Subdivision surface

In the field of 3D computer graphics, a subdivision surface is a method of representing a smooth surface via the specification of a coarser polygon mesh. The smooth surface can be calculated from the coarse mesh as the limit of an iterative process of subdividing each polygonal face into smaller faces that better approximate the smooth surface.

First three steps of Catmull-Clark subdivision of a cube with subdivision surface below

Overview

The subdivision surfaces algorithm is recursive in nature. The process starts with a given polygonal mesh. A Refinement Scheme is then applied to this mesh. This process takes that mesh and subdivides it, creating new vertices and new faces. The positions of the new vertices in the mesh are computed based on the positions of nearby old vertices. In some refinement schemes, the positions of old vertices might also be altered (possibly based on the positions of new vertices).

This process produces a denser mesh than the original one, containing more polygonal faces. This resulting mesh can be passed through the same refinement scheme again and so on.

The limit subdivision surface is the surface produced from this process being iteratively applied infinitely many times. In practical use however, this algorithm is only applied a limited, and usually fairly small, number of times.

Mathematically, the neighborhood of an extraordinary point (non-4-valent node for quad refined meshes) of a subdivision surface is a spline with a parametrically singular point [1].

Refinement schemes

Subdivision surface refinement schemes can be broadly classified into two categories: interpolating and approximating. Interpolating schemes are required to match the original position of vertices in the original mesh. Approximating schemes are not; they can and will adjust these positions as needed. In general, approximating schemes have greater smoothness, but the user has less overall control of the outcome. This is analogous to spline surfaces and curves, where Bézier curves are required to interpolate certain control points, while B-Splines are not.

There is another division in subdivision surface schemes as well: the type of polygon that they operate on. Some function for quadrilaterals (quads), while others operate on triangles.

Approximating schemes

Approximating means that the limit surfaces approximate the initial meshes and that after subdivision, the newly generated control points are not in the limit surfaces. There are four approximating subdivision schemes:

  • Catmull and Clark (1978) generalizes bi-cubic uniform B-spline knot insertion. For arbitrary initial meshes, this scheme generates limit surfaces that are C2 continuous everywhere except at extraordinary vertices where they are C1 continuous (Peters and Reif 1998) [2].
  • Doo-Sabin - The second subdivision scheme was developed by Doo and Sabin (1978) who successfully extended Chaikin's corner-cutting method (George Chaikin, 1974[3]) for curves to surfaces. They used the analytical expression of bi-quadratic uniform B-spline surface to generate their subdivision procedure to produce C1 limit surfaces with arbitrary topology for arbitrary initial meshes. An auxiliary point can improve the shape of Doo-Sabin subdivision [4].
  • Loop, Triangles - Loop (1987) proposed his subdivision scheme based on a quartic box-spline of six direction vectors to provide a rule to generate C2 continuous limit surfaces everywhere except at extraordinary vertices where they are C1 continuous (Zorin 1997).
  • Mid-Edge subdivision scheme - The mid-edge subdivision scheme was proposed independently by Peters-Reif (1997) [5] and Habib-Warren (1999) [6]. The former used the mid-point of each edge to build the new mesh. The latter used a four-directional box spline to build the scheme. This scheme generates C1 continuous limit surfaces on initial meshes with arbitrary topology.
  • √3 subdivision scheme - This scheme has been developed by Kobbelt (2000)[7] and offers several interesting features: it handles arbitrary triangular meshes, it is C2 continuous everywhere except at extraordinary vertices where it is C1 continuous and it offers a natural adaptive refinement when required. It exhibits at least two specificities: it is a Dual scheme for triangle meshes and it has a slower refinement rate than primal ones. (The slowest subdivision is Midedge subdivision which could be called √2 subdivision since two steps halve distances.)

Interpolating schemes

After subdivision, the control points of the original mesh and the new generated control points are interpolated on the limit surface. The earliest work was so-called butterfly scheme by Dyn, Levin and Gregory (1990), who extended the four-point interpolatory subdivision scheme for curves to a subdivision scheme for surface. Zorin, Schröder and Swelden (1996) noticed that the butterfly scheme cannot generate smooth surfaces for irregular triangle meshes and thus modified this scheme. Kobbelt (1996) further generalized the four-point interpolatory subdivision scheme for curves to the tensor product subdivision scheme for surfaces.

  • Doo-Sabin, Quads - generalization of bi-quadratic uniform B-splines
  • Butterfly, Triangles - named after the scheme's shape
  • Midedge, Quads
  • Kobbelt, Quads - a variational subdivision method that tries to overcome uniform subdivision drawbacks

Key developments
  • 1978: Subdivision surfaces were discovered simultaneously by Edwin Catmull and Jim Clark (see Catmull-Clark subdivision surface), and by Daniel Doo and Malcom Sabin (see Doo-Sabin subdivision surfaces.)
  • 1995: Ulrich Reif solved subdivision surface behaviour near extraordinary vertices [8].
  • 1998: Jos Stam contributed a method for exact evaluation for Catmull-Clark subdivision surfaces under arbitrary parameter values [9].

References
  1. J. Peters and U. Reif: Subdivision Surfaces, Springer series Geometry and Computing monograph 3, 2008, doi
  2. J. Peters and U. Reif: Analysis of generalized B-spline subdivision algorithms, SIAM J of Numer. Anal. 32 (2) 1998, p.728-748
  3. "Chaikin Curves in Processing".
  4. K. Karciauskas and J. Peters: Point-augmented biquadratic C1 subdivision surfaces, Graphical Models, 77, p.18-26
  5. J. Peters and U. Reif: The simplest subdivision scheme for smoothing polyhedra, ACM Transactions on Graphics 16(4) (October 1997) p.420-431, doi
  6. A. Habib and J. Warren: Edge and vertex insertion for a class of C1 subdivision surfaces, Computer Aided Geometric Design 16(4) (May 1999) p.223-247, doi
  7. L. Kobbelt: √3-subdivision, 27th annual conference on Computer graphics and interactive techniques, doi
  8. Ulrich Reif. 1995. A unified approach to subdivision algorithms near extraordinary vertices. Computer Aided Geometric Design. 12(2)153-174
  9. Jos Stam, "Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values", Proceedings of SIGGRAPH'98. In Computer Graphics Proceedings, ACM SIGGRAPH, 1998, 395-404
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