# Geodesics in the TPS Space

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## Abstract

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## 1. Introduction

- In Section 2, the general definitions of the two families of curves in Riemannian spaces are summarized.
- In Section 3, the main concept defining the TPS Space are recalled.
- In Section 4, the novel contribution of this paper is presented, that is, the construction of and comparison between autoparallel and geodesic lines in TPS Space.
- In Section 6, the numerical results are shown in order to discuss and compare the main features of autoparallel and geodesic lines in TPS Space.

## 2. Geodesics in Riemannian Manifolds

- A connection ∇ is torsion-free if and only if D is symmetric.
- A connection ∇ has the same geodesics as the Levi–Civita connection ${\nabla}^{g}$ if and only if D is skew-symmetric.

## 3. Geometry of the TPS Space

#### 3.1. Thin Plate Spline

#### 3.2. Energies

#### 3.3. TPS Metric

#### Alignments: OPA and MOPA Techniques

#### 3.4. TPS Direct Transport

- It is compatible with the TPS metric;
- It is compatible with the decomposition provided in (12);
- It is independent of the path.

- Perform a TPS analysis on X and find the ${S}_{X}$ and ${\mathsf{\Gamma}}_{11X}$;
- Perform an eigenvalue analysis on ${\mathsf{\Gamma}}_{11X}$ and obtain ${\mathsf{\Gamma}}_{11X}=\mathsf{\Gamma}\mathsf{\Lambda}{\mathsf{\Gamma}}^{T}$, where $\mathsf{\Gamma}$ is the $(k-1)\times (k-1)$ matrix containing the eigenvectors ${\gamma}_{i}$ in column and $\mathsf{\Lambda}$ is the diagonal $(k-1)\times (k-1)$ matrix of the eigenvalues ${\lambda}_{1},\dots ,{\lambda}_{k-1}$ ordered by increasing magnitude (the first m eigenvalues will be equal to 0);
- Drop the first m columns from $\mathsf{\Gamma}$ by obtaining the $(k-1)\times (k-1-m)$ matrix $\overline{\mathsf{\Gamma}}$, containing the principal warp eigenvectors by column;
- Drop the first m rows and the first m columns from $\mathsf{\Lambda}$ by obtaining the $(k-1-m)\times (k-1-m)$ matrix $\overline{\mathsf{\Lambda}}$;
- Define the $(k-1)\times (k-1-m)$ matrix ${E}_{X}={S}_{X}\overline{\mathsf{\Gamma}}{\overline{\mathsf{\Lambda}}}^{1/2}$.

## 4. Geodesics and Autoparallel Lines in TPS Space

#### 4.1. Analytical Solution for the Affine Subspace

#### 4.2. Geodesics Calculation: Objective Function Optimisation and Equality Constraints

Algorithm 1: geodesic algorithm. |

Result: Geodesic path with initial configuration X_{0} and final configuration X_{f} using n discretization steps. |

#### 4.3. Autoparallel Lines Calculation Algorithm via Shooting

Algorithm 2: shooting algorithm. |

Result: Shooting path with initial position X_{0} and initial velocity V_{0} using n discretization steps. |

## 5. Examples

- Affine case, spherical: in this simple case, the starting rectangle experiences only a size increase.
- Affine case, general: in this case, the rectangle undergoes only a pure affine transformation.
- Non-affine case bending: in this case, the rectangle experiences pure bending parameterized according to the parameters specified below.
- Non-affine case bending+size: in this case, the rectangle experiences pure bending parameterized according to the parameters specified below, with the addition of a size increase.
- Non-affine case bending+general affine component: in this case, the rectangle experiences pure bending parameterized according to the parameters specified below, with the addition of the same parameters of affine transformation as in case 2.

#### 5.1. Dataset

- A parametric trajectory $X\left(t\right)$ is generated by the mean of the (54). In this way, the initial and final points, $X\left(0\right)$ and $X\left(1\right)$, are identified.
- Linear interpolation between $X\left(0\right)$ and $X\left(1\right)$.
- The geodesic $Y\left(t\right)$, such that $Y\left(0\right)=X\left(0\right)$ and $Y\left(1\right)=X\left(1\right)$ are calculated following the procedure sketched in Section 4.2. The distance, $d\left(X\right(0),X(1\left)\right)$, and the initial tangent, ${Y}^{\prime}\left(0\right)$, are calculated.
- The autoparallel line $Z\left(t\right)$, starting from $X\left(0\right)$, is built by shooting ${Z}^{\prime}\left(0\right)={Y}^{\prime}\left(0\right)$ by means of Direct Transport for a distance of $\ell =d\left(X\right(0),X(1\left)\right)$, following the procedure sketched in Section 4.3.

- The trend of the $\mathsf{\Gamma}$-energy.
- The trend of the components of the $\mathsf{\Gamma}$-energy.

