Hypertextures are a useful modelling tool in that they can add three-dimensional detail to the surface of otherwise smooth objects. Hypertextures can be rendered as implicit surfaces, resulting in objects with a complex but well defined boundary. However, representing a hypertexture as an implicit surface often results in many small parts being detached from the main surface, turning an object into a disconnected set. Depending on the context, this can detract from the realism in a scene where one usually does not expect a solid object to have clouds of smaller objects floating around it. We present a topology correction technique, integrated in a ray casting algorithm for hypertextured implicit surfaces, that detects and removes all the surface components that have become disconnected from the main surface. Our method works with implicit surfaces that are C2 continuous and uses Morse theory to find the critical points of the surface. The method follows the separatrix lines joining the critical points to isolate disconnected components.
This paper was presented at the Shape Modelling International 2007 conference, that was held in Lyon, France, from the 13th to the 15th of June, 2007. The final paper is available from the IEEE Digital Library. A draft version of the paper is available below:
PDF format (442 K) (smi07.pdf)
The paper was later selected for publication in a special edition of The Visual Computer, containing the best papers from the SMI '07 conference. The new paper has been extended to include results for localised topology correction. It also includes the expressions for the gradient vector and Hessian matrix of the procedural noise functions used. The final paper is available at www.springerlink.com. A draft version of the paper is available below:
PDF format (568 K) (topcorrvisual.pdf)
A localised form of topology correction was later developed to handle very large implicit surfaces. The original model requires that all the critical points of the surface be located as a first step. This becomes impractical when the implicit surface represents a terrain with a potentially infinite number of critical points. The localised version of topology correction works by finding critical points only on a small neighbourhood around each ray-surface intersection point. The neighbourhood is grown, if necessary, until a definite answer about the connectivity state of the intersection point can be given. Caching is also used so that connectivity results can be reused for nearby intersection points. A paper on the localised topology correction method was presented at the EGUK's Theory and Practice of Computer Graphics 2008, held in Manchester from the 9th to the 11th of June, 2008. The paper won the prize for Best Student Paper for Technical Content. The paper is available below:
PDF format (3152 K) (eguk2008.pdf)
Here are four example animations illustrating several aspects of the removal of disconnected surface components from a hypertextured sphere. The animations can be visualised as an infinite cycle by turning on looping in the movie viewer.
A rotating spherical hypertexture, showing the disconnected components marked in red. It is these red components that have to be removed as part of the topological correction method in order to have a hypertextured surface that is simply connected.
The separatrix curves are shown in wireframe inside a hypertextured sphere. This network of separatrix curves is obtained with Morse Theory and encodes all the connectivity information about the surface.
A rotating hypertextured sphere with no topological correction (on the left) and after topological correction (on the right). All the disconnected components of the surface disappear on the right half of the animation.
An animated GIF showing the three stages in the topological correction of a hypertextured terrain. The first frame shows the original terrain with disconnected components. The second frame corresponds to connectivity querying and shows the disconnected components highlighted in green. The third and final frame corresponds to full topology correction and shows the terrain without disconnected components. Notice that shadows on the left part of the image, cast by disconnected components not directly visible, also disappear after topology correction. Click on the image for a larger version.