NHSJS Reports

Object Average Speed Calculation in XZ-Plane Using Monocular Depth Estimation and Vanishing Point Geometry

August 23, 2026

118

Abstract

In this paper, we propose an accurate method that utilizes monocular depth estimation (MDE) and vanishing point geometry to track an object’s average speed in the xz-plane on a flat environment over a period of time only requiring an estimate of the horizon line and a grounded object with known width and depth dimensions. Other methods that try to track an object’s speed either (a) work in only one dimension, (b) depend on camera calibration, (c) require camera intrinsic parameters, or (d) use more than one camera. First, we split this challenge into two parts: finding the depth component (z-axis) and width component (x-axis) of speed. Since speed is distance over time, we can further simplify the problem to the task of finding the distance that an object travels in the x and z dimensions over two frames. After using our method which utilizes a grounded reference object, we get the displacements in the x and z axes, combine them, and take the magnitude of the resulting vector to get the average speed in the xz-plane. We use Supervision and the You Only Look Once version 8 model (YOLOv8) to automate this process. In testing, we found the average speed estimations to be no more than 11% off, with the average being less than 6%. This low-cost yet accurate approach to object speed detection can not only be applied to autonomous driving with the development of self-driving vehicles, but also to sports analysis by determining the speed of moving objects such as runners or soccer balls on flat ground.

Keywords: Monocular Depth Estimation, YOLOv8, Vanishing Point Geometry, Object Tracking, Speed Estimation

Introduction

While there are many modern approaches that utilize computer vision to accurately estimate the average speed of an object in a video, a critical gap in the field remains in achieving this accuracy without relying on expensive hardware or impractical camera calibration setups. This paper addresses the gap by investigating this question: Would it be possible to achieve the same accuracy as these state-of-the-art (SOTA) approaches to speed estimation by only using an in-frame reference object and vanishing point geometry on the flat ground? In this paper, we came up with a cheaper, more accessible method for determining an object’s speed in the xz-plane and hypothesized that outside of extreme conditions such as a tilted camera angle or long-distance objects, the proposed method would be able to accurately determine the speed of an object in the xz-plane.

Monocular Depth Estimation (MDE) is a computer vision technique that estimates the distance of objects from a single image or video without relying on stereo or Light Detection and Ranging (LiDAR) data. Accurate depth estimation is critical for many real-world applications, including autonomous driving, sports analysis, and robotics. While models such as Depth Anything v2 have made significant progress¹, they still are often inaccurate due to a lack of real-world scale²^,³. Many of the current state-of-the-art solutions often require camera calibration⁴, rely on camera parameters⁵^,⁶, or use multiple cameras to accurately determine the depth of an object⁷ ^,⁸. These constraints may limit applications in circumstances when such prerequisites cannot be met.

We propose a simple yet effective solution that doesn’t require any additional information. Our goal is to just use monocular vision plus user-inputted metric calibrations to determine the average speed of an object in the xz-plane on flat ground, without needing any extra procedures or information that may be difficult to determine. For example, in a random online video, unless there are camera parameters recorded or a different perspective provided, it would not be possible to use any of the previously mentioned methods to analyze the video. However, the proposed method can be used in this case.

By using the model output to provide a relative depth combined with real-world measurements, we are able to accurately determine the displacement of depths of any two pixels in an image. This, combined with our width calculation method that only requires a ground truth and an estimate of the horizon, allows for taking any video, inputting 3 references (reference depth, reference width, and horizon), and obtaining an average speed estimation.

Methods

Accuracy Testing Depth Anything v2

Before we started, we had to test out how accurate Depth Anything v2 is. To do this, we tested the model by using regular photos on a football field so that distances such as 10 yards, 20 yards, and 30 yards were marked as ground truth within the image. The results revealed that absolute depth values were very inaccurate, but the relative depth was accurate and consistent. For example, a distance of 10 yards (9.14 meters) was predicted as 19.68 meters, while 20 yards was estimated at 38 meters — approximately double the earlier prediction. Similarly, at 30 yards, the model returned 58.69 meters, maintaining a near-perfect 3:1 ratio to the 10-yard estimate.

To further test this observation, we took 15 datapoints each from 4 different photographs featuring different camera angles and different zoom levels. For each image, we extracted depths on the left, at the center, and on the right at 5 different known depths, for a total of 15 points of data per image, to determine if lateral positioning would affect the accuracy of the relative depth estimation. The predicted depths were then plotted against the ground truth depths for each image, and a linear regression was taken for all 4 datasets to determine if the relative depth estimation held strong.

