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Abstract
Having laid down previously the foundations of the combined multiple-wavelength range-gated active imaging (WRAI) principle recording a moving object in a four-dimensional space represented by a single image, it was necessary to know if a Doppler effect could appear in the direction of the radial velocity of the object. This is due to the fact that this imaging principle requires the emission of laser pulses at a certain frequency in relation to this speed. To know the limits, the radial velocity of the object was supposed to go up to relativistic speeds. From a theoretical approach, some temporal behaviors leading to the Doppler effect were shown. To validate this effect experimentally, a setup based on the fictional positions (represented by mirrors) of an object moving at extreme speeds was performed. The results obtained with WRAI confirmed the presence of the Doppler effect, which must be taken into account when the radial velocity of the object is extreme. In this case, some conditions must be met as in the period and the time of the shutter aperture. Singular behaviors have also been shown in the case of relativistic speeds. So despite the Doppler effect, the distance of the fast-moving object and the different moments are well identified according to the different wavelengths in the combined WRAI principle.
© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
This direct visualization technique uses an image sensor array and its own illumination source (laser). In applications, the active imaging is used to scan in depth a scene and to restore this scene in 3D [1]. Every recorded space slice is selected according to the aperture delay of the camera and is visualized according to the light pulse width and the aperture time of the camera [2]. Furthermore, as this technique is range-gated, it is possible to observe an object behind scattering environments [3]. Recently, instead of using a single wavelength and being dependent of the video frequency by recording a visualized slice per image, it became possible with the multiple-wavelength range-gated active imaging (WRAI) in juxtaposed style to restore the 3D scene directly in a single image at the moment of recording with a video camera [4–6]. Thus, each emitted light pulse with a different wavelength corresponds to a visualized slice at a different distance in the scene. In the 3D restoration, the visualized zones corresponding to the wavelengths are juxtaposed one behind the other in the scene. However there is also the superimposed style where the wavelengths correspond to precise moments in time [7]. Each moment is directly recorded in a single image in which each wavelength frozen in time is a position of the moving object like a chronophotograp [8] but with the difference that it is possible to directly identify the chronological positions thanks to the chronological order of the wavelengths. Recently, by combining these two styles, it has been shown that it was possible to know the position of a moving object in the three dimensions of space at different moments by using the wavelengths of each style from a single image recorded independently of the video frequency, thus giving four dimensions in total [9]. Because in the WRAI principle several laser pulses are emitted to the moving object at a certain frequency, a Doppler effect should appear varying the frequency of the back light information at the observer place [10]. To know the limits of this effect in the WRAI principle, the speed of the object was assumed to be relativistic during the study to verify certain theoretical results [11,12]. Therefore in this paper, we propose to study the Doppler effect in multiple-wavelength range-gated active imaging.
In Section 2, we have shown the existence of the Doppler effect in the WRAI principle. To verify this behavior, the results of experimental tests based on the WRAI principle were evaluated and compared with the theoretical results in Section 3. From there on, the Doppler effect was taken into account in the combination of both styles of the WRAI principle in Section 4.
2. CONFIRMATION OF THE DOPPLER EFFECT IN THE WRAI PRINCIPLE
To show the Doppler effect in the WRAI method, its superimposed style been used inasmuch it relates the object movement as a function of time. The wavelengths freeze in time the different object positions in the recorded image. Two versions of the WRAI method were defined in superimposed style. One corresponds to the case when the camera shutter opens after the return of all the emitted wavelengths to record certain positions of the moving object. The other version corresponds to the case when the camera shutter opens after each emitted wavelength returns to record a position of the moving object. The choice of this first version avoided to manage the camera aperture according to each laser pulse return by including all returns [Fig. 1]. The aim was to observe each return relative to the moving object position to note the time parameters varying and those remaining constant. In this version, all pulses are emitted one behind the other with the same time shift before the start of the camera aperture. To give a quick idea of the result, we used the same graphic method as that of [13] in Fig. 1.
Fig. 1. Graphic determination of the intersection zone of the visualization slices of the wavelengths with the shutter aperture after the return of all the emitted wavelengths.
This graph is defined according to three axes (time, distance, and illuminance). The observer place is at the origin (${\rm{distance}} = {{0}}$). The light pulses emission and the camera aperture are defined by the time axe. The intersection area between the emission zone and the aperture zone corresponds to their convolution. The result of this convolution is given by the summation of the common surface according to the time direction. Its position is located on the distance axe and its value is given in function of the illuminance axis.
