10176

# Iterative Polygon Simplification

Polygons such as those created by geographic information systems are sometimes represented by data that is larger than necessary for relevant computations. This Demonstration shows how the representation of a polygon can be compressed. Although the compression is sometimes not lossless, for many practical purposes the shape or icon that results conveys adequate information for visualization, recognition, and computation. You select one of ten polygons that describe the shape of Texas counties using 50 points. You also select from two methodologies for determining the "relevance" of each vertex in the polygon. You can then drag the "removal iteration" slider to the right and see the least "relevant" points found by the selected method sequentially removed.

### DETAILS

"Method A" tends to result in more angular shapes than "method B", which tends to produce "curvier" shapes.
The algorithms used to determine the relevance of a vertex in a polygon are set forth and illustrated in D. J. Lee, S. Antani, and L. R. Long, "Similarity Measurement Using Polygon Curve Representation and Fourier Descriptors for Shape-Based Vertebral Image Retrieval," in Proceedings of SPIE--Volume 5032, Medical Imaging 2003: Image Processing (M. Sonka & J. M. Fitzpatrick, eds.), 2003 pp. 1283–1291.
The implementation of the simplification algorithm used in this Demonstration achieves needed speed by using the Mathematica compiler and by recognizing that most of the computed "relevance" values for each vertex are unaffected when a vertex is removed. Only the relevances of the neighboring vertices are potentially affected. Thus, the data structure used in this implementation stores a cache of computed relevance values and only computes new values when the removal of a vertex so requires. A compressed representation of the iterative simplification of the polygon is achieved by using an analogy to the GraphicsComplex command in Mathematica. The algorithm stores the coordinates of all the original points in the polygon, the currently active points, and the sequence of removed vertices. This data structure permits the Mathematica Fold or FoldList commands to readily reconstruct how a shape was simplified.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.