# The Quadrature Method: A Novel Dipole Localisation Algorithm for Artificial Lateral Lines Compared to State of the Art

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Dipole Flow Field

#### 2.2. Simulation Environment

#### 2.3. Performance Analyses

#### 2.4. Parameter Optimisation Approach

#### 2.5. Dipole Localisation Algorithms

#### 2.5.1. The Random Predictor (RND)

#### 2.5.2. Linear Constraint Minimum Variance (LCMV) Beamforming

#### 2.5.3. K-Nearest Neighbours (KNN)

#### 2.5.4. The Continuous Wavelet Transform (CWT)

#### 2.5.5. The Extreme Learning Machine (ELM)

#### 2.5.6. The Multi-Layer Perceptron (MLP)

#### 2.5.7. The Gauss–Newton (GN) Algorithm

#### 2.5.8. The Newton–Raphson (NR) Algorithm

#### 2.5.9. The Least Square Curve Fit (LSQ) Algorithm

#### 2.5.10. The Quadrature Method (QM) Algorithm

## 3. Results

#### 3.1. Analysis Method 1: Amount of Training and Optimisation Data

#### 3.2. Analysis Method 2: Sensor Sensitivity Axes

#### 3.3. Additional Results

## 4. Discussion

#### 4.1. The Gauss–Newton (GN) Algorithm

#### 4.2. The Newton–Raphson (NR) Algorithm

#### 4.3. The Multi-Layer Perceptron (MLP) and Extreme Learning Machine (ELM)

#### 4.4. The Quadrature Method (QM) Algorithm

#### 4.5. Future Research Directions and Possible Applications

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ALL | artificial lateral line |

AUV | autonomous underwater vehicle |

CNN | convolutional neural network |

CWT | continuous wavelet transform |

DFT | discrete Fourier transform |

ELM | extreme learning machine |

ESN | echo state network |

GN | Gauss–Newton |

KNN | k-nearest neighbours |

LCMV | linear constraint minimum variance |

LSQ | least square curve fit |

MAE | mean absolute error |

MSE | mean squared error |

MLP | multi-layer perceptron |

NR | Newton–Raphson |

OS-ELM | online sequential extreme learning machine |

QM | quadrature method |

RND | random |

SLFN | single layer feed-forward network |

SNR | signal to noise ratio |

## Appendix A. Final Hyperparameter Values

**Table A1.**All hyperparameter values for the first analysis optimised using the training set. This analysis varied the minimum distance ${D}_{s}$ between source states in the training and optimisation sets to show how the amount of training and optimisation data influences the dipole localisation algorithms’ performance. The (x + y) sensor configuration at the $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level was used in this analysis. Table 1 describes the respective data sets in more detail.

KNN | ${D}_{s}$ | k neighbours | |||

0.09 | 3 | ||||

0.05 | 3 | ||||

0.03 | 3 | ||||

0.01 | 5 | ||||

CWT | ${D}_{s}$ | threshold ${t}_{min}$ | threshold ${t}_{max}$ | $\phi $ factor ${c}_{x}$ | $\phi $ factor ${c}_{y}$ |

0.09 | 0.36 | 1.0 | 1.0 | 0.30 | |

0.05 | 0.18 | 0.92 | 0.89 | 0.89 | |

0.03 | 0.21 | 0.74 | 0.98 | 0.35 | |

0.01 | 0.26 | 0.57 | 0.82 | 0.38 | |

ELM | ${D}_{s}$ | $\overline{n}$ nodes | Analytical ${c}_{x}=0.6$ (higher value tunes | ||

0.09 | 120 | estimation for longer distances) | |||

0.05 | 75 | Analytical ${c}_{y}\approx 0.366$ (lower value tunes | |||

0.03 | 1383 | estimation for longer distances) | |||

0.01 | 11,169 | ||||

MLP | ${D}_{s}$ | learning rate $\u03f5$ | n layers | $\overline{n}$ nodes per layer | |

0.09 | $3.5\times {10}^{-3}$ | 1 | 990 | ||

0.05 | $3.4\times {10}^{-3}$ | 1 | 798 | ||

0.03 | $3.7\times {10}^{-3}$ | 1 | 1015 | ||

0.01 | $1.2\times {10}^{-3}$ | 4 | 993 | ||

GN | ${D}_{s}$ | initial distance ${d}_{0}$ ($\mathrm{c}\mathrm{m}$) | |||

0.09 | 5.0 | ||||

0.05 | 5.0 | ||||

0.03 | 2.5 | ||||

0.01 | 5.0 | ||||

NR | ${D}_{s}$ | initial distance ${d}_{0}$ ($\mathrm{c}\mathrm{m}$) | norm limit l | ||

