# Gap Reconstruction in Optical Motion Capture Sequences Using Neural Networks

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Optical Motion Capture Pipeline

#### 2.2. Functional Body Mesh

#### 2.3. Previous Works

## 3. Materials and Methods

#### 3.1. Proposed Regression Approach

#### 3.1.1. Feed Forward Neural Network

#### 3.1.2. Recurrent Neural Networks

#### 3.1.3. Employed Reconstruction Methods

- FFNN${}_{\mathrm{lin}}$, with 1 hidden fully connected (FC) layer—containing 8 linear neurons;
- FFNN${}_{\mathrm{tanh}}$, with 1 hidden FC layer—containing 8 sigmoidal neurons;
- LSTM followed by 1 FC layer containing 8 sigmoidal neurons;
- GRU followed by 1 FC layer containing 8 sigmoidal neurons;
- BILSTM followed by 1 FC layer containing 8 sigmoidal neurons.

#### 3.1.4. Implementation Details

- Initial Learn Rate: 0.01;
- Learn Rate Drop Factor: 0.9;
- Learn Rate Drop Period: 10;
- Gradient Threshold 0.7;
- Momentum: 0.8.

#### 3.2. Input Data Preparation

#### 3.3. Test Dataset

#### 3.4. Quality Evaluation

#### 3.5. Experimental Protocol

- We introduce two gaps of assumed length (on average) to the random markers at random moments; actual values are stored as testing data;
- The model is trained using the remaining part of the sequence (all but gaps);
- We reconstruct (predict) the gaps using the pool of methods;
- The resulting values are stored for evaluation.

#### Gap Generation Procedure

## 4. Results and Discussion

#### 4.1. Gap Reconstruction Efficiency

- It can be seen that, for the short gaps, interpolation methods outperform any of the NN-based methods.
- For gaps that are 50 samples long, the results become less obvious and NN results are no worse or (usually) better than interpolation methods.
- Linear FFNN usually performed better than any other methods (including non-linear FFNN${}_{\mathrm{tanh}}$), for gaps of 50 samples or longer, for most of the sequences.
- In very rare cases of short-gap cases, RNNs performed better than FFNN${}_{\mathrm{lin}}$, but, in general, simpler FFNN${}_{\mathrm{lin}}$ outperformed more complex NN models.
- There are two situations when the FFNN${}_{\mathrm{lin}}$, performed no better or worse than interpolation methods (walking and falling). This occurred for sequences with larger monotonicity values in Table 2. They have also increased velocity/acceleration/jerk values; the ‘running’ sequence has similar values for these, but FFNN${}_{\mathrm{lin}}$ perform the best in this case, so the kinematic/dynamic parameters should not be considered.

#### 4.2. Motion Factors Affecting Performance

## 5. Summary

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BILSTM | bidirectional LSTM |

CC | correlation coefficient |

FC | fully connected |

FBM | functional body mesh |

FFNN | feed forward neural network |

GRU | gated recurrent unit |

HML | Human Motion Laboratory |

IK | inverse kinematics |

KF | Kalman filter |

LS | least squares |

LSTM | long-short term memory |

Mocap | MOtion CAPture |

MSE | Mean Square Error |

NARX-NN | nonlinear autoregressive exogenous neural network |

NaN | not a number |

NN | neural network |

OMC | optical motion capture |

PCA | principal component analysis |

PJAIT | Polish-Japanese Academy of Information Technology |

RMSE | root mean squared error |

RNN | recurrent neural network |

STDDEV | standard deviation |

SVD | singular value decomposition |

## Appendix A. Performance Results for All Sequences

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD | |
---|---|---|---|---|---|---|---|---|---|---|---|

10 | RMSE | 14.222 | 26.428 | 8.844 | 9.932 | 7.004 | 5.088 | 1.287 | 2.464 | 2.507 | 5.088 |

mean (${\mathrm{RMSE}}_{k}$) | 12.398 | 23.213 | 7.659 | 9.014 | 6.495 | 3.442 | 0.810 | 1.621 | 1.697 | 3.442 | |

median (${\mathrm{RMSE}}_{k}$) | 10.865 | 21.290 | 6.327 | 8.262 | 5.956 | 2.051 | 0.511 | 1.087 | 1.180 | 2.051 | |

mode (${\mathrm{RMSE}}_{k}$) | 3.499 | 4.068 | 1.755 | 1.140 | 2.344 | 0.536 | 0.056 | 0.237 | 0.239 | 0.536 | |

stddev (${\mathrm{RMSE}}_{k}$) | 6.930 | 12.645 | 4.634 | 4.744 | 3.371 | 3.505 | 0.938 | 1.773 | 1.788 | 3.505 | |

iqr (${\mathrm{RMSE}}_{k}$) | 8.986 | 12.914 | 3.644 | 4.297 | 3.444 | 3.180 | 0.652 | 1.293 | 1.334 | 3.180 | |

20 | RMSE | 15.490 | 32.802 | 13.491 | 13.382 | 13.303 | 12.274 | 4.031 | 6.590 | 6.619 | 12.274 |

mean (${\mathrm{RMSE}}_{k}$) | 13.743 | 27.978 | 10.155 | 11.171 | 8.396 | 9.071 | 2.591 | 4.798 | 4.904 | 9.071 | |

median (${\mathrm{RMSE}}_{k}$) | 12.334 | 24.575 | 7.568 | 9.116 | 6.209 | 6.508 | 1.823 | 3.767 | 3.728 | 6.508 | |

mode (${\mathrm{RMSE}}_{k}$) | 2.654 | 5.774 | 3.242 | 5.247 | 2.352 | 0.401 | 0.314 | 0.316 | 0.382 | 0.401 | |

stddev (${\mathrm{RMSE}}_{k}$) | 6.723 | 16.042 | 8.161 | 6.609 | 8.827 | 8.020 | 2.828 | 4.290 | 4.175 | 8.020 | |

iqr (${\mathrm{RMSE}}_{k}$) | 7.454 | 15.726 | 4.545 | 5.667 | 2.491 | 6.791 | 1.571 | 3.308 | 3.921 | 6.791 | |

50 | RMSE | 21.907 | 40.375 | 24.343 | 23.833 | 23.434 | 42.517 | 21.474 | 26.332 | 25.995 | 42.517 |

mean (${\mathrm{RMSE}}_{k}$) | 19.168 | 36.769 | 19.788 | 19.867 | 18.831 | 33.944 | 16.673 | 21.757 | 21.607 | 33.944 | |

median (${\mathrm{RMSE}}_{k}$) | 16.432 | 32.752 | 15.196 | 15.655 | 14.926 | 23.652 | 12.952 | 16.134 | 15.996 | 23.652 | |

mode (${\mathrm{RMSE}}_{k}$) | 5.905 | 13.574 | 6.336 | 7.173 | 6.100 | 5.500 | 4.293 | 3.782 | 3.921 | 5.500 | |

stddev (${\mathrm{RMSE}}_{k}$) | 10.486 | 16.289 | 13.174 | 12.408 | 13.037 | 25.484 | 12.659 | 14.438 | 13.993 | 25.484 | |

iqr (${\mathrm{RMSE}}_{k}$) | 12.421 | 22.207 | 13.308 | 11.413 | 12.903 | 29.918 | 12.189 | 17.991 | 18.129 | 29.918 | |

