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Advanced Geometric ModelingCPSC789

Faramarz SamavatiFall 2004

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General information aboutthe course

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CPSC 789 – Advanced Geometric Modeling Fall 2004

Lecture Time and Place

ENF 334TR 9:30 – 10:45

Instructor : Faramarz Samavati

Office: MS 618

Office Hours: TR 11:00-12:00

Phone: 210 – 9454

Email: samavati@cpsc.ucalgary.ca

Home Page: www.cpsc.ucalgary.ca/~samavati

Faramarz Samavati

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CALGARYCourse Homepagehttp://pages.cpsc.ucalgary.ca/~samavati/cpsc789

Faramarz Samavati

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CALGARY General InformationCourse Subjects

Introduction to spline modeling( Splines, B-spline , NURBS, parametric surfaces and volumes)Subdivision modeling (Knot insertion , curves and tensor product subdivision rules, arbitrary topology surface subdivision, data structures, Loop, Doo-Sabin, Butterfly and Catmull-Clark schemes, convergence and smoothness analysis, Eigen-analysis)Multiresolution modeling( wavelets, Harr wavelets, general formulation, filter banks, B-spline wavelets, reverse subdivision, progressive models, data structure) some new research results in subdivision and multiresolutionComplementary Topics :??

Grading Project %60(Proposal, Reading papers, research, implementation, presentation, technical report)Proposal Due: October 12In class exam %20( Tuesday , November 16, basic and fundamental materials)Assignment %20 (three)

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CALGARY Text

Eric J. Stollnitz, Anthony D. DeRose, and David H. SalesinWavelets for Computer Graphics: Theory and ApplicationsThe Morgan Kaufmann Publishers,1996

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ReferencesGerald FarinCurves and Surfaces for CAGD: A Practical Guide, Fifth EditionThe Morgan Kaufmann Publishers, October 2001ISBN 1-55860-737-4

Online course: -myself-Subdivision for modeling and animation, course notes for SIGGRAPH 2000, D.Zorin, et.al.

J.Warren and H. Weimer,Subdivision methods for geometric design, The Morgan Kaufmann Publishers, 2002

Armin Iske (Editor), Ewald Quak (Editor), Michael S. Floater (Editor)Tutorials on Multiresolution in Geometric ModellingSpringer, 2002.

Les Piegl and Wayne TillerThe NURBS Book,Springer, 1995.

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Overview of the course

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geometric descriptions of objectsspecial case of mathematical modelingthe visual features are important

Geometric Modeling

Mathematical Models (geometrical)

?

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CALGARY Umm!? Mathematical Modeling!

S

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CALGARY Spline Modeling

First try : polynomial models

Piecewise polynomial

User Interaction!: Control points are used to define models

More general than Bezier curves and surfaces

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CALGARY B-spline Models

Input: control points

Output: piecewise polynomial

Basis functions evaluation

Support

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CALGARY B-spline ImportanceMost effective modeling technique in Computer

Graphics , CAD/CAM, approximation theory and

sampling[1]

[1] M. Unser, "Splines: A Perfect Fit for Medical Imaging,“

Keynote address, Proceedings of the SPIE International Symposium on Medical Imaging:

Image Processing (MI'02).

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CALGARY Issues : Basis function evaluation Basis function evaluation is not an efficient method (real time application)

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CALGARY Issues : General TopologyEasy for tensor-product and triangular patches

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CALGARY

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CALGARY

Faramarz Samavati

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CALGARY Issues : General TopologyHow about general topology surface?

Subdivision approach

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CALGARY Increase the resolution: A Simple Algorithm

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Increase the resolution: a Simple algorithm

Step Function : First order spline

Basis (Scaling) functions: first order Bsplines

Discontinuous and Blocky

No overall smoothness

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CALGARY Curve example

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Increase the resolution: Chaikinsubdivision

left: ¾ ¼

right: ¼ ¾

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Increase the resolution: Chaikinsubdivision

A better algorithm

Third order (quadratic ) spline

Scaling functions: third order Bspline

First level of smoothness

Corner cutting

Filter values (left: ¾ ¼ right: ¼ and ¾)

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CALGARY Analysis of the Subdivision methodsNecessary properties:

Affine invarianceConvergence Smoothness

Analysis of Subdivision schemes

convergence ?smoothness ?

