# A Survey of Methods for Symmetry Detection on 3D High Point Density Models in Biomedicine

^{*}

## Abstract

**:**

## 1. Introduction

- -
- 2D symmetry: symmetry line
- -
- 3D symmetry: symmetry plane

## 2. Symmetry Line

- -
- Cutaneous marking-based methods
- -
- Parallel sections-based methods
- -
- Adaptive sections-based methods

#### 2.1. Cutaneous Marking-Based Methods

#### 2.2. Parallel Sections-Based Methods

_{0}be a reference length defined by the measure of the radius of p neighbourhood. The length of reference value, L

_{0}, is crucial for evaluating the symmetry, especially if compared with the dimensions of the symmetrical portion of the back. Variation in the assignment of L

_{0}leads to different symmetry index values, as it checks symmetry in a narrower or wider range of the investigated zone. The single point asymmetry contribution a(u) is evaluated at the points of each slice symmetrically positioned to the left (subindex l) and right (subindex r) of the considered point p:

_{l}is H(ξ − u), H

_{r}is H(ξ + u), G is Gaussian curvature, G

_{l}is G(ξ − u), G

_{r}is G(ξ + u), ξ is the curvilinear abscissa of point p and $\epsilon ={\phi}_{l}-{\phi}_{r}$ is the difference between the angular orientation of the principal direction in l and r (k

_{1,l}and k

_{1,r}).

**p**, L

_{0}) has a value of zero for curve points that have a perfectly specular neighbourhood. The minimum value of A(

**p**, L

_{0}) identifies the position of the spine i.e., the point best representing the curve symmetry.

_{i}and β

_{i}are the weight factors for the different cost terms. The authors compared their method with the one proposed in [4] by applying it to the analysis of 33 patients with scoliosis, and the results were similarly accurate between the two methods. However, Huysmans et al. affirmed that their method was advantageous as it could be applied to several postures and biomechanical constraints taking into account information coming from previous measurements. Furthermore, due to the global optimization, the location of anatomical landmarks was more robust and reliable. Conversely, one drawback of this method was the need to evaluate six coefficients (α

_{1}, α

_{2}, β

_{1}, β

_{2}, β

_{3}, β

_{4}) to become accustomed to the study of average individuals.

_{0}is the reference length and where:

- -
- ${N}_{\alpha}(\nu ,u)=\frac{{n}_{\alpha}(\nu +u)+{n}_{\alpha}(\nu -u)}{\left|{n}_{\alpha}(\nu +u)+{n}_{\alpha}(\nu -u)\right|}$;
- -
- ${\overline{N}}_{\alpha}\left(\nu ,{L}_{0}\right)=\frac{1}{{L}_{0}}{\displaystyle \underset{u=0}{\overset{{L}_{0}/2}{\int}}{N}_{\alpha}(\nu ,u)du}$.

_{i}, b

_{i}and c

_{i}were calculated using a weighted least squares method. The position of the symmetrical points was associated with relative and not absolute maximum values of $S\left(\nu ,{L}_{0}\right)$, therefore the method opportunely estimates the symmetry line.

#### 2.3. Adaptive Sections-Based Methods

_{L}(t), ξ

_{L}(t), ψ

_{L}(t), ζ

_{L}(t)} is introduced (see Figure 4). O

_{L}(t) is a point of the symmetry line, ζ

_{L}(t) is the tangent in O

_{L}(t), ψ

_{L}(t) is the symmetry line perpendicular to O

_{L}(t) and ξ

_{L}(t) is normal to ζ

_{L}(t) and ψ

_{L}(t).

_{0}) defined according to the method proposed in [16], which was subsequently refined by the algorithm shown in Figure 5. The NEPA method identifies the set of planes [Π(t)] which define onto the back surface a set of profiles [Γ(t)] that exhibit the maximum possible symmetry according to the expression in Equation (6). The NEPA method converges if Π(t) sections the back in the most symmetrical profiles and if the symmetry line passes through the most symmetrical points of the back.

_{k}) orthogonal to the direction identified by the previous pair of symmetry points (each point being at a given distance or ‘step’ from the previous and following points as demonstrated in Figure 7).

## 3. Symmetry Plane

- Extended Gaussian image (EGI)
- Mirroring and registration

#### 3.1. Adaptive Sections-Based Methods

#### 3.2. Adaptive Sections-Based Methods

_{0}) and then mirror the original data PC with respect to Π

_{0}, thus defining a set of mirrored points PCm. PC and PCm are then registered by using the iterative closest point (ICP) algorithm [23]. The result is a registered point cloud PCm,r. The final estimation of the symmetry plane (Π

_{f}) is obtained as the least-square approximating plane through the middle points of the segments joining homologous points in PC and PCm,r. Many of the methods in the literature deal with symmetry plane retrieval of faces.

