Feature Sensitive Surface Extraction from Volume Data

Leif P. Kobbelt

Mario Botsch

Ulrich Schwanecke

Hans-Peter Seidel

Computer Graphics Group, RWTH-Aachen

Computer Graphics Group, MPI Saarbrucken

Proc. Of ACM SIGGRAPH 2001 , page 57 66

Abstract

• A new technique to extract high quality triangle meshes from volume data.

• The main two contributions are:– Enhanced distance field representation

– Extended Marching Cubes (EMC)

• About Standard Marching Cubes (MC)

Abstract

• The above figures show reconstructions of the well-known “fandisk” dataset.

Standard MC Standard MC +

Enhanced distance field

Extended MC Extended MC +

Enhanced distance field

uniform 65×65×65 grid

About Standard Marching Cubes

Introduction

• The volume data is usually sampled on a regular grid with a given step width.

• We often observe severe alias artifacts at sharp features on the extracted surfaces.

– Reduce these alias effect– Keep the simple algorithmic structure of the standard

MC algorithm

Alias artifacts

• The Marching-Cubes-type algorithm process discrete volume data.

• The sampling of the implicit surface f(x,y,z)=0 is performed on the basis of a uniform spatial grid.

Parametric surfaces v.s. Implicit surfaces

• Parametric surfaces– A mapping from R2(u,v) to R3(x,y,z)– Parametrized by u and v.

x=f(u,v) y=g(u,v) z=h(u,v)

– Allows easy enumeration of points. Just plug in values for u and v.

Parametric surfaces v.s. Implicit surfaces

• Implicit surfaces– Defined by f(x,y,z)=0

• Advantages– Easy to check whether a point is “inside” and “outside”

– Inside: f(x,y,z) < 0

• Disadvantages– One cannot easily enumerate points on the surface.

x2+y2+z2-R2=0

Introduction

• The central contributions of this paper are:– Enhanced representation of the distance field

• This allows us to find more accurate surface.

• Store directed distance in x, y, and z directions.

– Extended Marching Cubes algorithm• Reduce alias

(converge to the original surface’s normals.)

Distance field representation• For a given surface , a volume representation

consists of a scalar valued function such that

• Signed distance field function

3S3:f

0),,(],,[ zyxfSzyx

)],,,([:),,( Szyxdistzyxf

> 0 outside the surface< 0 inside the surface= 0 on the surface

Operation

A B

U21],,[ SSzyx 0)},,(),,,(max{ 21 zyxfzyxf

21],,[ SSzyx 0)},,(),,,(min{ 21 zyxfzyxf

21],,[ SSzyx 0)},,(),,,(max{ 21 zyxfzyxf

21],,[ SSzyx 0)},,(),,,(max{ 21 zyxfzyxfA - B

Distance field representation

• The standard way to sample f on a uniform spatial grid gi,j,k = [ ih, jh, kh ].

• The sampled distances di,j,k = f ( ih, jh, kh )

can be interpolated on each grid cell.

Ci,j,k (h) = [ ih, (i+1)h ] × [ jh, (j+1)h ] × [ kh, (k+1)h ]

Distance field representation

• The major limitation of this technique is that the samples on S* are not necessarily close to S in the vicinity of sharp features.

S

S*

Distance field representation

• To improve the approximation one could refine the discretization grid h h’ < h or switch to higher order polynomial interpolants within each cell Ci,j,k (h).

• First case: – output a larger number of triangles.

• Second case: – local computations are getting more complicated.

Distance field representation• Therefore we suggest a third alternative to avoid t

hese difficulties by using the directed distance field.

• We store at each grid point gi,j,k three directed distances in x, y, and z direction.

z

y

x

kji

dist

dist

dist

d ,,

> 0 outside the surface< 0 inside the surface= 0 on the surface

The three directed distances at one grid point always have the same sign.

( inside / outside status )

Distance field representation

• The intersection point (Interpolation)

• It is valid if and have opposite signs.

kjikjikjikji ghxdghxds ,,1,,,,,, )/|][(|)/|][|1(

][,, xd kji ][,,1 xd kji

gi,j,k gi+,j,kS

h

Distance field representation

Standard MC Standard MC +

Enhanced distance field

Extended marching cubes• Marching cubes in general cannot reconstruct very

sharp features and result in aliasing artifacts.– Problem : normals don’t converge

– Alias errors in surfaces generated by the MC algorithm.– By decreasing the grid size, the effect becomes less and

less visible.

Extended marching cubes• What we can do

– By using point and normal information on both sides of the sharp feature one can find a good estimate for the feature point at the intersection of the tangent elements.

Extended marching cubes

D2Y > 0

X

Y

D1X < 0D3X > 0

D1Y < 0

Surface

Exact intersectionpoint

Extended marching cubes

normal

tangent element

normal

tangent element

Reconstructedsurface

Extended marching cubes

• This works only if there is at most one sharp feature.

• Just like the standard MC, the extended algorithm processes each cell Ci,j,k (h) separately.

• If the cell does not contain a sharp feature– by using the standard Marching Cubes table.

• If a feature is present– We compute one new sample point close to the expected f

eature. (generate a triangle fan)

Extended marching cubes

• For each cell we first have to check if a feature is present and if yes, which type of feature.

Extended marching cubes

• Postprocessing step– Left : the cells/patches that contain a feature are identified.

– Center : one new sample is included per cell.

– Right : by using edge flipping to reconstruct the feature edges.

Extended marching cubes

Extended MC Extended MC +

Enhanced distance field

Result

– The execution times include only the running times for standard and extended MC, respectively.

– The (directed) distance fields and gradients have been generated in a pre-process.

Result - CSG

Result – Fan Disk

Standard MC Standard MC +

Enhanced distance field

Extended MC Extended MC +

Enhanced distance field

Result – Max Planck

Low pass filter

Result – CAD

129129129

Application – Milling simulation

Application – Surface reconstruction

– The original dataset consists 200K scattered points

Application - Remeshing

• Polygonal meshes that are generated at some intermediate stage of an industrial CAD process often have a bad quality.

– Convert a CAD model into a volume representation by sampling its distance field on a uniform grid.

– Apply the extended Marching Cubes algorithm to this volume gives a remeshed version.

Conclusions and future directions

• Adaptively refined octrees.– The problem is to fix the gaps

(different refinement levels meet)

• Parallelization– The algorithm processes each cell individually.

(like the standard Marching Cubes)