Camera Configuration	Coefficient of Determination (R²)	Root Mean Squared Error (RMSE)
Image 1: Front View, Zoomed Out, Small Roll	0.997	0.415
Image 2: Front View, Zoomed In, Big Roll	0.971	3.465
Image 3: Diagonal View, Zoomed Out, No Roll	0.998	0.598
Image 4: Front View, Zoomed In, No Roll	0.998	0.560

Table 1 | Linear Regression Parameters for Relative Depth Accuracy Testing

Overall, we found the correlation of the predicted depth to the actual depth to be very high in all 4 images, with the lowest coefficient of determination being 0.971 for image 2. Additionally, the root mean squared error was consistently low for all images other than image 2, with the highest being 0.598. Since the model overestimates the actual depth in an image with predicted values ranging from 10-80 meters, a root mean square error of < 1 is exceptionally small.⁹.

Looking at image 2’s configuration, we can attribute its decrease in precision compared to the other images to the significant roll of the image. This is also the reason why image 2 had the highest root mean squared error as well as the lowest coefficient of determination.

*Figure. 1 | Graph showing the linear regression of the predicted depth versus ground truth depth for image 2.*

As distances increased in image 2, the model became more uncertain about relative depths along the same distance. The relative depth increased from left to center and from the center to the right, which makes sense because the model is trying to correct the slant that it perceives. Other than image 2, the other images performed excellent in the test, and we can conclude that while roll can have a noticeable impact on the relative depth estimation, the camera angle and the zoom of the image does not.

Doing some further testing, we found that in images where an object lacked a concrete background such as a tall object or an object in the air, the depth prediction becomes inaccurate as demonstrated in Figure 2, which makes sense since the datasets that MDE models are trained on are primarily ground-based¹⁰.

*Figure. 2 | Example of picture (left) vs depth map (right) showing that depth estimation from Depth Anything v2 is less accurate with the lack of background.*

These findings show that Depth Anything v2 is the only accurate at estimating relative distances on or near ground level when the image has minimal roll, and we should focus on how to scale Depth Anything v2 estimations properly given these limitations.

Improving Depth Anything v2 to Calculate Depth Distance

We explored many methods to try to translate Depth Anything v2’s accurate relative depth perception into real-world depth, such as using stereo vision which uses multiple cameras and viewpoints, and deep learning methods that utilize camera parameters. However, we decided against these SOTA approaches because they required additional information or complex setups, which would limit their applications in real world scenarios.

Thus, we chose what we thought would be the best and most effective way which was by using a known reference object with known dimensions within the frame to calibrate the model. Essentially, we are using a reference object to find the scale factor and then applying it to the output to get a more accurate result. To obtain the scale factor, two points on an image are selected, and the corresponding raw relative depths at both points are provided by the depth map generated by Depth Anything v2, denoted as d₁ and d₂. Given a user-inputted measurement representing the real-world depth between the two selected points denoted as $D_{\mathrm{ref}}$ the depth scale factor $S_{\mathrm{depth}}$ is derived as:

(1) $\begin{equation*}S_{\mathrm{depth}}=\left|\frac{D_{\mathrm{ref}}}{d_1-d_2}\right|\end{equation*}$

The real-world depth between two arbitrary points can then be determined by multiplying their distance in relative depth map values with the depth scale factor in the following equation:

(2) $\begin{equation*}Z=\left|d_1-d_2\right|\times S_{\mathrm{depth}}\end{equation*}$

For example, by placing a known MacBook Air (30.41 cm), within the frame and clicking two endpoints (as seen in Figure 3), we can then compare the depth estimate to ground truth, and then use that as a scale factor to convert any estimated depth to the actual depth.

Figure. 3 | Example of placing object with known dimensions (MacBook Air) and selecting two points.

After evaluating 40 scaled depths versus unscaled depths over a variety of images was sampled, and the percent errors were compared as displayed in Figure 4. Since Depth Anything v2 has a max depth estimation of 80 meters, distances far away will be compressed and relative depths become non-linear. Taking this into account, we only sampled distances that Depth Anything predicted to be less than 60 meters, which translated to absolute depths of less than 30-50 meters depending on the image. Note that the unscaled error data has two “piles”, one at around 20% percent error and another at around 90% error, due to different images possess different scale factors because the relative depth values are disproportionate for different images.

Figure 4 | Histogram displaying the improvement in depth estimation (under 60 meters) after utilizing a scale factor.

The unscaled depths directly taken from the raw depth map of Depth Anything v2’s output had an average percent error of 32.9%, with a standard deviation of 24.2%. After scaling, both the mean and standard deviation of the percent error drastically dropped, to 3.8% and 2.8% respectively. This validates the accuracy of the proposed method to determine absolute depth using a scale factor.

