# Cuckoo Search Algorithm with Lévy Flights for Global-Support Parametric Surface Approximation in Reverse Engineering

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## Abstract

**:**

## 1. Introduction

#### 1.1. Surface Approximation in Reverse Engineering

#### 1.2. Aims and Structure of the Paper

## 2. Previous Work

## 3. Description of the Problem

#### 3.1. Basic Concepts and Definitions

#### 3.2. The Surface Approximation Problem

## 4. The Cuckoo Search Algorithm

#### 4.1. Nature-Inspired Algorithms

#### 4.2. Basic Principles

- Each cuckoo lays one egg at a time in a randomly chosen nest.
- The nests with the best eggs (i.e., high quality of solutions) will be carried over to the next generations, thus ensuring that good solutions are preserved over time.
- The number of available host nests is always fixed. A host can discover an alien egg with a probability ${p}_{a}\in [0,1]$. This rule can be approximated by the fact that a fraction ${p}_{a}$ of the n available host nests will be replaced by new nests (with new random solutions at new locations).

#### 4.3. The Algorithm

Algorithm 1: Cuckoo Search via Lévy Flights |

begin |

Objective function $f\left(\mathbf{x}\right)$, $\mathbf{x}={({x}_{1},\dots ,{x}_{d})}^{T}$ with $d=dim(\Omega )$ |

Generate initial population of N host nests ${\mathbf{x}}_{i}$ $(i=1,2,\dots ,N)$ |

while $(t<MaxGeneration)$ or (stop criterion) |

Get a cuckoo (say, i) randomly by Lévy flights |

Evaluate its fitness ${F}_{i}$ |

Choose a nest among N (say, j) randomly |

if (${F}_{i}>{F}_{j})$ |

Replace j by the new solution |

end |

A fraction (${p}_{a}$) of worse nests are abandoned and new ones are built via Lévy flights |

Keep the best solutions (or nests with quality solutions) |

Rank the solutions and find the current best |

end while |

Postprocess results and visualization |

end |

**x**and follows the normal distribution $N(0,1)$.

## 5. Method

#### 5.1. Overview of the Method

- data parameterization,
- surface fitting.

#### 5.2. Data Parameterization

#### 5.3. Data Fitting

## 6. Results

#### 6.1. Graphical and Numerical Results

#### 6.2. Parameter Tuning

- the population size ${N}_{p}$, and
- the probability ${p}_{a}$.

#### 6.3. Implementation Issues

#### 6.4. Computation Times

## 7. Discussion

#### 7.1. Comparative Work

- Our method improves the most classical parameterization methods described in the literature in our comparison. The error rate of the alternative approaches with respect to our method shows that it provides a significant improvement, not just incremental enhancements. This fact is also visible in Figure 7, Figure 8 and Figure 9, where the resulting Bézier surfaces for the uniform, chordal, and centripetal parameterizations, and with our method are displayed for easier visual inspection for the three examples in our benchmark, respectively.
- Among these parametrization methods, the centripetal parameterization yields the closest results to ours in all cases. In fact, it might be a competitive method for some applications, but fails to yield even near-optimal solutions. This fact is clearly noticeable from Table 4 by simple visual inspection of the corresponding numerical values.
- In general, both the uniform and the chordal parameterization yields approximation surfaces of moderate quality. We also remark that the chordal approximation performs even worse than uniform parameterization for Example I while it happens the opposite way for Example III and they perform more or less similarly for Example II. These results are related to the fact that data points for Example I and Example II are noisy but organized, while they are unorganized for Example III. The uniform parameterization does not perform well for such uneven distribution of points.
- The comparison of CSA and its variant ICSA shows that they perform very similarly for the three examples in the benchmark. In fact, the mean value is slightly better for ICSA while the best value is better for CSA for the three examples. This means that the method ICSA tends to have less variation for different executions (as confirmed by the smaller values for the variance and standard deviation than CSA), but CSA is better at approaching to the global minima. However, the differences between both methods are very small and none of them seems to dominate the other for our benchmark.

