# An Application of Manifold Learning in Global Shape Descriptors

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## Abstract

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## 1. Introduction

**first contribution**, inspired by the idea of Laplacian Eigenmaps [27], we learned the manifold of a 3D model and then, analogous to the approach taken by Shape-DNA, used the spectrum of the embedded manifold to build the global shape descriptor. This approach has two main advantages. First, it relies on the adjacency of the nodes, disregarding the fine details of the mesh structure. Therefore, it can be used for degenerate or non-uniform sampled meshes. Second, as manifold learning does not rely on the mesh structure and is not limited to a specific type of meshes, e.g., triangulated meshes, it can be applied easily to any other mesh types such as quadrilateral meshes.

**second contribution**, we presented a simple and straightforward normalization technique (motivated by the work in [20,28,29]) to obtain a scale-invariant global shape descriptor that is more robust to noise. To this end, we propose to subtract the first non-zero eigenvalue from the shape descriptor after taking the logarithm of the spectrum. One advantage of our approach over the idea of Bronstein et al. [28] is that we avoid taking the direct derivative; this advantage is significant since the differential operator amplifies the noise. Taking the logarithm additionally helps to suppress the effect of the noise that is present in higher order elements of the spectrum.

## 2. Background

#### 2.1. Spectral Shape Analysis

#### 2.2. Manifold Learning

## 3. Method

#### 3.1. Laplacian Eigenmap-Based Shape Descriptor

**1**are trivial solutions to Equation (10). The multiplicity of eigenvalue zero is associated with the number of connected components of the graph. Eigen-solvers often obtain very small, though not precisely zero, eigenvalues due to the computational approximations. If we may know the number of connected components of $\mathbf{L}$, we can discard all eigenvalues equal to zero, and form our shape fingerprint using the more informative section of the spectrum. This is easily done by Dulmage–Mendelsohn decomposition [47].

#### 3.2. Scale Normalization

#### 3.3. Algorithm

Algorithm 1: Laplacian Eigenmap-based scale-invariant global shape descriptor. |

## 4. Experiments

#### 4.1. Dataset

#### 4.2. Retrieval Results

#### 4.3. Multi-Class Classification Results

#### 4.4. Robustness

**Resistance to noise.**Multiple noisy versions of the TOSCA dataset were generated following the idea articulated in [57]. To this end, the surface meshes of all models were disturbed by changing the position of each point along its normal vector that was chosen randomly from an interval $(-L,L)$ with the 0 mean, where L determines the noise level and is a fraction of the diagonal length of the model bounding box. In this experiment, three noise levels $L=0.5\%$, $L=1\%$, and $L=2\%$ were tested, where the latter one represents a greater level of noise. Two-dimensional PCA projections of all descriptors with the presence of different levels of noise are plotted in Figure 7. Combining these with the results shown in Figure 3, where no noise is present, demonstrates that the LESI algorithm is highly noise-resistant while the performance of the Shape-DNA and cShape-DNA decreases as the level of noise increases. Moreover, GPS fails in separating different classes of models with the presence of noise. Figure 8 reflects the effect of noise on the discriminative power of the descriptors. The LESI algorithm shows consistent results as the level of noise increases from 0% (top row) to 2% (bottom row).

**Scale invariance.**To validate the insensitivity of the LESI descriptor to scale variations and compare the robustness of the proposed method with other descriptors, each model of the TOSCA dataset was scaled by a factor of 0.5, 0.875, 1.25, 1.625, or 2 randomly. Figure 9 shows that the LESI algorithm surpasses other methods in discerning different classes. Comparing the result of this experiment with the results shown in Figure 3 demonstrates the consistency of the LESI and cShape-DNA algorithms with the presence of scale variation. That is expected, as they can remove the parameter of scale and obtain scale-free descriptors. The distance matrices in Figure 10 show that the original Shape-DNA algorithm is very susceptible to scale variations. Even though the cShape-DNA has significantly improved scale sensitivity of the original Shape-DNA, it does not provide as accurate results as the LESI algorithm does.

**Resistance to the sampling rate.**To investigate the effect of sampling rates on the discriminative power of the shape descriptors, Bronstein et al. [8] proposed to reduce the number of vertices to 20% of its original size. Accordingly, the down-sampled version of the TOSCA dataset was generated, and shape descriptors associated with them were computed. The 2D PCA projections and distance matrices of descriptors are illustrated in Figure 11 and Figure 12, respectively. Although the original Shape-DNA shows a more accurate result than cShape-DNA, the separation of cat, dog, and wolf models is challenging. Although the performance of the LESI method is slightly affected, it still outperforms cShape-DNA and GPS methods.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The block diagram of the proposed Laplacian Eigenmaps based scale-invariant (LESI) global shape descriptor.