Using Vanishing Point Geometry to Calculating Width Distance

Figure 5 visualizes the camera coordinate system, where the x, y, and z axes represent the width, height, and depth respectively¹¹.

Figure. 5 | A camera’s coordinate system along with the rotations on each axis.

In the previous section, we have already established the process to determine the difference in depth between two arbitrary points in an image. Using this, we can calculate the depth component of an object’s speed (i.e., along the z-axis) by dividing the difference in depth by time over two distinct frames. As mentioned previously, a limiting factor of Depth Anything v2’s relative distance accuracy is when an object lacks a background reference (e.g., in the air), making our scale factor inadequate to accurately determine the real-world depth of the object. Due to this constraint, we choose to direct our focus to the case where the object is traveling on the ground, disregarding the y-component and only considering the xz-plane.

To determine an object’s average speed in the x-direction, we used vanishing point geometry along with a user-calibrated reference width, which is similarly produced and utilized as the reference depth but along the x-axis instead of the z-axis. Since Depth Anything v2’s depth estimation is the forward distance from the camera’s z-axis, we want lines projected from the vanishing point to be perpendicular to the camera z-axis in the 3D space. The farther the object is from the z-axis in 3D space, the closer the 2D x-value of the vanishing point is to the center of the image if the camera has no roll. Therefore, all lines parallel to the z-axis in the xz-plane appear to converge to the vanishing point in the image, which has an x-value at the center of the image. This behavior is consistent with established properties governing linear transformations on flat surfaces¹².

If the camera has some roll, then the x-value of the vanishing point would not be in the center of the image. Thus, minimizing roll is crucial as we assume that the x-value of the vanishing point lies in the center of the image. Additionally, Depth Anything v2’s relative depth is inaccurate for an image that is slanted. To determine the y-value of the vanishing point in the image, as exemplified in Figure 6, we can observe that in a flat plane, the vanishing point always lies on the horizon line¹³. Therefore, if the camera is pointed straight forward (i.e. no pitch), the horizon line will be at the center of the image, meaning that the y-value of the vanishing point will also be at the center of the image.

Figure. 6 | Example of vanishing point lying on the horizon.

Figure 7 shows how if the camera has some pitch, the horizon line will shift up or down dragging the vanishing point along with it¹⁴.

Figure. 7 | Effect of pitch angles on the horizon line.

There are many ways to track the horizon line of an image. The pitch of the camera and the vertical field of view (FOV) can be used to calculate the horizon line¹⁵, a horizon/vanishing point AI model can be utilized to estimate it¹⁶, or the user can input the horizon line manually and estimate the height where parallel lines converge. We did not choose to calculate the horizon line directly from the pitch and vertical FOV of the camera because it would violate our method’s advantage: we don’t need camera parameters. We chose not to use AI models to estimate the horizon line because their accuracy is biased towards images with specific features due to the dataset that they are trained on. For a universal setting, this meant that the user would also have to input an estimate of where the horizon lies. There will possibly be inaccuracies to the projection if the vanishing point position is off, but as long as the input horizon line is near the actual horizon line, there would be an insignificant error as seen in Figure 8.

The $x$ -coordinate of the projection of a pixel onto an arbitrary $y$ -value $y_2$ given the coordinate of the pixel $(x_1,y_1)$ and vanishing point $(vp_x,vp_y)$ is denoted as $x_{\mathrm{proj}}(x_1,y_1,y_2)$ and derived as:

(3) $\begin{equation*}x_{\mathrm{proj}}(x_1,y_1,y_2)=x_1+(y_2-y_1)\times\left(\frac{x_1-vp_x}{y_1-vp_y}\right)\end{equation*}$

Like the reference depth, the reference width is user-inputted by selecting two points on an image, denoted as $(a_x,a_y)$ and $(b_x,b_y)$ , and providing the corresponding real-world width $W_{\mathrm{ref}}$ . To determine the width scale factor at any $y$ -value in an image denoted as the function $S_{\mathrm{width}}(y)$ , we utilize both the reference width and the projection equation above as follows:

(4) $\begin{equation*}S_{\mathrm{width}}(y)=\frac{W_{\mathrm{ref}}}{\left|x_{\mathrm{proj}}(a_x,a_y,y)-x_{\mathrm{proj}}(b_x,b_y,y)\right|}\end{equation*}$

To determine the real world-width between two arbitrary points in an image $(x_1,y_1)$ and $(x_2,y_2)$ , we project the first point onto the $y$ -value of the second one and multiply the difference in $x$ -values with the appropriate width scale factor.

