# Hand Tracking and Gesture Recognition Using Lensless Smart Sensors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rambus Lensless Smart Sensor

## 3. Methodology

#### 3.1. Physical Setup

- (i)
- the longitudinal distance along the Z-axis between the LED and the sensor plane goes from 40 cm to 100 cm;
- (ii)
- the distance between the right and left sensors (Sen_R and Sen_L respectively, as shown in Figure 1) measured on the central points of the sensors, called baseline (b) is 30 cm;
- (iii)
- the combined FoV of 80°.

#### 3.2. Constraints for Multiple Points Tracking

- (i)
- Discrimination—When the distance between two light sources along the X- (or Y-) axis is less than 2 cm, the two sources will merge to a single point in the image frame, which makes them undistinguishable.

- (ii)
- Occlusion—Occlusions occur when the light source is moved away from the sensor focal point along the X-axis. In Figure 3, at extreme lateral positions when the FoV ≥ 40°, even when the LEDs are 3 cm away, one LED is occluded by the other.

#### 3.3. Placement of Light Points

#### 3.4. Hardware Setup

#### 3.5. Multiple Points Tracking

#### 3.5.1. Calibration Phase

_{ref}) for the tracking phase. When the five detected points are correctly identified during the process, every point is labeled in order to assign each to the right part of the hand. The reconstructed frames have their origin in the top-left corner, with a resolution of 320 × 480, which is half the size along the X-direction and the full size along the Y-direction compared to the original image frames, as shown in Figure 6b.

- The middle finger point (M) has the lowest row coordinate in both images.
- The lower palm point (LP) has the highest row coordinate in both images.
- The first upper palm point (UP1) has the lowest column coordinate in both images.
- The second upper palm point (UP2) has the second column coordinate in both images.
- The thumb point (T) has the highest column coordinate in both images.

#### 3.5.2. Tracking Phase

_{ref}, which was saved during the calibration phase. The tracking phase is developed in such a way that it is impossible to start tracking if at least three LEDs are not visible for the first 10 frames, and the number of points in the left and right frames is not equal, in order to always maintain a consistent accuracy. As such, if mL and mR represent the detected maxima for sen_L and sen_R respectively, the following different decisions are taken according to the explored image:

- |mL| ≠ |mR| and |mL|, |mR| > 3: a matrix of zeros is saved and a flag = 0 is set encoding “Unsuccessful Detection”.
- |mL|, |mR| < 3: a matrix of zeros is saved and a flag = 0 is set encoding “Unsuccessful Detection”.
- |mL| = |mR| and |mL|, |mR| > 3: the points are correctly identified, flag = 1 is set encoding “Successful Detection”, and the 2D coordinates of LED positions are saved.

_{0,i}, y

_{0,i}, z

_{0,i}] i = M, LP, T, UP1, UP2

_{−10}… t

_{−1}]

_{−10,i}, …, x

_{−1,i}], [y

_{−10,i}, …, y

_{−1,i}], [z

_{−10,i}, …, z

_{−1,i}]}

_{polyp,j}= a

_{p,i}t

^{2}+ b

_{p,i}t + c

_{p,i}, p = [x, y, z]

_{poly}estimates the coordinates of all five LED positions related to the current iteration, based on the last 10 iterations, as shown in Equation (7):

_{i}, where i = L, R, as the number of maxima detected by both sensors, the combinations without repetitions of three previously ranged points are computed. Indeed, if five points are detected, there will be 10 combinations: four combinations for four detected points, and one combination for three detected points. Thus, triple combinations of C points are generated. The matrix of relative distances for each candidate combination of points, as k ∈ [1, …, |C|], is calculated using Equation (1). The sum of squares of relative distances for each matrix is then determined by Equations (8) and (9) for both the calibration and tracking Phases, as k ∈ [1, …, |C|]:

_{k}and Sum

_{ref}can be associated with the palm coordinates. To avoid inaccurate results caused by environmental noise and possible failures in the local maxima detection, this difference is compared to a threshold to make sure that the estimate is sufficiently accurate. Several trials and experiments were carried out to find a suitable threshold to the value th

_{distance}= 30 cm. The closest candidate $\widehat{k}$ was then found using Equation (12):

_{plane}is the normal vector to the palm plane. It is calculated using the normalized cross-product of the 3D palm coordinates of the points using the right-hand rule, as in Equation (14):

