Subdivision Schemes Lee ByungGook Dongseo Univ http kowon

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 3 1 : 1 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 A control point The limit curve – Quadratic B-Spline Curve 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

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

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

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

Loop Subdivision • How Position New Vertices? – Choose locations for new vertices as weighted average of original vertices in local neighborhood Graphics Programming, Lee Byung-Gook, Dongseo Univ. , E-mail: lbg@dongseo. ac. kr

Loop Subdivision • Every new vertex is a weighted average of old ones. • The list of weights is called a Stencil • Is this scheme approximating or interpolating? The rule for vertex points 1 1 1 The rule for edge points 1 3 3 1 1 1 Graphics Programming, Lee Byung-Gook, Dongseo Univ. , E-mail: lbg@dongseo. ac. kr

Loop Subdivision • How Refine Mesh? – Refine each triangle into 4 triangles by splitting each edge and connecting new vertices Graphics Programming, Lee Byung-Gook, Dongseo Univ. , E-mail: lbg@dongseo. ac. kr

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

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

Loop - Results Loop’s scheme results in a limit surface which is of continuity everywhere except for a finite number of singular points, in which it is Graphics Programming, Lee Byung-Gook, Dongseo Univ. , E-mail: lbg@dongseo. ac. kr .

Butterfly Scheme • • Butterfly is an interpolatory scheme. Topology is the same as in Loop’s scheme. Vertex points use the location of the old vertex. Edge points use the following stencil: -w -w 2 w 1/2 -w 2 w -w Graphics Programming, Lee Byung-Gook, Dongseo Univ. , E-mail: lbg@dongseo. ac. kr

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

Butterfly - results The Butterfly Scheme results in a surface which is differentiable twice anywhere. Graphics Programming, Lee Byung-Gook, Dongseo Univ. , E-mail: lbg@dongseo. ac. kr but is not

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

Kobblet sqrt(3) Subdivision 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

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

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

Advantages & Difficulties • Advantages: – – – Simple method for describing complex surfaces Relatively easy to implement Arbitrary topology Smoothness guarantees Multiresolution • Difficulties: – Intuitive specification – Parameterization – Intersections Graphics Programming, Lee Byung-Gook, Dongseo Univ. , E-mail: lbg@dongseo. ac. kr

Subdivision Schemes • There are different subdivision schemes – Different methods for refining topology – Different rules for positioning vertices • Interpolating versus approximating Face Split Vertex Split Triangular Meshes Quad. Meshes Approximating Loop (C 2) Catmull-Clark (C 2) Interpolating Butterfly (C 1) Kobbelt (C 1) Graphics Programming, Lee Byung-Gook, Dongseo Univ. , E-mail: lbg@dongseo. ac. kr Doo-Sabin, Midege (C 1) Biquartic (C 2)