Gait analysis with curvature maps:
A simulation study

All four curvatures can be calculated at a given point on a surface based on the principal curvatures ($\kappa_{1}$ and $\kappa_{2}$) which are defined as the maximum and minimum values of the curvature of the curve resulting from the intersections of a normal plane with the surface at that point.

In practice, we need to compute discrete curvatures from a 3D mesh (e.g. reconstructed from a 3D point cloud generated with a depth camera). An effective way to do it is through the discrete Laplace-Beltrami operator [7]. It calculates curvature from the mesh surface of the object instead of principal curvature. With a 3D mesh surface S whose vertices are $v_{i}$, through the discrete Laplace-Beltrami operator [7], we can simply compute the mean curvature as:

On the other hand, if we suppose that $\theta_{j}$ is the angle created by 2 vectors $\vec{v_{i}v_{j}}$ which $v_{j}$ is a neighbor of $v_{i}$ and $A_{v_{i}}$ can be simply a $\frac{1}{3}$ of the areas of the triangles that form this ring, the Discrete Gaussian Curvature is defined as:

Knowing the mean curvature and Gaussian curvature, we can compute $\kappa_{1}$, $\kappa_{2}$ and $K_{rms}$ base on these curvatures.

Notice that the authors know that formulas (5) and (6) have some limitations but a complete analysis of discrete curvatures is beyond the scope of this paper.

To conduct our study we used MakeHuman [9] to simulate a 3D model of a 25-year-old male subject, 173 cm tall.

All walks, whether normal or abnormal, are a repetitive process of gait cycles. A gait cycle can be divided into two halves corresponding to the left and right half of the body [5]. Furthermore, each half cycle can also be divided into four basic postures including: contact, low, passing and high (see Fig. 1). Each gait has its own properties but all gaits must include these four types of posture. The animation of the simulated human body was done with Blender [10]

Each posture of walking can typically be identified according to the joints of the human body. For instance, we can notice that the knee and elbow joints are constantly active during walking to help the body move forward and balance. Therefore, we can expect that the curvature maps in these areas will change dramatically and could characterize the person’s gait through the 8 postures in the walking cycle.

In addition to assessing the change of curvature in a region of interest during the walk, we can compare the results between normal and abnormal gaits to demonstrate significant differences. We can evaluate an interesting curvature area based on three criteria:

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  The magnitude of the curvature in the area.

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  The size of the area with a curvature of interest.

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  The location of the area with an interesting curvature.

To evaluate all these factors, we conducted a simulation study on three gaits, including a normal gait and two abnormal gaits which are described as follows (see Fig. 2):

Discrete curvature maps are computed (section II.B) and displayed as a dynamic color map that changes from blue (negative) to green (zero) and to red (positive) depending on the magnitude of the specific curvature at a certain point on the human body during the motion (see Fig. 4 and 5).

To illustrate the quantitative change of curvature, we focused on one interesting area, which is the knee. We collected data from the knee section by looking at its anthropological position in the area of about 25-30% of body height. Points with maximum Gaussian curvature were considered to calculate the center point of the knee area. Then, all the points within a 5 cm radius were used to calculate the average curvature within the knee area as a function of time for the four basic curvatures described above.

The results are presented in Fig. 3 and show the magnitudes of the four curvatures in the left and right knee areas for two walking cycles. Because the subject was a humanoid simulation, the set of points in the knee area was not affected by noise for the two cycles. From theses results, we can draw a few comments as follows:

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  Because of its specific calculation emphasizing dome-like features, Gaussian curvature (blue curve) has high cyclic oscillation in the knee area compared to the other 3 types of curvature. We observed that this was the best type of curvature for showing interesting characteristics of the body during gait analysis. For this reason, We will use this curvature to illustrate curvature maps for the rest of the paper.

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  For a normal gait (first row in Fig. 3), the curvature graphs of the two knees are exactly the same, they only deviate from each other by half a cycle as expected.

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  The right knee area (second column in Fig. 3) for all three gait types displays similar behaviour. The extreme points appear usually in the same key frame. This is understandable because the subject’s right foot is relatively free to move in all three gait types. The difference only occurs by the nature of the constraints on the body motion when walking abnormally and walking normally.

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  The left knee curvature maps for the locked left leg (first column, second row) is much more stable. This was expected since in theory, the curvature values of the left knee should be almost constant. However, the act of extending the legs toward the side affected somewhat the detected area due to imperfect detection causing small curvature fluctuations.

In summary, we can infer that if the curvature graphs of both knees are similar and only deviate from each other by half a cycle, then the person is walking normally with a symmetric gait. Otherwise, further investigations are needed to establish the nature of the gait disorder.

Assuming that a normal walk is nearly symmetrical because of the equal role of each leg for walking (except for their half-a-cycle offset), we propose to evaluate the whole curvature map of the body by showing dynamically the left and right halves together for the same corresponding posture by aligning the postures that were originally offset by half a cycle. This means that the right contact posture is displayed with the left contact posture for comparison purpose and so on. An example is shown in Fig. 4 for the passing posture with the Gaussian curvature.

We can see that the curvature map is symmetrical with respect to the vertical red axis for the two halves of the body for a normal walk. As expected, for the two types of abnormal gait, at the passing pose, there are noticeable differences at the level of the legs between the two halves of the body. The curvature in the area of the right knee of both abnormal gaits is similar to the normal gait, but the amplitude of curvature for the left knee area is less, in other words the left knee is less bent than the right one. Moreover, the position of the knees is uneven in this passing pose.

In a nutshell, for these simulations, if we measure the position and curvature of the knee area in two postures, half-a-cycle apart, we could predict whether that gait is in the normal or abnormal class. Of course, other areas could be of interest for other types of abnormal gaits. Note that in the presence of noise, the average of several walking cycles can provide good curvature maps.

Another method to investigate the symmetry of curvature maps is to average the curvature across all frames of a gait cycle instead of studying one posture at a time. This way, a single image reveals the symmetry of the whole gait cycle.

Although the 3D model does not show many noisy points, in practice when a 3D mesh is computed form a point cloud obtained with a depth camera the noise could be an issue, the average of curvature maps could be extremely effective in this case which is another advantage of this mapping method.

From the color map shown in Fig. 5, we can see that some less interesting details are reduced by calculating the average curvature. Interesting positions such as knees, shoulders, wrists and ankles are highlighted. The calculation of the average curvature map not only significantly reduce the noise that exists on the object surface, it can also show the characteristic properties of the gait. Similarly to the previous analysis, the normal walk shows the expected symmetry of the curvature map with respect to the vertical axis.

The locked leg simulation exhibited low value of curvature at the left knee level due to the locked knee but also because of the averaging of several postures where the left leg deviates to the left side. For the other abnormal walk, we observed an height difference between the knees. This happens because the front leg always has to lift higher than the behind leg.

As a result, each gait gives an average curvature map of a completely different nature. This demonstrates the potential of this method to assess gait anomalies and could assist the clinicians in recognizing a particular disorder during walking.