User:Ruaridh88

’’’Spline Surface’’’ is a topic in computer vision which is discussed in CVonline

A Spline Surface, in mathematics, is a form of parametric surface and is the representation of a two dimensional curve that is extended into three dimensions; a surface. Spline surfaces describes the basic category to which existing spline techniques, in particular the cubic B-spline and cubic Bézier spline, can be easily extended to three dimensions.

Spline surfaces are popular in 3D computer graphics and computer vision for their economic traits: interpolating the curves using a sparse set of knot points, and, similar to splines, they can be used to approximate complex shapes through curve fitting.

Overview

For a surface, we start with a rectangular polygon mesh and for each point in the parametric space two blending functions are used; one in each parametric direction. These define the control points, or knots, and from this we can weight each control point by calculating the Cartesian product of the two blending functions used for the the spline curves.

For example, the blending function for bézier curves is given as follows:

$p\left(u,v\right)=\sum _{i=0}^{n}\sum _{j=0}^{m}B_{i}^{n}\left(u\right)B_{j}^{m}\left(v\right)k_{i,j}$

where k(i,j) is a control point, $B_{i}^{n}\left(u\right)$ and $B_{j}^{m}\left(v\right)$ are the blending functions.

Spline surfaces share maintain many of the traits of splines, including:

1. The surface passes through the end (corner) points.
2. The surface lies within the convex hull of the control points.

An example of a surface:

• 
  Sample Bézier surface

Technique Overview

The following are common uses of splines, mainly in computer graphics.

Bézier surface

See: bézier surface

Bézier surfaces, as above, use a set of control points where each control point influences the shape of the mesh rather than directly manipulating the mesh.

Non-uniform rational b-spline (NURBS)

See: non-uniform rational b-spline (NURBS)

NURBS are very common in computer graphics, where the NURBS surfaces are obtained by generalisations of both B-Splines and Béziers. They are useful for their flexibility and computational simplicity.

Subdivision surfaces

See: subdivision surface

A subdivision surface is a method used in computer graphics to smooth an object (mesh). Subdivision surface techniques make use of spline functions to either approximate or interpolate the surface.

Common Applications

Spline surfaces have many uses, of which include:

1. Designing car bodies,
2. Representing ship hulls,
3. Aircraft exteriors

And are available in many graphics packages, including:

1. Autodesk Maya
2. Autodesk 3ds Max
3. Blender
4. Cinema4D

See also

1. Spline
2. Surface
3. B-spline
4. Non-uniform rational b-spline
5. Bézier curve
6. Bézier surface
7. Subdivision surface
8. Parametric surface

Bibliography

• Datta, Ritendra (2008). "Image Retrieval: Ideas, Influences, and Trends of the New Age". ACM Computing Surveys. 40 (2): 160. doi:10.1145/1348246.1348248. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); line feed character in |coauthors= at position 14 (help); line feed character in |title= at position 40 (help)
• Datta, Ritendra. "Introduction to Surfaces". {{cite web}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Datta, Ritendra. "Surfaces Based on Splines". Computer Vision. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Datta, Ritendra. "The Simplified Surface Spline" (PDF). {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Datta, Ritendra. "Non-Rigid Alignment". {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Datta, Ritendra (1990). "Generalized B-spline Surfaces of Arbitrary Topology". ACM Computer Graphics. 24 (4). {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• Advanced Vision homepage
• Introduction to Surfaces (Andy Salter)
• Spline Surfaces in 3D (with C implementation) (Paul Bourke)

Category:Image processing Category:Artificial intelligence Category:3D computer graphics