_{f}) was assessed by using an iterative procedure aiming at determining the minimum of a properly build objective function (given by the sum of the weighted distances between the points reflected with respect to Π

_{f}and the corresponding nearest points of the cloud). Wights were expressed as the Leclerc function [35] to avoid the presence of an asymmetrically sampled area. However, the procedure appeared to still be sensitive to sampling inconsistency. Two perfectly symmetrical homologous surfaces characterised by different sampling densities turned out to be asymmetrical exclusively due to the distance between the originally acquired point cloud and the mirrored one.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Clinical image of a patient affected by a severe thoracic scoliosis. (

**a**) The vertebral spinous processes are marked on the skin. (

**b**) A radiograph image from the patient. (

**c**) Topographic representation of the back surface [7].

**Figure 4.**Local reference systems [11].

**Figure 5.**Refinement algorithm of the NEPA method [11].

**Figure 6.**Example of NEPA results [11].

**Figure 7.**Graphical description of the refinement algorithm provided in [13].

**Figure 8.**Results using the traditional and the refinement approaches for some significant analysed cases [13].

**Figure 9.**Results using the traditional and refinement approaches for some significant analysed cases. Mushroom model consisting of quadrilateral and triangular patches (

**a**), the corresponding orientation histogram (

**b**) [14].

**Figure 10.**Original and mirrored face registered [18].

**Figure 11.**Preprocessing scans for use with the FRGC PCA algorithm is a two-step process. The scan is initially processed into its Canonical Face Depth Map CFDM format before the FRGC normalisation algorithm ‘face2norm’ masks out unwanted regions and normalises the depth and colour [22].

**Figure 12.**Example of nose region used for locating the symmetry plane in [23].

**Figure 13.**Refined symmetry planes from [23].

**Figure 14.**The MarkSkirt operator. (

**a**) 3D facial mesh characterized by partial/incomplete boundary. (

**b**) Mirrored mesh. (

**c**) Alignment of (

**a**) and (

**b**) by means of ICP algorithm. (

**d**) region between the boundary and the dashed curve on the mirrored mesh. (

**e**) Alignment of the non-skirt region on Sm and original mesh, S. (

**f**–

**h**) Examples where part of the forehead is missing due to the hair occlusion. The red vertices in (

**f**) are S. (

**g**) Evaluated symmetry plane without the use of the MarkSkirt operator. (

**h**) Computed symmetry plane obtained by using the MarkSkirt operator [30].

**Figure 15.**Cloud-Registration by means of the vertical symmetry plane, slope of the nose bridge and nose tip, taken from [35].

Authors | Description | Advantages | Limitations—Aspects to Be Improved | Performance | |
---|---|---|---|---|---|

CUTANEOUS MARKING BASED METHODS | Turner-Smith [12] | 3D position of landmarks manually acquired. Symmetry line evaluated as the broken line jointing the barycentre of each marker. | Early approach for determining the symmetry line based on the 3D data of the back. Good accuracy (estimated in 5 mm). | Need to manually identify the apophyses and apply markers. | Asymmetry correlated with Cobb angle with sample correlation coefficient r = [0.77 – 0.94] and p-value p < 0.0001. r values depending on the number of patients used for the experimentation |

Sotoca et al. [13] | Spine curve obtained by: - Cutaneous markers positioned in correspondence of the vertebral spinous processes, from C7 to the lumbar vertebra L5; - Projecting the x-ray images on the topographic representation of the surface. Markers location approximated with a polynomial curve. | High correlation (r = 0.89) between this method and the X-ray based one. Good accuracy (estimated in 5 mm). | Asymmetry correlated with Cobb angle with sample correlation coefficient r = 0.89 and p-value p < 0.0001 | ||

PARALLEL SECTIONS BASED METHODS | Drerup-Hierholzer [10] | Coordinate system associated to the subject’s back. Slicing of the back surface using parallel planes normal to the vertical axis. Position of the spine associated to the minimum value of the lateral asymmetry function. | First methodology based on symmetry properties of the horizontal sections of the subject’s back. | Results could be not compatible with biomechanical constraints. Thousands of instances to be explored to select the most promising set of points, with respect of the work proposed by Santiesteban et al. | Asymmetry correlated with Cobb angle with sample correlation coefficient r = 0.9 and p-value p < 0.0001 |

