Isosurfaces

Also known asMetaballs, blobs, soft objects, implicit surfaces, …

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Outline

Marching cube algorithmIsosurfaceMetaballsImplicit modelingVolume rendering

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Introduction

Surface-based visualization technique for scalar data field treat isosurface as one special instance o

f implicit function: f(x,y,z) = cPrevailing technology: marching cube (patented)Related: marching tetrahedra

iso-surface: equal

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Basic Idea(Marching Square)

References: 1, 2

value = 5

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Marching Square

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Continuation vs. Exhaustive Search (3D)

Continuation: O(n2) function evaluationsNeed a starting point for each piece of disjoint surfaceDesigned for continuous, real-valued functions

Exhaustive search: O(n3) function evaluationsCan find all pieces of surfacesDesigned to process 3D arrays (CT, MRI)

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Marching Cube

Similar ambiguity can occurNeed to resolve the topological inconsistency (more cases devised; see reference)

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Determining normals at vertices of triangular faces

It is often necessary to create normals for each vertex of the triangular faces for smooth shading. After the facets have been created, compute the average of the normals of all the faces that share a triangle vertex. A common approach is at each vertex to use a weighted average of normals of the polygons sharing the vertex. The weight is the inverse of the area of the polygon, so small polygons have greater weight. The idea is that small polygons may occur in regions of high surface curvature. The original Siggraph paper computes normals at vertices by interpolating the normals at the cube vertices. These cube vertex normals are computed using Central Differences of the volumetric data.

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Smooth vs. Flat Shading

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Interpolating

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Grid Resolution

One very desirable control when polygonizing a field where the values are known or can be interpolated anywhere in space is the resolution of the sampling grid. This allows coarse or fine approximation to the isosurface to be generated depending on the smoothness required and/or the processing power available to display the surface.

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Marching Tetrahedra

Jules Bloomenthal (Graphics Gems IV)

Decompose a cube into six tetrahedra

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MT (cont)

Pros No ambiguous cases simpler implementation

Cons Number of triangles quality of triangles

Shirley90 decomposes a cube into 5 tetrahedra (smallest number of tetrahedra possible)

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MT vs. MC

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Regularized MT (Treece98)

MT: overcome topological problem (false positive/false negative) of MC But problem still exists: poor triangle quality; MT has many triangles“regularized”: combine with vertex clustering (but still maintain topological property) result: good fewer triangles

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MC Codes

References and codes for marching cube and marching tetrahedra are on Paul Bourke’s page

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A Brief History (ref)People have known for a long time that if you have two implicit surfaces f(x,y,z)=0 and g(x,y,z)=0 that are fairly continuous, with a common sign convention (f and g positive on the inside, negative on the outside, say) then the implicit surface defined by f+g=0 is a blend of the shapes. See [Ricci 1983] for a variant of this. The van der Waals surfaces of molecules (roughly speaking, iso-potentials of the electron density) are described in Chemistry and Physics books and [Max 1983]. To create animation of DNA for Carl Sagan's COSMOS TV Series, Jim Blinn proposed approximating each atom by a Gaussian potential, and using superposition of these potentials to define a surface. He ray traced these [Blinn 1982], and called them "blobby models". Shortly thereafter, people at Osaka University and at Toyo Links in Japan began using blobby models also. They called theirs "metaballs" (or, when misspelled, "meatballs"). Yoichiro Kawaguchi became a big user of their software and their Links parallel processor machine to create his "Growth" animations which have appeared in the SIGGRAPH film show over the years. The graduate students implementing the metaball software under Koichi Omura at Osaka used a piecewise quadratic approximation to the Gaussian, however, for faster ray-surface intersection testing (no need for iterative root finders; you just solve a quadratic). I don't know of any papers by the Japanese on their blobby modeling work, which is too bad, because they have probably pushed the technique further than anyone. Bloomenthal has discussed techniques for modeling organic forms (trees, leaves, arms) using blobby techniques [Bloomenthal 1991] (though he prefers the term "implicit modeling") and for polygonizing these using adaptive, surface- tracking octrees [Bloomenthal 1988]. The latter algorithm is not limited to blobby models, but works for any implicit model, not just blobs. Polygonization allows fast z-buffer renderers to be used instead of ray tracers, for interactive previewing of shapes. A less general variant of this algorithm was described in the "marching cubes" paper by [Lorensen 87] and some bugs in this paper have been discussed in the scientific visualization community in the years since. In the sci-vis community, people call them "iso-surfaces" not "implicit surfaces". Meanwhile, in Canada and New Zealand, the Wyvill brothers, and grad students, were doing investigating many of the same ideas: approximations of Gaussians, animation, and other ideas. See their papers listed below. Rather than "blobbies" or "metaballs", they called their creations "soft objects". But it's really the same idea. Bloomenthal and Wyvill collected many good papers on blobby and implicit modeling for a recent SIGGRAPH tutorial (1991?).

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Metaball ModelingEach metaball as a positively charged particle, with a suitable (differentiable) potential function Implicit function f(x,y,z): the potential field formed by summing contribution of all metaballsRendering isosurfaces Surface-based: evaluate the potential at grid points;

use marching cube Ray-casting: for each ray, find the point of intersecti

on (and corresponding normal for Phong shading)

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Example Potential Function (ref)

0)( ,0)(

0)0( ,25.0)0(

24)(

25.0)(

21

21

3

24

gg

gg

rrrg

rrrg

0)( ,0)(

0)0( ,25.0)0(

24)(

25.0)(

21

21

3

24

gg

gg

rrrg

rrrg

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Remark

The influence of each charged particle is negligible outside the 1/sqrt(2) concept of “bounding sphere”When casting a ray, only find the metaballs that are activeFor all the active metaballs along a ray, inch along the ray, sum up all active balls to find out the exact point (if any) where the potential exceeds the thresholdDetermine the normal at such points by summing up the normal vectors of the active balls

)(

)()(

xff

xfxf

i

i

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Metaballs

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More Rendering

Texture coordinate generationMaterial …

Threshold

zero

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Equipotential Lines

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Examples

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PSP Game [Hg]

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Metaballs

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Snowman

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Human Modeling Using Metaballs

35 metaballs Levels of control Mid: thigh, chest, … High: weight, sex, age,

muscle, …

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Rendering Metaballs

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MC Speed Up specifically for Metaballs (Ref)

Do not evaluate each cellStart from each ball, move along one direction until it reaches the “boundary”If the cell is not yet evaluated, do so and propagate to its adjacent cells

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Related: Implicit Modeling

F(x,y,z) = cF(x,y,z) = x2 + y2 + z2 − R2 Simple geometric description (plane, spheres, cylinders, …)Region separationCan be combined to create complex objects using Boolean operators

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Implicit Modeling

At a point x0,y0,z0 the value of

F G = min(F(x0,y0,z0), G(x0,y0,z0))

F G = max(F(x0,y0,z0), G(x0,y0,z0))

F − G = max(F(x0,y0,z0), −G(x0,y0,z0))

F(x,y)=x2+y2 − 22

G(x,y)=(x-3)2+(y-2)2 − 32

F G = 0

To create CSG objects

Volume Rendering

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Volume Rendering

512x512x12-bit, upto 50 slices in a stackEach slice is spaced 1-5mm apart

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Volume Rendering

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Ray Casting (ref)

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Example

Jaw with skin Jaw without skin

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Related Work

Iso-surface extraction Implicit surface/metaballs Thresholded data (medical imaging) Surface from cross-sections

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