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

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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.

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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

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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.

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Meshless Wavelet Tight-Frames

• Charles Chui
• Wenjie He
• University of Missouri-St. Louis
• March 29, 2005 Savannah, Georgia

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Stationary Wavelets
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Stationary wavelet notation
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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.
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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.

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Wavelet tight-frames associated with
Multiresolution Analysis (MRA)

• Refinable function
• Frame generators
• Two-scale symbols
• Vanishing moments of order K

is divisible by
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Unitary matrix extension (UEP) for MRA tight
frames

• Let
• Then
  is a normalized tight frame.

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Equivalent matrix formulation
on
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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
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Full characterization of MRA tight frames
Oblique Extension Principle (OEP)
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Minimum-supported VMR functions for cardinal
B-splines

• For achieving vanishing moments for all
  tight-frame generators with symbols

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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

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Wavelet decomposition and reconstruction

• Decomposition and perfect reconstruction scheme
  for computing DFWT

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FIR schemes

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

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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

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Non-Stationary Wavelets
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Non-stationary MRA (NMRA) wavelets

• Let and be the two-scale
  matrices of the refinable functions
  and the wavelets
• , respectively that is,
• where

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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 .
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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,

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NMRA tight-frame conditions

• (1) For a finite interval I,
• For an infinite interval I, each is
  bounded
• on and
• (2)

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Non-stationary filters

• Non-stationary DFWT decomposition and perfect
  reconstruction

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Matrix factorization for stationary tight frames
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Matrix factorization for non-stationary tight
frames
where we use the notations
and
the even rows of
the odd rows of
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FIR filters for non-stationary perfect
reconstruction
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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.

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Interior wavelets with simple knots
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Boundary wavelets with simple interior knots
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Interior wavelets with double knots
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Boundary wavelets with double interior knots
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• Meshless Spline Wavelets

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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 .
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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

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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
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Refinement matrices

• can be derived by the knot insertion" identity

where
and
with
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Tight-frame wavelets with maximum order of
vanishing moments

• Wavelets
• Define operators
• that associate with some symmetric matrices
  s
• Tight wavelet frames

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Tight frame condition imposed on the
nonstationary wavelets
and
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VMR matrices s construction
is the row-vector of approximate duals for
,
that is,
where P is the polar form of
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k-Voronoi diagrams simplex spline
interpolation
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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
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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
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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 )

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Simplex spline interpolation

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

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Simplex spline interpolation

• What knot sets to use?

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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
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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.

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What do you see in this map?
Contour mapfor fishing (Imagine theboaters
map)
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Management

• POC Jack Snoeyink
• UMSL - Mathematical development
• UNC - Algorithmic development
• Coupled by project wiki visits

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Four phases
Perf period Primary focus Cost
Phase 1 Mathematical devel feasibility 759,569
18 months
Phase 2 Application and prototype devel 843,787
18 mo
Phase 3 Intensive devel of key applications 389,793
12 months
Phase 4 Transition to industry 351,114
12 months

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

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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)

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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

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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