Subdivision Schemes

Subdivision Schemes Lee Byung-Gook Dongseo Univ. http://kowon.dongseo.ac.kr/~lbg/ Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

What is Subdivision? • Subdivision is a process in which a poly-line/mesh is recursively refined in order to achieve a smooth curve/surface. • Two main groups of schemes: • Approximating - original vertices are moved • Interpolating – original vertices are unaffected Is the scheme used here interpolating or approximating? Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Geri’s Game Frame from “Geri’s Game” by Pixar Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Subdivision Curves • How do You Make a Smooth Curve? Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Chaikin’s Algorithm Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Corner Cutting Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Corner Cutting 3 : 1 1 : 3 Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Corner Cutting Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Corner Cutting – Limit Curve Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Corner Cutting The limit curve – Quadratic B-Spline Curve A control point The control polygon Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Linear B-spline Subdivision Scheme Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Quadratic B-spline Subdivision Scheme Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Cubic B-spline Subdivision Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

N. Dyn 4-points Subdivision Scheme Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

4-Point Scheme Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

4-Point Scheme 1 : 1 1 : 1 Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

4-Point Scheme 1 : 8 Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

4-Point Scheme Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

4-Point Scheme A control point The limit curve The control polygon Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Comparison • Non interpolatory subdivision schemes • Corner Cutting • Interpolatory subdivision schemes • The 4-point scheme Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Theoretical Questions • Given a Subdivision scheme, does it converge for all polygons? • If so, does it converge to a smooth curve? • Better? • Does the limit surface have any singular points? • How do we compute the derivative of the limit surface? Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Subdivision Surfaces Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Subdivision Surfaces Geri’s hand as a piecewise smooth Catmull-Clark surface Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Subdivision Surfaces • A surface subdivision scheme starts with a control net (i.e. vertices, edges and faces) • In each iteration, the scheme constructs a refined net, increasing the number of vertices by some factor. • The limit of the control vertices should be a limit surface. • a scheme always consists of 2 main parts: • A method to generate the topology of the new net. • Rules to determine the geometry of the vertices in the new net. Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

General Notations • There are 3 types of new control points: • Vertex points - vertices that are created in place of an old vertex. • Edge points - vertices that are created on an old edge. • Face points – vertices that are created inside an old face. • Every scheme has rules on how (if) to create any of the above. • If a scheme does not change old vertices (for example - interpolating), then it is viewed simply as if Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Doo-Sabin Subdivision Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Doo-Sabin subdivision Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Catmull-Clark Subdivision Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Catmull-Clark Subdivision • The mesh is the control net of a tensor product B-Spline surface. • The refined mesh is also a control net, and the scheme was devised so that both nets create the same B-Spline surface. • Uses face points, edge points and vertex points. • The construction is incremental – • First the face points are calculated, • Then using the face points, the edge points are computed. • Finally using both face and edge points, we calculate the vertex points. Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Catmull-Clark - results Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Catmull-Clark - results Catmull-Clark Scheme results in a surface which is almost everywhere Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Loop Subdivision • Based on a triangular mesh • Loop’s scheme does not create face points Vertex points New face Old face Edge points Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Subdivision Schemes