100 | RMSE | 39.346 | 75.817 | 61.641 | 60.420 | 60.823 | 76.058 | 58.357 | 62.302 | 62.419 | 76.058 |

mean (${\mathrm{RMSE}}_{k}$) | 32.287 | 66.701 | 50.195 | 49.019 | 49.453 | 63.445 | 46.476 | 50.803 | 50.693 | 63.445 | |

median (${\mathrm{RMSE}}_{k}$) | 23.318 | 56.329 | 38.960 | 37.001 | 39.074 | 51.683 | 35.447 | 40.065 | 40.418 | 51.683 | |

mode (${\mathrm{RMSE}}_{k}$) | 8.122 | 22.940 | 14.125 | 15.094 | 14.334 | 12.943 | 12.407 | 12.074 | 12.493 | 12.943 | |

stddev (${\mathrm{RMSE}}_{k}$) | 22.397 | 35.709 | 35.107 | 34.707 | 34.685 | 41.564 | 34.503 | 35.371 | 35.700 | 41.564 | |

iqr (${\mathrm{RMSE}}_{k}$) | 18.933 | 41.446 | 39.727 | 40.427 | 40.813 | 63.062 | 39.062 | 49.784 | 50.440 | 63.062 | |

200 | RMSE | 112.933 | 134.121 | 127.416 | 132.150 | 124.566 | 79.741 | 105.237 | 79.407 | 80.457 | 79.741 |

mean (${\mathrm{RMSE}}_{k}$) | 87.084 | 121.229 | 108.733 | 111.164 | 107.192 | 75.307 | 91.826 | 69.585 | 70.031 | 75.307 | |

median (${\mathrm{RMSE}}_{k}$) | 59.288 | 104.710 | 91.987 | 89.523 | 91.019 | 68.567 | 80.427 | 63.559 | 61.704 | 68.567 | |

mode (${\mathrm{RMSE}}_{k}$) | 26.007 | 46.150 | 23.032 | 23.675 | 22.813 | 42.408 | 21.841 | 21.984 | 21.602 | 42.408 | |

stddev (${\mathrm{RMSE}}_{k}$) | 71.160 | 57.197 | 66.944 | 71.470 | 63.693 | 26.502 | 53.401 | 39.296 | 40.746 | 26.502 | |

iqr (${\mathrm{RMSE}}_{k}$) | 61.864 | 71.116 | 90.839 | 90.285 | 90.685 | 42.057 | 88.500 | 65.862 | 66.873 | 42.057 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD | |
---|---|---|---|---|---|---|---|---|---|---|---|

10 | RMSE | 11.702 | 25.988 | 8.748 | 8.666 | 7.066 | 3.001 | 0.701 | 1.291 | 1.259 | 3.001 |

mean (${\mathrm{RMSE}}_{k}$) | 9.939 | 23.049 | 7.675 | 7.581 | 6.105 | 2.221 | 0.476 | 0.985 | 0.942 | 2.221 | |

median (${\mathrm{RMSE}}_{k}$) | 8.661 | 20.122 | 6.973 | 6.485 | 5.540 | 1.743 | 0.346 | 0.831 | 0.720 | 1.743 | |

mode (${\mathrm{RMSE}}_{k}$) | 1.933 | 6.022 | 1.838 | 1.236 | 1.106 | 0.234 | 0.079 | 0.149 | 0.151 | 0.234 | |

stddev (${\mathrm{RMSE}}_{k}$) | 5.919 | 11.837 | 4.214 | 4.245 | 3.797 | 1.714 | 0.439 | 0.692 | 0.691 | 1.714 | |

iqr (${\mathrm{RMSE}}_{k}$) | 7.005 | 15.692 | 5.106 | 4.850 | 3.513 | 1.835 | 0.286 | 0.835 | 0.799 | 1.835 | |

20 | RMSE | 12.141 | 27.729 | 11.594 | 11.232 | 9.321 | 7.397 | 1.742 | 3.401 | 3.439 | 7.397 |

mean (${\mathrm{RMSE}}_{k}$) | 10.331 | 25.124 | 9.324 | 9.440 | 6.919 | 5.676 | 1.274 | 2.601 | 2.589 | 5.676 | |

median (${\mathrm{RMSE}}_{k}$) | 8.695 | 23.641 | 7.664 | 7.948 | 5.424 | 4.496 | 0.968 | 1.988 | 1.853 | 4.496 | |

mode (${\mathrm{RMSE}}_{k}$) | 2.547 | 6.946 | 2.438 | 1.512 | 1.953 | 0.661 | 0.237 | 0.453 | 0.438 | 0.661 | |

stddev (${\mathrm{RMSE}}_{k}$) | 6.215 | 11.425 | 6.552 | 5.753 | 5.787 | 4.017 | 1.010 | 1.889 | 2.021 | 4.017 | |

iqr (${\mathrm{RMSE}}_{k}$) | 8.168 | 12.490 | 4.481 | 5.442 | 3.111 | 3.995 | 1.017 | 2.154 | 2.061 | 3.995 | |

50 | RMSE | 23.573 | 39.084 | 31.147 | 24.057 | 23.597 | 34.144 | 12.857 | 19.473 | 21.328 | 34.144 |

mean (${\mathrm{RMSE}}_{k}$) | 14.767 | 31.801 | 17.835 | 15.504 | 14.637 | 27.624 | 8.608 | 14.842 | 16.431 | 27.624 | |

median (${\mathrm{RMSE}}_{k}$) | 9.523 | 25.412 | 10.904 | 10.501 | 8.853 | 25.122 | 6.834 | 12.894 | 13.844 | 25.122 | |

mode (${\mathrm{RMSE}}_{k}$) | 3.229 | 9.379 | 4.119 | 2.888 | 3.306 | 2.559 | 0.896 | 1.291 | 1.737 | 2.559 | |

stddev (${\mathrm{RMSE}}_{k}$) | 18.345 | 22.596 | 25.456 | 18.049 | 18.231 | 18.865 | 8.914 | 11.837 | 12.760 | 18.865 | |

iqr (${\mathrm{RMSE}}_{k}$) | 6.432 | 16.838 | 6.719 | 7.811 | 7.903 | 20.224 | 6.920 | 9.883 | 11.590 | 20.224 | |

100 | RMSE | 38.173 | 61.656 | 68.606 | 54.639 | 58.223 | 94.347 | 45.740 | 58.606 | 62.724 | 94.347 |

mean (${\mathrm{RMSE}}_{k}$) | 25.165 | 49.288 | 44.780 | 40.344 | 42.251 | 83.854 | 37.303 | 51.072 | 55.958 | 83.854 | |

median (${\mathrm{RMSE}}_{k}$) | 18.493 | 41.944 | 33.811 | 31.168 | 32.177 | 77.220 | 32.103 | 46.438 | 50.903 | 77.220 | |

mode (${\mathrm{RMSE}}_{k}$) | 4.901 | 11.780 | 8.178 | 5.555 | 4.181 | 4.989 | 4.549 | 3.554 | 3.884 | 4.989 | |

stddev (${\mathrm{RMSE}}_{k}$) | 27.594 | 35.231 | 50.041 | 35.158 | 38.271 | 41.350 | 25.286 | 27.575 | 27.272 | 41.350 | |

iqr (${\mathrm{RMSE}}_{k}$) | 13.060 | 29.863 | 24.844 | 24.922 | 25.449 | 47.512 | 25.816 | 26.432 | 29.725 | 47.512 | |