#### 7.2. Statistical Analysis

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Application of our method to Example I: (

**top**) given cloud of noisy data points; (

**bottom**) noisy data points and best approximating Bézier surface.

**Figure 3.**Application of our method to Example II: (

**top**) given cloud of noisy data points; (

**bottom**) noisy data points and best approximating Bézier surface.

**Figure 5.**Application of our method to Example III: (

**top**) given cloud of noisy data points; (

**bottom**) noisy data points and best approximating Bézier surface.

**Figure 7.**Visual comparison of different methods for Example I: (

**top-left**) uniform parameterization; (

**top-right**) chordal parameterization; (

**bottom-left**) centripetal parameterization; (

**bottom-right**) our method.

**Figure 8.**Visual comparison of different methods for Example II: (

**top-left**) uniform parameterization; (

**top-right**) chordal parameterization; (

**bottom-left**) centripetal parameterization; (

**bottom-right**) our method.

**Figure 9.**Visual comparison of different methods for Example III: (

**top-left**) uniform parameterization; (

**top-right**) chordal parameterization; (

**bottom-left**) centripetal parameterization; (

**bottom-right**) our method.

**Table 1.**Experimental setup used in this paper. For the three examples in the benchmark (in columns), the table reports (in rows): number of data points, DOFs (degrees of freedom), SNR (signal-to-noise ratio) value, and degree of the approximating Bézier surface.

Iterm | Example I | Example II | Example III |
---|---|---|---|

$\kappa $ | 841 | 1681 | 628 |

$DOFs$ | 1754 | 3562 | 1328 |

SNR | 20.75 | 12.5 | 18 |

Degree | $(5,5)$ | $(9,9)$ | $(5,5)$ |

**Table 2.**Mean value, variance and standard deviation of the X, Y and Z coordinates (in rows) of the estimated data points for the three examples (in columns) of our benchmark.

Iterm | Example I | Example II | Example III |
---|---|---|---|

Mean value (X): | 1.323329 × 10${}^{-2}$ | 1.170657 × 10${}^{-2}$ | 4.035762 × 10${}^{-2}$ |

var (X): | 4.513201 × 10${}^{-7}$ | 1.233936 × 10${}^{-6}$ | 9.392598 × 10${}^{-6}$ |

std (X): | 9.500738 × 10${}^{-5}$ | 1.570946 × 10${}^{-4}$ | 4.334189 × 10${}^{-4}$ |

Mean value (Y): | 1.324084 × 10${}^{-2}$ | 9.279757 × 10${}^{-3}$ | 3.775183 × 10${}^{-2}$ |

var (Y): | 5.311829 × 10${}^{-8}$ | 2.451310 × 10${}^{-7}$ | 3.939023 × 10${}^{-6}$ |

std (Y): | 3.259395 × 10${}^{-5}$ | 7.001871 × 10${}^{-5}$ | 2.806785 × 10${}^{-4}$ |

Mean value (Z): | 2.281740 × 10${}^{-2}$ | 6.965113 × 10${}^{-3}$ | 2.538050 × 10${}^{-2}$ |

var (Z): | 5.682211 × 10${}^{-8}$ | 7.304708 × 10${}^{-10}$ | 1.648420 × 10${}^{-6}$ |

std (Z): | 3.371115 × 10${}^{-5}$ | 3.822226 × 10${}^{-10}$ | 1.815720 × 10${}^{-4}$ |

**Table 3.**Fitting errors for the examples (arranged in columns) of the benchmark used in this paper. The table reports (in rows): mean, best, variance and standard deviation of the $\mathbf{Y}$ error, and mean and best $RMSE$ (root-mean-square error) from 50 independent executions.