In Fig. 1, the light pulses are emitted with the same pulse width and the same frequency. Each period corresponds to a different wavelength. In case the object movement is perpendicular to the distance axis, the visualized zone of each wavelength appears on the distance axis superimposing with the others and creating an intersection region. Because in our case the object radial velocity $V$ is taken into account, the visualized zone of each wavelength appears staggered in depth. The whole of these visualized zones is delimited by the camera aperture time. Beginning from a general point of view, we get for the backward movement
where ${d_{\textit{StartB}}}$ represents the starting distance of the object moving backward onto the distance axis, $\Delta {d_{\textit{back}}}$ is the distance traveled by moving backward from the starting position to the place where the laser pulse meets the object, ${T_{\textit{back}}}$ is the round trip time of a laser pulse in the case of an object recoil, and $c$ is the light speed.From Eqs. (1) and (2), the round trip time for the first wavelength with the first period corresponds to
or with constant $V$, the visualized position of the object in the scene at each laser illumination period ${\textit{period}_{\textit{laser}}}$ relative to the starting point:Concerning the forward moving, we get
where ${d_{\textit{StartF}}}$ represents the starting distance of the object moving forward onto the distance axis, $\Delta {d_{\textit{for}}}$ is the distance traveled by moving forward from the starting position to the place where the laser pulse meets the object, and ${T_{\textit{for}}}$ is the round trip time of a laser pulse in the case of a forward movement of the object.With Eqs. (7) and (8), the round trip time for the first wavelength with the first period corresponds to
with constant $V$, the visualized position of the object in the scene at each laser illumination period relative to the starting point:The progression of the visualized position of the object in the scene at each period is linear regardless of the moving direction [Eqs. (5) and (11)], which means that the object position frequency in the scene remains constant for a constant illumination frequency and a constant object speed [Fig. 2(a)]. A difference between the two movements appears showing as in Fig. 1 that the frequency ${f_{\textit{Forward}}}$ is higher than the frequency ${f_{\textit{Backward}}}$ for the same speed value.
Fig. 2. (a) Object position frequency in the scene and round trip frequency of a laser pulse as a function of (b) the used wavelengths number and of (c) the radial velocity of the object.
At the observer place [Fig. 2(b)], the round trip time of a laser pulse varies at each new pulse. The frequency related to the recoil decreases with the period number. Inversely, the frequency related to the advance increases with the period number. Concerning their limit frequencies [Fig. 2(c)] for the recoil, the faster the object speed moves towards the light speed, the faster the frequency will move towards zero. For the advance, the more the object speed tends towards the light speed, the more the frequency tends towards
To determine other temporal parameters and to take into account the case of relativistic speed, we used the Bondi $k$-calculus [14] that we integrated into the previous graph [13] (Fig. 3). So we have for the backward movement [Fig. 3(a)] the distance between the observer and the place where the laser pulse meets the object:
with $k$ the ${{k}}$-calculus parameter and $T$ the period between the moment when the object is supposed to be at the observer place and the starting moment of the first laser pulse:Fig. 3. Bondi diagram combined with the active imaging graph (a) for the backward moving object and (b) for the forward moving object.
The radial velocity with Eq. (15) is
The radial velocity with Eq. (16) is
By combining Eqs. (17) and (18),
Given that
Eq. (19) can be writtenBy including Eq. (3) in Eq. (21), we identified again the known equation of $k$-calculus [12,14]:
Likewise, to make the link with the Bondi $k$-calculus for the forward movement [Fig. 3(b)], the distance between the observer and the place where the laser pulse meets the object is given by
The radial velocity with Eq. (23) is
The radial velocity with Eq. (24) is
By combining Eqs. (25) and (26),
Given that
Eq. (27) can be writtenBy including Eq. (9) in Eq. (29), we identified again the known equation of ${{k}}$-calculus [Eq. (22)].
Concerning the pulse return period at the observer place for the backward movement ($Perio{d_{\textit{back}}}$) and for the forward movement ($Perio{d_{\textit{for}}}$), a difference is clearly visible between both movements [Fig. 1]. To estimate this difference, each observer period was determined according to the laser illumination period Period with the Bondi $k$-calculus. It is noted that the observer period for each movement remains constant regardless of the period number.
Therefore, based on Fig. 4(a), we get for the backward movement
Fig. 4. Period of information feedback at the observer place according to the laser illumination period for (a) the backward movement and for (b) the forward movement.
By including Eqs. (30) and (31) in Eq. (32),
Taking into account Eq. (16) in Eq. (33),
Therefore, based on Fig. 4(b), we get for the forward movement,
By including Eq. (36) and (37) in Eq. (38) and taking into account Eq. (24), the laser illumination period can be written
By isolating $Perio{d_{for}}$ from Eq. (40),
Comparing the two periods [Eqs. (35) and (41)], the period $Perio{d_{\textit{back}}}$ is higher than the period $Perio{d_{\textit{for}}}$ as long as $k$ is strictly greater than 1. If $k$ is equal to 1, the object speed and its movement are zero. By including these conditions into Eqs. (35) and (41), both periods appear to be equal. The more the object speed will tend towards the light speed, the more $Perio{d_{\textit{back}}}$ will increase and will tend to infinity shifting its frequency towards zero. Likewise, the more the object speed will tend towards the light speed, the more $Perio{d_{\textit{for}}}$ will decrease and will tend to zero shifting its frequency towards infinity.
Concerning the visualized time of the object for the backward movement ($\Delta {t_{\textit{back}}}$) and for the forward movement ($\Delta {t_{\textit{for}}}$), a difference is clearly visible between both movements [Fig. 1]. To estimate this difference, each visualized time was determined according to the laser pulse width $\Delta {t_{\textit{laser}}}$ with the Bondi $k$-calculus. It is noted that the value of the visualized time for each movement remains constant regardless of the period number.
