Discrete Calculus by Leo J. Grady & Jonathan R. Polimeni

Author:Leo J. Grady & Jonathan R. Polimeni
Language: eng
Format: epub
Publisher: Springer London, London

5.9.2 Three-Dimensional Mesh Filtering

Filtering of geometric data is an important process in computer graphics and the processing of data obtained from various three-dimensional scanners. Is this context, the node data (0-cochain) is a tuple of coordinates assigned to each node. Therefore, the nodal variable in the above algorithms corresponds to an n×K set of K-dimensional filtered coordinates assigned to each node, with corresponding to the n×K noisy coordinate values acquired for each node. The output of a filtering procedure is therefore a new set of coordinates for each node. The edge structure of the graph is generally given via a surface extraction preprocessing step. Since the most common method for rendering three-dimensional data requires a list of faces for the surface, the faces are usually extracted via a triangulation process.

We begin this section with a synthetic example of filtering coordinates obtained by generating a circle and adding noise to the coordinates of each point on the circle. Figure 5.8 gives an example of a lowpass filter obtained via the Basic Energy Model with p=2 (i.e., a mean filter) and a lowpass filter given by Taubin’s method. By subtracting the lowpass coordinates from the original coordinates, a highpass filter of this ring is obtained.

Fig. 5.8Filtering coordinate data on a ring graph. (A) A noisy ring graph produced by adding Gaussian random noise to the radius of nodes arranged in a perfect circle. (B) The effect of applying the Basic Energy Model with p=2 (i.e., a mean filter) to the coordinates of the graph in (A). (C) The low-pass filter of Taubin [371] applied to the coordinates of the graph in (A). (D) A high-pass filter of the coordinates in (C), produced by differencing the low-pass signal of (C) with the original

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