_{1}and UL

_{2}onto the palm plane are calculated by finding the inner product of X

_{plane}and (${X}_{U{L}_{i}}$ − ${X}_{UP2}$). It is then multiplied by X

_{plane}and subtracted from the unlabeled LED positions, as shown in Equation (15):

_{i}, which exposes the minimum angle, while the thumb is selected as the maximum one, as shown in Equations (21) and (22):

_{UL}and the segment X

_{ref}in the same way, as shown in Equation (19). The decision is made according to an empirically designed threshold th

_{angle}= 30°, as shown in Equation (23):

_{distance}constraint given in Equation (12), proceed in the same way as that of Section 3.5.2(a). Thus, the proposed novel multiple points live tracking algorithm labels and tracks all of the LEDs placed on the hand.

#### 3.5.3. Orientation Estimation

_{plane}in Equation (14) and the direction given by the middle finger. If the palm orientation, the distance between initial and final segments (S

_{0}and S

_{3}) of middle finger (d), and the segment lengths (${S}_{il}$, i = 0,1,2,3) are assigned as known variables, the orientation of all of the segments can be estimated using a pentagon approximation model, as shown in Figure 7.

_{plane}, as previously calculated in Equation (14), and the middle finger segment (${X}_{UP2}$ − X

_{M}), is calculated as shown in Equation (24). The angle between S

_{0}and d (=α) is its complementary angle is shown in Equation (25).

_{0}and d is equal to the angle between S

_{3}and d. Thus, knowing that the sum of the internal angles of a pentagon is 540°, the value of the other angles (=β) is also computed assuming that they are equal to each other, as shown in Equation (26).

_{plane}is then computed as shown in Equation (27), and the rotation matrix R is built using angle α around the derived vector according to Euler’s rotation theorem [49]. This rotation matrix is used to calculate the orientation of segment S

_{0}in Equation (28) and its corresponding 2D location in Equation (29).

_{1}, S

_{2}, and S

_{3}), as well as their 2D locations, is calculated. The same method is applied for finding the orientation of all of the thumb segments using the segment lengths and 2D location of lower palm LED (X

_{LP}), but with a trapezoid approximation for estimating the angles.

#### 3.6. 3D Rendering

#### 3.7. Gesture Recognition

## 4. Results and Discussion

#### 4.1. Validation of Rambus LSS

_{i}) are plotted. The corresponding plots are shown in Figure 9a,b, respectively. The results are provided in Table 3.

#### 4.2. Validation of Multiple Points Tracking

_{3}and d (=α) is estimated and plotted with respect to the actual angles in Figure 13d. Here also, the possible three readings are plotted, based on the visibility of the LED fitted on the middle finger. The other joint angles are derived from this angle itself, as explained in Section 3.5.3.