Iterm | Example I | Example II | Example III |
---|---|---|---|

$\mathbf{Y}$ (mean) | $5.880665$ × 10${}^{-2}$ | $3.868251$ × 10${}^{-2}$ | $1.390415$ × 10${}^{-1}$ |

$\mathbf{Y}$ (best) | $2.073165$ × 10${}^{-2}$ | $1.541309$ × 10${}^{-2}$ | $8.072414$ × 10${}^{-2}$ |

$\mathbf{Y}$ (var) | $2.915582$ × 10${}^{-4}$ | $1.688052$ × 10${}^{-4}$ | $1.203554$ × 10${}^{-3}$ |

$\mathbf{Y}$ (std) | $1.707507$ × 10${}^{-2}$ | $1.296349$ × 10${}^{-2}$ | $3.468796$ × 10${}^{-2}$ |

RMSE (mean) | $8.362097$ × 10${}^{-3}$ | $4.797041$ × 10${}^{-3}$ | $1.487963$ × 10${}^{-2}$ |

RMSE (best) | $4.964996$ × 10${}^{-3}$ | $3.028035$ × 10${}^{-3}$ | $1.133762$ × 10${}^{-2}$ |

**Table 4.**Comparative analysis of different methods (in columns) for the three examples of this paper (in rows). Best results are highlighted in bold.

Surface Example | Fitting Error | Uniform Param. | Chordal Param. | Centripetal Param. | Cuckoo Search (CSA) | Improved CSA (ICSA) |
---|---|---|---|---|---|---|

Example I | $\mathbf{Y}$ (mean) | $1.120627$ × 10${}^{-1}$ | $1.480965$ × 10${}^{-1}$ | $8.845724$ × 10${}^{-2}$ | 5.880665 × 10${}^{-2}$ | 5.758827 × 10${}^{-\mathbf{2}}$ |

E.R. (in %) | (193.6) | (255.9) | (152.8) | (101.6) | − | |

$\mathbf{Y}$ (best) | $5.611987$ × 10${}^{-2}$ | $6.956061$ × 10${}^{-2}$ | $4.960015$ × 10${}^{-2}$ | 2.073165 × 10${}^{-2}$ | $2.114396$ × 10${}^{-2}$ | |

E.R. (in %) | (270.7) | (335.5) | (239.2) | − | (102.0) | |

$\mathbf{Y}$ (var) | $2.568297$ × 10${}^{-2}$ | $2.420068$ × 10${}^{-5}$ | $5.946705$ × 10${}^{-4}$ | $2.915582$ × 10${}^{-4}$ | $1.912503$ × 10${}^{-5}$ | |

$\mathbf{Y}$ (std) | $4.111509$ × 10${}^{-1}$ | $4.919419$ × 10${}^{-3}$ | $2.438586$ × 10${}^{-2}$ | $1.707507$ × 10${}^{-2}$ | $4.373208$ × 10${}^{-2}$ | |

RMSE (mean) | $1.154337$ × 10${}^{-2}$ | $1.327011$ × 10${}^{-2}$ | $1.025578$ × 10${}^{-2}$ | $8.362097$ × 10${}^{-3}$ | 8.275019 × 10${}^{-\mathbf{3}}$ | |

E.R. (in %) | (139.5) | (160.4) | (123.9) | (101.1) | − | |

RMSE (best) | $8.168839$ × 10${}^{-3}$ | $9.094602$ × 10${}^{-3}$ | $7.679687$ × 10${}^{-3}$ | 4.964996 × 10${}^{-3}$ | $5.014126$ × 10${}^{-3}$ | |

E.R. (in %) | (164.5) | (183.2) | (154.6) | − | (101.0) | |

Example II | $\mathbf{Y}$ (mean) | $4.781280$ × 10${}^{-2}$ | $4.751447$ × 10${}^{-2}$ | $4.297031$ × 10${}^{-2}$ | $3.868251$ × 10${}^{-2}$ | 3.775440 × 10${}^{-2}$ |