200 | RMSE | 110.196 | 145.641 | 145.387 | 143.360 | 145.050 | 248.552 | 138.231 | 167.249 | 199.417 | 248.552 |

mean (${\mathrm{RMSE}}_{k}$) | 88.708 | 129.262 | 125.767 | 123.634 | 125.213 | 235.787 | 119.848 | 146.780 | 185.085 | 235.787 | |

median (${\mathrm{RMSE}}_{k}$) | 70.845 | 113.902 | 108.387 | 105.181 | 107.987 | 233.618 | 103.952 | 128.657 | 171.109 | 233.618 | |

mode (${\mathrm{RMSE}}_{k}$) | 20.092 | 53.434 | 39.113 | 39.722 | 38.728 | 96.336 | 38.444 | 36.027 | 74.145 | 96.336 | |

stddev (${\mathrm{RMSE}}_{k}$) | 63.969 | 65.135 | 70.990 | 70.695 | 71.285 | 73.293 | 66.963 | 77.021 | 70.628 | 73.293 | |

iqr (${\mathrm{RMSE}}_{k}$) | 67.200 | 73.343 | 87.747 | 82.080 | 89.947 | 77.986 | 83.010 | 64.869 | 47.085 | 77.986 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD | |
---|---|---|---|---|---|---|---|---|---|---|---|

10 | RMSE | 3.701 | 3.792 | 1.664 | 1.954 | 1.373 | 1.697 | 0.711 | 0.841 | 0.839 | 1.697 |

mean (${\mathrm{RMSE}}_{k}$) | 3.272 | 3.386 | 1.463 | 1.737 | 1.210 | 1.218 | 0.478 | 0.617 | 0.606 | 1.218 | |

median (${\mathrm{RMSE}}_{k}$) | 2.996 | 2.987 | 1.351 | 1.682 | 1.108 | 0.948 | 0.339 | 0.475 | 0.429 | 0.948 | |

mode (${\mathrm{RMSE}}_{k}$) | 0.437 | 0.558 | 0.197 | 0.212 | 0.249 | 0.072 | 0.059 | 0.041 | 0.043 | 0.072 | |

stddev (${\mathrm{RMSE}}_{k}$) | 1.896 | 1.767 | 0.806 | 0.846 | 0.642 | 1.094 | 0.483 | 0.530 | 0.537 | 1.094 | |

iqr (${\mathrm{RMSE}}_{k}$) | 2.282 | 2.025 | 0.991 | 1.301 | 0.702 | 1.049 | 0.260 | 0.467 | 0.480 | 1.049 | |

20 | RMSE | 3.464 | 3.829 | 2.060 | 2.025 | 1.688 | 3.902 | 1.285 | 1.904 | 2.029 | 3.902 |

mean (${\mathrm{RMSE}}_{k}$) | 3.106 | 3.429 | 1.708 | 1.797 | 1.475 | 3.057 | 0.942 | 1.515 | 1.559 | 3.057 | |

median (${\mathrm{RMSE}}_{k}$) | 2.911 | 3.319 | 1.519 | 1.572 | 1.318 | 2.434 | 0.739 | 1.230 | 1.169 | 2.434 | |

mode (${\mathrm{RMSE}}_{k}$) | 0.522 | 0.497 | 0.300 | 0.240 | 0.271 | 0.211 | 0.126 | 0.155 | 0.161 | 0.211 | |

stddev (${\mathrm{RMSE}}_{k}$) | 1.577 | 1.750 | 1.122 | 0.962 | 0.812 | 2.415 | 0.838 | 1.153 | 1.311 | 2.415 | |

iqr (${\mathrm{RMSE}}_{k}$) | 2.233 | 2.263 | 1.038 | 1.069 | 0.934 | 2.762 | 0.781 | 0.979 | 0.995 | 2.762 | |

20 | RMSE | 4.901 | 6.291 | 6.392 | 5.952 | 6.255 | 15.596 | 6.334 | 9.332 | 10.056 | 15.596 |

mean (${\mathrm{RMSE}}_{k}$) | 4.383 | 5.355 | 5.064 | 4.697 | 4.895 | 12.767 | 4.902 | 7.260 | 7.710 | 12.767 | |

median (${\mathrm{RMSE}}_{k}$) | 3.982 | 4.831 | 4.007 | 3.623 | 3.803 | 11.036 | 3.652 | 5.788 | 6.343 | 11.036 | |

mode (${\mathrm{RMSE}}_{k}$) | 0.482 | 0.417 | 0.313 | 0.422 | 0.277 | 0.267 | 0.332 | 0.267 | 0.240 | 0.267 | |

stddev (${\mathrm{RMSE}}_{k}$) | 2.276 | 3.254 | 3.793 | 3.568 | 3.778 | 8.741 | 3.880 | 5.667 | 6.265 | 8.741 | |

iqr (${\mathrm{RMSE}}_{k}$) | 2.978 | 3.833 | 5.160 | 4.098 | 4.999 | 11.116 | 5.269 | 6.546 | 6.801 | 11.116 | |

20 | RMSE | 15.716 | 21.780 | 23.727 | 23.023 | 23.575 | 38.083 | 23.547 | 28.358 | 28.813 | 38.083 |

mean (${\mathrm{RMSE}}_{k}$) | 11.904 | 16.468 | 18.440 | 17.539 | 18.222 | 33.439 | 18.245 | 23.435 | 24.033 | 33.439 | |

median (${\mathrm{RMSE}}_{k}$) | 8.596 | 13.132 | 15.903 | 14.109 | 15.147 | 30.517 | 15.467 | 20.365 | 20.691 | 30.517 | |

mode (${\mathrm{RMSE}}_{k}$) | 0.643 | 0.711 | 0.927 | 0.743 | 0.950 | 1.324 | 1.170 | 1.139 | 1.121 | 1.324 | |

stddev (${\mathrm{RMSE}}_{k}$) | 9.980 | 13.839 | 14.495 | 14.484 | 14.524 | 17.840 | 14.459 | 15.569 | 15.542 | 17.840 | |

iqr (${\mathrm{RMSE}}_{k}$) | 7.816 | 11.087 | 13.380 | 12.476 | 13.201 | 23.419 | 13.054 | 15.405 | 14.280 | 23.419 | |

20 | RMSE | 37.101 | 48.909 | 51.388 | 50.842 | 51.274 | 72.745 | 51.478 | 59.839 | 59.857 | 72.745 |

mean (${\mathrm{RMSE}}_{k}$) | 31.439 | 41.811 | 44.331 | 43.711 | 44.219 | 66.280 | 44.321 | 54.030 | 54.156 | 66.280 | |

median (${\mathrm{RMSE}}_{k}$) | 26.422 | 36.792 | 40.178 | 39.257 | 40.099 | 71.201 | 39.395 | 55.235 | 54.311 | 71.201 | |

mode (${\mathrm{RMSE}}_{k}$) | 1.783 | 2.342 | 2.592 | 2.372 | 2.558 | 0.972 | 2.819 | 0.875 | 0.912 | 0.972 | |

stddev (${\mathrm{RMSE}}_{k}$) | 20.198 | 25.924 | 26.514 | 26.496 | 26.480 | 30.443 | 26.659 | 26.183 | 26.001 | 30.443 | |

iqr (${\mathrm{RMSE}}_{k}$) | 22.947 | 30.188 | 29.510 | 29.617 | 29.241 | 37.209 | 29.316 | 29.572 | 28.094 | 37.209 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD | |
---|---|---|---|---|---|---|---|---|---|---|---|

10 | RMSE | 2.603 | 3.006 | 1.217 | 1.467 | 1.008 | 1.175 | 0.986 | 0.668 | 0.735 | 1.175 |

mean (${\mathrm{RMSE}}_{k}$) | 2.321 | 2.697 | 1.087 | 1.316 | 0.885 | 0.848 | 0.484 | 0.461 | 0.507 | 0.848 | |

median (${\mathrm{RMSE}}_{k}$) | 2.036 | 2.476 | 1.001 | 1.173 | 0.783 | 0.666 | 0.276 | 0.317 | 0.322 | 0.666 | |

mode (${\mathrm{RMSE}}_{k}$) | 0.505 | 0.309 | 0.270 | 0.303 | 0.218 | 0.036 | 0.043 | 0.035 | 0.034 | 0.036 | |

stddev (${\mathrm{RMSE}}_{k}$) | 1.174 | 1.354 | 0.521 | 0.613 | 0.456 | 0.712 | 0.765 | 0.420 | 0.473 | 0.712 | |

iqr (${\mathrm{RMSE}}_{k}$) | 1.449 | 1.769 | 0.504 | 0.705 | 0.542 | 0.709 | 0.307 | 0.341 | 0.490 | 0.709 | |