#### 5.2. Affine Case: Spherical

#### 5.3. Affine Case: General Case

#### 5.4. Non Affine Case: Bending

#### 5.5. Non-Affine Case: Bending and Scaling

#### 5.6. Non-Affine Case: Bending and General Affine Component

## 6. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

TPS | Thin Plate Spline |

PT | Parallel Transport |

DT | Direct Transport |

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**Figure 1.**Example of a 2D configuration made by six landmarks, with the first five lying on the boundary and one, ${x}_{6}$, inside the region $\mathsf{\Omega}$.

**Figure 2.**Affine spherical case results. (

**a**) Left panel: geodesic trajectory shapes (black) plotted against the original parametric shapes (red). (

**b**) Center panel: geodesic trajectory shapes (black) plotted against the shapes found via linear interpolation between the first and last shapes of the parametric dataset (red). (

**c**) Right panel: geodesic trajectory shapes (black) plotted against autoparallel trajectory (red) built via shooting of the first two configurations of the geodesic.

**Figure 3.**Trend of the energies for different interpolations: (

**a**) Left: linear interpolation (

**b**) Center: geodesics interpolation. (

**c**) Right: autoparallel interpolation. Blue refers to ${\mathbf{d}}_{\mathbf{u}}$, red to ${\mathbf{d}}_{\mathbf{b}}$, grey to ${\mathbf{d}}_{\mathbf{tot}}$.

**Figure 4.**Affine general case results. (

**a**) Left panel: geodesic trajectory shapes (black) plotted against the original parametric shapes (red). (

**b**) Center panel: geodesic trajectory shapes (black) plotted against the shapes found via linear interpolation between the first and last shapes of the parametric dataset (red). (

**c**) Right panel: geodesic trajectory shapes (black) plotted against autoparallel trajectory (red) built via shooting of the first two configurations of the geodesic.

**Figure 5.**Trend of the energies for different interpolations: (

**a**) Left: linear interpolation (

**b**) Center: geodesics interpolation. (

**c**) Right: autoparallel interpolation. Blue refers to ${\mathbf{d}}_{\mathbf{u}}$, red to ${\mathbf{d}}_{\mathbf{b}}$, grey to ${\mathbf{d}}_{\mathbf{tot}}$.

**Figure 6.**PC1-PC2 scatterplot of the PCA perfomed on the four set of shapes resulting from the general affine case. PC1 explains 97.2% of total variance, while PC2 explains 2.66%. Black refers to the optimized shapes, red to the linearized, green to the original shapes, and cyan to the shooted shapes.

**Figure 7.**Non-affine case bending-only results. (

**a**) Left panel: geodesic trajectory shapes (black) plotted against the original parametric shapes (red). (

**b**) Center panel: geodesic trajectory shapes (black) plotted against the shapes found via linear interpolation between the first and last shapes of the parametric dataset (red). (

**c**) Right panel: geodesic trajectory shapes (black) plotted against autoparallel trajectory (red) built via shooting of the first two configurations of the geodesic.

**Figure 8.**Trend of the energies for different interpolations: (

**a**) Left: linear interpolation (

**b**) Center: geodesics interpolation. (

**c**) Right: autoparallel interpolation. Blue refers to ${\mathbf{d}}_{\mathbf{u}}$, red to ${\mathbf{d}}_{\mathbf{b}}$, grey to ${\mathbf{d}}_{\mathbf{tot}}$.

**Figure 9.**PC1-PC2 scatterplot of the PCA perfomed on the four set of shapes resulting from the pure bending case. PC1 explains 91.11% of total variance, while PC2 explains 6.50%. Black refers to the optimized shapes, red to the linearized, green to the original shapes and cyan to the shooted shapes.

**Figure 10.**Non-affine case bending+size results. (

**a**) Left panel: geodesic trajectory shapes (black) plotted against the original parametric shapes (red). (

**b**) Center panel: geodesic trajectory shapes (black) plotted against the shapes found via linear interpolation between the first and last shapes of the parametric dataset (red). (

**c**) Right panel: geodesic trajectory shapes (black) plotted against autoparallel trajectory (red) built via shooting of the first two configurations of the geodesic.

**Figure 11.**Trend of the energies for different interpolations: (

**a**) Left: linear interpolation (

**b**) Center: geodesics interpolation. (

**c**) Right: autoparallel interpolation. Blue refers to ${\mathbf{d}}_{\mathbf{u}}$, red to ${\mathbf{d}}_{\mathbf{b}}$, grey to ${\mathbf{d}}_{\mathbf{tot}}$.

**Figure 12.**PC1-PC2 scatterplot of the PCA perfomed on the four set of shapes resulting from the bending+size case. PC1 explains 79.36% of total variance, while PC2 explains 17.54%. Black refers to the optimized shapes, red to the linearized, green to the original shapes and cyan to the shooted shapes.

