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Application:Multiresolution Curves
Jyun-Ming ChenSpring 2001
Introduction
• Plays fundamental roles in– Animation, 2D design, …– CAD: cross section design
• A good representation should support– Continuous level of
smoothing (fig)– Editing
• LOD; direct manipulation– Data fitting
• We use B-spline wavelets to develop multiresolution curve
• All algorithms are simple, fast and require no extra storage (we shall see)
Curve Smoothing• Construct an approximate
curve with fewer control points
• Assume end-point interpolating cubic B-spline curve
• Discrete nature:– m = 4, 5, 7, 11, 19, …
• Trivially done using analysis filters
• Fast– Linear with banded LU
Fractional-level Curve
• Resolving discrete nature
• Smoothing …• Editing (p.113-116)• Direct manipulation• Local change
Example
Direct Manipulation
MR Editing
• Changing overall sweep• Alter detailed characterstic• (eq on p.113 is quite flexible, depending on j)
MR Editing• Curve character library
Contains different detail functionsCan be extracted from hand-drawn strokes, or procedurally generated
Application: Variational Modeling
Introduction• In geometry design, instead
of direct manipulating the mathematical representation, sometimes we set up an objective function (typically as a minimization of some functional) and subject to some constraints; and let the computer determine the “best” shape satisfying the conditions
• Minimizing the integral is in the domain of variational calculus (and so named variational modeling)
• Wavelets are useful in speeding the computations required for variational modeling
Example Problem
• Design a “smooth” curve that passes through some particular points
• The curve (here: a functional curve)
• Formulate “smoothness” as a variational problem• (minimize total curvature)
Aside: for surface problems
• Smoothness/fairing– Energy-minimizing surface
Solution Method: the finite-element method
• Choose a set of basis functions (called finite element)– discretize and parameterize
the problem space• Represent the unknown fu
nction as a linear combination of the finite elements
• Substitute back to the original problem
Back to the Problem
• If we choose to represent the curve as a quartic function
Problem (cont)
Ab
Algebraic Manipulation
The problem becomes …
• Problem in the form of quadratic programming• Use the method of Lagrange multipliers
– Works well for quadratic programming problems
Discussion• In general, the matrix in the li
near system is quite large; therefore, iterative solvers (e.g., Gauss-Seidel or conjugate gradient) are used
• Unlike the previous demonstration, usually B-spline basis is chosen (instead of the monomial basis)– The computation result can be d
irectly used in geometric representation
• However, B-spline basis converges slowly in the iterative solver– compact support of the basi
s prohibits broad changes• Gortler and Cohen (1995)
uses B-spline wavelets for the finite element (instead of B-splines themselves) and works better
• Mathematically, they solve
where W represents the wavelet transform and is the set of the wavelet coefficient for the solution
Gortler and Cohen (cont)• Intuitively, the wavelet
basis allows changes in the curve to propagate much more quickly from one region to another by allowing the effects to “bubble up” the hierarchy to basis functions with broader support and then descend back down to hierarchy to the narrower basis functions for the regions affected
x̂
Application: Tiling
Skipped for now
Tiling: The Problem• General requirement: matched (linked)
indentations• “correct” tiling: depends on the nature of the
problem
Solution Methods
• Optimization– Formulate as graph
searching problem; solved by dynamic programming
– High complexity O(n2log n)
– Too expensive for interactive applications with thousands of vertices
• Greedy methods– Linear time– Do not work well
Challenging Case (Contours from Human Brain)
Input:A pair of contours
Results from greedy algorithmsResults from
optimizing algorithm;Still require user
interaction
Multiresolution Tiling Meyers et al. (1992)
Details (MR Tiling)
Compare: MR Tiling and Optimizing Method
MR Tiling Optimization
Application:Surfaces
• Polyhedral compression• continuous level-of-detail• progressive transmission• Multiresolution editig