Fall detection for multiple pedestrians using a PCA approach to 3-D inclination

Abstract

Aging populations in developing countries have driven increased research interest in issues related to “eldercare,” with a particular focus on early fall detection. This article describes an approach to processing images captured using a depth camera to predict the probable inclination of an imminent fall by a pedestrian. First, the 3-D object reconstruction method is explained and realized. Then, the 3-D object inclination analysis is given. Experimental results show prediction rates achieved accuracy up to 98.11%, making this approach promising for use in applications in hospital and home environments in the near future.

Keywords

Early fall detection 3-D inclination calculation principle component analysis 3-D object reconstruction depth camera

Introduction

Fall detection technologies can be divided into two categories,^1,2 wearable sensors^3

–10 and image-based mechanisms.^11
–13 The increasing power and ubiquity of smart phones and the emergence of other wearable devices have broadly increased the application of wearable-type sensors in fall detection. Tang et al.⁹ used a motion capture system to capture the 3-D coordinates of a reflective marker so as to calculate the angular displacement and velocity of the ankle to effectively detect leg sprain and falls caused by the sudden rotation of the ankle joint. However, wearable devices may not provide an ideal solution for elderly users or medical patients, because such users may forget to wear them, protect them from water, or keep them charged. Hospitals and elder care facilities thus primarily rely on image-based fall detection systems, and the present research only considers this type of approach.

Image-based fall detection technologies are divided into three categories based on their specific hardware characteristics.

2-D images: Images are taken from a single camera to determine falls.^10

–15 Wang et al.¹⁴ performed tracing and limb motion analysis by performing feature marking on the user. Liao et al.¹⁵ applied a Bayesian belief network model to pedestrian characteristics to detect slips and falls.

3-D images: Two or more cameras are used to capture 3-D images for fall detection in an ideal hardware configuration.^16,17

Depth images: Depth information for each pixel in a 2-D image is used to produce a 3-D image. Structured light cameras are used to obtain image depth, including Microsoft’s Kinect and time-of-flight cameras.

2-D image-based fall detection is generally less accurate than 3-D detection. In addition, 2-D approaches require the use of markers¹⁴ or are restricted by the need to learn pedestrian characteristics.¹⁵ 3-D images provide more complete information, but depend on the use of time-consuming algorithms. Image depth techniques are frequently used as a compromise.

The Kinect depth camera uses a built-in human skeletal analysis algorithm or 3-D bounding box to detect falls.^18

–21 However, the algorithm is imperfect and can produce incorrect results, such as when the target figure is lying prone.^22,23

All fall detection techniques must determine the subject’s inclination relative to the vertical. In 2-D image-based techniques,^13,15 curve fitting is used to create an ellipse which approximates the subject’s outline. The angle between the ellipse’s major axis and the vertical is then calculated as the subject’s inclination.

Current methods do not adequately detect subject inclination, which is the key to detecting falls.^24,25 Rougier et al.¹³ and Liao et al.¹⁵ use a 2-D image which cannot produce the actual inclination of a 3-D object. To address this issue, the present article proposes a method to obtain the inclination of a 3-D object.

Yang and Lin¹ proposed a fall determination flow using a depth camera. Although the fall determination technique is powerful, the subject’s inclination may be incorrect in some cases.

In the work of Yang and Lin,¹ the subject’s centroid is selected as the new origin. A line passing through the origin is found to minimize the sum of the distance between the line and the edges of the subject’s outline.

Earlier works¹ ^, ^13,15 all rely on 2-D images and are thus cannot calculate the subject’s correct inclination when the fall direction is parallel to the camera.

Obtaining the subject’s correct inclination entails first reconstructing the 3-D object and then calculating its inclination. The minimization of the sum of the 3-D distance is actually an orthogonal regression or total least squares problem.²⁶

Lay²⁷ mentioned that principal component analysis (PCA) is equivalent to orthogonal regression methods, thus this article utilized PCA to derive the inclination of the 3-D object which is the result of orthogonal regression (i.e. the sum of the squares of the orthogonal distances to the spatial line is minimized).

This article focuses on 3-D object reconstruction and object inclination calculation (see “Fall detection system assessment” and “Experimental results” sections). The fall determination flow is implemented through six processes: foreground extraction, overlapping segmentation, object reconstruction and detection, object tracking, calculating (object inclination calculation Δθ_t ), and fall event determining (Figure 1).

Figure 1.

Fall determination flow.¹

This article proposes a new method to calculate 3-D inclination using information extracted from a Kinect unit. PCA is then used to calculate the 3-D inclination of a human subject. The proposed method significantly improves on fall detection accuracy when the subject falls in a direction which is parallel to the camera, with detection accuracy approaching 100%.

