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Meshless wavelets and their application to terrain modeling

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Title: Meshless wavelets and their application to terrain modeling


1
Meshless wavelets and their application to
terrain modeling
  • A DARPA GEO project
  • Jack Snoeyink, Leonard McMillan, Marc Pollefeys,
    Wei Wang (UNC-CH)
  • Charles Chui, Wenjie He (UMSL)

2
Outline
  • Project Team, Motivation, Objectives
  • Meshless wavelets
  • CK Chui Compactly supported, refinable spline
    fcns
  • Y Liu Order-k Voronoi diagrams simplex splines
  • Simplification/compression for applications
  • Mobility elevation slope mapping
  • Feature identification and matching
  • Management, Risks Rewards

3
Team Introduction
  • U of Missouri, St. Louis
  • Charles K Chui wavelets splines
  • Wenjie He splines
  • UNC Chapel Hill
  • Jack Snoeyink computational geometry
  • Marc Pollyfeys computer vision
  • Leonard McMillan computer graphics
  • Wei Wang spatial databases
  • Yuanxin (Leo) Liu Henry McEuen

4
Self-evident truths
  • Terrain data volumes are increasing.
  • NIMA In only 9 days and 18 hours, SRTM
    collected elevation data for 80 of the world's
    landmass to enable the production of DTED Level
    2.
  • Old data formats were chosen for ease of
    computation more than completeness of
    representation.
  • Consider USGS raster DEMs use of integer
    identifiers.
  • Terrain is irregular and multi-scale its
    representation should be, too.
  • breaklines, multiple sources sensors, viewer
    level of interest
  • Consistency is a virtue in multi-(use,
    resolution, sensor, spectral...)
  • Example of elevation and slope mentioned in BAA
  • Image compression schemes are designed to look
    good.
  • TIFF, JPEG, JPEG2000,
  • The GIS industry cannot innovate on data reps.
  • Backward compatibility trumps even algorithmic
    improvements
  • It is a good time to look at new options for
    terrain representation.

5
Key research question
  • What compact representations of terrain still
    support interesting queries?
  • Elevation slope for mobility visibility
  • Feature identification across imaging modes and
    viewing conditions for localization, change
    detection, and terrain construction

6
Bivariate meshless wavelets
  • We propose
  • a new compact representation for geospatial data
    that is optimized for specific geometric and
    image queries.
  • meshless'' bivariate wavelets defined over
    scattered point sets allow a flexible description
    since the point set can be specified without
    connectivity and each point's influence is local,
    while still supporting the multiscale analysis
    afforded by wavelets.
  • Objectives
  • complete the theory of bivariate meshless
    wavelets
  • point/knot selection algorithms optimized for
    specific geometric tasks and data queries
  • demonstration implementation showing the
    advantages of our modeling approach.

7
Meshless Wavelet Tight-Frames
  • Charles Chui
  • Wenjie He
  • University of Missouri-St. Louis
  • March 29, 2005 Savannah, Georgia

8
Stationary Wavelets
9
Stationary wavelet notation
10
Definition of stationary wavelet tight-frames
A family
is a stationary wavelet frame of
, if there exist constants
such that
If , the frame is called a normalized
tight frame.
11
Characterization of wavelet tight-frames
  • Theorem. Frazier-Garrigós-Wang-Weiss 1996,
    Ron-Shen 1997, Chui-Shi 1999.
  • Let .
    The family is a normalized tight
    frame of , if and only if
  • and


  • odd.

12
Wavelet tight-frames associated with
Multiresolution Analysis (MRA)
  • Refinable function
  • Frame generators
  • Two-scale symbols
  • Vanishing moments of order K

is divisible by
13
Unitary matrix extension (UEP) for MRA tight
frames
  • Let
  • Then
    is a normalized tight frame.

14
Equivalent matrix formulation
on
15
Limitations of UEP
  • Applicable only if
  • For , i.e.,
    cardinal B-spline of order m,
  • at least one of the has only the
    factor of
  • but not a higher power, (i.e., only one
    vanishing moment
  • for the corresponding frame generator).

on
16
Full characterization of MRA tight frames
Oblique Extension Principle (OEP)
17
Minimum-supported VMR functions for cardinal
B-splines
  • For achieving vanishing moments for all
    tight-frame generators with symbols

18
Orders of vanishing moments
  • Each has at least K vanishing moments,
    i.e.
  • has vanishing moments of order at least K, if
    and only if

19
Wavelet decomposition and reconstruction
  • Decomposition and perfect reconstruction scheme
    for computing DFWT

20
FIR schemes
  • New FIR filters for perfect reconstruction from
    DFWT with higher order of vanishing moments.

21
Existence of perfect reconstruction FIR filters
  • (Chui and He) Suppose that
    are Laurent polynomials, and
    that the matrix

  • has full rank for
  • Then there exist
    such that

22
Non-Stationary Wavelets
23
Non-stationary MRA (NMRA) wavelets
  • Let and be the two-scale
    matrices of the refinable functions
    and the wavelets
  • , respectively that is,
  • where

24
Vanishing moment condition
  • is an approximate dual of order
    L.
  • If I is a finite interval, the above condition is
    equivalent to

the space of all polynomials of
degree up to .
25
NMRA wavelet tight-frames
  • VMR matrices are symmetric positive
    semi-definite banded matrices
  • If I is a finite interval,
  • If I is an infinite interval,

26
NMRA tight-frame conditions
  • (1) For a finite interval I,
  • For an infinite interval I, each is
    bounded
  • on and
  • (2)

27
Non-stationary filters
  • Non-stationary DFWT decomposition and perfect
    reconstruction

28
Matrix factorization for stationary tight frames
29
Matrix factorization for non-stationary tight
frames
where we use the notations
and
the even rows of
the odd rows of
30
FIR filters for non-stationary perfect
reconstruction
31
Two-scale matrix
  • Consider two nested knot vectors
  • we have the refinement equation
  • where the matrix has
    non-negative entries, with each row summing to 1.
  • can be derived by a sequence of knot
    insertions.