20 | RMSE | 2.581 | 3.298 | 1.446 | 1.591 | 1.200 | 3.534 | 1.157 | 1.648 | 2.021 | 3.534 |

mean (${\mathrm{RMSE}}_{k}$) | 2.295 | 3.030 | 1.341 | 1.458 | 1.070 | 2.818 | 0.797 | 1.309 | 1.519 | 2.818 | |

median (${\mathrm{RMSE}}_{k}$) | 2.022 | 2.780 | 1.242 | 1.353 | 0.934 | 2.282 | 0.608 | 0.983 | 1.071 | 2.282 | |

mode (${\mathrm{RMSE}}_{k}$) | 0.826 | 0.700 | 0.326 | 0.402 | 0.303 | 0.273 | 0.106 | 0.125 | 0.126 | 0.273 | |

stddev (${\mathrm{RMSE}}_{k}$) | 1.161 | 1.308 | 0.541 | 0.606 | 0.549 | 1.965 | 0.819 | 0.930 | 1.249 | 1.965 | |

iqr (${\mathrm{RMSE}}_{k}$) | 1.415 | 1.704 | 0.736 | 0.732 | 0.494 | 2.153 | 0.491 | 1.038 | 1.333 | 2.153 | |

50 | RMSE | 4.045 | 5.038 | 4.965 | 4.067 | 4.295 | 14.095 | 3.956 | 7.248 | 9.171 | 14.095 |

mean (${\mathrm{RMSE}}_{k}$) | 3.211 | 4.183 | 3.609 | 3.109 | 3.306 | 11.957 | 3.262 | 6.083 | 7.562 | 11.957 | |

median (${\mathrm{RMSE}}_{k}$) | 2.546 | 3.503 | 2.661 | 2.500 | 2.634 | 10.384 | 2.736 | 5.271 | 6.318 | 10.384 | |

mode (${\mathrm{RMSE}}_{k}$) | 0.699 | 1.102 | 0.699 | 0.538 | 0.480 | 0.444 | 0.542 | 0.513 | 0.546 | 0.444 | |

stddev (${\mathrm{RMSE}}_{k}$) | 2.460 | 2.788 | 3.404 | 2.614 | 2.747 | 7.236 | 2.235 | 3.802 | 4.994 | 7.236 | |

iqr (${\mathrm{RMSE}}_{k}$) | 1.743 | 1.595 | 1.968 | 1.540 | 2.062 | 9.821 | 2.059 | 5.074 | 7.302 | 9.821 | |

100 | RMSE | 10.134 | 16.216 | 21.424 | 19.275 | 21.386 | 36.436 | 21.538 | 27.723 | 30.374 | 36.436 |

mean (${\mathrm{RMSE}}_{k}$) | 8.175 | 13.241 | 17.438 | 15.384 | 17.357 | 31.336 | 17.608 | 23.779 | 26.421 | 31.336 | |

median (${\mathrm{RMSE}}_{k}$) | 6.398 | 11.337 | 14.702 | 12.285 | 14.627 | 27.834 | 14.823 | 22.008 | 24.825 | 27.834 | |

mode (${\mathrm{RMSE}}_{k}$) | 0.864 | 1.156 | 1.085 | 0.973 | 1.090 | 0.514 | 0.912 | 0.632 | 0.490 | 0.514 | |

stddev (${\mathrm{RMSE}}_{k}$) | 5.837 | 9.220 | 12.123 | 11.372 | 12.169 | 18.465 | 12.075 | 14.128 | 14.876 | 18.465 | |

iqr (${\mathrm{RMSE}}_{k}$) | 6.261 | 12.033 | 16.415 | 16.672 | 16.414 | 25.577 | 16.361 | 18.637 | 19.008 | 25.577 | |

200 | RMSE | 42.833 | 60.847 | 71.465 | 70.625 | 71.514 | 64.829 | 72.201 | 60.721 | 61.704 | 64.829 |

mean (${\mathrm{RMSE}}_{k}$) | 36.693 | 54.330 | 64.743 | 63.732 | 64.805 | 61.507 | 65.477 | 56.493 | 57.666 | 61.507 | |

median (${\mathrm{RMSE}}_{k}$) | 33.631 | 50.764 | 61.017 | 60.170 | 61.057 | 60.782 | 62.218 | 55.492 | 57.030 | 60.782 | |

mode (${\mathrm{RMSE}}_{k}$) | 4.592 | 9.116 | 10.042 | 9.788 | 9.974 | 8.998 | 10.077 | 8.616 | 8.609 | 8.998 | |

stddev (${\mathrm{RMSE}}_{k}$) | 21.768 | 26.954 | 29.620 | 29.819 | 29.609 | 20.171 | 29.798 | 22.097 | 21.740 | 20.171 | |

iqr (${\mathrm{RMSE}}_{k}$) | 21.992 | 31.408 | 36.039 | 35.205 | 36.075 | 24.945 | 36.485 | 29.814 | 28.505 | 24.945 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD | |
---|---|---|---|---|---|---|---|---|---|---|---|

10 | RMSE | 19.193 | 17.106 | 8.537 | 9.585 | 6.720 | 5.763 | 1.601 | 2.872 | 3.365 | 5.763 |

mean (${\mathrm{RMSE}}_{k}$) | 15.455 | 15.022 | 7.818 | 8.772 | 6.166 | 3.827 | 0.994 | 1.851 | 1.968 | 3.827 | |

median (${\mathrm{RMSE}}_{k}$) | 13.186 | 13.571 | 6.947 | 8.341 | 5.616 | 2.359 | 0.618 | 1.107 | 1.145 | 2.359 | |

mode (${\mathrm{RMSE}}_{k}$) | 2.760 | 3.139 | 2.310 | 2.880 | 2.110 | 0.244 | 0.105 | 0.145 | 0.149 | 0.244 | |

stddev (${\mathrm{RMSE}}_{k}$) | 11.270 | 8.163 | 3.494 | 3.852 | 2.555 | 4.023 | 1.138 | 2.039 | 2.551 | 4.023 | |

iqr (${\mathrm{RMSE}}_{k}$) | 9.203 | 10.174 | 3.101 | 4.009 | 3.520 | 3.723 | 0.789 | 1.795 | 1.813 | 3.723 | |

20 | RMSE | 18.496 | 17.762 | 10.664 | 11.914 | 9.057 | 15.278 | 6.073 | 9.213 | 9.596 | 15.278 |

mean (${\mathrm{RMSE}}_{k}$) | 16.206 | 16.199 | 8.940 | 10.261 | 7.897 | 10.937 | 3.694 | 5.981 | 6.392 | 10.937 | |

median (${\mathrm{RMSE}}_{k}$) | 14.108 | 14.897 | 8.130 | 9.530 | 7.106 | 7.613 | 2.089 | 3.687 | 4.319 | 7.613 | |

mode (${\mathrm{RMSE}}_{k}$) | 4.659 | 2.388 | 2.143 | 1.822 | 2.821 | 0.953 | 0.339 | 0.756 | 0.832 | 0.953 | |

stddev (${\mathrm{RMSE}}_{k}$) | 8.511 | 7.184 | 6.211 | 5.915 | 4.455 | 9.596 | 4.383 | 6.169 | 6.567 | 9.596 | |

iqr (${\mathrm{RMSE}}_{k}$) | 9.496 | 8.219 | 4.321 | 5.520 | 3.642 | 10.133 | 3.402 | 5.298 | 5.154 | 10.133 | |