Methods

3-D object reconstruction

In addition to providing depth information in millimeter units, the Kinect depth camera also creates 2-D images measured in pixels. Yang and Lin¹ propose transforming a depth image into a point cloud based on the 3-D coordinates of the depth camera and transforming the 2-D image into real measurements in millimeter. It is also necessary to convert the camera’s coordinates into earth coordinates to completely reconstruct the 3-D object and thus obtain a more accurate angle of the object’s inclination.

Correctly tracking these objects requires separating this overlap and then calculating the fall characteristics. These tasks use coordinate information which must be calculated using a geodetic coordinate system. The camera’s coordinate information must be converted from the projection coordinate system to a real-world coordinate system, and then to an earth coordinate system.

The method described in the previous section operates at a right angle earth coordinate system. Therefore, this section explores how the information obtained from the depth camera is then converted to a spatial position in a geodetic coordinate system.

The depth camera’s image plane is a projection coordinate system (see Figure 2), using pixels as units. Assume that the resolution of the image plane is w × h pixel², and the plane size is $ℓ_{w} \times ℓ_{h}$ mm². In addition, pixel coordinates in a projection coordinate system are converted into a real-world coordinate system in millimeter. The origin of the pixel coordinates is in the upper left corner of the image plane and the origin of the real world coordinates is at the center of the image plane. Note that the real world coordinate system discussed in the work of Yang and Lin¹ uses the left-hand rule, while the real-world coordinate system in Figure 2 uses the right-hand rule.

Figure 2.

Relationship between the camera’s projection coordinate system and the real-world coordinate system.

If a given pixel’s coordinate value is $(p_{w}, p_{h})$ , then the distance from the origin of the pixel coordinate is $p_{w} \times (ℓ_{w} / w)$ , $p_{h} \times (ℓ_{h} / h)$ mm. Then, the real-world coordinate system provides the position (x_v , y_v ) as follows

x_{v} = (\frac{p_{w}}{w} \times ℓ_{w}) - \frac{ℓ_{w}}{2}

y_{v} = (\frac{p_{h}}{h} \times ℓ_{h}) - \frac{ℓ_{h}}{2}

Then, assume the camera’s horizontal and vertical fields of view are, respectively, θ_w and θ_h , and the camera’s focal length is f mm. The following relationships can be derived from Figure 3

Figure 3.

Triangular relationship between the projected coordinate system and the real-world coordinate system.

tan \frac{θ_{w}}{2} = \frac{ℓ_{w} / 2}{f}

tan \frac{θ_{h}}{2} = \frac{ℓ_{h} / 2}{f}

\begin{array}{l} \frac{x_{k}}{z_{k}} = \frac{x_{v}}{f} \Rightarrow x_{k} = \frac{z_{k} tan (θ_{w} / 2)}{ℓ_{w} / 2} x_{v} \\ = \frac{z_{k} tan (θ_{w} / 2)}{ℓ_{w} / 2} \times (\frac{p_{w}}{w} - 0.5) ℓ_{w} = 2 (\frac{p_{w}}{w} - 0.5) z_{k} tan (θ_{w} / 2) \end{array}

\begin{array}{l} \frac{y_{k}}{z_{k}} = \frac{y_{v}}{f} \Rightarrow y_{k} = \frac{z_{k} tan (θ_{h} / 2)}{ℓ_{h} / 2} y_{v} \\ = \frac{z_{k} tan (θ_{h} / 2)}{ℓ_{h} / 2} \times (\frac{p_{h}}{h} - 0.5) ℓ_{h} = 2 (\frac{p_{h}}{h} - 0.5) z_{k} tan (θ_{h} / 2) \end{array}

These two equations use only four constants: w, h, θ_w , and θ_h .

The camera is installed on the wall near the ceiling with different real-world and earth coordinate systems. To calculate the angle of inclination, the real-world coordinate system must be rotated to coincide with the earth coordinate system.

The relationship between the two right-angled coordinate systems can be expressed as Euler angles: The z-axis is rotated ϕ, after which the new coordinate system y-axis is rotated θ. Finally, the new coordinate system z-axis is rotated ϕ. This result is the product of three rotation matrices: Rot(z, ϕ), Rot(y, θ), and Rot(z, ϕ). Because we only aim to calculate the inclination against the vertical from the ground to the sky (i.e. against gravity) as the earth coordinate system’s y-axis and do not consider the x-axis, only the plane XZ must be horizontal. Therefore, only two rotations are required to align the real-world and earth coordinate systems: rotate the z-axis ϕ such that the x-axis falls on the horizontal plane, and rotate the x-axis θ, such that the y-axis is a plumb line.

