Meshless wavelets and their application to terrain modeling
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Transcript of Meshless wavelets and their application to terrain modeling
Meshless wavelets and their Meshless wavelets and their application to terrain modelingapplication to terrain modeling
A DARPA GEO* projectA DARPA GEO* project
Jack Snoeyink, Leonard McMillan, Jack Snoeyink, Leonard McMillan, Marc Pollefeys, Wei Wang (UNC-CH)Marc Pollefeys, Wei Wang (UNC-CH)
Charles Chui, Wenjie He (UMSL)Charles Chui, Wenjie He (UMSL)
OutlineOutline
• 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
• 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
Team IntroductionTeam Introduction
Self-evident truths …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 DEM’s 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.
Key research questionKey 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
Bivariate meshless waveletsBivariate 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.
Meshless Wavelet Tight-Frames
Charles Chui
Wenjie He
University of Missouri-St. Louis
March 29, 2005 Savannah, Georgia
Stationary Wavelets
Stationary wavelet notation
2( ) 2 (2 )j jj k x x k j k Z
2( )L R
Definition of stationary wavelet tight-frames
2 2 2; , 2
1
|| || || || ( )n
i j ki j k Z
A f f B f f L R
; ,{ 1 , ,}i j k i n j k Z
2( )L R 0 A B
A family is a stationary wavelet
frame of , if there exist constants
such that
If , the frame is called a normalized tight frame.
1A B
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.
2
1
ˆ (2 ) 1 a en
ji
j Z i
0 1
ˆ (2 ) (2 ( 2 )) 0 a en
j jii
j i
k k
2( ),i L R 1 i n { }i j k
2( )L R
Wavelet tight-frames associated with
Multiresolution Analysis (MRA)
• Refinable function:• Frame generators:• Two-scale symbols:
• Vanishing moments of order K:
2
1
( ) (2 )N
kk N
x p x k
( )( ) (2 )k
k
x q x k
12( ) ,k
kkP z p z ( )1
2( ) ,kkk
Q z q z
iz e
( )Q z is divisible by (1 ) .Kz
Unitary matrix extension (UEP) for MRA tight frames
Let
Then is a normalized tight frame.
2 2
1
| ( ) ( ) 1,n
P z Q z
1
( ) ( ) ( ) ( ) 0,n
P z P z Q z Q z
| | 1.z
{ 1 }j k n j k Z
Equivalent matrix formulation
1 1
( ) ( )
( ) ( )The matrix has orthonormal columns
( ) ( )n n
P z P z
Q z Q z
Q z Q z
| | 1.z on
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).
2 2( ) ( ) 1P z P z | | 1.z on
1( )
2
mz
P z
11
2 2
1 0
1 1( ) 1 ( ) 1,
2 2 2
kn m
ii k
x x z zQ z P z x z
( )iQ z (1 )z
Full characterization of MRA tight frames
2 2 2
1( ) ( ) ( ) ( )
n
iiS z P z Q z S z
2
1( ) ( ) ( ) ( ) ( ) 0 1
n
i iiS z P z P z Q z Q z z
Oblique Extension Principle (OEP)
Minimum-supported VMR functions for cardinal B-splines
For achieving vanishing moments for all tight-frame generators with symbols ( ) 1 ,iQ z i … n
11 12 20
1 110 04 1
( )
1 ( 1) 4j
m jm m x z zjj
jm m j mj
S z s x
ms s s
j
( )mS z
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
2 2ˆ1 ( ) ( ) ( ) for 0i KS e O
Wavelet decomposition and reconstruction
Decomposition and perfect reconstruction scheme for computing DFWT
FIR schemes
New FIR filters for perfect reconstruction from DFWT with higher order of vanishing moments.
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
1( ), ( ), , ( )nP z Q z Q z
1
1
( )( )( )
( )( )( )n
n
Q zQ zP z
Q zQ zP z
0.z
1( ), ( ), , ( ),nP z Q z Q z
1
1
( )( )( )
( )( )( )n
n
Q zQ zP z
Q zQ zP z
1 1
1 11 1
1 1
( ) ( )
( ) ( )
( ) ( )n n
P z P z
Q z Q z
Q z Q z
2I
Non-Stationary Wavelets
Non-stationary MRA (NMRA) wavelets
Let and be the two-scale matrices of the “refinable” functions and the wavelets
, respectively; that is,
where
1 1j j j j j jP Q
{ }jP { }jQ
{ }j
{ }j
[ ],j j k [ ].j j k
Vanishing moment condition
• is an approximate dual of order L.