**Figure 13.**Non-affine case, bending+general affine component, results. (

**a**) Left panel: geodesic trajectory shapes (black) plotted against the original parametric shapes (red). (

**b**) Center panel: geodesic trajectory shapes (black) plotted against the shapes found via linear interpolation between the first and last shapes of the parametric dataset (red). (

**c**) Right panel: geodesic trajectory shapes (black) plotted against autoparallel trajectory (red) built via shooting of the first two configurations of the geodesic.

**Figure 14.**Trend of the energies for different interpolations: (

**a**) Left: linear interpolation (

**b**) Center: geodesics interpolation. (

**c**) Right: autoparallel interpolation. Blue refers to ${\mathbf{d}}_{\mathbf{u}}$, red to ${\mathbf{d}}_{\mathbf{b}}$, grey to ${\mathbf{d}}_{\mathbf{tot}}$.

**Figure 15.**PC1-PC2 scatterplot of the PCA perfomed on the four set of shapes resulting from the bending+general affine component case. PC1 explains 62.61% of total variance, while PC2 explains 25.48%. Black refers to the optimized shapes, red to the linearized, green to the original shapes and cyan to the shooted shapes.

**Table 1.**Values of the energies for the affine spherical case; ${\mathbf{d}}_{\mathbf{u}}$ represents the affine component of the $\mathsf{\Gamma}$-energy, ${\mathbf{d}}_{\mathbf{b}}$ the non-affine component and ${\mathbf{d}}_{\mathbf{tot}}$ the total $\mathsf{\Gamma}$-energy.

LINEAR | GEODESIC | A-PARALLEL | ||||||
---|---|---|---|---|---|---|---|---|

${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ | ${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ | ${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ |

0.082 | 0.000 | 0.082 | 0.029 | 0.000 | 0.029 | 0.029 | 0.000 | 0.029 |

0.049 | 0.000 | 0.049 | 0.029 | 0.000 | 0.029 | 0.029 | 0.000 | 0.029 |

0.033 | 0.000 | 0.033 | 0.029 | 0.000 | 0.029 | 0.029 | 0.000 | 0.029 |

0.024 | 0.000 | 0.024 | 0.029 | 0.000 | 0.029 | 0.029 | 0.000 | 0.029 |

0.018 | 0.000 | 0.018 | 0.029 | 0.000 | 0.029 | 0.029 | 0.000 | 0.029 |

0.014 | 0.000 | 0.014 | 0.029 | 0.000 | 0.029 | 0.029 | 0.000 | 0.029 |

0.011 | 0.000 | 0.011 | 0.030 | 0.000 | 0.030 | 0.029 | 0.000 | 0.029 |

**Table 2.**Values of the energies for the affine general case; ${\mathbf{d}}_{\mathbf{u}}$ represents the affine component of the $\mathsf{\Gamma}$-energy, ${\mathbf{d}}_{\mathbf{b}}$ the non-affine component, and ${\mathbf{d}}_{\mathbf{tot}}$ the total $\mathsf{\Gamma}$-energy.

LINEAR | GEODESIC | A-PARALLEL | ||||||
---|---|---|---|---|---|---|---|---|

${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ | ${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ | ${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ |

0.020 | 0.000 | 0.020 | 0.010 | 0.000 | 0.010 | 0.010 | 0.000 | 0.010 |

0.015 | 0.000 | 0.015 | 0.010 | 0.000 | 0.010 | 0.010 | 0.000 | 0.010 |

0.011 | 0.000 | 0.011 | 0.010 | 0.000 | 0.010 | 0.010 | 0.000 | 0.010 |

0.009 | 0.000 | 0.009 | 0.010 | 0.000 | 0.010 | 0.010 | 0.000 | 0.010 |

0.007 | 0.000 | 0.007 | 0.010 | 0.000 | 0.010 | 0.010 | 0.000 | 0.010 |

0.006 | 0.000 | 0.006 | 0.010 | 0.000 | 0.010 | 0.010 | 0.000 | 0.010 |

0.005 | 0.000 | 0.005 | 0.010 | 0.000 | 0.010 | 0.010 | 0.000 | 0.010 |

**Table 3.**Values of the energies for the non-affine case of bending: ${\mathbf{d}}_{\mathbf{u}}$ represents the affine component of the $\mathsf{\Gamma}$-energy, ${\mathbf{d}}_{\mathbf{b}}$ the non-affine component, and ${\mathbf{d}}_{\mathbf{tot}}$ the total $\mathsf{\Gamma}$-energy.