(5) $\begin{equation*}X=\left|x_{\mathrm{proj}}(x_1,y_1,y_2)-x_2\right|\times S_{\mathrm{width}}(y_2)\end{equation*}$

It is true that the farther away objects are, the closer they are to the vanishing point, and therefore the difference in the estimated projection versus the real projection would be much greater. However, it is already established that Depth Anything v2 cannot accurately determine the relative depth of objects far away, meaning that our method can only work accurately for closer objects. Closer objects are farther away from the vanishing point, so the difference in estimated projection versus the actual projection is negligible as illustrated in Figure 8.

Figure. 8 | Example of a slight inaccurate estimation of the vanishing point.

Figure 9 breaks down the entire process of determining the real-life horizontal distance between any two pixels into 4 steps.

Figure. 9 | Steps to calculate the real-world width between two arbitrary points in an image.

Calculating Average Speed Over Two Frames

After determining the real-world depth distance $(Z)$ from equation 2 and the real-world width distance $(X)$ from equation 5, we can use the Pythagorean theorem to calculate the total Euclidian distance travelled in the $xz$ -plane, and divide it by the change in time $(\Delta t)$ over two frames to get the average speed $(v)$ of the object in the following equation:

(6) $\begin{equation*}v=\frac{\sqrt{X^2+Z^2}}{\Delta t}\end{equation*}$

Data Acquisition and Camera Specifications

The videos we used for evaluations were captured using a monocular camera system fixed at a stabilized position overlooking the flat ground with minimal roll. The videos were captured with a resolution of 1920 x 1080 pixels and with a constant frame rate of 30 frames per second.

The time interval $\Delta t$ utilized in the denominator of the speed calculation in Equation 6 is derived directly from the following equation:

(7) $\begin{equation*}\Delta t=\frac{N_{\mathrm{frames}}}{FPS}\end{equation*}$

where $N_{\mathrm{frames}}$ represents the number of frames between the two time frames selected for speed evaluation, and $FPS = 30$ .

where $N_{\mathrm{frames}}$ represents the number of frames between the two time frames selected for speed evaluation, and FPS = 30.

Object Detection Using YOLOv8

To automate the speed calculation process instead of having the user choose two frames and pick out the specific position of the object, we decided to use computer vision to handle this task. For object detection in a video, we used Supervision¹⁷, which utilized the YOLOv8 framework to detect and identify objects in individual frames¹⁸, and the ByteTrack tracking algorithm¹⁹. This builds upon foundational real-time regression-based detector architectures that revolutionized single-shot localization networks²⁰.

To optimize object tracking continuity for high-speed objects, the tracking parameters were deliberately configured to handle the rapid displacements. Specifically, the tracking activation threshold (track_activation_threshold) was set at a low value of 0.05 to maintain the identity of a tracked object even if the detector’s confidence is lowered by blur due to the speed of the object. To prevent false-positive association, the minimum matching threshold (minimum_matching_threshold) was constrained to 0.95, meaning that the frame-to-frame bounding boxes must match at least 95% to maintain identity. Finally, the tracking frame buffer (lost_track_buffer) was set to 60 frames to accommodate any faults in detection or obstruction of object for up to 2 seconds at a frame rate of 30 fps, maintaining tracking continuity throughout the entire sequence. This configuration optimizes modern spatial box association metrics under rapid, real-time motion states²¹.

However, just calibrating the parameters alone could not solve the issue that really fast objects would not get tracked at all. When looking into the problem, we realized that YOLOv8 was still detecting the objects in individual frames, so we figured the issue lies in Supervision’s code. The problem was that the bounding box of the object was too small, so it would not be recognized as the same object by Supervision. To fix this, we increased the bounding box’s width by 5 times and height by 3 times, allowing for smooth object tracking.

Although increasing the bounding box’s size raises some concerns about its impact on the precision, the expansion did not shift the centroid of the bounding box, which is what we are basing off of to determine an object’s position in an image. Since we uniformly expanded the bounding box to maintain symmetry on all sides, this tracking modification merely ensures tracking continuity across quick frames while preserving absolute spatial positioning.

Using the user’s inputs (e.g. reference height and width, start and end time, tracked object’s ID number, horizon estimate), we can run the two frames through Depth Anything v2 and use the calculation methods explained in the methodology section to calculate the average speed of any trackable object in the xz-plane.

Procedure Summary

Step 1: Video runs through object detection.

Step 2: User inputs the time frame.

Step 3: Runs the two frames through Depth Anything v2 to get relative depth.

Step 4: User identifies object to perform the speed calculation on.