E.R. (in %) | (126.6) | (125.8) | (113.8) | (102.5) | − | |

$\mathbf{Y}$ (best) | $3.788349$ × 10${}^{-2}$ | $3.911705$ × 10${}^{-2}$ | $2.957763$ × 10${}^{-2}$ | 1.541309 × 10${}^{-2}$ | $2.093271$ × 10${}^{-2}$ | |

E.R. (in %) | (245.8) | (253.8) | (191.9) | − | (135.8) | |

$\mathbf{Y}$ (var) | $2.866010$ × 10${}^{-5}$ | $2.420068$ × 10${}^{-5}$ | $6.589306$ × 10${}^{-5}$ | $1.688052$ × 10${}^{-4}$ | $5.547001$ × 10${}^{-6}$ | |

$\mathbf{Y}$ (std) | $5.353513$ × 10${}^{-3}$ | $4.919419$ × 10${}^{-4}$ | $8.117478$ × 10${}^{-3}$ | $1.296349$ × 10${}^{-2}$ | $2.35520$ × 10${}^{-3}$ | |

RMSE (mean) | $5.333204$ × 10${}^{-3}$ | $5.316540$ × 10${}^{-3}$ | $5.055922$ × 10${}^{-3}$ | $4.797041$ × 10${}^{-3}$ | 4.739144 × 10${}^{-3}$ | |

E.R. (in %) | (112.5) | (112.2) | (106.7) | (101.2) | − | |

RMSE (best) | $4.747239$ × 10${}^{-3}$ | $4.823909$ × 10${}^{-3}$ | $4.194670$ × 10${}^{-3}$ | 3.028035 × 10${}^{-3}$ | $3.528814$ × 10${}^{-3}$ | |

E.R. (in %) | (156.7) | (159.3) | (138.5) | − | (116.5) | |

Example III | $\mathbf{Y}$ (mean) | $7.528725$ × 10${}^{-1}$ | $5.691582$ × 10${}^{-1}$ | $4.241297$ × 10${}^{-1}$ | $1.390415$ × 10${}^{-1}$ | 1.367894 × 10${}^{-1}$ |

E.R. (in %) | (550.4) | (416.1) | (310.1) | (101.6) | − | |

$\mathbf{Y}$ (best) | $4.988371$ × 10${}^{-1}$ | $3.455608$ × 10${}^{-1}$ | $1.003214$ × 10${}^{-1}$ | 8.072414 × 10${}^{-2}$ | $1.025736$ × 10${}^{-1}$ | |

E.R. (in %) | (617.9) | (428.1) | (124.3) | − | (127.1) | |

$\mathbf{Y}$ (var) | $1.526136$ × 10${}^{-2}$ | $5.720855$ × 10${}^{-1}$ | $3.06609$ × 10${}^{-2}$ | $1.203554$ × 10${}^{-3}$ | $2.485324$ × 10${}^{-4}$ | |

$\mathbf{Y}$ (std) | $1.235368$ × 10${}^{-3}$ | $1.250304$ × 10${}^{-1}$ | $1.751028$ × 10${}^{-1}$ | $3.468796$ × 10${}^{-2}$ | $1.576491$ × 10${}^{-2}$ | |

RMSE (mean) | $3.462429$ × 10${}^{-2}$ | $3.010486$ × 10${}^{-2}$ | $2.598780$ × 10${}^{-2}$ | $1.487963$ × 10${}^{-2}$ | 1.475864 × 10${}^{-2}$ | |

E.R. (in %) | (234.6) | (204.0) | (176.1) | (100.9) | − | |

RMSE (best) | $2.818380$ × 10${}^{-2}$ | $2.345753$ × 10${}^{-2}$ | $1.263912$ × 10${}^{-2}$ | 1.133762 × 10${}^{-2}$ | $1.278020$ × 10${}^{-2}$ | |

E.R. (in %) | (248.6) | (206.9) | (111.5) | − | (112.7) |

**Table 5.**Pairwise nonparametric statistical tests of the methods in our comparison (in rows) for the three examples in this paper (in columns). Unless otherwise stated, the level of significance for the tests is assumed to be $\alpha =0.05$.