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CALGARY SubdivisionCoarse points fine points

Ck+1= PkCk

Pk has a regular banded structure

No need to the basis functions

subdivision

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From Chaikin to Doo-Sabin Subdivision3D corner cutting method

Generalization of Chaikin

Each step =2 stages

Contracting and adjoining

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CALGARY One Step of Doo Subdivision

One Step of Subdivision (F,V) (NF,NV)

Line Face Curve Surface

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CALGARY Edge-to-faces

New Contracted Face

New Contracted Face

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CALGARY Mask DetailsInterior vertices

9/16 3/16

3/16 1/16

n= number of vertices on face

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Examples

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3D Studio Max

Example of Doo Subdivision

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CALGARY Applications

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CALGARY Multiresolution decreasing and increasing of the resolution at the same time

Time and space efficiencies (non-redundancy)

decomposition reconstruction

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CALGARY ApplicationsImage pyramid: refine solution progressively

Compression

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CALGARY Multiresolution filter matrices

decomposition and reconstruction filters (matrices)

F CD

A

BFC

D

P

Q

C=A*FD=B*F

F=P*C+Q*D

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CALGARY Wavelets A good approach to find the filters

F CD

A

B

FCD

P

Q

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CALGARY Wavelets as Underlying FunctionUsing underlying

functions

B-spline

Wavelets

Our idea : constructing a multiresolution by reversing subdivision

Avoiding underlying functions

Subdivision

Reversing subdivision

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CALGARY Multiresolution filter matrices : Haar

Haar wavelets : extension on of the very simple algorithm (duplicating of pixels)simple to implement but …

F CD

A

BFC

D

P

Q

C=A*FD=B*F

F=P*C+Q*D

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CALGARY Bspline waveletsnot simple as Haar, however better result

[1] M. Unser, "Splines: A Perfect Fit for Medical Imaging," Keynote address, Proceedings

of the SPIE International Symposium on Medical Imaging: Image Processing (MI'02).

[2] A.Aldroubi, M. Eden, M. Unser, “Discrete spline filters for multiresolutions and

wavelets of L2”, SIAM J,Math. Anal. 25(1994) 1412-1432.

[3] F.F.Samavati and R.H.Bartels,”Multiresolution curve and surface representation by reversing subdivision rules”. Computer Graphics Forum, 18(2), 97-119, June 1999.

[4] R.H. Bartels and F. F. Samavati,” Reversing Subdivision Rules: Local Linear Conditions and Observations on Inner Products”, Journal of Computational and Applied Mathematics, Vol. 119, Issue 1-2, pp. 29-67, 2000.

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CALGARY A filter for Chaikin (width 4)

Typical row of A is

Same magic numbers !!

The simplest result :

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CALGARY B Filter (width 4)

Typical row of B is

Again the same numbers !!

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CALGARY Q Filter (width 4)

Typical column

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Results

HaarPSNR=36.86Level of decomposition from : 2All details are removed

ChaikinPSNR=37.91Level of decomposition from : 2All details are removed

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1)original : 514 x 3862) Haar

one level of decomposition50% of details,PSNR=45.91

3) Chaikinone level of decomposition50% of detailsPSNR=48.2

4) Haartwo levels of decomposition

0% of details,PSNR=36.86

5) Chaikin, two levels of decomposition0% of detailsPSNR=37.91

12 3

4 5

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CALGARY Movie, VolumeA movie is a sequence of frames.

Manipulate resolution of time t

Transformation along a set of rays

that the value of a pixel changing over time

Image :

I(x,y)=intensity of pixel at (x,y)

Volume:

I(x,y,z)= intensity (density of material) at (x,y,z)

u

v t

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CALGARY Details as characteristic of the curves!

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CALGARY Result

Use captured details for simple base paths

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CALGARY Scan conversion

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CALGARY Digitizer to the control points

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CALGARY

scanned deformed

Application example –editing

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CALGARY Application: Flexible Editing

Stollnitz’s Paper

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CALGARY Result (flexible editing)

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CALGARY Texture Mapping

Carry texture from coarse mesh over all finer meshes

Add texture coordinates (t,s) to vertex coordinates

Use subdivision scheme for control vertices in R5

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