Step 5: Asks the user for reference object width, reference object depth, and estimate of horizon.

Step 6: Use the algorithm to determine displacement in x-axis and z-axis.

Step 7: Calculate and output the average speed in the xz-plane.

Ethical Considerations

Because the proposed speed estimation framework relies on real-time object tracking with the YOLOv8 model, it is necessary to examine potential risks of using this technology. In fact, it was due to the ethical concerns about the misusages of YOLO in military applications such as autonomous weapons that, in 2020, one of the original founders of YOLO Joseph Redmon decided to step away from any further development²².

To mitigate these risks within our research, several guardrails are maintained. First, we made sure that the object tracking method operates strictly at the geometric level, simply utilizing bounding box calculations to track object positions rather than applying facial recognition, license plate recognition, or any other type of identification mechanism that potentially threatens the security of subjects being surveyed in the video. Second, since the method relies on specific, manual calibration for reference depth, reference width, and horizon line estimation, it is very difficult to implement mass and dynamic surveillance. Ultimately, while acknowledging the ethical challenges in real-time computer vision, we ensure that the applications for this framework are intended to be a localized tool for promoting traffic safety or sports analysis only. Other applications that utilize similar object tracking frameworks emphasize the privacy protection of this approach²³^,²⁴.

Results

We used a speed gun to obtain the actual speed of a soccer ball and ran the video of the ball through our program to determine the accuracy of our method.

Calculated (mph)	Actual (mph)	Percent Error (%)
40.1	42	4.5
48.0	49	2.1
34.8	39	10.7
33.7	34	0.8
34.7	35	1.0

Table 2 | Depth Speed

Average Error: 3.8%

Calculated (mph)	Actual (mph)	Percent Error (%)
43.6	44	0.8
43.4	45	3.6
34.3	36	4.6
24.8	27	8.1
35.3	39	9.4

Table 3 | Width Speed

Average Error: 5.3%

Calculated (mph)	Actual (mph)	Percent Error (%)
36.3	36	1.0
27.7	29	4.6
27.5	27	1.8
27.6	27	2.1
28.6	28	2.2

Table 4 | Depth and Width Speed

Average Error: 2.3%

Discussion

Accuracy Comparison to Other Methods

To evaluate the relative merit of the proposed framework, our average speed precision was compared directly to traditional monocular baselines. Specifically, the vehicle speed measurement framework proposed by Lian et al. maps 2D image coordinates to a 3D ground plane for tracking, reporting that 93.3% of their tested vehicles fell under a 5% speed error margin, with average errors ranging between 3.6% and 4.5%⁴^. However, their coordinate transformation matrix is entirely dependent on rigid, pre-calibrated camera parameters, specifically requiring precise measurements of the camera’s mounting height and tilt angle. In contrast, our method relies on an in-frame grounded reference object to scale the relative depth of Depth Anything v2, achieving a competitive average speed error of just 2.3% in combined tracking sequences, while maintaining complete compatibility with uncalibrated, casual, or single-view videos as long as a reference object is present and an estimate of the horizon is given.

Limitations and Reflection

Notable constraints of our test for the accuracy of the proposed method are the lack of sample size and the limited variety in environments in which the videos were taken, relying on sets of five trials per test that tracked a soccer ball under consistent conditions. While five data points are insufficient to establish a meaningful generalization for the accuracy of the method under other circumstances such as different weather, lighting, or camera angles, the results serve as a proof-of-concept that verified our coordinate-based geometry driven methodology. The low average error margins across these tests indicate that the geometric logic functions under basic conditions. However, we acknowledge that increasing the sample sizes and test varieties to include diverse objects, farther distances, and different camera positionings is necessary to fully validate the method’s adaptability for other applications such as autonomous driving.

Conclusion

In this study, we propose an accurate and easy-to-use method that requires only ground truths of the dimension of a reference object and the estimation of the horizon line to precisely determine the average speed of an object in the xz-plane on flat ground. Using Depth Anything v2’s state-of-the-art relative depth estimation with the proposed calibration using the dimensions of a reference object, along with the proposed method to estimate x-axis displacement using the vanishing point and input horizon location, and finally Supervision’s object detection method, which utilizes the YOLOv8 model to automatically determine object positions, we were able to achieve an average speed estimation error of less than 6%.

Our method is accurate, accessible, and can be applied to many different situations in everyday life without relying on expensive equipment such as LiDAR or multiple cameras. We hope this proposed design can open the door to or encourage more to acknowledge the possibility of using a reference object based three-dimensional reconstruction, instead of focusing on focal length and other camera parameters that, without context, may be hard to attain.

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