50 | RMSE | 38.618 | 43.058 | 50.077 | 47.367 | 47.474 | 60.232 | 46.220 | 42.945 | 44.782 | 60.232 |

mean (${\mathrm{RMSE}}_{k}$) | 28.149 | 30.795 | 32.292 | 30.356 | 30.213 | 43.423 | 28.314 | 29.543 | 31.603 | 43.423 | |

median (${\mathrm{RMSE}}_{k}$) | 18.927 | 18.873 | 16.214 | 15.491 | 14.312 | 29.262 | 14.172 | 17.724 | 19.705 | 29.262 | |

mode (${\mathrm{RMSE}}_{k}$) | 5.585 | 3.883 | 3.061 | 4.112 | 2.789 | 4.507 | 1.345 | 2.710 | 2.615 | 4.507 | |

stddev (${\mathrm{RMSE}}_{k}$) | 25.916 | 29.395 | 37.233 | 35.345 | 35.587 | 42.053 | 35.660 | 31.333 | 31.914 | 42.053 | |

iqr (${\mathrm{RMSE}}_{k}$) | 15.417 | 17.126 | 31.871 | 21.094 | 29.043 | 38.828 | 27.009 | 24.239 | 28.168 | 38.828 | |

100 | RMSE | 70.671 | 89.650 | 100.005 | 95.523 | 100.282 | 125.495 | 98.770 | 92.878 | 97.573 | 125.495 |

mean (${\mathrm{RMSE}}_{k}$) | 55.641 | 72.172 | 81.983 | 76.503 | 81.814 | 104.667 | 81.794 | 76.620 | 81.261 | 104.667 | |

median (${\mathrm{RMSE}}_{k}$) | 42.728 | 57.532 | 66.468 | 62.277 | 66.311 | 86.119 | 68.990 | 59.710 | 66.757 | 86.119 | |

mode (${\mathrm{RMSE}}_{k}$) | 7.967 | 8.688 | 10.247 | 7.912 | 9.749 | 7.809 | 9.146 | 6.892 | 7.449 | 7.809 | |

stddev (${\mathrm{RMSE}}_{k}$) | 43.593 | 53.283 | 57.286 | 57.268 | 58.033 | 69.796 | 55.712 | 52.857 | 54.185 | 69.796 | |

iqr (${\mathrm{RMSE}}_{k}$) | 52.533 | 72.029 | 82.980 | 85.218 | 86.618 | 85.060 | 92.529 | 71.859 | 75.211 | 85.060 | |

200 | RMSE | 192.371 | 224.989 | 240.459 | 237.068 | 240.104 | 219.332 | 238.962 | 182.390 | 177.973 | 219.332 |

mean (${\mathrm{RMSE}}_{k}$) | 168.542 | 199.701 | 214.118 | 209.626 | 213.731 | 198.998 | 212.497 | 165.908 | 161.330 | 198.998 | |

median (${\mathrm{RMSE}}_{k}$) | 145.399 | 185.636 | 190.954 | 187.446 | 191.565 | 196.704 | 189.560 | 169.458 | 163.676 | 196.704 | |

mode (${\mathrm{RMSE}}_{k}$) | 43.924 | 47.226 | 58.636 | 49.156 | 60.128 | 38.386 | 60.706 | 33.396 | 32.207 | 38.386 | |

stddev (${\mathrm{RMSE}}_{k}$) | 92.157 | 103.406 | 108.703 | 110.129 | 108.684 | 94.898 | 108.542 | 77.915 | 77.491 | 94.898 | |

iqr (${\mathrm{RMSE}}_{k}$) | 102.432 | 114.007 | 119.186 | 116.515 | 119.440 | 153.708 | 117.857 | 121.289 | 120.480 | 153.708 |

## Appendix B. Correlations between RMSE an Sequence Parameters

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD |
---|---|---|---|---|---|---|---|---|---|---|

10 | 0.741 | 0.878 | 0.890 | 0.849 | 0.890 | 0.614 | 0.261 | 0.552 | 0.520 | 0.614 |

20 | 0.760 | 0.827 | 0.852 | 0.842 | 0.790 | 0.608 | 0.466 | 0.550 | 0.533 | 0.608 |

50 | 0.744 | 0.851 | 0.740 | 0.678 | 0.670 | 0.660 | 0.503 | 0.603 | 0.608 | 0.660 |

100 | 0.639 | 0.719 | 0.724 | 0.649 | 0.662 | 0.742 | 0.576 | 0.661 | 0.679 | 0.742 |

200 | 0.658 | 0.691 | 0.667 | 0.665 | 0.664 | 0.777 | 0.626 | 0.756 | 0.812 | 0.777 |

10 | 0.092 | 0.021 | 0.017 | 0.033 | 0.017 | 0.195 | 0.617 | 0.256 | 0.290 | 0.195 |

20 | 0.080 | 0.042 | 0.031 | 0.036 | 0.061 | 0.200 | 0.352 | 0.258 | 0.276 | 0.200 |

50 | 0.090 | 0.032 | 0.093 | 0.139 | 0.146 | 0.153 | 0.309 | 0.205 | 0.200 | 0.153 |

100 | 0.172 | 0.108 | 0.104 | 0.163 | 0.152 | 0.091 | 0.231 | 0.153 | 0.138 | 0.091 |

200 | 0.155 | 0.129 | 0.148 | 0.150 | 0.150 | 0.069 | 0.184 | 0.082 | 0.050 | 0.069 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD |
---|---|---|---|---|---|---|---|---|---|---|

10 | 0.775 | 0.997 | 0.950 | 0.928 | 0.956 | 0.729 | 0.419 | 0.668 | 0.586 | 0.729 |

20 | 0.823 | 0.986 | 0.969 | 0.943 | 0.948 | 0.688 | 0.505 | 0.595 | 0.556 | 0.688 |

50 | 0.736 | 0.924 | 0.703 | 0.661 | 0.645 | 0.718 | 0.479 | 0.627 | 0.614 | 0.718 |

100 | 0.673 | 0.833 | 0.755 | 0.708 | 0.698 | 0.747 | 0.611 | 0.718 | 0.707 | 0.747 |

200 | 0.696 | 0.719 | 0.649 | 0.667 | 0.641 | 0.648 | 0.570 | 0.659 | 0.694 | 0.648 |

10 | 0.070 | 0.000 | 0.004 | 0.007 | 0.003 | 0.100 | 0.408 | 0.147 | 0.222 | 0.100 |

20 | 0.044 | 0.000 | 0.001 | 0.005 | 0.004 | 0.131 | 0.307 | 0.213 | 0.252 | 0.131 |

50 | 0.095 | 0.008 | 0.119 | 0.153 | 0.167 | 0.108 | 0.336 | 0.183 | 0.195 | 0.108 |

100 | 0.143 | 0.040 | 0.083 | 0.115 | 0.123 | 0.088 | 0.198 | 0.108 | 0.116 | 0.088 |

200 | 0.125 | 0.107 | 0.163 | 0.148 | 0.170 | 0.164 | 0.237 | 0.155 | 0.126 | 0.164 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD |
---|---|---|---|---|---|---|---|---|---|---|

10 | 0.768 | 0.983 | 0.943 | 0.915 | 0.950 | 0.701 | 0.419 | 0.640 | 0.564 | 0.701 |

20 | 0.812 | 0.962 | 0.950 | 0.927 | 0.916 | 0.669 | 0.486 | 0.576 | 0.540 | 0.669 |

50 | 0.749 | 0.921 | 0.724 | 0.672 | 0.657 | 0.716 | 0.478 | 0.624 | 0.619 | 0.716 |

