# A Method for Determining the Shape Similarity of Complex Three-Dimensional Structures to Aid Decay Restoration and Digitization Error Correction

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## Abstract

**:**

## 1. Introduction

## 2. Prior Work

## 3. Preliminaries

#### 3.1. Gaussian and Mean Curvature Descriptors

**n**. We define

**e**as a unit direction in the tangent plane and the normal curvature ${\kappa}_{N}$ as a curvature of the curve that belongs to both the surface $M$ and the tangent plane that contains both

**n**and

**e**. The average value and product of both principal curvatures ${\kappa}_{1}$ and ${\kappa}_{2}$ of the surface defines the mean curvature ${\kappa}_{H}=({\kappa}_{1}+{\kappa}_{2})\u22152$ and Gaussian curvature ${\kappa}_{G}={\kappa}_{1}{\kappa}_{2}$. The mean curvature $2{\kappa}_{H}({v}_{i})$ and unit normal $n({v}_{i})$ at the vertex ${v}_{i}$ are given by mean curvature normal (Laplace–Beltrami operator) $K({v}_{i})=2{\kappa}_{H}({v}_{i})n({v}_{i})$. Using the derived expression for the discrete mean curvature normal operator [29],

#### 3.2. Fitting Quadric Curvature Estimation

**n**at the point

**v**either by simple or weighted averaging or by finding the least-squares fitted plane to the point and its neighbors. Then, we position a local coordinate system $({x}^{\prime},{y}^{\prime},{z}^{\prime})$ at the point

**v**and align axis ${z}^{\prime}$ along the estimated normal. We use the McIvor and Valkenburg suggestion [30] for aligning of the ${x}^{\prime}$ coordinate axis with a projection of the global $x$ axis onto the tangent plane defined by

**n**. If we use the suggested improvements and fit the mapped data to extended quadric: $\widehat{z}=a{\widehat{x}}^{2}+{b}^{\prime}\widehat{x}\widehat{y}+{c}^{\prime}{\widehat{y}}^{2}+\text{}{d}^{\prime}\widehat{x}+{e}^{\prime}\widehat{y}$, and solve the resulting system of linear equations, we finally obtain the estimation for the Gaussian and mean curvature:

#### 3.3. Mesh Quantization

#### 3.4. Ordered Statistics Vertex Extraction

## 4. Our Algorithm

#### 4.1. Mesh Processing

#### 4.2. Ordered Statistics Algorithm

#### 4.3. Adaptive Mesh Quantization

#### 4.4. Similarity Matching Procedure

#### 4.5. Neural Networks for 3D Feature Ranking

## 5. Numerical Results

#### 5.1. Mesh Processing Performance

- angelo-1L.obj—the sculpture of an angel on the top-left,
- angelo-1R.obj—the symmetrical pair at the right,
- angelo-2L.obj—the left angel ornament,
- angelo-2R.obj—the symmetrical pair at the right, and
- camino degli angeli.obj—the whole fireplace 3D model.

#### 5.2. Ordered Statistics Vertex Extraction Performance

#### 5.3. Results of the Quantization

#### 5.4. Matching Shapes Performance

## 6. Discussion, Conclusions and Future Work

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Case Study: Description and Data Survey

Feature | Sensor Dimension | Value | Unit | Bounding Box | Value | Unit |
---|---|---|---|---|---|---|

w | Length | 35.6 | mm | Length | 423 | mm |

h | Height | 23.8 | mm | Height | 420 | mm |

D | Target distance | 1000 | mm | Sidelap | 60 | % |

f | Focal length | 24 | mm | Overlap | 60 | % |

w | Horizontal image size | 6000 | pix | Sidelap | 89 | cm |

h | Vertical image size | 3376 | pix | Overlap | 60 | cm |

Camera resolution | (Mpix) | Displacement x | 59 | cm | ||

pix h | Sensor pixel size (horiz.) | 0.006 | mm | Displacement y | 40 | cm |

pix v | Sensor pixel size (vert.) | 0.007 | mm | Shooting Stations | - | |

s | Magnification | 41.667 | no. of stations along the x axis | 7 | - | |

Artifact Dimension | no. of stations along the y axis | 11 | - | |||

w | Length | 148 | cm | Total of nadiral photos | 75 | - |

h | Height | 99.17 | cm | no. of oblique x axis stations | 4 | - |

Pixel dimension | no. of oblique y axis stations | 5 | - | |||

w | Length | 0.247 | mm | Total of side photos | 62 | - |

h | Height | 0.294 | mm | Total photos | 137 | - |

## Appendix B

**Figure A2.**The input models ((

**a**) angelo-1L.obj, (

**b**) angelo-2L.obj, (

**c**) camino degli angeli.obj, (

**d**) angelo-2R.obj, (

**e**) angelo-1R.obj) rendered and texturized by the color map in accordance with all computed criteria from Table 1 and Section 4.2. Blue corresponds to low values and red to high values of the computed criteria.

**Figure A3.**Illustration of quantization results using specified quantization levels and corresponding adaptive levels: (

**a**) k = 2: (9 $\times 28\times 14)$, (

**b**) $k=3$: $(14\times 42\times 20)$, (

**c**) $k=4$: $(18\times 55\times 27)$. The red, black and magenta markers in all illustrations denote the tolerance levels 1/2, 1/6, and 1/12, respectively.