Comparison | Index | Example I | Example II | Example III |
---|---|---|---|---|

CSA vs. Uniform | p-value (Wilcoxon sign): | 8.663083 × 10${}^{-8}$ | 1.572960 × 10${}^{-4}$ | 7.556929 × 10${}^{-10}$ |

signed rank: | 1192 | 1029 | 1275 | |

h: | 1 | 1 | 1 | |

p-value (Wilcoxon sum): | 3.191585 × 10${}^{-7}$ | 1.115515 × 10${}^{-4}$ | 7.032679 × 10${}^{-18}$ | |

rank sum: | 3267 | 3086 | 3775 | |

h: | 1 | 1 | 1 | |

CSA vs. Chordal | p-value (Wilcoxon sign): | 8.534226 × 10${}^{-10}$ | 4.000165 × 10${}^{-5}$ | 1.110095 × 10${}^{-9}$ |

signed rank: | 1273 | 1063 | 1225 | |

h: | 1 | 1 | 1 | |

p-value (Wilcoxon sum): | 8.003963 × 10${}^{-17}$ | 3.919684 × 10${}^{-5}$ | 1.032414 × 10${}^{-17}$ | |

rank sum: | 3734 | 3122 | 3675 | |

h: | 1 | 1 | 1 | |

CSA vs. Centripetal | p-value (Wilcoxon sign): | 6.394483 × 10${}^{-6}$ | 8.626159 × 10${}^{-2}$ | 1.087244 × 10${}^{-8}$ |

signed rank: | 1105 | 460 | 1269 | |

h: | 1 | 1 ($\alpha =0.1$) | 1 | |

p-value (Wilcoxon sum): | 8.317099 × 10${}^{-6}$ | 0.148521 × 10${}^{-1}$ | 1.109774 × 10${}^{-15}$ | |

rank sum: | 3172 | 2325 | 3688 | |

h: | 1 | 1 ($\alpha =0.15$) | 1 | |

CSA vs. ICSA | p-value (Wilcoxon sign): | 4.784510 × 10${}^{-1}$ | 8.280511 × 10${}^{-1}$ | 4.900625 × 10${}^{-1}$ |

signed rank: | 441 | 615 | 709 | |

h: | 0 | 0 | 0 | |

p-value (Wilcoxon sum): | 3.883964 × 10${}^{-1}$ | p = 7.275182 × 10${}^{-1}$ | 3.024247 × 10${}^{-1}$ | |

rank sum: | 2125 | 2475 | 2675 | |

h: | 0 | 0 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Iglesias, A.; Gálvez, A.; Suárez, P.; Shinya, M.; Yoshida, N.; Otero, C.; Manchado, C.; Gomez-Jauregui, V. Cuckoo Search Algorithm with Lévy Flights for Global-Support Parametric Surface Approximation in Reverse Engineering. *Symmetry* **2018**, *10*, 58.
https://doi.org/10.3390/sym10030058

**AMA Style**

Iglesias A, Gálvez A, Suárez P, Shinya M, Yoshida N, Otero C, Manchado C, Gomez-Jauregui V. Cuckoo Search Algorithm with Lévy Flights for Global-Support Parametric Surface Approximation in Reverse Engineering. *Symmetry*. 2018; 10(3):58.
https://doi.org/10.3390/sym10030058

**Chicago/Turabian Style**

Iglesias, Andrés, Akemi Gálvez, Patricia Suárez, Mikio Shinya, Norimasa Yoshida, César Otero, Cristina Manchado, and Valentin Gomez-Jauregui. 2018. "Cuckoo Search Algorithm with Lévy Flights for Global-Support Parametric Surface Approximation in Reverse Engineering" *Symmetry* 10, no. 3: 58.
https://doi.org/10.3390/sym10030058