A%20Simple,%20Fast,%20and%20Effective%20Scattered%20Data%20Interpolation%20using%20Multilevel%20Local%20B-spline%20Approximation

1
A Simple, Fast, and Effective Scattered Data
Interpolation using Multilevel Local B-spline
Approximation

• KSIAM 2004, Gyungju, Dec. 34
• Lee Byung Gook, Lee Joon Jae
• Dongseo University, Busan, Korea
• 4th Dec. 2004

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Outline

• Overview
• B-spline approximation
• Global approximation
• Quasi-interpolants
• Multilevel B-spline approximation
• Multiresolution applications
• Image representation and compression
• Surface approximation

3
Overview

• Uniform data(image, range data)
• Scatter data(3D scanner, PCB)
• Algorithms
• B-spline approximation
• Radial basis function interpolation
• Moving least square method
• Thin plate spline
• Multilevel B-spline approximation with
  quasi-interpolation

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B-spline Approximation

• Cubic B-spline approximation
• Scatter data points
• Rectangular domain
• Set hx,hy as the knot interval, where
• x-direction
• y-direction
• Set (nx 3)x(ny 3) control lattice, ?
• Let ?ij as control point on lattice ?, located at
  (ihx, jhy) for
• i -1,0,1,,mx1 and j -1,0,1,,my1

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Initial Approximation Function

• Formulate approximate function f as a uniform
  bicubic B-spline function, defined by a control
  lattice ?
• Approximation function f is defined as
• Bi and Bj are uniform cubic B-spline basis
  functions, where the knot vector are

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Cubic B-spline Basis Function

• B-spline basis function of degree 3 on a uniform
  partition (-2,-1,0,1,2) is defined as the
  piecewise polynomial,

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Control Lattice Configuration
Legend
Control Point
Data Point
Domain
Knot Vector
In x-y axis mxhxnx myhyny
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Using Global curve case

• Linear system solution B? Z
• Least-square solution to find control points
• ? (BTB)-1BTZ

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Using Global surface case

• Linear system solution involves all data points
  B? Z
• Least-square solution to find control points
• ? (BTB)-1BTZ
• Becomes complex to solve the least-square
  solution when number of data points increase

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B-Spline Algorithm Result
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Quasi-Interpolation

• Introduce by deBoor and Fix
• Local property
• Constructs a B-spline curve to approximate
  another curve
• Exact where possible, otherwise best-fitting

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Quasi-interpolation

• We shall approximate f by an approximation Pd f
  in a spline space Sd,? on the form
• where Bj,d is a sequence of Bspline of order
  d for the knot vector
• ?j f are appropriate linear functional chosen so
  that
• Pd f can be applied to a large class of functions
  including, for example, continuous functions
• Pd f is local in the sense that (Pd f )(x)
  depends only on values of f in a small
  neighborhood of x

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How to construct ?j f

• We shall construct ?j f on the form
• where are given data points in the
  vicinity of the support ?j , ?jd1 of the
  B-spline Bj,d
• for all f in the spline space

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Using Quasi-Interpolation curve case

• To compute the ?i get sub linear system, Bi?i
  Zi
• Least-square solution, ?i (BiTBi)-1BiTZi
• Choose the middle control point ?i from the set
  of ?i

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Uniform data Curve Case

• when h1, weight w
• when h2, weight w

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Using Quasi-Interpolation surface case
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Uniform data Surface Case

• when h1,weighted w
• when h2, weighted w

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Multilevel B-spline Approximation
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Multilevel B-Spline Approximation

• Hierarchy control lattices, ?0, ?1, , ?h
  overlaid on the domain ?
• Spacing for ?0 is given (m3)x(n3) and the
  spacing from one lattice to the next lattice is
  halved where (2m3)x(2n3) lattice
• The ij-th control point in ?k coincides with that
  of the (2i, 2j)-th control point in ?k1
• Begin from coarsest ?0 to finest ?h serve to
  approximate and remove the residual error

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Multilevel B-Splines Lattices

• Set h 6
• ?0 m0 n0 1
• ?1 m1 n1 2
• ?2 m2 n2 4
• ?3 m3 n3 8
• ?4 m4 n4 16
• ?5 m5 n5 32
• ?6 m6 n6 64

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Optimization with B-Spline Refinement

• In Multilevel B-Spline Approximation, the
  approximation function f is the sum of each fk
  from each ?k up to h level
• B-Spline Refinement is apply progressively to the
  control lattice hierarchy to overcome the
  computation overhead

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Optimization of Multilevel B-Splines Algorithm
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Refinement of Bicubic C2 Continuous Splines

• Refinement operator R1/2 take fk by inserting
  grid lines halfway between the ?k to obtain ?k1
• Let denote Ri(t), i0,1,2,3, the univariate
  B-Spline basis functions in refined space
• Ri(t) Bi(2t) on the interval 0 t lt 0.5

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Control points in ?

• ?2i,2j 1/64?i-1,j-1?i-1,j1?i1,j-1?i1,j
  1
• 6(?i-1,j?i,j-1?i,j1?i1,j)36?ij
• ?2i,2j1 1/16?i-1,j?i-1,j1?i1,j?i1,j1
  6(?ij?i,j1)
• ?2i1,2j 1/16?i,j-1?i,j1?i1,j-1?i1,j1
  6(?ij?i1,j)
• ?2i1,2j1 1/4?ij?i,j1?i1,j?i1,j1
• References Ø. Hjelle. Approximation of
  Scattered Data with Multilevel B-splines.
  Technical Report STF42 A01011, SINTEF 2001

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Experimental Results
26
Experimental Results
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Reference and Future Work

• Reference
• S. Y. Lee, G. Wolberg, and S. Y. Shin. Scattered
  Data Interpolation with Multilevel B-Splines,
  IEEE Transactions on Visualization and Computer
  Graphics, 3(3) 229-244, 1997.
• Ø. Hjelle. Approximation of Scattered Data with
  Multilevel B-splines. Technical Report STF42
  A01011, SINTEF 2001
• Lyche, T. and Knut M?rken, Spline Methods Draft,
  2003, pp. 112177.
• Byung-Gook Lee, Tom Lyche and Knut Morken, Some
  Examples of Quasi-Interpolants Constructed from
  Local Spline Projectors, Mathematical Methods for
  Curves and Surfaces Oslo 2000, T. Lyche, and L.L.
  Schumaker, (eds.), Vanderbilt Press, Nashville,
  pp. 243-252, 2001.
• Future work
• Non-uniform knot vector
• Different degree of B-spline Approximation