation Overview Errors
- http://web.cs.wpi.edu/~matt/courses/cs563/talks/march_cub.html