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

Is the scheme used here interpolating or approximating? 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

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

Comparison Non interpolatory subdivision schemes Corner Cutting 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

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

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 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 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 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 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 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

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

The rule for edge points 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 3 1 1 1 The rule for edge points 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 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 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 1/2 2w 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 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 Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

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

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

Kobblet sqrt(3) Subdivision 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 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 Doo-Sabin Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr

Advantages & Difficulties 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 Doo-Sabin, Midege (C1) Approximating Loop (C2) Catmull-Clark (C2) Biquartic (C2) Interpolating Butterfly (C1) Kobbelt (C1) Graphics Programming, Lee Byung-Gook, Dongseo Univ., E-mail:lbg@dongseo.ac.kr