#### 4.3. Validation of Latency Improvements

#### 4.4. Validation of Gesture Recognition

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**3D Rendering of hand (

**a**); calculated LED positions (

**b**); palm plane reconstructed using Three LEDs on Palm (LP, UP1, and UP2) (

**c**); middle finger reconstructed with M LED (

**d**); index, ring, and little fingers reconstructed using the properties of M LED (

**e**); thumb reconstructed with T LED.

**Figure 10.**(

**a**) LP; (

**b**) UP1; (

**c**) UP2; (

**d**) M; (

**e**) T. Tracked plane with respect to the reference plane for all LED positions.

**Figure 12.**Different orientations of hand in front of LSSs: (

**a**) hand held straight; (

**b**) hand held upside down at an inclination; (

**c**) fingers bend.

**Figure 13.**Calculated vs. Actual Orientation: (

**a**) Along X-Axis; (

**b**) Along Y-Axis; (

**c**) Along Z-Axis; (

**d**) Along Middle Finger between S3 and d (α).

**Figure 14.**Image Frames and Reconstructed Frames (

**a**) before Reducing the Region of Interest (ROI); (

**b**) after Reducing the ROI.

**Figure 15.**(

**a**) Classification accuracy as a function of LED positions; (

**b**) confusion matrix for the dataset.

LP | UP1 | UP2 | |
---|---|---|---|

LP | d_{LP, LP} | d_{LP, UP1} | d_{LP, UP2} |

UP1 | d_{UP1, LP} | d_{UP1, UP1} | d_{UP1, UP2} |

UP2 | d_{UP2, LP} | d_{UP2, UP1} | d_{UP2, UP2} |

Gesture Label | Gesture Description |
---|---|

0 – Forward | Forward Movement along Z axis |

1—Backward | Backward Movement along Z axis |

2—Triangle | Triangle performed on X-Y plane: Basis parallel to the X axis |

3—Circle | Circle performed on X-Y plane |

4—Line Up → Down | Line Up/ Down direction on X-Y plane |

5—Blank | None of the previous gestures |

Precision | Centre | +40 deg | −40 deg | +60 deg | −60 deg |
---|---|---|---|---|---|

RMSE (cm) | 0.2059 | 0.2511 | 0.2740 | 0.3572 | 0.4128 |

Repeatability (cm) | 0.0016 | 0.0058 | 0.0054 | 0.0210 | 0.0313 |

Temporal Noise (cm) | 0.0027 | 0.0082 | 0.0078 | 0.0435 | 0.0108 |

RMSE (cm) | LP | UP1 | UP2 | M | T |
---|---|---|---|---|---|

X | 0.5054 | 0.6344 | 0.5325 | 0.7556 | 0.7450 |

Y | 0.3622 | 0.3467 | 0.5541 | 0.9934 | 0.5222 |

Z | 0.8510 | 1.0789 | 0.9498 | 1.2081 | 0.7903 |

Total | 1.0540 | 1.2987 | 1.1457 | 1.7370 | 1.2051 |

RMSE (cm) | X-Axis | Y-Axis | ||
---|---|---|---|---|

With Respect to LP | With Respect to Mean | With Respect to LP | With Respect to Mean | |

UP1 | 0.5054 | 0.6344 | 0.5325 | 0.7556 |

UP2 | 0.3622 | 0.3467 | 0.5541 | 0.9934 |

M | 0.8510 | 1.0789 | 0.9498 | 1.2081 |

T | 1.0540 | 1.2987 | 1.1457 | 1.7370 |

No. of Frames Averaged | Full Image Frames (480 × 320) | Image Frames with Reduced ROI (200 × 320) | ||
---|---|---|---|---|

Dt (s) | EFPs | Dt (s) | EFPs | |

5 | 0.0553 ± 0.0086 | ≈ 18 | 0.0482 ± 0.0134 | ≈ 21 |

2 | 0.0439 ± 0.0182 | ≈ 23 | 0.0296 ± 0.0085 | ≈ 34 |

1 | 0.0368 ± 0.0036 | ≈ 27 | 0.0248 ± 0.0047 | ≈ 40 |

No. of Frames Averaged | Full Image Frames (480 × 320) | Image Frames with Reduced ROI (200 × 320) | ||||
---|---|---|---|---|---|---|

RMSE (cm) | Repeatability (cm) | Temporal Noise (cm) | RMSE (cm) | Repeatability (cm) | Temporal Noise (cm) | |

5 | 0.4814 | 0.0025 | 0.0031 | 0.5071 | 0.0018 | 0.0036 |

2 | 0.5333 | 0.0047 | 0.0052 | 0.5524 | 0.0041 | 0.0049 |

1 | 0.5861 | 0.0086 | 0.0074 | 0.6143 | 0.0088 | 0.0091 |

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**MDPI and ACS Style**

Abraham, L.; Urru, A.; Normani, N.; Wilk, M.P.; Walsh, M.; O’Flynn, B.
Hand Tracking and Gesture Recognition Using Lensless Smart Sensors. *Sensors* **2018**, *18*, 2834.
https://doi.org/10.3390/s18092834

**AMA Style**

Abraham L, Urru A, Normani N, Wilk MP, Walsh M, O’Flynn B.
Hand Tracking and Gesture Recognition Using Lensless Smart Sensors. *Sensors*. 2018; 18(9):2834.
https://doi.org/10.3390/s18092834

**Chicago/Turabian Style**

Abraham, Lizy, Andrea Urru, Niccolò Normani, Mariusz P. Wilk, Michael Walsh, and Brendan O’Flynn.
2018. "Hand Tracking and Gesture Recognition Using Lensless Smart Sensors" *Sensors* 18, no. 9: 2834.
https://doi.org/10.3390/s18092834