0.09 | 2.5 | 0.14 | |||

0.05 | 2.9 | 0.10 | |||

0.03 | 2.5 | 0.10 | |||

0.01 | 2.5 | 0.10 |

**Table A2.**All hyperparameters for the second analysis optimised using the training set. This analysis varied the sensor sensitivity axes to show how the velocity components contribute to the predictions while keeping ${D}_{s}$ constant. The configurations were: (x + y) both components on all sensors, (x|y) alternating ${v}_{x}$ and ${v}_{y}$ for subsequent sensors, (x) only ${v}_{x}$ on all sensors, (y) only ${v}_{y}$ on all sensors. The $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level was used and with the ${D}_{s}=0.01$ training and optimisation set.

KNN | sensor | k neighbours | |||

(x + y) | 5 | ||||

(x|y) | 5 | ||||

(x) | 6 | ||||

(y) | 6 | ||||

CWT | sensor | threshold ${t}_{min}$ | threshold ${t}_{max}$ | $\phi $ factor ${c}_{x}$ | $\phi $ factor ${c}_{y}$ |

(x + y) | 0.23 | 0.60 | 0.71 | 0.33 | |

(x|y) | 0.22 | 0.40 | 0.76 | 0.64 | |

(x) | 0.21 | 0.53 | 0.61 | 0.40 | |

(y) | 0.12 | 0.79 | 0.58 | 0.87 | |

ELM | sensor | $\overline{n}$ nodes | Analytical ${c}_{x}=0.6$ (higher value tunes | ||

(x + y) | 11,169 | estimation for longer distances) | |||

(x|y) | 5165 | Analytical ${c}_{y}\approx 0.366$ (lower value tunes | |||

(x) | 5579 | estimation for longer distances) | |||

(y) | 5579 | ||||

MLP | sensor | learning rate $\u03f5$ | n layers | $\overline{n}$ nodes per layer | |

(x + y) | $1.4\times {10}^{-3}$ | 4 | 1014 | ||

(x|y) | $2.3\times {10}^{-3}$ | 4 | 982 | ||

(x) | $2.0\times {10}^{-3}$ | 4 | 1024 | ||

(y) | $2.0\times {10}^{-3}$ | 4 | 1016 | ||

GN | sensor | initial distance ${d}_{0}$ ($\mathrm{c}\mathrm{m}$) | |||

(x + y) | 5.0 | ||||

(x|y) | 2.5 | ||||

(x) | 7.5 | ||||

(y) | 2.5 | ||||

NR | sensor | initial distance ${d}_{0}$ ($\mathrm{c}\mathrm{m}$) | norm limit l | ||

(x + y) | 2.5 | 0.10 | |||

(x|y) | 7.8 | 0.13 | |||

(x) | 9.3 | 0.10 | |||

(y) | 2.5 | 0.10 |

## Appendix B. Potential Flow Wavelets

#### Appendix B.1. The Even Wavelet

#### Appendix B.2. The Odd Wavelet

## Appendix C. Movement Direction Estimation with the Continuous Wavelet Transform (CWT)CWT

#### Appendix C.1. Movement Direction Estimation with the Parallel Velocity Component

#### Appendix C.2. Movement Direction Estimation with the Perpendicular Velocity Component

## Appendix D. The Quadrature Method (QM)

**Figure A1.**Velocity profiles of a continuous sensor array for five source movement directions φ: (

**a**) v

_{x}and (

**b**) v

_{y}. The envelopes of the velocity profiles ψ

_{x,env}(ρ) and ψ

_{y,env}(ρ) are shown in green.

**Figure A2.**Quadrature profiles ${\psi}_{quad,norm}$ (left panel) of a continuous sensor array for five source directions $\phi $ (same as Figure A1). Furthermore, the two constituting functions ${\mathrm{\Phi}}_{sym}(\rho ,\phi )$ (middle panel) and ${\mathrm{\Phi}}_{skew}(\rho ,\phi )$ (right panel) are shown. These functions are, respectively, even and odd in $\rho $. At the secondary anchor points $\pm {\rho}_{anch}$ the function ${\mathrm{\Phi}}_{sym}(\rho ,\phi )$ provides angle independent values which may be employed to determine the source distance d before the source direction of motion $\phi $ is known.