LINEAR | GEODESIC | A-PARALLEL | ||||||
---|---|---|---|---|---|---|---|---|

${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ | ${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ | ${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ |

0.033 | 0.042 | 0.075 | 0.005 | 0.072 | 0.077 | 0.005 | 0.106 | 0.111 |

0.034 | 0.070 | 0.104 | 0.003 | 0.072 | 0.075 | 0.005 | 0.106 | 0.111 |

0.033 | 0.215 | 0.248 | 0.002 | 0.071 | 0.073 | 0.005 | 0.107 | 0.112 |

0.001 | 0.084 | 0.085 | 0.001 | 0.068 | 0.069 | 0.005 | 0.110 | 0.115 |

0.006 | 0.084 | 0.090 | 0.001 | 0.062 | 0.063 | 0.005 | 0.114 | 0.119 |

0.020 | 0.195 | 0.214 | 0.002 | 0.059 | 0.061 | 0.005 | 0.115 | 0.120 |

0.008 | 0.129 | 0.137 | 0.003 | 0.061 | 0.064 | 0.005 | 0.114 | 0.119 |

**Table 4.**Values of the energies for the non-affine case of bending and scaling; ${\mathbf{d}}_{\mathbf{u}}$ represents the affine component of the $\mathsf{\Gamma}$-energy, ${\mathbf{d}}_{\mathbf{b}}$ the non-affine component, and ${\mathbf{d}}_{\mathbf{tot}}$ the total $\mathsf{\Gamma}$-energy.

LINEAR | GEODESIC | A-PARALLEL | ||||||
---|---|---|---|---|---|---|---|---|

${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ | ${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ | ${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ |

0.060 | 0.37 | 0.431 | 0.017 | 0.120 | 0.137 | 0.017 | 0.150 | 0.167 |

0.003 | 0.41 | 0.412 | 0.003 | 0.128 | 0.131 | 0.017 | 0.150 | 0.167 |

0.147 | 0.42 | 0.571 | 0.007 | 0.115 | 0.122 | 0.017 | 0.150 | 0.168 |

0.028 | 0.04 | 0.073 | 0.021 | 0.101 | 0.122 | 0.017 | 0.159 | 0.177 |

0.021 | 0.03 | 0.056 | 0.029 | 0.087 | 0.116 | 0.017 | 0.155 | 0.172 |

0.036 | 0.07 | 0.104 | 0.025 | 0.091 | 0.115 | 0.017 | 0.176 | 0.193 |

0.012 | 0.03 | 0.040 | 0.044 | 0.065 | 0.110 | 0.017 | 0.175 | 0.193 |

**Table 5.**Values of the energies for the non-affine case of bending and general affine deformation; ${\mathbf{d}}_{\mathbf{u}}$ represents the affine component of the $\mathsf{\Gamma}$-energy, ${\mathbf{d}}_{\mathbf{b}}$ the non-affine component, and ${\mathbf{d}}_{\mathbf{tot}}$ the total $\mathsf{\Gamma}$-energy.

LINEAR | GEODESIC | A-PARALLEL | ||||||
---|---|---|---|---|---|---|---|---|

${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ | ${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ | ${\mathbf{d}}_{\mathbf{u}}$ | ${\mathbf{d}}_{\mathbf{b}}$ | ${\mathbf{d}}_{\mathbf{tot}}$ |

0.037 | 0.196 | 0.233 | 0.004 | 0.096 | 0.100 | 0.004 | 0.110 | 0.114 |

0.017 | 0.453 | 0.470 | 0.005 | 0.097 | 0.101 | 0.004 | 0.110 | 0.114 |

0.039 | 0.689 | 0.729 | 0.005 | 0.100 | 0.104 | 0.004 | 0.113 | 0.117 |

0.010 | 0.158 | 0.168 | 0.001 | 0.108 | 0.110 | 0.004 | 0.120 | 0.123 |

0.012 | 0.152 | 0.164 | 0.013 | 0.101 | 0.114 | 0.004 | 0.114 | 0.117 |

0.025 | 0.289 | 0.315 | 0.028 | 0.089 | 0.117 | 0.004 | 0.115 | 0.119 |

0.009 | 0.124 | 0.132 | 0.017 | 0.103 | 0.120 | 0.004 | 0.128 | 0.132 |

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**MDPI and ACS Style**

Varano, V.; Gabriele, S.; Milicchio, F.; Shlager, S.; Dryden, I.; Piras, P.
Geodesics in the TPS Space. *Mathematics* **2022**, *10*, 1562.
https://doi.org/10.3390/math10091562

**AMA Style**

Varano V, Gabriele S, Milicchio F, Shlager S, Dryden I, Piras P.
Geodesics in the TPS Space. *Mathematics*. 2022; 10(9):1562.
https://doi.org/10.3390/math10091562

**Chicago/Turabian Style**

Varano, Valerio, Stefano Gabriele, Franco Milicchio, Stefan Shlager, Ian Dryden, and Paolo Piras.
2022. "Geodesics in the TPS Space" *Mathematics* 10, no. 9: 1562.
https://doi.org/10.3390/math10091562