IJK: Isosurface Jeneration Kode

IJK is a set of C++ classes, routines and programs for generating isosurface. It includes programs for generating isosurface lookup tables for arbitrary convex polyhedra in arbitrary dimensions. It also contains an implementation of the Marching Cubes Algorithm, the 4D Marching Cubes Algorithm, the Marching Cubes Algorithm using negative, equals and positive lookup tables and SnapMC which produces quality isosurface triangles.

References:

"Isosurfacing in higher dimensions" by Bhaniramka, Wenger and Crawfis, Visualization 2000, Salt Lake City, Utah: IEEE Computer Society Press;
"Isosurface construction in any dimension using convex hulls'' by Bhaniramka, Wenger and Crawfis, IEEE Trans. on Vis. and Computer Graphics, 10, 2004;
"Quality isosurface mesh generation using an extended Marching Cubes lookup table" by Raman and Wenger, Eurographics/IEEE Symposium on Visualization, 2008, to appear;

Downloadable software/tables:

ijktable.v0.2.5.tar: Source code and documentation for isosurface table generation including isotable C++ classes and program ijkgentable. Includes K. Clarkson's program hull for generating convex hulls.
isotable_xit.v0.1.0.tar.gz: Sample cube, simplex and simplicial prism isosurface tables in various dimensions. Simplex vertices are ordered using program orientisotable so that adjacent simplices have matching orientation.
ijkorient.v0.2.5.tar: Source code for reordering vertices of isosurface simplices in an isosurface table so that adjacent simplices have matching orientation. For convenience, this tar file contains duplicates of the files ijktable.cxx and ijktable.h. Requires CGAL library.
ijkmcube.v0.2.7.tar: Source code for 2D/3D/4D Marching Cubes algorithm. Requires ITKNrrdIO.a library.
ijkslice.v0.2.5.tar: Source code to slice a set of d-simplices in (d+1)-space with a hyperplane and triangulate the slices.
NrrdIO.v1.9.0.tar: Source code NrrdIO. NrrdIO is a subset of the teem library. (See teem.sourceforge.net/nrrd.) This is an exact copy of NrrdIO-1.9.9-src.tar.gz from the teem library with the exception of some modifications to CMakeLists.txt, the control file for cmake.

Old versions of the software, isosurface tables and documentation.

Documentation:

Isosurface table generation:
- ijkgentable (html): Program for generating isosurface tables of hypercubes and simplices.
- ijktableinfo (html): Program for reporting some information about an isosurface table.
- ijkdifftable (html): Program for reporting differences between isosurface tables.
- ijkorient (html): Program for reordering simplex vertices in an isosurface table so that adjacent simplices have matching orientations.
- ijktable (html): C++ classes for representing isosurface tables of convex polyhedra in arbitrary dimensions.
- ijkgenpatch (html): Routine for generating an isosurface in a convex polyhedron in arbitrary dimensions.
XML Isosurface Table:
- xit (html): File format for storing isosurface tables of convex polyhedra in arbitrary dimensions.
Isosurface generation:
- ijkmcube (html): Marching cubes algorithm and variants (including 4D, use of negative-equals-positive lookup tables and SnapMC for quality mesh generation.)
Change log: CHANGES.

Time-varying data:

Time varying volumetric data can be visualized by treating time as the fourth dimension and building a three-dimensional isosurface in 4D. The isosurface can be sliced along the time axis to construct a time-varying animation or it can be sliced along other axes to visualize how subsets of the data vary in time. For 10 time steps of the Jet ShockWave data set, an isovalue of 37 generated an isosurface with 8,021,739 tetrahedra and 1,394,104 vertices in 1109 seconds on an SGI Octane. The isosurface intersected 1,317,975 hypercubes, giving an average of around 6 tetrahedra per hypercube. The total number of triangles generated for the same 10 time steps by Marching Cubes was 1,796,350.


Slice along X-axis	Slice along Y-axis	Slice along Z-axis

Slices of time-varying isosurface for the Jet Shockwave data set along different axes.

Isovalue = 37, Timesteps = 56-65.

Isosurface Morphing:

An isosurface in 3D can be morphed into another isosurface in 3D by constructing an appropriate isosurface in 4D. Let f₁(x,y,z) = 0 and f₂(x,y,z) = 0 define two isosurfaces, S₁ and S₂, respectively. Define F(x,y,z,t) = f₁(x,y,z)(1-t) + f₂(x,y,z)t. Note that F(x,y,z,0) = f₁(x,y,z) and F(x,y,z,1) = f₂(x,y,z). Sample F along a four dimensional grid G(x,y,z,t) whose last value t is either 0 or 1. Build an isosurface from the grid G and slice for various values of t to get a time-varying animation of the morph from S₁ to S₂.

Morphing using isosurfaces in 4D.

Interval Volumes

An interval volume is the set of points lying between two isosufaces. Three dimensional interval volumes can be constructed using a four dimensional isosurface. Let f(x,y,z) = a₁ and f(x,y,z) = a₂ define two isosurfaces, S₁ and S₂, respectively. Define F(x,y,z,w) = (f(x,y,z) - a₁)(1-w) + (f(x,y,z) - a₂)w. Note that the isosurface F(x,y,z,0) = 0 equals S₁ and the isosurface F(x,y,z,1) = 0 is S₂. Sample F along a four dimensional grid G(x,y,z,w) whose last value w is either 0 or 1. Build the isosurface with isovalue 0 from the grid G and project the tetrahedra into the plane w = 0 to form the interval volume S₁ and S₂.


Interval volume	Wireframe

Interval volume for the sphere function constructed using 4D isosurfaces.

Visualizing functions:

A surface given by the equation F(x₁, x₂, ..., x_d) = 0 can be visualized by evaluating F at the vertices of a d-dimensional grid, building an isosurface from that grid, and then slicing the isosurface by three-dimensional subspaces.


Slice along W-axis	Slice along X-axis

Slices of the function F(x,y,z,w) = x²+y²+z²-w along different coordinate axes.

Last updated by Dr. Rephael Wenger, May 16, 2008.