**Figure A3.**The width between the secondary anchor points $\pm {\rho}_{anch}$ of the normalised quadrature curve ${\psi}_{quad,norm}$ are almost constant $1.78\pm 0.011$ with respect to the movement direction angle $\phi $ and is well approximated by the analytically determined value $4/\sqrt{5}$.

## Appendix E. Additional Figures

**Figure A4.**Spatial contours of the median position error (Equation (4)) for each localisation algorithm in all conditions of Analysis Method 1. This analysis varied the minimum distance ${D}_{s}$ between sources in the training and optimisation sets (Table 1) while using the (x + y) sensor configuration at the $\sigma =1\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The median position error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A5.**Spatial contours of the median movement direction error (Equation (5)) for each localisation algorithm in all conditions of Analysis Method 1. This analysis varied the minimum distance ${D}_{s}$ between sources in the training and optimisation sets (Table 1) while using the (x + y) sensor configuration at the $\sigma =1\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The median movement direction error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A6.**Polar contours of the median position error (Equation (4)) for each localisation algorithm in all conditions of Analysis Method 1. These figures indicate how the movement direction $\phi $ and distance d of a source influence the error. This analysis varied the minimum distance ${D}_{s}$ between sources in the training and optimisation sets (Table 1) while using the (x + y) sensor configuration at the $\sigma =1\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The median position error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A7.**Polar contours of the median movement direction error (Equation (4)) for each localisation algorithm in all conditions of Analysis Method 1. These figures indicate how the movement direction $\phi $ and distance d of a source influence the error. This analysis varied the minimum distance ${D}_{s}$ between sources in the training and optimisation sets (Table 1) while using the (x + y) sensor configuration at the $\sigma =1\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The median movement direction error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A8.**Spatial contours of the median position error (Equation (4)) for each localisation algorithm in all conditions of Analysis Method 2. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring ${v}_{x}$ and ${v}_{y}$ for subsequent sensors, (x) measured only ${v}_{x}$ at all sensors, (y) measured only ${v}_{y}$ at all sensors. The ${D}_{s}=0.01$ training and optimisation set and $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level were used. The median position error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A9.**Spatial contours of the median movement direction error (Equation (5)) for each localisation algorithm in all conditions of Analysis Method 2. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring ${v}_{x}$ and ${v}_{y}$ for subsequent sensors, (x) measured only ${v}_{x}$ at all sensors, (y) measured only ${v}_{y}$ at all sensors. The ${D}_{s}=0.01$ training and optimisation set and $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level were used. The median movement direction error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A10.**Polar contours of the median position error (Equation (4)) for each localisation algorithm in all conditions of Analysis Method 2. These figures indicate how the movement direction $\phi $ and distance d of a source influence the error. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring ${v}_{x}$ and ${v}_{y}$ for subsequent sensors, (x) measured only ${v}_{x}$ at all sensors, (y) measured only ${v}_{y}$ at all sensors. The ${D}_{s}=0.01$ training and optimisation set and $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level were used. The median position error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A11.**Polar contours of the median movement direction error (Equation (5)) for each localisation algorithm in all conditions of Analysis Method 2. These figures indicate how the movement direction $\phi $ and distance d of a source influence the error. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring ${v}_{x}$ and ${v}_{y}$ for subsequent sensors, (x) measured only ${v}_{x}$ at all sensors, (y) measured only ${v}_{y}$ at all sensors. The ${D}_{s}=0.01$ training and optimisation set and $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level were used. The median movement direction error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A12.**An overview of the movement direction error E

_{ϖ}Equation (5) of QM, GN, and MLP using simulated sensors with higher velocity equivalent noise levels. (

**a**) Total areas with a median movement direction error E

_{φ}below 1 cm, 3 cm, 5 cm, and 9 cm. (

**b**) Boxplots of the movement direction error distributions, whiskers indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. The values for $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ are based on the ${D}_{s}=0.01$ condition in Analysis Method 1. The (x + y) sensor configuration was used. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the ${D}_{s}=0.01$ condition of Analysis Method 1.