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**Figure 2.**Render of the input 3D models: (

**a**) angelo-1L.obj, (

**b**) angelo-2L.obj, (

**c**) camino degli angeli.obj, (

**d**) angelo-2R.obj, (

**e**) angelo-1R.obj.

**Figure 3.**The same input models as in Figure 2 rendered and texturized by the color map in accordance with the computed features, respectively: (

**a**) Gaussian curvature, (

**b**) mean curvature, (

**c**) gradient of the Gaussian curvature, (

**d**) maximal dihedral angle, and (

**e**) theta angle. In the used color map, blue corresponds to low values and red to high values of the computed feature.

**Figure 4.**Illustration of our algorithm’s accuracy: (

**a**) visual example of the angelo-2L.obj renderings multiplied by ranked criteria, (

**b**) spatial position of the top-60 ordered vertices out of 4056.

**Figure 5.**Red dots illustrate the invariance of vector ${v}^{\mathrm{I}}$ to different strengths of optimization: (

**a**) original 3D mesh without deletions, (

**b**) 80% deleted vertices and (

**c**) 95% deleted vertices.

**Figure 6.**The number of matched vertices using quantization levels $k=\left[2,3,4\right]$. The shown values vary depending on the selected tolerance and the total number of selected vertices (Table 2).

**Figure 7.**The cropped view of the 3D mesh with extracted matching vertices using the quantization level of 14 × 42 × 20 and tolerance 1/6: all extracted quantized vertices in this area (black markers), and matched vertices (red markers).

**Figure 8.**Illustration of a matching distance descriptor calculated using quantization level $k=3$ for the sample area quantization and corresponding $14\times 42\times 12$ number of the whole mesh quantization. (

**a**) Set of vertices selected in the sample area, (

**b**) matched area with calculated distance descriptors.

Criterion | Description | Weight | Criterion | Description | Weight |
---|---|---|---|---|---|

${\kappa}_{G}>0$ | Positive ${\kappa}_{G}$ | 0.4 | ${\kappa}_{G1}<0$ | Negative ${\kappa}_{G1}$ ^{1} | 0.7 |

${\kappa}_{G}<0$ | Negative ${\kappa}_{G}$ | 0.8 | ${\kappa}_{H1}>0$ | Positive ${\kappa}_{H1}$ ^{1} | 1.0 |

${\kappa}_{H}>0$ | Positive mean curvature | 0.6 | ${\kappa}_{H1}<0$ | Negative ${\kappa}_{H1}$ ^{1} | 1.0 |

${\kappa}_{H}<0$ | Negative mean curvature | 0.7 | ${\psi}_{max}\ge 0$ | Maximal dihedral | 0.8 |

$\theta <2\pi $ | Small theta angle | 0.8 | ${\psi}_{min}\ge 0$ | Minimal dihedral | 0.7 |

$\theta >2\pi $ | Big theta angle | 0.2 | $\Delta {\kappa}_{G}>0$ | Gradient ${\kappa}_{G}$ | 0.7 |

${\kappa}_{G1}>0$ | Positive ${\kappa}_{G1}$ ^{1} | 1.0 | $\Delta {\kappa}_{H}>0$ | Gradient ${\kappa}_{H}$ | 0.4 |

^{1}Fitting quadric method of curvature estimation.

Whole Mesh Quants ^{1} | Sample Quants | Tolerance Values ^{2} | Extracted Vertices ^{3} | Sample Vertices ^{4} | Matched Vertices |
---|---|---|---|---|---|

9 × 28 × 14 | 2 × 2 × 2 | 1/2 | 737 | 6 | 3 |

1/6 | 653 | 3 | |||

1/12 | 595 | 2 | |||

14 × 42 × 12 | 3 × 3 × 3 | 1/2 | 1597 | 10 | 10 |

1/6 | 1316 | 8 | |||

1/12 | 1146 | 6 | |||

18 × 55 × 27 | 4 × 4 × 4 | 1/2 | 2687 | 15 | 14 |

1/6 | 2099 | 11 | |||

1/12 | 1777 | 9 |

^{1}Resulting number of quantization levels per dimension for the given input level $k$;

^{2}tolerance values;

^{3}the total number of quantized coordinates of important vertices ${v}^{\mathrm{I}}$;

^{4}the number of selected vertices in the quantized mesh sample.

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**MDPI and ACS Style**

Vasic, I.; Quattrini, R.; Pierdicca, R.; Frontoni, E.; Vasic, B.
A Method for Determining the Shape Similarity of Complex Three-Dimensional Structures to Aid Decay Restoration and Digitization Error Correction. *Information* **2022**, *13*, 145.
https://doi.org/10.3390/info13030145

**AMA Style**

Vasic I, Quattrini R, Pierdicca R, Frontoni E, Vasic B.
A Method for Determining the Shape Similarity of Complex Three-Dimensional Structures to Aid Decay Restoration and Digitization Error Correction. *Information*. 2022; 13(3):145.
https://doi.org/10.3390/info13030145

**Chicago/Turabian Style**

Vasic, Iva, Ramona Quattrini, Roberto Pierdicca, Emanuele Frontoni, and Bata Vasic.
2022. "A Method for Determining the Shape Similarity of Complex Three-Dimensional Structures to Aid Decay Restoration and Digitization Error Correction" *Information* 13, no. 3: 145.
https://doi.org/10.3390/info13030145