**Figure 2.**An example showing the proposed normalization method of the shape descriptor. (

**a**) A Teddy Bear model and its scaled version (scale factor 0.7). (

**b**) Two different glasses that vary in size. (

**c**) The spectrum of the original Teddy Bear, scaled Teddy Bear, and both glasses with $t=0.5$ and $t={d}_{max}^{2}$. Please note that the spectrum of the original and scaled Teddy Bear models coincide when $t={d}_{max}^{2}$. (

**d**) The spectrum of all models after proposed normalization.

**Figure 3.**2D PCA projection of shape descriptors computed from: (

**a**) original Shape-DNA; (

**b**) cShape-DNA; (

**c**) GPS; and (

**d**) LESI algorithms on TOSCA dataset.

**Figure 4.**Experiment on separating different models of men: (

**a**) a sample of Michael (left) and David (right) models; and (

**b**) PCA projection of LESI descriptors of different men models using $t=2{d}_{max}^{2}$, $t=5$, and $t=15$.

**Figure 5.**2D PCA projection of shape descriptors computed from: (

**a**) original Shape-DNA; (

**b**) cShape-DNA; (

**c**) GPS; and (

**d**) LESI algorithms on McGill dataset.

**Figure 6.**Confusion matrix obtained from linear multi-class SVM for McGill dataset using LESI descriptors.

**Figure 7.**2D PCA projection of shape descriptors computed by (from left to right) Shape-DNA, cShape-DNA, GPS, and LESI algorithms from perturbed TOSCA dataset with (from top to bottom) 0.5%, 1%, and 2% noise level, respectively.

**Figure 8.**The Euclidean pairwise distance matrix of shape descriptors computed by (from left to right) Shape-DNA, cShape-DNA, GPS, and LESI algorithms from perturbed TOSCA dataset by (from top to bottom) 0%, 0.5%, 1%, 2% noise levels.

**Figure 9.**2D PCA projection of shape descriptors computed by: (

**a**) original Shape-DNA; (

**b**) cShape-DNA; (

**c**) GPS; and (

**d**) LESI algorithms over scaled version of the TOSCA dataset by a randomly chosen factor of 0.5, 0.875, 1.25, 1.625, or 2.

**Figure 10.**The Euclidean pairwise distance matrix of shape descriptors computed by (from left to right) Shape-DNA, cShape-DNA, GPS, and LESI algorithms over scaled version of the TOSCA dataset by a randomly chosen factor of 0.5, 0.875, 1.25, 1.625, or 2.

**Figure 11.**2D PCA projection of shape descriptors computed by: (

**a**) original Shape-DNA; (

**b**) cShape-DNA; (

**c**) GPS; and (

**d**) LESI algorithms from down sampled TOSCA dataset by rate of 20%.

**Figure 12.**The Euclidean pairwise distance matrix of shape descriptors computed by (from left to right) Shape-DNA, cShape-DNA, GPS, and LESI algorithms from from down sampled TOSCA dataset by rate of 20%.

Dataset | Method | NN | FT | ST | E | DCG |
---|---|---|---|---|---|---|

TOSCA | ShapeDNA | 1.0000 | 0.8091 | 0.9391 | 0.4486 | 0.9584 |

cShapeDNA | 0.9474 | 0.7748 | 0.8984 | 0.4748 | 0.9241 | |

GPS | 0.4868 | 0.4244 | 0.6320 | 0.3614 | 0.6787 | |

LESI | 0.8684 | 0.8456 | 0.9430 | 0.4860 | 0.9244 | |

McGill | ShapeDNA | 0.7922 | 0.3452 | 0.4977 | 0.3411 | 0.7192 |

cShapeDNA | 0.7882 | 0.3943 | 0.5483 | 0.3852 | 0.7470 | |

GPS | 0.3843 | 0.2508 | 0.4066 | 0.2588 | 0.6020 | |

LESI | 0.9647 | 0.7046 | 0.8739 | 0.6644 | 0.9251 |

Method | Average Accuracy |
---|---|

Shape-DNA | 21.02% |

Shape-DNA (Normalized) | 90.60% |

cShape-DNA | 71.37% |

GPS | 50.11% |

LESI | 95.69% |

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**MDPI and ACS Style**

Bashiri, F.S.; Rostami, R.; Peissig, P.; D’Souza, R.M.; Yu, Z. An Application of Manifold Learning in Global Shape Descriptors. *Algorithms* **2019**, *12*, 171.
https://doi.org/10.3390/a12080171

**AMA Style**

Bashiri FS, Rostami R, Peissig P, D’Souza RM, Yu Z. An Application of Manifold Learning in Global Shape Descriptors. *Algorithms*. 2019; 12(8):171.
https://doi.org/10.3390/a12080171

**Chicago/Turabian Style**

Bashiri, Fereshteh S., Reihaneh Rostami, Peggy Peissig, Roshan M. D’Souza, and Zeyun Yu. 2019. "An Application of Manifold Learning in Global Shape Descriptors" *Algorithms* 12, no. 8: 171.
https://doi.org/10.3390/a12080171