**Figure A13.**Spatial contours of the median position error (Equation (4)) for QM, GN, and MLP using simulated sensors with higher velocity equivalent noise levels. The ${D}_{s}=0.01$ training and optimisation set and (x + y) sensor configuration were used. The values for $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ are based on the ${D}_{s}=0.01$ condition in Analysis Method 1. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the ${D}_{s}=0.01$ condition of Analysis Method 1. The median position error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A14.**Spatial contours of the median movement direction error (Equation (5)) for QM, GN, and MLP using simulated sensors with higher velocity equivalent noise levels. The ${D}_{s}=0.01$ training and optimisation set and (x + y) sensor configuration were used. The values for $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ are based on the ${D}_{s}=0.01$ condition in Analysis Method 1. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the ${D}_{s}=0.01$ condition of Analysis Method 1. The median movement direction error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A15.**Polar contours of the median position error (Equation (4)) for QM, GN, and MLP using simulated sensors with higher velocity equivalent noise levels. These figures indicate how the movement direction $\phi $ and distance d of a source influence the error. The ${D}_{s}=0.01$ training and optimisation set and (x + y) sensor configuration were used. The values for $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ are based on the ${D}_{s}=0.01$ condition in Analysis Method 1. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the ${D}_{s}=0.01$ condition of Analysis Method 1. The median position error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

**Figure A16.**Polar contours of the median movement direction error (Equation (5)) for each localisation algorithm in all conditions of Analysis Method 2. These figures indicate how the movement direction $\phi $ and distance d of a source influence the error. The ${D}_{s}=0.01$ training and optimisation set and (x + y) sensor configuration were used. The values for $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ are based on the ${D}_{s}=0.01$ condition in Analysis Method 1. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the ${D}_{s}=0.01$ condition of Analysis Method 1. The median movement direction error was computed in $2\times 2$ ${\mathrm{cm}}^{2}$ cells. The sensors were equidistantly placed between $x=\pm 0.2$ $\mathrm{m}$.

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**Figure 1.**Normalised continuous velocity profiles for five movement directions (indicated) of a source at d = 1 m from the sensor array’s centre: (

**a**) v

_{x}and (

**b**) v

_{y}. The sensors are located along the x axis and b is the source sphere is x position in m.

**Figure 2.**A schematic view of the simulated environment. The source sphere (green) has a radius of 1 $\mathrm{c}$$\mathrm{m}$ and is shown to scale. A possible movement direction is shown by the arrow (not in scale). The sensor locations are shown in blue. Parallel ${v}_{x}$ and perpendicular ${v}_{y}$ velocity components are indicated at the right-most sensor (not to scale). The area in which the source sphere is positioned is offset by 25 mm from the array location, ensuring a minimal distance of 15 mm between the source’s edge and closest sensor’s centre.

**Figure 3.**The signal to noise ratio (SNR) of both velocity components measured by the fifth sensor ($x=2.86$ $\mathrm{c}\mathrm{m}$). The top row shows contours of the median SNR in cells of $2\times 2$ ${\mathrm{cm}}^{2}$. The bottom row shows the median SNR’s polar contours in cells of $0.02\pi $ $\mathrm{rad}$ × 2 cm for source states with an x-coordinate between $x=-7.14$ $\mathrm{c}\mathrm{m}$ and $x=12.86$ $\mathrm{c}\mathrm{m}$, indicating how the movement direction of a dipole $\phi $ influences the SNR. Simulated potential flow measurements (Equation (1)) and the same measurements with additive Gaussian distributed noise values ($\sigma =1.5\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$, $\mu =0$ $\mathrm{m}/\mathrm{s}$) were used to compute the SNR. Specifically, the SNR was computed as the frequency power ratio between the noisy measurements and noise floor at the source frequency ($f=45$ $\mathrm{Hz}$). The frequency power was computed by a discrete Fourier transform (DFT) using a Hamming window.

**Figure 4.**Graphical illustration of the movement direction estimation from the measured velocity and the source’s position

**p**= 〈b, d〉 (

**a**) A view of ψ

_{e}(ρ) and ψ

_{o}(ρ) along the sensor array. (

**b**) The values of the wavelets can be interpreted as vectors (${\overrightarrow{\psi}}_{e}$ and ${\overrightarrow{\psi}}_{o}$) in a 3D ψ

_{e}–ψ

_{e}–ρ space. Their vector combination ${\overrightarrow{\psi}}_{env}$ = ${\overrightarrow{\psi}}_{e}$ + ${\overrightarrow{\psi}}_{o}$ is a fixed 3D wavelet structure that can be constructed solely from the source’s previously determined position