Huysmans et al. [14] | Lateral asymmetry function integrated with, blending, curvatures, torsions and biomechanical constraints. | Compared to the work of Drerup—Hierholzer: - Avoids results that could be not compatible with biomechanical constraints; - Applicable to different postures; - Biomechanical constraints and information from previous measurements; - More robust and reliable location of the anatomical landmarks. | Evaluation of six coefficients required to adjust the procedure for analysing average individuals. | Mean r.m.s. error of 0.9 mm for the lateral deviation and 0.4° for the axial rotation when compared with the Drerup-Hierholzer method [10]. | |

Santiesteban et al. [15] | Principal curvatures directions are used as local shape descriptors of the surface. The cutting plane is defined from the set of centroids and from profile directions. | Fewer instances required, with respect to the method of Drerup-Hierholzer. | Need to estimate and quantify asymmetries in the whole back. | Authors provides a method for estimating and modelling human spine from 3D data but no quantitative assessment is provided. | |

Di Angelo et al. [16] | Symmetry index defined from normal unit vectors of horizontal sections orientation of the back surface. | With respect to the method proposed by Drerup-Hierholzer, lower errors in the lumbar and thoracic tracts and fewer instances required. | Estimation of symmetry line in the cervical tract is not trivial. | Mean error in mm (w.r.t. Drerup-Hierholzer [10]) for patients: - upright standing: 3.2 (lumbar), 3.5 (thoracic), 4.7 (cervical) - sitting: 2.8 (lumbar), 3.6 (thoracic), 5.2 (cervical) | |

Final Considerations | Suitable to detect the symmetry line in case of erect postures. | Adequate to analyse postures producing spine configurations protruding outside from the sagittal plane. | |||

ADAPTIVE SECTIONS BASED METHODS | Di Angelo et al. [17,18] | Starting from a first attempt symmetry line (C_{0}), the NEPA method, iteratively, finds the set of planes Π(t) which define onto the back surface a set of profiles Γ(t) exhibiting the maximum possible symmetry according to the expression in the Equation (4). | Suitable for asymmetric postures with spine configurations lying far outside the sagittal plane. Improving the symmetry line detection of about 6–7%, with respect to the previous method of the authors. | The method finds the symmetry line under two hypothesis: (1) symmetry line spans from the “most symmetric” points of the back; (2) plane Π(t) slices the back in the “most symmetric” profiles. | Mean error reduction w.r.t. the method in [16] equal to: 1.53% (lumbar), 7.59% (thoracic) |

Di Angelo et al. [19] | This new method analyses the profiles given by the intersection of the back surface with a set of planes Π_{k} orthogonal to the direction identified by the previous pair of symmetry points (each point being at a given distance (“step”) from the previous and the following) | Correct evaluation of the symmetry line even for extremely asymmetric postures. | Strong influence of the body morphology of the subject, especially by those features that produce asymmetry such as gibbosities or other alterations. In those cases, the method could fail and false symmetries could be detected. | Mean error reduction w.r.t. first-attempt symmetry line equal to: 2.2% (lumbar), 21.8% (thoracic), 34.5% (cervical) |

Authors | Description | Advantages | Limitations—Aspects to Be Improved | Performance | |
---|---|---|---|---|---|

EGI | Sun-Sherrah [14] | Analysis of the recurrence histogram built from the orientation of normal unit vectors from EGI and examination of the EGI map around the principal axes of inertia. | Early EGI-based approach for determining symmetry plane of solids. Effort to reduce computational costs are made. | Not much robust when noisy data are acquired. | Reflectional symmetry evaluated in 1 min. Rotational symmetry evaluated in 1–5 min. Complex medicalimages requires 100 min. Accuracy comparable to the one obtained using the “sphere resolution” |

Pan et al. [15] | Computation of the orientation histogram by the inverse of the Gaussian curvature. | Increasing robustness in presence of noisy data. | The proposed method should be further tested. | More than 95% of the models have good detection results. Computational time less than 1s (on a Pentium IV 2.0 GHz) | |

Final Considerations | Results depend on the symmetry of the acquired data. | EGI-based methods are not able to analyse the object symmetry. | |||

Mirroring and registration model | Benz et al. [24] | Mirroring and registration method applied to support surgical facial reconstruction from an aesthetical point of view. | Computation of symmetry plane using registration algorithms where the symmetry plane is retrieved even in case of asymmetric geometries. The 3D eyes position is evaluated by a provided procedure. | Need to cover a higher number of experimental tests to validate the results. | Mean deviation (mm) of the mirrored position from actual position equal to 1.3 (along x-axis), −0.75 along y-axis and −0.25 along z-axis. |

