# Characteristic Number: Theory and Its Application to Shape Analysis

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## Abstract

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## 1. Introduction

## 2. Characteristic Number

**Figure 1.**Geometric imaging process. Three points, $P,U,Q$, of a line in the 3D space are projected as $p,u,q$ and ${p}^{{}^{\prime}}$, ${u}^{{}^{\prime}}$, ${q}^{{}^{\prime}}$ into two imaging planes with the focal centers, ${O}_{1}$ and ${O}_{2}$, respectively.

#### 2.1. Characteristic Ratio: An Affine Invariant

**Definition 1**.

**Theorem 2**.

**Proof**.

#### 2.2. Characteristic Number: A Projective Invariant

**Definition 3**.

**Theorem 4**.

#### 2.3. Intrinsic Properties of a Hypersurface and Curve

**Theorem 5**.

**Corollary 6**.

**Figure 3.**Theorem 5 reveals the intrinsic properties of a hypersurface or curve. (

**a**) An example for a surface in the three-dimensional Euclid space; (

**b**) the characteristic number of three collinear points is $-1$; (

**c**) the characteristic number on any six points derived from Theorem 5.

**Corollary 7**.

#### 2.4. Generalization of Pascal’s Theorem

**Theorem 8**.

## 3. Application I: A Perspective Invariant Shape Descriptor

**Figure 4.**Comparison between characteristic ratio spectrum (CHARS) and cross-ratio spectrum (CRS). (a) Two symbols with subtle inner differences; (b) The cross-ratio spectra of the symbols; (c) The characteristic ratio spectra of the symbols; (d) and (e) illustrate the inner points used to calculate CHARS (circles) and CRS (red dots) of the symbols.

#### 3.1. Descriptor Construction

- $\kappa ({P}_{i},{P}_{j},{P}_{k})=\kappa ({P}_{i},{P}_{j})\xb7\kappa ({P}_{j},{P}_{k})$ if there are at least two intersections on both sides, ${P}_{i}{P}_{j}$ and ${P}_{j}{P}_{k}$.
- $\kappa ({P}_{i},{P}_{j},{P}_{k})=\kappa ({P}_{j},{P}_{k})$(or $\kappa ({P}_{i},{P}_{j})$) if there are at least two intersections on the side, ${P}_{j}{P}_{k}$ (or ${P}_{i}{P}_{j}$), and no more than one intersection on the side, ${P}_{i}{P}_{j}$ (or ${P}_{j}{P}_{k}$).
- $\kappa ({P}_{i},{P}_{j},{P}_{k})=0$ if there is at most one intersection on either ${P}_{i}{P}_{j}$ or ${P}_{j}{P}_{k}$.

- The characteristic number on a triangle is permutable, i.e., $\kappa ({P}_{i},{P}_{j},{P}_{k})=\kappa ({P}_{j},{P}_{k},{P}_{i})=\kappa ({P}_{k},{P}_{i},$${P}_{j})$. This can be be readily verified by Equation (8).
- The choice of initial point (triangle) does not change individual values in $\mathcal{D}\left(\mathcal{S}\right)$, but determines the order in which CN values appear in $\mathcal{D}\left(\mathcal{S}\right)$. It is also straightforward to derive this property from the above and Equation (10).
- Slight fluctuations of the vertices on the convex hull of $\mathcal{S}$ bring gradual changes on $\mathcal{D}\left(\mathcal{S}\right)$. It is assumed that three pairs of points, (${P}_{i}$, ${P}_{i}^{\prime}$), (${P}_{j}$, ${P}_{j}^{\prime}$) and (${P}_{k}$, ${P}_{k}^{\prime}$), are neighbors on the smooth part of convex hull. We have $\kappa ({P}_{i},{P}_{j},{P}_{k})\approx \kappa ({P}_{i}^{\prime},{P}_{j}^{\prime},{P}_{k}^{\prime})$, since each side of the triangles to calculate CN values is also close to each other.
- The descriptor presents fluctuations in the case of affine transformations due to jags on the inner intersections. Severe perspective deformations would make it worse. A dynamic programming algorithm, i.e., dynamic time warping (DTW), is employed to align the shape descriptors of query and template shapes, as done in CRS and CHARS. This process can alleviate the deviations brought by the choice of the starting point.

#### 3.2. Performance Evaluation

**Figure 6.**Experimental set: (

**a**) 32 logos of television networks; and (

**b**) an example subject to perspective transformations with the perspective factors $(\alpha ,\beta )=(0.5,0)$ and various ($az$, $el$) (azimuth (az) and elevation (el)) parameters.

**Table 1.**Recognition rates on the query set when $\alpha =0.5$ and $\beta =0$. CNF, descriptor derived from the characteristic number; CRS, cross-ratio spectrum.