**p**. This vector ${\overrightarrow{\psi}}_{env}$ has a magnitude ${\psi}_{env}$ = $\sqrt{{\Psi}_{e}^{2}+{\Psi}_{o}^{2}}$ and angle ψ

^{’}= atan ψ

_{o}/ψ

_{e}. The measured velocities—which are linear combinations of ψ

_{e}and ψ

_{o}—can be viewed as a 2D projection of this 3D wavelet. For instance, a projection on the ρ–ψ

_{e}plane (bottom plane) yields ψ

_{e}for φ = 0 rad. For a general angle φ, the measured velocity profile is a projection on a plane through the ρ axis subtending an angle φ with the ρ–φ

_{e}plane. (

**c**) Diagram illustrating the geometric relation between a measured v

_{x}, the angles α and φ

^{’}which are constrained via Ψ

_{env}, and the movement orientation φ. We show a slice of ${\overrightarrow{\psi}}_{env}$ (green) in the ψ

_{e}–ψ

_{o}plane for a fixed value of ρ. The velocity value at this fixed ρ is a vector ${\overrightarrow{v}}_{x}$ (black) in this space. It has a length v

_{x}∝ ψ

_{e}cos φ + ψ

_{o}sin φ and has angle φ. The contributions of ${\overrightarrow{\psi}}_{e}$ (blue) and ${\overrightarrow{\psi}}_{o}$ (orange) to v

_{x}are shown in yellow and purple. The angle φ

^{’}of ~ ${\overrightarrow{\psi}}_{env}$ can be computed directly from an estimated source position. Given that the difference between φ and φ

^{’}is α = acos v

_{x}/ψ

_{env}, we can compute an estimate of φ at every sensor using only measured velocity values and a position estimate.

**Figure 5.**Total areas with a median position error ${E}_{\mathit{p}}$ (Equation (4)) below 1 $\mathrm{c}$$\mathrm{m}$, 3 $\mathrm{c}$$\mathrm{m}$, 5 $\mathrm{c}$$\mathrm{m}$, and 9 $\mathrm{c}$$\mathrm{m}$ for the training and optimisation sets with a varying minimum distance between source states ${D}_{s}$ (see Section 2.3 and Table 1) and the (x + y) sensor configuration at the $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The median position error was computed for $2\times 2$ $\mathrm{c}{\mathrm{m}}^{2}$ cells. Note, the bar for LSQ* is based the (x + y) condition in Analysis Method 2.

**Figure 6.**Total areas with a median movement direction error ${E}_{\phi}$ (Equation (5)) below $0.01\pi $ $\mathrm{rad}$, $0.03\pi $ $\mathrm{rad}$, $0.05\pi $ $\mathrm{rad}$ for the training and optimisation sets with a varying minimum distance between source states ${D}_{s}$ (see Section 2.3 and Table 1) and the (x + y) sensor configuration at the $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The median movement direction error was computed for $2\times 2$ $\mathrm{c}{\mathrm{m}}^{2}$ cells. Note, the bar for LSQ* is based on the (x + y) condition in Analysis Method 2.

**Figure 7.**Boxplots of the position error distributions for all dipole localisation algorithms in each condition of first analysis method. This analysis varied the minimum distance ${D}_{s}$ between source states in the training and optimisation set (see Section 2.3 and Table 1) and used the (x + y) sensor configuration at the $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The whiskers of the boxplots indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. LSQ* is based on the (x + y) condition in Analysis Method 2.

**Figure 8.**Boxplots of the movement direction error distributions for all dipole localisation algorithms in each condition of first analysis method. This analysis varied the minimum distance ${D}_{s}$ between source states in the training and optimisation set (see Section 2.3 and Table 1) and used the (x + y) sensor configuration at the $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The whiskers of the boxplots indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. LSQ* is based on the (x + y) condition in Analysis Method 2.

**Figure 9.**Total area with a median position error ${E}_{\mathit{p}}$ (Equation (4)) below 1 $\mathrm{c}$$\mathrm{m}$, 3 $\mathrm{c}$$\mathrm{m}$, 5 $\mathrm{c}$$\mathrm{m}$, and 9 $\mathrm{c}$$\mathrm{m}$ for varying sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring ${v}_{x}$ and ${v}_{y}$ for subsequent sensors, (x) measured only ${v}_{x}$ at all sensors, (y) measured only ${v}_{y}$ at all sensors. This analysis method used the ${D}_{s}=0.01$ training and optimisation set and the $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The median position error was computed in $2\times 2$ $\mathrm{c}{\mathrm{m}}^{2}$ cells. Note, the bar for QM* is based on the ${D}_{s}=0.01$ condition in Analysis Method 1.

