Geodesics on Implicit Surfaces and Point Clouds

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Geodesics on Implicit Surfaces and Point Clouds. Guillermo Sapiro Electrical and Computer Engineering University of Minnesota guille@ece.umn.edu. Supported by NSF, ONR, NGA, NIH. IPAM 2005. Overview. Motivation Distances and Geodesics Implicit surfaces Point clouds. Key Motivation.

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### Geodesics on Implicit Surfaces and Point Clouds

Guillermo Sapiro

Electrical and Computer Engineering

University of Minnesota

guille@ece.umn.edu

Supported by NSF, ONR, NGA, NIH

IPAM 2005

Overview
• Motivation
• Distances and Geodesics
• Implicit surfaces
• Point clouds
Key Motivation
• We can’t do much if we do not know the distance between two points!
Representation Motivation
• Implicit surfaces
• Facilitate fundamental computations
• Natural representation for many algorithms (e.g., medical imaging)
• Part of the computation very often (distance functions)
• Point clouds
• Natural representation for 3D scanners
• Natural representation for manifold learning
• Dimensionality independent
• Pure geometry (no artificial meshes, etc)
Background: Distance and Geodesic Computation via Dijkstra
• Complexity: O(n log n)
• Advantage: Works in any dimension and with any geometry (graphs)
• Problems:
• Not consistent
• Unorganized points?
• Noise?
• Implicit surfaces?
Background: Distance and Geodesic Computation via Dijkstra
• Complexity: O(n log n)
• Advantage: Works in any dimension and with any geometry (graphs)
• Problems:
• Not consistent
• Unorganized points?
• Noise?
• Implicit surfaces?
Background: Distance and Geodesic Computation via Dijkstra
• Complexity: O(n log n)
• Advantage: Works in any dimension and with any geometry (graphs)
• Problems:
• Not consistent
• Unorganized points?
• Noise?
• Implicit surfaces?
Background: Distance and Geodesic Computation via Dijkstra
• Complexity: O(n log n)
• Advantage: Works in any dimension and with any geometry (graphs)
• Problems:
• Not consistent
• Unorganized points?
• Noise?
• Implicit surfaces?
Background: Distance and Geodesic Computation via Dijkstra
• Complexity: O(n log n)
• Advantage: Works in any dimension and with any geometry (graphs)
• Problems:
• Not consistent
• Unorganized points?
• Noise?
• Implicit surfaces?
Background: Distance Functions as Hamilton-Jacobi Equations
• g = weight on the hyper-surface
• The g-weighted distance function between two points p and x on the hyper-surface S is:

d

Slope = 1

• Solved in O(n log n) by Tsitsiklis, by Sethian, and by Helmsen, only for Euclidean spaces and Cartesian grids
• Solved only for acute 3D triangulations by Kimmel and Sethian

^

^

T4=T0+4Δ

10

11

12

^

^

T5=T0+5Δ

13

^

T0

1

2

3

4

Pr

Pmin

^

^

T1=T0+Δ

5

^

^

T2=T0+2Δ

^

^

T3=T0+3Δ

6

7

8

9

O(N) Implementation of the Fast Marching Algorithm (Yatziv, Bartesaghi, S.)
• Untidy Priority queue
• Cyclic Bucket sort
• Calendar queue
• Rounding off priorities
• Error bounded O(h)
• Size of array can be small and unrelated to N

### A real time example(following Cohen-Kimmel)

The Problem
• How to compute intrinsic distances and geodesics for
• General dimensions
• Implicit surfaces
• Unorganized noisy points (hyper-surfaces just given by examples)
Our Approach
• We have to solve
Basic Idea

Theorem (Memoli-S.):

Implicit Form Representation

Figure from G. Turk

Data extension
• Embed M:
• Extend I outside M:

M

I

I

I

Randomly sampled manifolds(with noise)

Theorem (Memoli - S. 2002):

Intermezzo: Tenenbaum, de Silva, et al...
• Main Problem:
• Doesn’t address noisy examples/measurements: Much less robust to noise!
Intermezzo: de Silva, Tenenbaum, et al...
• Problems:
• Doesn’t address noisy examples/measurements: Much less robust to noise!
• Only convex surfaces
• Uses Dijkestra (back to non consistency)
• Doesn’t work for implicit surface representations
Concluding remarks
• A general computational framework for distance functions and geodesics.
• Implicit hyper-surfaces and un-organized points

End of part 1

Thank You