CNF | CRS | SIFT | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

az | $-{75}^{\xb0}$ | $-{30}^{\xb0}$ | ${15}^{\xb0}$ | ${60}^{\xb0}$ | $-{75}^{\xb0}$ | $-{30}^{\xb0}$ | ${15}^{\xb0}$ | ${60}^{\xb0}$ | $-{75}^{\xb0}$ | $-{30}^{\xb0}$ | ${15}^{\xb0}$ | ${60}^{\xb0}$ |

$el={15}^{\xb0}$ | 43.75 | 50 | 50 | 59.38 | 50 | 62.5 | 46.88 | 46.88 | 6.25 | 9.38 | 6.25 | 15.63 |

$el={60}^{\xb0}$ | 93.75 | 100 | 96.88 | 96.88 | 87.5 | 84.38 | 87.5 | 87.5 | 28.13 | 65.63 | 53.13 | 25 |

$el={105}^{\xb0}$ | 100 | 100 | 100 | 100 | 93.75 | 90.63 | 90.63 | 87.5 | 12.5 | 71.88 | 71.88 | 21.88 |

$el={150}^{\xb0}$ | 71.88 | 75 | 87.5 | 75 | 68.75 | 84.38 | 84.38 | 68.75 | 6.25 | 18.75 | 25 | 3.13 |

CNF | CRS | SIFT | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

az | $-{75}^{\xb0}$ | $-{30}^{\xb0}$ | ${15}^{\xb0}$ | ${60}^{\xb0}$ | $-{75}^{\xb0}$ | $-{30}^{\xb0}$ | ${15}^{\xb0}$ | ${60}^{\xb0}$ | $-{75}^{\xb0}$ | $-{30}^{\xb0}$ | ${15}^{\xb0}$ | ${60}^{\xb0}$ |

$el={15}^{\xb0}$ | 37.5 | 62.5 | 43.75 | 50 | 53.13 | 53.13 | 34.38 | 34.38 | 6.25 | 6.25 | 0 | 6.25 |

$el={60}^{\xb0}$ | 90.63 | 100 | 100 | 78.13 | 84.38 | 84.38 | 81.25 | 75 | 12.5 | 31.25 | 25 | 9.38 |

$el={105}^{\xb0}$ | 96.88 | 100 | 93.75 | 87.5 | 87.5 | 90.63 | 87.5 | 71.88 | 18.75 | 46.88 | 40.63 | 9.38 |

$el={150}^{\xb0}$ | 56.25 | 75 | 62.5 | 53.13 | 75 | 78.13 | 65.63 | 65.63 | 12.5 | 12.5 | 12.5 | 15.63 |

CNF | CRS | SIFT | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

az | $-{75}^{\xb0}$ | $-{30}^{\xb0}$ | ${15}^{\xb0}$ | ${60}^{\xb0}$ | $-{75}^{\xb0}$ | $-{30}^{\xb0}$ | ${15}^{\xb0}$ | ${60}^{\xb0}$ | $-{75}^{\xb0}$ | $-{30}^{\xb0}$ | ${15}^{\xb0}$ | ${60}^{\xb0}$ |

$el={15}^{\xb0}$ | 31.25 | 31.25 | 37.5 | 31.25 | 40.63 | 50 | 34.38 | 21.88 | 6.25 | 9.38 | 6.25 | 15.63 |

$el={60}^{\xb0}$ | 87.5 | 84.38 | 84.38 | 62.5 | 87.5 | 78.13 | 78.13 | 59.38 | 28.13 | 65.63 | 53.13 | 25 |

$el={105}^{\xb0}$ | 87.5 | 93.75 | 90.63 | 75 | 78.13 | 87.5 | 81.25 | 71.88 | 12.5 | 68.75 | 68.75 | 21.88 |

$el={150}^{\xb0}$ | 43.75 | 62.5 | 53.13 | 46.88 | 65.13 | 68.75 | 62.5 | 56.25 | 6.25 | 18.75 | 25 | 3.13 |

## 4. Application III: Shape Matching with Characteristic Number

#### 4.1. Shape Priors Using Characteristic Number

**Figure 8.**CN values for subsets with three (blue), five (red) and six (green) points: (

**a**) histograms of CN values on the subsets of fiducial points whose locations are annotated in the frontal face images in (

**b**), and (

**c**) histograms of CN values on the same combinations of points as (a). The point coordinates for (c) are extracted from images in (

**d**), significantly different from (b). Horizontal axes of the histograms are CN values, and the vertical axes are the number of faces.

#### 4.2. Performance Evaluation

**Figure 9.**Fiducial point localization with pose changes, as well as variations on age, expression and resolution.

**Figure 10.**Cumulative error distributions of localizations with collinearity (three-point CN, red dots), all CN constraints (green solid and no shape constraints (blue dash dots). The x-axis is the normalized mean error (NME), and the y-axis indicates the percentage of images on which localization NMEs are lower than the x-value.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Fan, X.; Luo, Z.; Zhang, J.; Zhou, X.; Jia, Q.; Luo, D.
Characteristic Number: Theory and Its Application to Shape Analysis. *Axioms* **2014**, *3*, 202-221.
https://doi.org/10.3390/axioms3020202

**AMA Style**

Fan X, Luo Z, Zhang J, Zhou X, Jia Q, Luo D.
Characteristic Number: Theory and Its Application to Shape Analysis. *Axioms*. 2014; 3(2):202-221.
https://doi.org/10.3390/axioms3020202

**Chicago/Turabian Style**

Fan, Xin, Zhongxuan Luo, Jielin Zhang, Xinchen Zhou, Qi Jia, and Daiyun Luo.
2014. "Characteristic Number: Theory and Its Application to Shape Analysis" *Axioms* 3, no. 2: 202-221.
https://doi.org/10.3390/axioms3020202