Surface Reconstruction Approach Overview of important methods - - PowerPoint PPT Presentation

Surface Reconstruction

Approach

• Overview of important methods
• Properties needed
• Possible solution

– Method – Current work – Next Steps

• Conclusions and Discussion

Overview of different methods

Voronoi diagrams and Delaunay triangulation / tetrahedrization

Fig a Fig b Voronoi cells creation p Fig d Triangles removal Fig c Links creation Point set

Delaunay Triangulation algorithms

Complexity = O(n2) in 3D Can be decreased to O(n) with uniform sampling and other assumptions Different algorithms :

• Divide and conquer
• Convex hull sculpting
• Alpha shapes
• Incremental construction

Surface Deformation

• Energy minimization process
• Start from an initial rough shape
• Deform it until a local minimum is

reached

Example

The liver template (Johan Montagnat, Inria) Evolution of a model (on one slice of the whole 3D image).

• Rough initialization
• Successive deformations
• Energy deformation is

functions of liver's contour points

Properties

+ Fast methods + initial guess easy to find

• Closed surface
• Local minimum
• Importance of initial guess

Implicit functions

• Concept : surface = zeroes of a function

– Iso-surface – Space partition (inside/outside/on the object)

• Lots of possible (and famous) solutions

– Hoppe, Amenta … F(X) = 0 F(X) < 0 F(X)>0 Ex : equation of sphere x2 + y2 + z2 – 1 = 0

Example of skeleton extraction

`frame" over which the ``meat" of the shape hangs locus of the centers of all tangent discs contained in the shape.

Polygonization

• Marching cubes (or triangles)

Images from Lakshman Prasad newsletter

Properties

+ Important Data size reduction + Primitive + Adaptable polygonization + Objects can have holes + n-dimension function

• Approximating
• Important computation time :

Recent improvement (RBF) decreased complexity to O(n2)

• - Closed surface (c’est pas tout a fait vrai mais)

Patches

Numerous approaches : Idea : locally find the “best fitting” surface patch to reconstruct the surface Least square data-fitting

• Spatial Recursive subdivision and curve fitting
• Subdivide the space to reduce the number of points.
• Surface fitting faster

Idea : subdivide the space to fit lower degree splines.

Recursive spatial subdivision

Images from Benjamin Gregorski, Bernd Hamann, and Kenneth I. Joy

Patch joining

Images from Benjamin Gregorski, Bernd Hamann, and Kenneth I. Joy

Properties?

both Approximating (can be both) approximating interpolating Scheme type Continuity between patches Yes Adjustable None patches No boundary Yes Strong Closed

• bjects, no

boundaries implicit Very simple

• bjects

Yes Strong Simple

• bjects (no

holes) deformation No primitive extraction No Weak None delaunay Main drawback Primitive extraction Robustness to noise Topological restrictions

Goals

• Speed and important data size
• Primitive extraction (higher abstraction)
• General approach
• Manage boundaries

Hierarchical Patch Approach

Possible Solution

Principal direction for each point “Iso-value” lines Minimizing curvature variation

Image from Meyer and Desbrun 2002

Local energy extrema

Hierarchical Patch extraction

t I1 I2 many parametric functions + control meshes t 1 t 1 Initial function F0(t)

Curvature energy Local curvature energy maxima Local curvature energy minima

Result?

Profile example Profile reconstructed

Patch 0-0 Isolated point 0A Patch 1-C Patch 1-B Patch 1-A Isolated point 1B Isolated point 1A Patch 2-A Patch 3-A Isolated point 3A

What Now?

• Segmentation of the point cloud based on curvature

– “Good” Uniform sampling – Pseudo-Uniform sampling with holes or cracks – New model with non uniform sampling

Next steps

Work on surface theory

• What kind of surface?
• Uniform or non-uniform
• Quadratic, cubic, else
• How to combine them?
• How to merge them?
• Subdivision surfaces
• R-functions

Conclusions and discussion

Hierarchical patch reconstruction

• 1. No topological restriction (boundaries)
• 2. Parametric approach adaptable meshing
• 3. Different levels of details
• 4. Size reduction