• If I is a finite interval, the above condition is equivalent to
21|| || 0j LT f f f j
: the space of all polynomials of degree up to . 1L 1L
jSjS
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,
jS
T
j j k j j kT f f S f
2 20 2
0
|| || ( )j
j kj k
T f f f f f L I
2 22|| || ( )
j
j kj Z k
f f f L I
NMRA tight-frame conditions
(1) For a finite interval I,
For an infinite interval I, each is bounded
on and
(2)
22lim || || ( )j
jT f f f L I
1T T
j j j j j jS P S P Q Q
2lim || || and lim 0.j jj j
T f f T f
jT
2( ),L I
Non-stationary filters
Non-stationary DFWT decomposition and perfect reconstruction
Matrix factorization for stationary tight frames
1 12 2 2 11 2 1 1
1 12 1 2 21 2 2 2
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
Q z Q z Q z Q zS z S z P z S z P z P z
Q z Q z Q z Q zS z P z P z S z S z P z
Matrix factorization for non-stationary tight frames
1 1
1 1
Tee e e T ee eeeo e o T eo eoj j j j j jj j j j j j
oe o e T oo o o T oe oo oe ooj j j j j j j j j j j j
S P S P S P S P Q Q Q Q
S P S P S P S P Q Q Q Q
2 2 2 2 1[ ] [ ],ee eok kM m M m 2 1 2[ ],oe
kM m 2 1 2 1[ ].ookM m
where we use the notations
and
:ejP :o
jPthe even rows of ,jPjP the odd rows of .jP
FIR filters for non-stationary perfect reconstruction
1 1 1
1
# #e o
# #e e ne e o
No no o
# #e n o n
P P
P Q Q Q QI
P Q Q
Q Q
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.
,t t
t m t m t t mP
t t mP
t t mP
Interior wavelets with simple knots
Boundary wavelets with simple interior knots
Interior wavelets with double knots
Boundary wavelets with double interior knots
Meshless Spline Wavelets
Simplex spline
{ }J jX j J T x
3J#X k 2
2
{ }volvol
IR( ) k
k
JM X
v v xx
T: a knot set in D
D: a bounded convex polygonal domain in 2IR
such that the projection of the set of vertices of simplex to is . 2IR JX
Neamtu’s 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
2k
{( )}k B IX X 1
2( ) area[ ] ( )
2B I B B I
kN X X X M X X
( )
( ) { ( ) 3}B K k K I
I B K k B I k BX X X X
B N X X I I I X X #X
A multi-level approximation by bivariate B-splines
(0) (1) ( )jT T T Let
be a nested sequence of knot sets.
Let denote the Delaunay configuration associated with the knot set .
( )jk
( )jT[ IN ]j j j
[ ]I kB I I represent bivariate B-splines
corresponding to ( )jk
Refinement matrices
can be derived by the “knot insertion" identity
1j j jP
0
( ) ( { })r
J Ji i
i
M X c M X
y
where { }J J i i JiX X \ X x x
0
r iii
c
y x0
1r
iic
and with
Tight-frame wavelets with maximum order of vanishing moments
• Wavelets• Define operators
that associate with some symmetric matrices ’s • Tight wavelet frames
1j j jQ T
j j j jT f f S f
jS
2 20 2
0
|| || ( )jj
T f f f f L D
Tight frame condition imposed on the nonstationary wavelets
22lim || || ( )j
jT f f f L D
and
1T T
j j j j j jS P S P Q Q
VMR matrices ’s construction
1[ ]j j
j
S Sj j r j jS
jS
is the row-vector of approximate duals for , j
that is,
( ) for1 and 0jSj I jp P X r j
where P is the polar form of 2kp
k-Voronoi diagrams k-Voronoi diagrams & &
simplex spline interpolationsimplex spline interpolation
k-Voronoi diagramsk-Voronoi diagrams A set of knots X in 2D
A family of (i+3) subsets of X ( features in (i+1)-Voronoi diagram )
A set degree-k of simplex spline basis
A set of terrain samples P in 2D
Simplex spline surface
k-Voronoi diagramsk-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
k-Voronoi diagramsk-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 )
Simplex spline interpolationSimplex spline interpolation• Problem: Given a set of terrain sample points,
reconstruct the terrain with simplex splines.
Simplex spline interpolationSimplex spline interpolation• What knot sets to use?
k-Voronoi diagramsk-Voronoi diagrams A set of knots X in 2D
A family of (i+3) subsets of X ( features in (i+1)-Voronoi diagram )
A set degree-k of simplex spline basis
A set of terrain samples P in 2D
Simplex spline surface
Simplify, preserving essentialsSimplify, 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.
Contour mapfor fishing…
(Imagine theboaters’ map)
What do you see in this map?What do you see in this map?
ManagementManagement
• POC: Jack Snoeyink
• UMSL - Mathematical development
• UNC - Algorithmic development
• Coupled by project wiki & visits
Four phasesFour phases
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.
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
RisksRisks
• 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)
RewardsRewards
• 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
29 Mar 05 Snoeyink, McMillian, Polyfeys, Wang; Chui, He
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
UNC CH & UMSL GEO* BAA 04–12, 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
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