100 | 0.681 | 0.825 | 0.772 | 0.715 | 0.709 | 0.771 | 0.615 | 0.728 | 0.723 | 0.771 |

200 | 0.712 | 0.742 | 0.679 | 0.694 | 0.673 | 0.710 | 0.609 | 0.714 | 0.755 | 0.710 |

10 | 0.074 | 0.000 | 0.005 | 0.011 | 0.004 | 0.121 | 0.409 | 0.172 | 0.243 | 0.121 |

20 | 0.050 | 0.002 | 0.004 | 0.008 | 0.010 | 0.146 | 0.328 | 0.231 | 0.269 | 0.146 |

50 | 0.087 | 0.009 | 0.104 | 0.143 | 0.157 | 0.110 | 0.338 | 0.186 | 0.190 | 0.110 |

100 | 0.136 | 0.043 | 0.072 | 0.111 | 0.115 | 0.072 | 0.194 | 0.101 | 0.104 | 0.072 |

200 | 0.112 | 0.091 | 0.138 | 0.126 | 0.143 | 0.114 | 0.199 | 0.111 | 0.083 | 0.114 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD |
---|---|---|---|---|---|---|---|---|---|---|

10 | 0.901 | 0.867 | 0.917 | 0.928 | 0.922 | 0.879 | 0.775 | 0.853 | 0.806 | 0.879 |

20 | 0.923 | 0.853 | 0.916 | 0.932 | 0.894 | 0.870 | 0.754 | 0.806 | 0.779 | 0.870 |

50 | 0.896 | 0.952 | 0.870 | 0.858 | 0.846 | 0.909 | 0.740 | 0.845 | 0.844 | 0.909 |

100 | 0.886 | 0.960 | 0.928 | 0.916 | 0.907 | 0.914 | 0.858 | 0.926 | 0.916 | 0.914 |

200 | 0.918 | 0.929 | 0.884 | 0.901 | 0.879 | 0.699 | 0.830 | 0.789 | 0.745 | 0.699 |

10 | 0.014 | 0.025 | 0.010 | 0.008 | 0.009 | 0.021 | 0.070 | 0.031 | 0.053 | 0.021 |

20 | 0.009 | 0.031 | 0.010 | 0.007 | 0.016 | 0.024 | 0.083 | 0.053 | 0.068 | 0.024 |

50 | 0.016 | 0.003 | 0.024 | 0.029 | 0.034 | 0.012 | 0.093 | 0.034 | 0.034 | 0.012 |

100 | 0.019 | 0.002 | 0.008 | 0.010 | 0.012 | 0.011 | 0.029 | 0.008 | 0.010 | 0.011 |

200 | 0.010 | 0.007 | 0.019 | 0.014 | 0.021 | 0.122 | 0.041 | 0.062 | 0.089 | 0.122 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD |
---|---|---|---|---|---|---|---|---|---|---|

10 | 0.784 | 0.711 | 0.752 | 0.785 | 0.760 | 0.823 | 0.861 | 0.818 | 0.765 | 0.823 |

20 | 0.811 | 0.723 | 0.778 | 0.798 | 0.784 | 0.810 | 0.720 | 0.750 | 0.720 | 0.810 |

50 | 0.772 | 0.813 | 0.736 | 0.749 | 0.737 | 0.833 | 0.674 | 0.770 | 0.766 | 0.833 |

100 | 0.806 | 0.881 | 0.826 | 0.846 | 0.827 | 0.797 | 0.800 | 0.855 | 0.830 | 0.797 |

200 | 0.843 | 0.843 | 0.791 | 0.816 | 0.785 | 0.502 | 0.736 | 0.625 | 0.546 | 0.502 |

10 | 0.065 | 0.113 | 0.084 | 0.064 | 0.080 | 0.044 | 0.028 | 0.047 | 0.076 | 0.044 |

20 | 0.050 | 0.104 | 0.068 | 0.057 | 0.065 | 0.051 | 0.107 | 0.086 | 0.106 | 0.051 |

50 | 0.072 | 0.049 | 0.095 | 0.086 | 0.095 | 0.040 | 0.142 | 0.073 | 0.076 | 0.040 |

100 | 0.053 | 0.020 | 0.043 | 0.034 | 0.042 | 0.057 | 0.056 | 0.030 | 0.041 | 0.057 |

200 | 0.035 | 0.035 | 0.061 | 0.048 | 0.064 | 0.311 | 0.095 | 0.185 | 0.262 | 0.311 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD |
---|---|---|---|---|---|---|---|---|---|---|

10 | 0.918 | 0.533 | 0.722 | 0.781 | 0.709 | 0.952 | 0.866 | 0.971 | 0.993 | 0.952 |

20 | 0.898 | 0.529 | 0.694 | 0.759 | 0.703 | 0.971 | 0.999 | 0.993 | 0.996 | 0.971 |

50 | 0.883 | 0.774 | 0.857 | 0.914 | 0.918 | 0.937 | 0.974 | 0.965 | 0.953 | 0.937 |

100 | 0.908 | 0.873 | 0.853 | 0.908 | 0.904 | 0.817 | 0.951 | 0.897 | 0.890 | 0.817 |

200 | 0.892 | 0.858 | 0.866 | 0.871 | 0.862 | 0.441 | 0.842 | 0.612 | 0.476 | 0.441 |

10 | 0.010 | 0.276 | 0.106 | 0.067 | 0.115 | 0.003 | 0.026 | 0.001 | 0.000 | 0.003 |

20 | 0.015 | 0.281 | 0.126 | 0.080 | 0.119 | 0.001 | 0.000 | 0.000 | 0.000 | 0.001 |

50 | 0.020 | 0.071 | 0.029 | 0.011 | 0.010 | 0.006 | 0.001 | 0.002 | 0.003 | 0.006 |

100 | 0.012 | 0.023 | 0.031 | 0.012 | 0.013 | 0.047 | 0.003 | 0.015 | 0.018 | 0.047 |

200 | 0.017 | 0.029 | 0.026 | 0.024 | 0.027 | 0.381 | 0.036 | 0.196 | 0.340 | 0.381 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD |
---|---|---|---|---|---|---|---|---|---|---|

10 | −0.795 | −0.937 | −0.913 | −0.906 | −0.922 | −0.781 | −0.532 | −0.729 | −0.645 | −0.781 |

20 | −0.837 | −0.931 | −0.936 | −0.920 | −0.919 | −0.733 | −0.568 | −0.644 | −0.599 | −0.733 |

50 | −0.763 | −0.914 | −0.730 | −0.703 | −0.687 | −0.770 | −0.544 | −0.685 | −0.670 | −0.770 |

100 | −0.744 | −0.878 | −0.802 | −0.775 | −0.758 | −0.787 | −0.682 | −0.780 | −0.759 | −0.787 |

200 | −0.754 | −0.769 | −0.692 | −0.714 | −0.685 | −0.637 | −0.618 | −0.673 | −0.675 | −0.637 |

10 | 0.059 | 0.006 | 0.011 | 0.013 | 0.009 | 0.067 | 0.278 | 0.100 | 0.167 | 0.067 |

20 | 0.038 | 0.007 | 0.006 | 0.009 | 0.010 | 0.097 | 0.239 | 0.167 | 0.209 | 0.097 |

50 | 0.078 | 0.011 | 0.099 | 0.119 | 0.131 | 0.074 | 0.265 | 0.134 | 0.145 | 0.074 |

100 | 0.090 | 0.021 | 0.055 | 0.070 | 0.081 | 0.063 | 0.135 | 0.067 | 0.080 | 0.063 |

200 | 0.083 | 0.074 | 0.128 | 0.111 | 0.133 | 0.174 | 0.191 | 0.143 | 0.141 | 0.174 |

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**Figure 1.**Stages of the motion capture pipeline: actor (

**a**); registered markers (

**b**); body mesh (

**c**); mesh matched skeleton (

**d**).

**Figure 2.**Outline of the body model (

**a**), and corresponding parts hierarchy annotated with parents and siblings (

**b**).

**Figure 4.**Usage of recurrent NNs in sequence to sequence task: (

**a**) folded, (

**b**) unfolded unidirectional variant, (

**c**) unfolded bidirectional variant.

**Figure 7.**Results for most of the quality measures for all the test sequences. Bars denote $RMSE$; for $RMS{E}_{k}$: ⋄ denotes mean value, × denotes median, ∘ denotes mode, whiskers indicate IQR; standard deviation is not depicted here; dash-outlined areas are zoomed in Figure 8.