**Figure 10.**Total area with a median movement direction error ${E}_{\mathit{p}}$ (Equation (4)) below 1 $\mathrm{c}$$\mathrm{m}$, 3 $\mathrm{c}$$\mathrm{m}$, 5 $\mathrm{c}$$\mathrm{m}$, and 9 $\mathrm{c}$$\mathrm{m}$ for varying sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring ${v}_{x}$ and ${v}_{y}$ for subsequent sensors, (x) measured only ${v}_{x}$ at all sensors, (y) measured only ${v}_{y}$ at all sensors. This analysis method used the ${D}_{s}=0.01$ training and optimisation set and the $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The median movement direction error was computed in $2\times 2$ $\mathrm{c}{\mathrm{m}}^{2}$ cells. Note, the bar for QM* is based on the ${D}_{s}=0.01$ condition in Analysis Method 1.

**Figure 11.**Boxplots of the position error distributions for all algorithms in the second analysis method. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring ${v}_{x}$ and ${v}_{y}$ for subsequent sensors, (x) measured only ${v}_{x}$ at all sensors, (y) measured only ${v}_{y}$ at all sensors. The ${D}_{s}=0.01$ training and optimisation set and $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level were used. The whiskers of the boxplots indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. QM* is based on the ${D}_{s}=0.01$ condition in Analysis Method 1.

**Figure 12.**Boxplots of the movement direction error distributions for all algorithms in the second analysis method. This analysis varied the sensitivity axes of the sensors: (x + y) measured both velocity components at all sensors, (x|y) alternated measuring ${v}_{x}$ and ${v}_{y}$ for subsequent sensors, (x) measured only ${v}_{x}$ at all sensors, (y) measured only ${v}_{y}$ at all sensors. The ${D}_{s}=0.01$ training and optimisation set and $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level were used. The whiskers of the boxplots indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. QM* is based on the ${D}_{s}=0.01$ condition in Analysis Method 1.

**Figure 13.**Spatial contours of the median position error ${E}_{\mathit{p}}$ (blue) (Equation (4)) and median movement direction error ${E}_{\phi}$ (orange) (Equation (5)) of the predictors using the largest training and optimisation set (${D}_{s}=0.01$) and 2D sensitive sensors (x + y) at the $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The algorithms are ordered with an increasing overall median position error. The median errors were computed in $2\times 2$ $\mathrm{c}{\mathrm{m}}^{2}$ cells.

**Figure 14.**These figures indicate how the movement direction $\phi $ and distance d of a source state influence the median position error ${E}_{p}$ (blue) (Equation (4)) and median movement direction error ${E}_{\phi}$ (orange) (Equation (5)). The quadrature method (QM), Gauss–Newton (GN), and least square curve fit (LSQ) predictors were used with three sensor configurations: (x + y), (x), (y) at the $\sigma =1.0\times {10}^{-5}$ $\mathrm{m}/\mathrm{s}$ noise level. The median errors were computed in cells of $0.01\pi $ $\mathrm{rad}$× 1 cm.

**Figure 15.**An overview of the position error E

_{p}(Equation (4)) of QM, GN, and MLP using simulated sensors with higher velocity equivalent noise levels. (

**a**) Total areas with a median position error E

_{p}below 1 cm, 3 cm, 5 cm, and 9 cm. (

**b**) Boxplots of the position error distributions, whiskers indicate the 5th and 95th percentiles of the distributions. Predictions with errors outside these percentiles are shown individually. The values for σ = 1.0 × 10

^{‒5}m s

^{‒1}are based on the D

_{s}= 0.01 condition in Analysis Method 1. The (x + y) sensor configuration was used. The MLP was re-trained for each noise level. Both the MLP and GN used the optimal hyperparameter values from the D

_{s}= 0.01 condition of Analysis Method 1.

**Table 1.**An overview of the data sets used in this study. The minimum distance between samples ${D}_{s}$ controls the number of source states. A source state is specified by a position $\mathit{p}$ and orientation $\phi $. The distance between two states was computed as the Euclidean distance in the combined x–y–$\phi /2\pi $ space containing all possible combinations of source positions and movement directions. The orientation dimension was divided by $2\pi $ to balance the number of positions and orientations. The testing data set has a different number of source states than the training set with ${D}_{s}=0.01$, due to the randomness of Poisson Disc sampling [37]. The average distance to the closest neighbour within each data set is indicated for both the position and orientation to support the interpretation of ${D}_{s}$.