The geodetic coordinate system is defined as E_G = {g _x , g _y , g _z }, and the real-world coordinates of the camera are E_K = {k _x , k _y , k _z }. Let $R_{K - > G}$ be the coordinate transformation matrix from E_K to E_G

\begin{array}{l} R_{K - > G} = {[\begin{matrix} k_{x} & k_{y} & k_{z} \end{matrix}]}^{- 1} [\begin{matrix} g_{x} & g_{y} & g_{z} \end{matrix}] \\ = Rot (z, ϕ) Rot (x, θ) = [\begin{matrix} cos ϕ & - sin ϕ & 0 \\ sin ϕ & cos ϕ & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & cos θ & - sin θ \\ 0 & sin θ & cos θ \end{matrix}] \end{array}

If the space in the camera’s real-world coordinate system is (x_k , y_k , z_k ), and the earth coordinate system is expressed by (x_g , y_g , z_g ), then

[\begin{matrix} k_{x} & k_{y} & k_{z} \end{matrix}] [\begin{matrix} x_{k} \\ y_{k} \\ z_{k} \end{matrix}] = [\begin{matrix} g_{x} & g_{y} & g_{z} \end{matrix}] [\begin{matrix} x_{g} \\ y_{g} \\ z_{g} \end{matrix}]

can obtain

\begin{array}{l} [\begin{matrix} x_{g} \\ y_{g} \\ z_{g} \end{matrix}] = {[\begin{matrix} g_{x} & g_{y} & g_{z} \end{matrix}]}^{- 1} [\begin{matrix} k_{x} & k_{y} & k_{z} \end{matrix}] [\begin{matrix} x_{k} \\ y_{k} \\ z_{k} \end{matrix}] \\ = {(R_{K - > G})}^{- 1} [\begin{matrix} x_{k} \\ y_{k} \\ z_{k} \end{matrix}] = R_{G - > K} [\begin{matrix} x_{k} \\ y_{k} \\ z_{k} \end{matrix}] \end{array}

Since R_K->G is an orthogonal matrix, (R_K->G )⁻¹ = (R_K->G ) ^T .

The angles ϕ and θ are difficult to measure and must be obtained using system identification. The camera orientation identification method is used to record the depth camera image using a vertical line perpendicular to the ground and a horizontal line parallel to the ground to obtain the camera’s position in the real-world coordinate system. The laser light is directly projected onto the screen to obtain information needed for experimental setup.

First, using the real-world coordinate system, we select all vertical laser lines, with a total of n pixels above the vertical laser line. These points are referred to as $p_{k i}$ , and the horizontal laser line are referred to as q _ki as shown in Figure 4.

Figure 4.

Vertical and horizontal laser reference point.

Then, the laser projected on the screen to create a point cloud map. Intuitively, the laser beam should be straight, but the point cloud shows jitter due to the depth camera resolution, as shown in Figure 5.

Figure 5.

Laser point cloud map.

If we directly take any two point vectors within the laser point cloud diagram, the selection of different locations may result in deviation between the calculated linear vector and the actual laser line. To solve this problem, this article analyzes the principal components of vertical and horizontal lines and obtains the main vector to represent the two laser lines. The approximate line obtained from PCA calculations of the laser points as shown in Figure 6.

Figure 6.

Approximate line obtained from PCA. PCA: principal component analysis.

PCA operation is applied to all points of the vertical laser line to obtain the first principal component $u_{k 1}$ of the real-world coordinate system to represent the vertical laser line vector. $u_{g 1}$ is the principal component of the earth coordinate system’s vertical laser line. Since the vertical lasers are perpendicular to the horizontal plane, the $u_{g 1}$ vector’s x and z components are both zero. Based on this fact, the relationship between this vector in the real-world and earth coordinate systems can be expressed as

[\begin{matrix} u_{k 1, x} \\ u_{k 1, y} \\ u_{k 1, z} \end{matrix}] = [\begin{matrix} cos ϕ & - sin ϕ cos θ & sin ϕ sin θ \\ sin ϕ & cos ϕ cos θ & - cos ϕ sin θ \\ 0 & sin θ & cos θ \end{matrix}] [\begin{matrix} 0 \\ u_{g 1, y} \\ 0 \end{matrix}]

By reducing the second column from the first, we obtain $ϕ$ as

ϕ = {tan}^{- 1} (\frac{- u_{k 1, x}}{u_{k 1, y}}), when (p_{g 1, y} - p_{g n, y}) > 0

Similarly, using the horizontal laser line, we can obtain θ as follows

θ = {tan}^{- 1} (\frac{v_{k 1, x} sin ϕ - v_{k 1, y} cos ϕ}{v_{k 1, z}})

where $v_{k 1}$ is the first principal component of the horizontal laser.

