Application: Multiresolution Curves Jyun-Ming Chen Spring 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 ) 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 functions Can 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 function 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) A b

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 linear 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 directly used in geometric representation However, B-spline basis converges slowly in the iterative solver –compact support of the basis 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

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(n 2 log 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 algorithms Results 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