Type | Min. Sample Distance (${\mathit{D}}_{\mathit{s}}$) | Num. States | Avg. Min ${\mathit{D}}_{\mathit{s},\mathit{p}}$(m) | Avg. Min $|2\mathit{\pi}{\mathit{D}}_{\mathit{s},\mathit{\phi}}|$ ($\mathit{rad}$) |
---|---|---|---|---|

training | 0.09 | 169 | $2.76\times {10}^{-2}$ | $1.81\times {10}^{-2}$ |

training | 0.05 | 874 | $1.21\times {10}^{-2}$ | $3.55\times {10}^{-3}$ |

training | 0.03 | 3796 | $5.72\times {10}^{-3}$ | $8.28\times {10}^{-4}$ |

training | 0.01 | 90,435 | ||

testing | 0.01 | 90,502 |

**Table 2.**Properties of the dipole localisation algorithms. The ‘Limited to domain’ column indicates whether the algorithm can produce predictions outside the simulated domain (see Figure 2). The ‘Limited to sample’ column indicates whether the algorithm is able to produce a prediction that is not present in the training set.

Algorithm | Type | Training | Hyperparameters | Limited to Domain | Limited to Sample |
---|---|---|---|---|---|

RND | — | no | no | yes | no |

LCMV [12,13] | template-based | yes | no | yes | yes |

KNN | template-based | yes | yes | yes | no |

CWT [5] | template-based | yes | yes | yes | no |

ELM [9,10] | neural network | yes | no | no | no |

MLP [7,9] | neural network | yes | no | no | no |

GN [8] | model-based | no | yes | yes | no |

NR [8] | model-based | no | yes | yes | no |

LSQ | model-based | no | yes | yes | no |

QM | model-based | no | yes | yes | no |

**Table 3.**Training and prediction time measurements of all dipole localisation algorithms. The (x + y) sensor configuration was used combined with the largest training and optimisation set ${D}_{s}=0.01$.

Algorithm | Avg. Prediction Time | Relative to MLP | Total Training Time |
---|---|---|---|

RND | $3.2\times {10}^{-4}$$\mathrm{s}$ | 0.9 | |

MLP | $3.6\times {10}^{-4}$$\mathrm{s}$ | 1.0 | 12 $\mathrm{min}$ 0 $\mathrm{s}$ |

ELM | $4.3\times {10}^{-4}$$\mathrm{s}$ | 1.2 | 1 $\mathrm{min}$ 52 $\mathrm{s}$ |

KNN | $9.7\times {10}^{-4}$$\mathrm{s}$ | 2.7 | |

GN | $1.4\times {10}^{-3}$$\mathrm{s}$ | 3.9 | |

LSQ | $2.8\times {10}^{-3}$$\mathrm{s}$ | 7.8 | |

QM | $3.4\times {10}^{-3}$$\mathrm{s}$ | 9.6 | |

LCMV | $1.3\times {10}^{-2}$$\mathrm{s}$ | 37.1 | |

NR | $6.3\times {10}^{-2}$$\mathrm{s}$ | 176.3 | |

CWT | $1.1\times {10}^{-1}$$\mathrm{s}$ | 311.1 |

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**MDPI and ACS Style**

Bot, D.M.; Wolf, B.J.; van Netten, S.M.
The Quadrature Method: A Novel Dipole Localisation Algorithm for Artificial Lateral Lines Compared to State of the Art. *Sensors* **2021**, *21*, 4558.
https://doi.org/10.3390/s21134558

**AMA Style**

Bot DM, Wolf BJ, van Netten SM.
The Quadrature Method: A Novel Dipole Localisation Algorithm for Artificial Lateral Lines Compared to State of the Art. *Sensors*. 2021; 21(13):4558.
https://doi.org/10.3390/s21134558

**Chicago/Turabian Style**

Bot, Daniël M., Ben J. Wolf, and Sietse M. van Netten.
2021. "The Quadrature Method: A Novel Dipole Localisation Algorithm for Artificial Lateral Lines Compared to State of the Art" *Sensors* 21, no. 13: 4558.
https://doi.org/10.3390/s21134558