The earth coordinate system transition results are shown in Figure 7.

Figure 7.

Earth coordinate system laser line.

As seen in Figure 7, the proposed 3-D object reconstruction method can reconstruct the intersection of the two laser beams, where the vertical line and the X_gZ_g plane (earth) is the vertical and the horizontal line and the X_gZ_g plane is the horizontal.

3-D object inclination calculation

This article uses a 3-D object inclination calculation method to monitor changes between the inclination of a pedestrian and the vertical as a basis for detecting falls.

Inclination calculation

Figure 8 shows a point cloud for a 3-D object. Assume a total of n pixels for p _ri . First, find the centroid p _M . Then, move the coordinate origin to the centroid, and the new coordinate value for each pixel is p _i . The process is as follows

p_{M} = \frac{1}{n} (p_{r 1} + ⌊ + p_{r n})

p_{i} = p_{r i} - p_{M}

Figure 8.

Point cloud pixels for a 3-D object.

This n point vector p _i can form a 3 × n matrix B, and then a covariance matrix S can be obtained as follows

B = [\begin{matrix} p_{1} & p_{2} & ⌊ & p_{n} \end{matrix}]

S = \frac{1}{n - 1} B B^{T}

Clearly, S is a symmetric matrix, so an orthogonal matrix Q can be found to diagonalize it. This orthogonal matrix Q is composed of the eigenvectors u _i of the matrix S, and the diagonal elements of the diagonalized matrix D are eigenvalues λ_I corresponding to the eigenvectors u _i

D = Q^{T} S Q = [\begin{matrix} λ_{1} & 0 & 0 \\ 0 & λ_{2} & 0 \\ 0 & 0 & λ_{3} \end{matrix}], where Q = [\begin{matrix} u_{1} & u_{2} & u_{3} \end{matrix}]

The diagonal elements of the matrix D are arranged in descending order. The largest eigenvalue λ ₁ corresponds to the eigenvector u ₁, also known as the first principal component, while u ₂ and u ₃ are, respectively, called the second and third principal components. The line along the first principal component u ₁ is the line that minimizes the square of the distance of all 3-D pixels. This distance minimizes the direction of the line u ₁, which we call PCA solution: u _PCA = u ₁.

Orthogonal conversion does not change vector lengths. That is p _i = (u _1T p _i )² + (u _2T p _i )² + (u _3T p _i )², so that Σp _iT p _i = λ ₁ + λ ₂ + λ ₃ is a fixed value. We can deduce that: λ ₁ = Σ(u _1T p _i)² maximization will result in λ ₂ + λ ₃ = Σ(u _2T p _i )² + (u _3T p _i )² minimization. (u _2T p _i )² + (u _3T p _i )² is the square of the distance from point p _i to line u ₁, such that u ₁, u ₂, and u ₃ are mutually orthogonal. Thus, u _PCA is the only line for which the square of the distance from all points to the line is minimized. Once u _PCA is obtained, we can obtain the angle to the vertical.

Inclination verification

In this subsection, we verify the PCA calculation results in “3-D object reconstruction” section, where the subject is standing, squatting, sitting down, bending over, or falling in the direction parallel to the camera view and validate results for when the direction of the fall is not parallel to the camera view.

Figure 9 shows the PCA calculation for the 3-D inclination of a subject, with PCA used to obtain the subject’s axis through PCA.

Figure 9.

PCA 3-D inclination verification original image and result: (a) standing, (b) squatting, (c) sitting down, (d) bending over, (e) falling parallel to the camera view, and (f) falling nonparallel to the camera view. PCA: principal component analysis.

These six different types of validation show that, regardless of posture, PCA can be used to correctly determine the subject’s axis. Thus, compared to the work of Yang and Lin,¹ the proposed approach offers a more accurate calculation of the subject’s actual inclination, thus increasing detection accuracy.

Experimental results and discussion

Fall detection system assessment

To assess system performance in accurately determining falls, experiments were conducted with falling and non-falling subjects to determine system results in terms of true positive (TP), false positive (FP), false negative (FN), and true negative (TN; see Table 1).

Table 1.

Behavioral states.

