3d surfaces - cs.cmu.edu

3D SurfacesAdrien Treuille

Carnegie Mellon University

source: http://iparla.labri.fr/publications/2007/BS07b/sketch_teaser.jpg

Outline

• Homework 1

• Height Fields- Normals

• Implicit and Explicit Surfaces- Numerical vs. Analytic- Implicit vs. Explicit- Conversions

• Subdivision Surfaces

source: http://www.cs.umd.edu/class/spring2005/cmsc828v/thumbnails/thMPU.gif

source: http://graphics.cs.lth.se/theses/projects/projgrid/demo11.jpg


Homework 1

Due on Thursday.

No late days!

Outline

• Homework 1

• Height Fields

• Normals

• Implicit and Explicit Surfaces- Numerical vs. Analytic- Implicit vs. Explicit- Conversions





2D Scalar Field2D Scalar Field

• z = f(x,y)

How do you visualize this function?

!"# <+!!

=1 if ,1

),(

2222yxyx

yxf 0

Height FieldHeight Field

• Visualizing an explicit function

• Adding contour curves

z = f(x,y)

f(x,y) = c

MeshesMeshes

• Function is sampled (given) at xi, yi, 0 ! i, j ! n

• Assume equally spaced

• Generate quadrilateral or triangular mesh

• [Asst 1]

Height Field Normals

how to compute them?

Outline• Homework 1

• Height Fields

• Normals

• Implicit and Explicit Surfaces

• Numerical vs. Analytic

• Implicit vs. Explicit

• Conversions





(Dis)Advantages?• shape matching

• morphing

• implementation in the opengl pipeline

• CSG

• surface reflection

• fluid simulation


• Height Fields

• Normals




• Conversions





Implicit → Explicit 2D(Marching Squares Algorithm)

Marching SquaresMarching Squares

• Sample function f at every grid point xi, yj

• For every point fi j = f(xi, yj) either fi j ! c or fi j > c

Cases for Vertex LabelsCases for Vertex Labels

16 cases for vertex labels

4 unique mod. symmetries

Ambiguities of LabelingsAmbiguities of Labelings

Ambiguous labels

Different resulting

contours

Resolution by subdivision

(where possible)

Marching Squares ExamplesMarching Squares Examples

Can you do better?

Interpolating IntersectionsInterpolating Intersections

• Approximate intersection

– Midpoint between xi, xi+1 and yj, yj+1

– Better: interpolate

• If fi j = a is closer to c than b = fi+1 j then

intersection is closer to (xi, yj):

• Analogous calculation

for y directionfi j = a < c c < b = fi+1 j

xi xi+1x


Adaptive Subdivision

Implicit → Explicit 3D(Marching Cubes Algorithm)

3D Scalar Fields3D Scalar Fields

• Volumetric data sets

• Example: tissue density

• Assume again regularly sampled

• Represent as voxels

• Two rendering methods

–Isosurface rendering

–Direct volume rendering (use all values [next])

IsosurfacesIsosurfaces

• Generalize contour curves to 3D

• Isosurface given by f(x,y,z) = c

– f(x, y, z) < c inside

– f(x, y, z) = c surface

– f(x, y, z) > c outside

Marching CubesMarching Cubes

• Display technique for isosurfaces

• 3D version of marching squares

• How many possible cases?

28 = 256

…

Marching CubesMarching Cubes

• 14 cube labelings (after elimination symmetries)

Marching Cube TessellationsMarching Cube Tessellations

• Generalize marching squares, just more cases

• Interpolate as in 2D

• Ambiguities similar to 2D

Explicit → Implicit 3D(Fast Marching Algorithm)

Signed Distance Field

What a line’s SDF?

(interior)

(exterior)

+

-

(y = 0)

φ(x, y) = y∇φ = [0, 1]T ||∇φ|| = 1

What a circle’s SDF?

+

-

1

||∇φ|| = 1

φ(x, y) =�

x2 + y2 − 1

∇φ =1�

x2 + y2[x, y]T

What do these cases have in common?

11−1

−1

1

||∇φ|| = 1

What about General Case?

http://www.cse.ohio-state.edu/~chenyis/Markerless_Motion_Capture/

(sorry for change in color scheme)

Fast Marching Algorithm

Start Along the Boundary

Boundary Cases

D. Adalsteinsson, J. A. Sethian. The Fast Construction of Extension Velocities in Level Set Methods. Journal of

Computational Physics (1999)

Push Adjacent Cells

Estimating The Distance||∇φ|| = 1Use:

�φi,j − φi−1,j

h

�2

+�

φi,j − φi,j−1

h

�2

= 1

φi,jφi−1,j

φi,j−1

Pop Nearest Cell

Add It’s Neighbors to the Heap

Continue Propogating...

Further and Further...

Until All Cells Have Been Processed

Reminder: Why would we want to do this?


• Height Fields

• Normals




• Conversions

• Meshes




Mesh Representations & Subdivision Surfaces

Tom FunkhouserPrinceton University

COS 426, Spring 2007

Polygon Meshes• How should we represent a mesh in a computer?

oEfficient traversal of topologyoEfficient use of memoryoEfficient updates

• Mesh RepresentationsoIndependent facesoVertex and face tablesoAdjacency listsoWinged-EdgeoHalf-Edgeoetc.

Zorin & Schroeder, SIGGRAPH 99, Course Notes

Independent Faces• Each face lists vertex coordinates

oRedundant verticesoNo adjacency information

Vertex and Face Tables• Each face lists vertex references

oShared verticesoStill no adjacency information

Possible Queries

•Which faces use this vertex?

•Which edges use this vertex?

•Which faces border this edge?

•Which edges border this face?

•Which faces are adjacent to this face?

Adjacency Lists• Store all vertex, edge, and face adjacencies

oEfficient adjacency traversaloExtra storage

Partial Adjacency Lists• Can we store only some adjacency relationships

and derive others?

Winged Edge• Adjacency encoded in edges

oAll adjacencies in O(1) timeoLittle extra storage (fixed records)oArbitrary polygons

Winged Edge• Example:

Half Edge• Adjacency encoded in edges

oAll adjacencies in O(1) timeoLittle extra storage (fixed records)oArbitrary polygons

• Similar to winged-edge,except adjacency encoded in half-edges

e heinv

vbegin

Fleft

henext

3d surfaces - cs.cmu.edu

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