32
Interior wavelets with simple knots
33
Boundary wavelets with simple interior knots
34
Interior wavelets with double knots
35
Boundary wavelets with double interior knots
36
  • Meshless Spline Wavelets

37
Simplex spline
D a bounded convex polygonal domain in
T a knot set in D
such that the projection of the set of vertices
of simplex to is .
38
Neamtus work  on bivariate splines
  • The space of bivariate polynomials of
    (total) degree k is locally generated by
    simplex splines defined on the Delaunay
    configuration of degree
    k

39
A multi-level approximation by bivariate
B-splines
Let
be a nested sequence of knot sets.
Let denote the Delaunay configuration
associated with the knot set .
represent bivariate B-splines
corresponding to
40
Refinement matrices
  • can be derived by the knot insertion" identity

where
and
with
41
Tight-frame wavelets with maximum order of
vanishing moments
  • Wavelets
  • Define operators
  • that associate with some symmetric matrices
    s
  • Tight wavelet frames

42
Tight frame condition imposed on the
nonstationary wavelets
and
43
VMR matrices s construction
is the row-vector of approximate duals for
,
that is,
where P is the polar form of
44
k-Voronoi diagrams simplex spline
interpolation
45
k-Voronoi diagrams
  • A set of knots X in 2D
  • A family of (i3) subsets of X (
    features in (i1)-Voronoi diagram )
  • A set degree-k of simplex spline basis

A set of terrain samples P in 2D
Simplex spline surface
46
k-Voronoi diagrams
  • Definition A k-Voronoi diagram in 2D partitions
    the plane into cells such that points in each
    cell have the same closest k neighbors.

Order 1
Order 3
47
k-Voronoi diagrams
  • Computation - Theory O(n log(n)) time
    O(n) space - Practice O(n) time
  • Engineering challenges
  • speed
  • memory (streaming )
  • robustness ( degeneracy, round-off errors )

48
Simplex spline interpolation
  • Problem Given a set of terrain sample points,
    reconstruct the terrain with simplex splines.

49
Simplex spline interpolation
  • What knot sets to use?

50
k-Voronoi diagrams
  • A set of knots X in 2D
  • A family of (i3) subsets of X (
    features in (i1)-Voronoi diagram )
  • A set degree-k of simplex spline basis

A set of terrain samples P in 2D
Simplex spline surface
51
Simplify, preserving essentials
  • BAA says that GEO emphasizes the development of
    math and algorithms that enable parsimonious
    representations coupled to end user
    applications image to DEM, targeting, route
    planning, and motion mobility simulations.
  • Key question who defines end user application?
  • General compression schemes are good. To be
    better, we need a user, even if the user is us.

52
What do you see in this map?
Contour mapfor fishing (Imagine theboaters
map)
53
Management
  • POC Jack Snoeyink
  • UMSL - Mathematical development
  • UNC - Algorithmic development
  • Coupled by project wiki visits

54
Four phases
  • mathematics of meshless wavelets and finding key
    points for applications to include compression,
    registration, route planning, and visibility.
  • developing prototypes for these applications on
    top of the meshless wavelets and key points
    representations,
  • Option to develop one or more applications in
    detail,
  • Option for additional focused efforts by the PIs
    to transition technology to an industrial or
    military partner.

55
Risks
  • The mathematics is challenging
  • Goal is meshless wavelets, but can begin with
    tensor-product constructions
  • The implementation is complex
  • Order-k Voronoi simplex splines wavelets
    interpolation will initially be dominated by
    regular grids
  • Need data and user contacts
  • Contact with Dr. Alexander Reid, terrain modeling
    project leader, U.S. Army TACOM Lab (Warren, MI)

56
Rewards
  • Wavelet analysis of surfaces from irregular data
    samples.
  • Compression that can be tuned to a particular
    application of the terrain
  • Feature identification across imaging modalities,
    conditions, and scales

57
UNC CH UMSL GEO BAA 0412, Add 2 Meshless
wavelets and their application to terrain
modeling
  • Description / Objectives / Methods
  • Wavelet analysis for smooth terrain on
    irregularly sampled data
  • Construct compactly supported, refineable spline
    functions
  • Tensor product splines wavelets
  • Order-k Voronoi, simplex splines, VIP
  • Compact level-of-detail representations with
    consistent analysis
  • Feature identification in multimodal
  • Analysis for shortest paths, visibility
  • Schedule
  • Phase I mathematical development
  • 6 mo tensor product representation order-k
    Voronoi for simplex splines point importance
    orders
  • 18mo wavelet analysis for simplex splines
    initial feature identification
  • Phase II application development
  • Mobility, visibility, feature matching,
    localization
  • Further work on applications transition to
    military
  • Military Impact / Sponsorship
  • Compact, yet accurate terrain reprsntns for
    mobility and multimodal feature analysis give
    better planning and positioning
  • Seek DARPA help to obtain terrain data from Army
    TACOM Lab (contact Dr. A. Reid)
  • Seek multimodal data same area under various
    sensors conditions
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