geodesics on implicit surfaces and point clouds n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Geodesics on Implicit Surfaces and Point Clouds PowerPoint Presentation
Download Presentation
Geodesics on Implicit Surfaces and Point Clouds

Loading in 2 Seconds...

play fullscreen
1 / 49

Geodesics on Implicit Surfaces and Point Clouds - PowerPoint PPT Presentation


  • 112 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Geodesics on Implicit Surfaces and Point Clouds' - yetty


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
geodesics on implicit surfaces and point clouds

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
Overview
  • Motivation
  • Distances and Geodesics
    • Implicit surfaces
    • Point clouds
key motivation
Key Motivation
  • We can’t do much if we do not know the distance between two points!
representation motivation
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
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 dijkstra1
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 dijkstra2
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 dijkstra3
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 dijkstra4
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
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:
slide18

d

Slope = 1

background computing distance functions as hamilton jacobi equations
Background: Computing Distance Functions as Hamilton-Jacobi Equations
  • 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
o n implementation of the fast marching algorithm yatziv bartesaghi s

^

^

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
the problem
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
Our Approach
  • We have to solve
basic idea2
Basic Idea

Theorem (Memoli-S.):

implicit form representation
Implicit Form Representation

Figure from G. Turk

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

M

I

I

I

randomly sampled manifolds with noise
Randomly sampled manifolds(with noise)

Theorem (Memoli - S. 2002):

intermezzo tenenbaum de silva et al
Intermezzo: Tenenbaum, de Silva, et al...
  • Main Problem:
    • Doesn’t address noisy examples/measurements: Much less robust to noise!
intermezzo de silva tenenbaum et al1
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
Concluding remarks
  • A general computational framework for distance functions and geodesics.
  • Implicit hyper-surfaces and un-organized points
slide49

End of part 1

Thank You