**Figure 8.**Results of the most of the quality measures for all the test sequences—zoomed variant for gaps 10, 20, and 50. Bars denote $RMSE$; for $RMS{E}_{k}$: ⋄ denotes mean value, × denotes median, ∘ denotes mode, whiskers indicate IQR; standard deviation is not depicted here.

**Figure 9.**Influence of training sequence length on the quality of obtained results for NN methods: Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and MSE.

No. | Name | Scenario | Duration | Difficulty |
---|---|---|---|---|

1 | Static | Actor stands in the middle of scene, looking around and shifting from one foot to another, freely swinging arms | 32 s | varied motions |

2 | Walking | Actor stands still at the edge of the scene, then walks straight for 6 m, then stands still | 7 s | low dynamics, easy |

3 | Running | Actor stands in the middle of scene, then goes backwards to the edge of the scene and runs for 6 m, then goes backwards to the middle of the scene | 16 s | moderate dynamics |

4 | Sitting | Actor stands in the middle of scene, then sits on a stool, and, after a few seconds, stands again | 15 s | occlusions |

5 | Boxing | Actor stands in the middle of scene, and performs some fast boxing punches | 14 s | high dynamics |

6 | Falling | Actor stands on 0.5 m elevation in the middle of scene, the walks to edge of platform, then falls on the mattress, lies for 2 s and stands | 16 s | high dynamics, occlusions |

No | Entropy ($\mathit{H}\left(\mathit{X}\right)$) | Stddev (${\mathit{\sigma}}_{\mathit{X}}$) | Velocity ($\frac{\mathit{\partial}\mathit{X}}{\mathit{\partial}\mathit{t}}$) | Acc. ($\frac{{\mathit{\partial}}^{2}\mathit{X}}{\mathit{\partial}{\mathit{t}}^{2}}$) | Jerk ($\frac{{\mathit{\partial}}^{3}\mathit{X}}{\mathit{\partial}{\mathit{t}}^{3}}$) | Monotonicity | Complexity |
---|---|---|---|---|---|---|---|

[Bits/Mark.] | [mm/Coordinate] | [m/s/Mark.] | $[\mathbf{m}/{\mathbf{s}}^{2}/\mathbf{Mark}.]$ | $[\mathbf{m}/{\mathbf{s}}^{3}/\mathbf{Mark}.]$ | [-] | [-] | |

1 | 12.697 | 129.705 | 0.208 | 1.561 | 64.817 | 0.352 | 0.027 |

2 | 13.943 | 941.123 | 0.773 | 6.476 | 829.271 | 0.582 | 0.000 |

3 | 15.710 | 982.342 | 0.895 | 6.176 | 643.337 | 0.379 | 0.001 |

4 | 10.231 | 135.356 | 0.190 | 2.863 | 452.142 | 0.347 | 0.016 |

5 | 11.356 | 121.094 | 0.259 | 3.557 | 507.975 | 0.323 | 0.023 |

6 | 14.152 | 601.140 | 0.589 | 6.703 | 799.039 | 0.745 | 0.007 |

Len | FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD | |
---|---|---|---|---|---|---|---|---|---|---|---|

10 | RMSE | 3.830 | 5.375 | 2.410 | 2.494 | 1.801 | 1.267 | 0.348 | 0.610 | 0.737 | 1.267 |

mean (${\mathrm{RMSE}}_{k}$) | 3.280 | 4.869 | 2.175 | 2.290 | 1.708 | 0.971 | 0.243 | 0.468 | 0.512 | 0.971 | |

median (${\mathrm{RMSE}}_{k}$) | 2.746 | 4.399 | 2.035 | 2.120 | 1.614 | 0.893 | 0.205 | 0.406 | 0.391 | 0.893 | |

mode (${\mathrm{RMSE}}_{k}$) | 0.993 | 1.821 | 0.626 | 0.861 | 0.455 | 0.099 | 0.000 | 0.045 | 0.036 | 0.099 | |

stddev (${\mathrm{RMSE}}_{k}$) | 1.893 | 2.209 | 0.939 | 0.989 | 0.573 | 0.695 | 0.216 | 0.336 | 0.458 | 0.695 | |

iqr (${\mathrm{RMSE}}_{k}$) | 2.123 | 2.905 | 0.881 | 0.901 | 0.684 | 0.692 | 0.235 | 0.370 | 0.434 | 0.692 | |

20 | RMSE | 3.474 | 5.114 | 2.559 | 2.527 | 2.082 | 3.366 | 1.191 | 1.914 | 2.354 | 3.366 |

mean (${\mathrm{RMSE}}_{k}$) | 3.187 | 4.775 | 2.371 | 2.351 | 1.903 | 2.694 | 0.933 | 1.525 | 1.738 | 2.694 | |

median (${\mathrm{RMSE}}_{k}$) | 2.828 | 4.709 | 2.274 | 2.235 | 1.779 | 2.147 | 0.764 | 1.251 | 1.287 | 2.147 | |

mode (${\mathrm{RMSE}}_{k}$) | 0.605 | 0.584 | 0.540 | 0.381 | 0.415 | 0.052 | 0.005 | 0.026 | 0.023 | 0.052 | |

stddev (${\mathrm{RMSE}}_{k}$) | 1.442 | 1.871 | 0.891 | 0.898 | 0.826 | 1.831 | 0.664 | 1.045 | 1.483 | 1.831 | |

iqr (${\mathrm{RMSE}}_{k}$) | 1.841 | 2.394 | 1.103 | 1.013 | 0.813 | 1.983 | 0.866 | 1.173 | 1.437 | 1.983 | |

50 | RMSE | 3.813 | 5.910 | 5.001 | 4.041 | 4.777 | 10.363 | 5.517 | 6.928 | 7.677 | 10.363 |

mean (${\mathrm{RMSE}}_{k}$) | 3.401 | 5.434 | 4.233 | 3.445 | 3.958 | 9.207 | 4.572 | 6.027 | 6.573 | 9.207 | |

median (${\mathrm{RMSE}}_{k}$) | 2.906 | 5.154 | 3.776 | 3.118 | 3.496 | 8.733 | 3.888 | 5.512 | 5.733 | 8.733 | |

mode (${\mathrm{RMSE}}_{k}$) | 1.326 | 1.393 | 0.831 | 1.066 | 1.000 | 1.169 | 0.400 | 0.800 | 0.793 | 1.169 | |

stddev (${\mathrm{RMSE}}_{k}$) | 1.688 | 2.168 | 2.430 | 1.921 | 2.448 | 4.464 | 2.852 | 3.174 | 3.764 | 4.464 | |

iqr (${\mathrm{RMSE}}_{k}$) | 1.421 | 2.216 | 2.169 | 1.642 | 2.282 | 6.078 | 2.418 | 3.770 | 4.373 | 6.078 | |