De Momi et al. [26] | Early attempt for the estimation of the symmetry plane from manually selected areas. | Applicable to all 3D models related to any anatomical body area. | Areas manually selected. | Time required to obtain a satisfactory result (on average) equal to 10 min for each skull, including computation of the symmetry plane. Mean deviation (mm) of the mirrored position from actual position equal to 1.5 orbital, 1.4 zygomatic, 1.7 maxillary | |

Colbry-Stockman [27] | First-attempt symmetry plane determined by applying the PCA method. | Fully automatic method. | This approach tends to produce unreliable results when asymmetrically scanned data are used as input (Tang et al.). | Computational time equal to 4 s for 320 × 240 pixel images and to 12 s for 640 × 480 pixel images. Improvement of mid-line normalization over database roll, pitch, and yaw differences equal to, respectively, 0.01° ± 0.58° 0.01° ± 2.01° 0.00° ± 0.79° 2.90 mm ± 7.81 mm | |

Tang et al. [29] | - Initially guessed symmetry plane passing through the centroid; - Registration performed by analyzing a symmetric rectangular region selected around the nose. | ICP registration insensitive to asymmetrical data. | - It allows a correct estimation of the symmetry plane only in case the actual plane is aligned with the yz—plane of the 3D scanner used for acquisition; - Valid approach only in the case of undistorted noses - rise in terms of computational costs due to the ICP algorithm. | When compared with the FSP method, the improvements scores from 7.1% to 5.5% for the symmetry curve and from 12.0% to 8.9% for the cheek curve | |

Zhang et al. [30] | MarkSkirt operator is used to assess the registration on the whole point clouds with the exception of the points belonging to the 10–ring of the boundary. | ICP algorithm does not fail in case irregularities in the face boundary are present. | - Texture information are used as an additional clue during face comparison tasks; - expanded database, managed in a more efficient way; - use of the MarkSkirt operator in the recognition process. | 2.8 s to obtain the symmetry profile representation from a facial mesh (1GHz Pentium IV). 10.8% equal error rate and 87.5% rank-one recognition rate. False results mainly caused by extreme expression | |

Combès et al. [31,32,33] | Estimation of the symmetry plane without the need of intermediate pre-processing operations such as roto–translation and registration. | Solves the problem of non-symmetrical sampling of the face. | Highly sensitive to non uniform sapling. | Angular and linear errors of the estimated symmetry plane compared to the ground truth solution less than 10 ^{−2} deg. | |

Spreeuwers [35] | Not based on ICP algorithm. Symmetry plane estimated by varying a set of parameters in a given range. | Computationally more efficient than the ICP. | The registration method could be further improved and more advanced approaches to select the best configuration of regions classifiers and to evaluate appropriate weights for the voting process could be applied. | Performance of proposed approach w.r.t. Tang et al. [29] shows equal error rates equal to 0.7% against 7.1% (6.1% using manual procedure). Computational time is equal to 2.5. | |

Di Angelo et al. [37] | First-attempt estimation of the symmetry plane performed by PCA. | Not sensitive to data asymmetry resulting from the scanning process. | Computationally more onerous of the previous ones. | Computational time equal to 11.2 s (average) against 245 s (average) evaluated in Pan et. Al. [21]. Robustness for reproducibility w.r.t. Pan et al. [21] equal to 1.86° (mean value). | |

Final Considerations | More recent contributions are insensitive to both asymmetrically 3D data and non-uniformity in terms of point cloud density. | Exception made for contributions. The others are sensitive to asymmetries. |

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**MDPI and ACS Style**

Bartalucci, C.; Furferi, R.; Governi, L.; Volpe, Y.
A Survey of Methods for Symmetry Detection on 3D High Point Density Models in Biomedicine. *Symmetry* **2018**, *10*, 263.
https://doi.org/10.3390/sym10070263

**AMA Style**

Bartalucci C, Furferi R, Governi L, Volpe Y.
A Survey of Methods for Symmetry Detection on 3D High Point Density Models in Biomedicine. *Symmetry*. 2018; 10(7):263.
https://doi.org/10.3390/sym10070263

**Chicago/Turabian Style**

Bartalucci, Chiara, Rocco Furferi, Lapo Governi, and Yary Volpe.
2018. "A Survey of Methods for Symmetry Detection on 3D High Point Density Models in Biomedicine" *Symmetry* 10, no. 7: 263.
https://doi.org/10.3390/sym10070263