Actual conditionTest outcome	Fall	No fall
Fall detected	True positive (TP)	False positive (FP)
No fall detected	False negative (FN)	True negative (TN)

TP indicates that a fall event was correctly recognized, FP indicates that a fall event was recognized despite no fall having occurred, FN indicates that a fall occurred but was not detected, and TN indicates that the system correctly found no fall had occurred.

From the above data, we can define system accuracy, precision, sensitivity, and F-score. Accuracy is defined as the ratio of correct detection results to all detection results as follows

Accuracy = \frac{TP + TN}{TP + TN + FP + FN}

Precision is defined as the ratio of falls to correct fall detection as follows

Precision = \frac{TP}{TP + FP}

Sensitivity is defined as the ratio of correctly detected falls to actual falls as follows

Sensitivity = \frac{TP}{TP + FN}

F-score is defined as the harmonic mean of precision and sensitivity as follows

F-score = 2 \times \frac{Precision \times Sensitivity}{Precision + Sensitivity} = \frac{2TP}{2TP + FP + FN}

F-score is used to illustrate the ideal degree of experimental methods, with a higher harmonic mean indicating a more ideal method.

Experimental results

Identical experiments were performed in three contexts: an empty classroom, a laboratory, and a corridor. In the tests, subjects walked back and forth three times, sat three times, or stooped to pick up an object on the floor three times. During the tests, subjects either fell or did not. Falling tests were conducted with falling in a direction either parallel or nonparallel to the camera view, each repeated six times.

Figure 10 shows non-fall behavior detection results in different contexts: walking, sitting, and picking up objects, with the green frame in each image indicting non-fall behavior detected. The three subjects completed a total of 81 non-fall actions, with detection results presented in Table 2, and analysis results presented in Table 3. Table 2 presents that FP is zero, indicating that the system did not generate any FP outcomes, with an accuracy of 100%.

Figure 10.

Detection of non-fall behavior: (a) walking, (b) sitting down, and (c) stooping to pick up objects.

Table 2.

Non-fall detection test results.

Actual conditionTest outcome	Fall	No fall
Fall detected	0	0
No fall detected	0	81

Table 3.

Non-fall detection result analysis.

	Proposed method (%)
Accuracy	100.0

Figures 11 and 12, respectively, show fall and non-fall behavior parallel and nonparallel to the camera view. In the figures, the red frame shows fall behavior detected by the system. Three subjects engaged in 108 instances of fall behavior. Test results are given in Table 4 and are analyzed in Table 5.

Figure 11.

Detection of falls walking parallel to the camera: (a) prior to fall and (b) after fall.

Figure 12.

Detection of falls walking perpendicular to the camera: (a) prior to fall and (b) after fall.

Table 4.

Fall detection test results.

Actual conditionTest outcome	Fall	No fall
Fall detected	104	0
No fall detected	4	0

Table 5.

Fall detection result analysis.

	The proposed method (%)	Method in the work of Yang and Lin¹ (%)
Precision	100.0	94.31
Sensitivity	96.30	85.57
F-score	98.11	89.73

Table 4 presents an FN count of 4, indicating that the system missed several falls, but these instances all occurred when the fall action was blocked from the camera’s view by another subject. Under such conditions, the system treats both subjects as a single object, resulting in failure to correctly acquire the object features. The proposed method has a precision of 100%, sensitivity of 96.30%, and an F-score of 98.11%, while the results in the work of Yang and Lin¹ provide a precision of 94.31%, sensitivity of 85.57%, and F-score of 89.73%.

Discussion

Based on these experimental results, the proposed 3-D inclination calculation approach improves on the fall detection mechanism proposed in the work of Yang and Lin,¹ raising the sensitivity from 85.57% to 96.30%, thus reducing the incidence of FN detection results, especially when the subject falls in a direction parallel to the camera.

This article presents a fall detection system using a depth camera. The proposed solution differs from others using 2-D images in that the depth camera provides depth information for each pixel, allowing us to determine the object’s position in 3-D space, thus allowing for object tracking and fall detection.

Conclusion

The proposed method first reconstructs the 3-D object to correct for errors when the subject falls parallel to the camera, and then uses a PCA approach to calculate the 3-D inclination as a key feature for determining falls. The proposed system detects non-fall behavior with 100% precision, while fall behavior is detected with 100% precision, 96.30% sensitivity, and an F-score of 98.11%. The system uses a depth camera device, making it inexpensive and easy to use, and thus suitable for use in hospital and home environments for hospital patients and those with reduced mobility, thus increasing the safety of living environments while reducing the cost of nursing care.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This research was supported by the Ministry of Science and Technology Grant ID (MOST 105-2221-E-106).

ORCID iD

Chien-Feng Wu

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