100 | RMSE | 4.759 | 7.805 | 10.798 | 7.678 | 10.716 | 24.634 | 12.548 | 15.231 | 18.746 | 24.634 |

mean (${\mathrm{RMSE}}_{k}$) | 4.233 | 7.134 | 9.460 | 6.721 | 9.302 | 21.812 | 11.236 | 13.587 | 16.108 | 21.812 | |

median (${\mathrm{RMSE}}_{k}$) | 3.658 | 6.329 | 8.333 | 5.953 | 8.198 | 21.129 | 10.345 | 12.875 | 14.785 | 21.129 | |

mode (${\mathrm{RMSE}}_{k}$) | 1.517 | 2.252 | 1.377 | 1.465 | 1.400 | 3.266 | 2.546 | 1.986 | 1.937 | 3.266 | |

stddev (${\mathrm{RMSE}}_{k}$) | 2.132 | 3.143 | 5.114 | 3.692 | 5.230 | 11.305 | 5.472 | 6.825 | 9.556 | 11.305 | |

iqr (${\mathrm{RMSE}}_{k}$) | 2.215 | 3.473 | 5.650 | 4.217 | 5.700 | 14.536 | 6.850 | 8.029 | 11.019 | 14.536 | |

200 | RMSE | 9.959 | 18.970 | 33.147 | 27.987 | 33.104 | 62.786 | 34.481 | 47.259 | 56.570 | 62.786 |

mean (${\mathrm{RMSE}}_{k}$) | 9.062 | 17.303 | 30.204 | 24.837 | 30.135 | 55.099 | 31.616 | 41.676 | 48.789 | 55.099 | |

median (${\mathrm{RMSE}}_{k}$) | 8.683 | 16.200 | 28.352 | 22.655 | 28.462 | 49.641 | 29.914 | 38.410 | 42.155 | 49.641 | |

mode (${\mathrm{RMSE}}_{k}$) | 2.404 | 3.973 | 5.523 | 4.263 | 5.010 | 8.510 | 6.518 | 6.459 | 6.033 | 8.510 | |

stddev (${\mathrm{RMSE}}_{k}$) | 4.013 | 7.631 | 13.450 | 12.743 | 13.503 | 29.934 | 13.511 | 22.022 | 28.463 | 29.934 | |

iqr (${\mathrm{RMSE}}_{k}$) | 5.084 | 9.413 | 18.231 | 16.895 | 18.436 | 48.864 | 17.125 | 36.315 | 46.222 | 48.864 |

NN Type | Number of Learnable Parameters | Value for Exemplary Case |
---|---|---|

FFNN: | $hiddenLayerSize\times inputvectorSize+hiddenLayerSize$ | 275 |

$+3\times hiddenLayerSize+3$ | ||

LSTM: | $4\times hiddenRecurrentNeurons\times inputvectorSize$ | 22,023 |

$+4\times hiddenRecurrentNeurons\times hiddenRecurrentNeurons$ | ||

$+4\times hiddenRecurrentNeurons$ | ||

$+3\times hiddenRecurrentNeurons+3$ | ||

GRU: | $3\times hiddenRecurrentNeurons\times inputvectorSize$ | 16,563 |

$+3\times hiddenRecurrentNeurons\times hiddenRecurrentNeurons$ | ||

$+3\times hiddenRecurrentNeurons$ | ||

$+3\times hiddenRecurrentNeurons+3$ | ||

BILSTM: | $8\times hiddenRecurrentNeurons\times inputvectorSize$ | 47,043 |

$+8\times hiddenRecurrentNeurons\times hiddenRecurrentNeurons$ | ||

$+8\times hiddenRecurrentNeurons$ | ||

$+3\times 2\times hiddenRecurrentNeurons+3$ |

FFNN${}_{\mathbf{lin}}$ | FFNN${}_{\mathbf{tanh}}$ | LSTM | GRU | BILSTM | LIN | SPLINE | MAKIMA | PCHIP | mSVD | |
---|---|---|---|---|---|---|---|---|---|---|

Entropy | 0.708 | 0.793 | 0.775 | 0.736 | 0.735 | 0.680 | 0.486 | 0.624 | 0.630 | 0.680 |

Stddev | 0.741 | 0.892 | 0.805 | 0.781 | 0.778 | 0.706 | 0.517 | 0.653 | 0.631 | 0.706 |

Velocity | 0.744 | 0.886 | 0.813 | 0.784 | 0.781 | 0.713 | 0.521 | 0.656 | 0.640 | 0.713 |

Acceleration | 0.905 | 0.912 | 0.903 | 0.907 | 0.890 | 0.854 | 0.791 | 0.844 | 0.818 | 0.854 |

Jerk | 0.803 | 0.794 | 0.777 | 0.799 | 0.779 | 0.753 | 0.758 | 0.763 | 0.725 | 0.753 |

Monotonicity | 0.900 | 0.713 | 0.798 | 0.847 | 0.819 | 0.824 | 0.926 | 0.888 | 0.862 | 0.824 |

Complexity | −0.779 | −0.886 | −0.815 | −0.804 | −0.794 | −0.742 | −0.589 | −0.702 | −0.670 | −0.742 |

Entropy | Stddev | Velocity | Acceleration | Jerk | Monotonicity | Complexity | |
---|---|---|---|---|---|---|---|

Entropy | 1.000 | 0.869 | 0.898 | 0.730 | 0.459 | 0.465 | −0.712 |

Stddev | 0.869 | 1.000 | 0.992 | 0.879 | 0.732 | 0.501 | −0.949 |

Velocity | 0.898 | 0.992 | 1.000 | 0.890 | 0.731 | 0.477 | −0.929 |

Acceleration | 0.730 | 0.879 | 0.890 | 1.000 | 0.941 | 0.735 | −0.913 |

Jerk | 0.459 | 0.732 | 0.731 | 0.941 | 1.000 | 0.695 | −0.847 |

Monotonicity | 0.465 | 0.501 | 0.477 | 0.735 | 0.695 | 1.000 | −0.560 |

Complexity | −0.712 | −0.949 | −0.929 | −0.913 | −0.847 | −0.560 | 1.000 |

p-values | |||||||

Entropy | 1.000 | 0.025 | 0.015 | 0.100 | 0.360 | 0.353 | 0.112 |

Stddev | 0.025 | 1.000 | 0.000 | 0.021 | 0.098 | 0.311 | 0.004 |

Velocity | 0.015 | 0.000 | 1.000 | 0.017 | 0.099 | 0.338 | 0.007 |

Acceleration | 0.100 | 0.021 | 0.017 | 1.000 | 0.005 | 0.096 | 0.011 |

Jerk | 0.360 | 0.098 | 0.099 | 0.005 | 1.000 | 0.125 | 0.033 |

Monotonicity | 0.353 | 0.311 | 0.338 | 0.096 | 0.125 | 1.000 | 0.248 |

Complexity | 0.112 | 0.004 | 0.007 | 0.011 | 0.033 | 0.248 | 1.000 |

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**MDPI and ACS Style**

Skurowski, P.; Pawlyta, M. Gap Reconstruction in Optical Motion Capture Sequences Using Neural Networks. *Sensors* **2021**, *21*, 6115.
https://doi.org/10.3390/s21186115

**AMA Style**

Skurowski P, Pawlyta M. Gap Reconstruction in Optical Motion Capture Sequences Using Neural Networks. *Sensors*. 2021; 21(18):6115.
https://doi.org/10.3390/s21186115

**Chicago/Turabian Style**

Skurowski, Przemysław, and Magdalena Pawlyta. 2021. "Gap Reconstruction in Optical Motion Capture Sequences Using Neural Networks" *Sensors* 21, no. 18: 6115.
https://doi.org/10.3390/s21186115