# Tsallis Entropy for Geometry Simplification

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## Abstract

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## 1. Introduction

**Figure 1.**Simplification example. (a) Original Model. 49694 triangles. (b) Model simplified to 12% with our algorithm using TVMI ($\alpha =1$). 6285 triangles.

## 2. Background

#### 2.1. Related Work

#### 2.2. Viewpoint Information Measures

- Conditional probability matrix $p\left(Z\right|V)$, where each element $p\left(z\right|v)={a}_{z}\left(v\right)/{a}_{t}$ is defined by the normalized projected area of polygon z over the sphere of directions centered at viewpoint v, where ${a}_{z}\left(v\right)$ is the projected area of polygon z at viewpoint v and ${a}_{t}$ is the total projected area of all polygons over the sphere of directions. Conditional probabilities fulfill ${\sum}_{z\in \mathcal{Z}}p\left(z\right|v)=1$. The background can be taken into account as any other polygon.
- Input distribution $p\left(V\right)$, which represents the probability of selecting each viewpoint. This probability can be obtained, for instance, from the normalization of the projected area of the object at each viewpoint or assigning the same probability to each viewpoint. In this paper we have adopted the second alternative, that is, we have assigned the same importance to each viewpoint v.
- Output distribution $p\left(Z\right)$, given by$$p\left(z\right)=\sum _{v\in \mathcal{V}}p\left(v\right)p\left(z\right|v)$$

## 3. Simplification Algorithm

**Figure 2.**The two possible half edge collapses for the edge highlighted with a thicker line. Triangles in grey will be removed.

#### 3.1. Error Metrics

#### 3.1.1. Viewpoint Generalized Entropy

#### 3.1.2. Viewpoint Generalized Mutual Information

#### 3.2. Edge Collapse Cost

#### 3.3. Implementation Issues

## 4. Results and Discussion

**Table 1.**Visual error (RMSE) and simplification time (seconds) for all models simplified with QSlim, IDS, TVE and TVMI.

Model | Triangles | QSlim | IDS | TVE | TVMI | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Original | Final | RMSE | Time | RMSE | Time | α | RMSE | Time | α | RMSE | Time | |

Galo | 6592 | 600 | 11.82 | 0.07 | 8.65 | 162.44 | 1.5 | 7.53 | 271.26 | 0.5 | 7.88 | 296.97 |

Skull | 9934 | 1784 | 11.06 | 0.08 | 11.05 | 360.78 | 1 | 10.31 | 285.80 | 0.5 | 10.37 | 343.74 |

Brush | 20698 | 1200 | 15.49 | 0.11 | 14.86 | 863.13 | 1 | 13.56 | 683.75 | 0.5 | 13.47 | 822.36 |

Junk | 61242 | 6212 | 13.66 | 0.50 | 11.73 | 2436.04 | 1 | 10.58 | 1929.76 | 0.5 | 10.73 | 2320.98 |

**Figure 3.**Galo model. (a) Original model. T = 6592. (b) QSlim. T = 600. (c) IDS. T = 600. (d) TVE($\alpha =1.5$). T = 600. (e) TVMI($\alpha =0.5$). T = 600. T indicates the number of triangles. Different approximations of Galo model obtained with QSlim, IDS, TVE and TVMI. The original model is shown in (a). TVE and TVMI preserve the comb and tail better than QSlim and IDS.

**Figure 4.**RMSE for Galo model. (a) RMSE vs. different alpha values. T = 600. (b) Decimation %. High percentage values indicate that the model has been simplified slightly. Low values correspond to a very coarse model.

**Figure 5.**Skull model. (a) Original model. T = 9934. (b) QSlim. T = 1784. (c) IDS. T = 1784. (d) TVE ($\alpha =1$). T = 1783. (e) TVMI ($\alpha =0.5$). T = 1784. Different approximations of Skull model obtained with QSlim, IDS, TVE and TVMI. The original model is shown in (a).

**Figure 6.**Close-ups of Skull model. (a) Original model. (b) QSlim. T = 1784. RMSE = 47.58. (c) IDS. T = 1784. RMSE = 42.05. (d) TVE ($\alpha =1$). T = 1783. RMSE = 42.53. (e) TVMI ($\alpha =0.5$). T = 1784. RMSE = 40.24. These images show that the region around the mouth, especially the teeth in the lower junk, is preserved better in TVMI, IDS and TVE than in QSlim. In the bottom row, difference images are shown. These difference images were produced by superimposing the simplified image over the original image. Here black signifies no difference, while red corresponds to maximum difference.

**Figure 8.**Brush model. (a) Original model. T = 20698. (b) QSlim. T = 1200. (c) IDS. T = 1199. (d) TVE ($\alpha =1$). T = 1199. (e) TVMI ($\alpha =0.5$). T = 1199. Different approximations of Brush model obtained with QSlim, IDS, TVE and TVMI. The original model is shown in (a). The results show that our measures (TVE and TVMI) are capable of retaining more polygons in the brush pins than IDS and QSlim.

**Figure 10.**Junk model. (a) Original model. T = 61242. (b) QSlim. T = 6212. (c) IDS. T = 6211. (d) TVE ($\alpha =1$). T = 6218. (e) TVMI ($\alpha =0.5$). T = 6219. Different approximations of Junk model obtained with QSlim, IDS, TVE and TVMI. The original model is shown in (a). All the visual simplifications (IDS, TVE and TVMI) preserve the ropes better than the purely geometric simplification (QSlim). The silhouette of the model is retained better with TVE and TVMI than with IDS, see for example the sail at the ship’s stern in (c).

**Figure 11.**Close-ups of Junk model. (a) Original model. (b) QSlim. T = 6212. RMSE = 30.15. (c) IDS. T = 6211. RMSE = 28.18. (d) TVE ($\alpha =1$). T = 6218. RMSE = 27.46. (e) TVMI ($\alpha =0.5$). T = 6219. RMSE = 26.59. The ropes, some masts and poles are retained better in TVE, TVMI and IDS than in QSlim. TVE (see (e)) achieves an improvement about 6% over IDS (see (c)) and over 14% over QSlim (see (b)).

## 5. Conclusions and Future Work

## Acknowledgments

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**MDPI and ACS Style**

Castelló, P.; González, C.; Chover, M.; Sbert, M.; Feixas, M.
Tsallis Entropy for Geometry Simplification. *Entropy* **2011**, *13*, 1805-1828.
https://doi.org/10.3390/e13101805

**AMA Style**

Castelló P, González C, Chover M, Sbert M, Feixas M.
Tsallis Entropy for Geometry Simplification. *Entropy*. 2011; 13(10):1805-1828.
https://doi.org/10.3390/e13101805

**Chicago/Turabian Style**

Castelló, Pascual, Carlos González, Miguel Chover, Mateu Sbert, and Miquel Feixas.
2011. "Tsallis Entropy for Geometry Simplification" *Entropy* 13, no. 10: 1805-1828.
https://doi.org/10.3390/e13101805