Therefore, based on Fig. 5(a), we get for the backward movement,
Fig. 5. Visualized time of the object for (a) the backward movement and for (b) the forward movement according to the laser pulse width.
By including Eqs. (42) and (43) in Eq. (44) and taking into account Eq. (16), the visualized time can be written
Likewise, based on Fig. 5(b), we get for the forward movement
By including Eqs. (47) and (48) in Eq. (49) and taking into account Eq. (24), the laser pulse width can be written
By isolating $\Delta {t_{\textit{for}}}$ from Eq. (51),
Comparing the two visualized times of the object [Eqs. (46) and (52)], the time $\Delta {t_{\textit{back}}}$ is longer than the time $\Delta {t_{\textit{for}}}$ as long as ${{k}}$ is strictly greater than 1. If ${{k}}$ is equal to 1, the object speed and its movement are zero. By including these conditions into Eqs. (46) and (52), both visualized times appear to be equal. The more the object speed will tend towards the light speed, the more $\Delta {t_{\textit{back}}}$ will increase and will tend to infinity, and the more $\Delta {t_{\textit{for}}}$ will decrease and will tend to zero.
Fig. 6. Graphic determination of the intersection zone of the visualization slices of the wavelengths with shutter aperture after each emitted wavelength return.
Finally, this version of superimposed style confirmed that the WRAI principle is subject to the Doppler effect when the observed object moves in the direction of the line-of-sight (radial velocity). For that reason, it is important to take into account this effect when calculating the time of the light information return of each wavelength to correctly open the camera shutter. Compared to the second version of the superimposed style when the camera shutter opens at each emitted wavelength return to record an object position, the laser period integrates the acquisition period [Fig. 6]. Nevertheless, the equations remain identical. On the other hand, it is important that the times ${T_{\textit{back}}}$ and ${T_{\textit{for}}}$ take into account the aperture delay and the aperture time of the camera shutter. Otherwise the object image will not appear in the image recorded by the camera. Concerning the behaviors observed previously in the first version of the superimposed style, they are also found in this second version. Likewise, the link with the Bondi $k$-calculus remains the same. On the other hand in this version where the shutter opens several times in succession, there can be some artifacts under some conditions (high frequency) visualizing the objects in the background at a specific distance [15,16].
3. EXPERIMENTAL RESULTS
To verify experimentally the Doppler effect in the multiple-wavelength range-gated active imaging in superimposed style and the validity of the results of the previously proposed equations, we performed a system based only on the model with the shutter aperture after the return of all emitted wavelengths. The second version differs from the first only in the way to record the information but the propagation phenomena of laser pulses remain the same. In our case, the difference of the wavelengths did not intervene in the time and the distances. This is why the use of a single-wavelength laser was sufficient to demonstrate the validity assuming that each pulse simulated a different wavelength [Fig. 7]. To experience the relativistic part without having an object moving at extreme speeds, we simulated the speeds by adjusting the different positions of the object that it should have if it had moved at such extreme speeds. Each assumed position of the object was materialized by a mirror facing the observer. Likewise, to adjust the emission period of the laser pulses to adapt it to the scene, we separated the laser beam into three paths. Each path was delayed for the purpose of emitting at the same frequency at the start position a series of three pulses.
Fig. 8. Measured values from the experimental setup and transcribed into the graph for (a) a rapidly moving backward object and for (b) a rapidly moving forward object.
After measuring the distances of the different positions (A, B, and C), the different times were measured according to the output signal of the laser, which is synchronous with the light pulse. To begin with, the start of each pulse (${{\rm{T}}_1},\;{{\rm{T}}_2}$, and ${{\rm{T}}_3}$) representing each wavelength was recorded by placing the photodiode in front of the beams at the start position. The second set of measures (${{\rm{T}}_4},\;{{\rm{T}}_5}$, and ${{\rm{T}}_6}$) was performed by placing the photodiode at the position of the mirrors A, B, and C in the scene. The last set of measures (${{\rm{T}}_7},\;{{\rm{T}}_8}$, and ${{\rm{T}}_9}$) was made by placing the photodiode at the observer place facing the scene. The arrangement of the mirrors in the scene was carried out in two different ways. The first arrangement to simulate an object moving rapidly backward from the observer and the second to simulate an object moving rapidly forward from the observer. However, the simulated speed of the object remained the same regardless of the moving direction. For that, the spacing between the mirrors of the scene, in the case of the approaching object, has been adjusted to have the same speed as the opposite moving. The temporal measurements for a rapidly moving backward object have been transcribed in Fig. 8(a). The temporal measurements for a rapidly forward moving object have been transcribed in Fig. 8(b). To validate the tests, a comparison was made between the theoretical values given by the previous equations and the measured values, taking into account first the distances and then the different durations. The recoil speed was estimated at ${3.44} \times {{1}}{{{0}}^7}\;{\rm{m}}/{\rm{s}}$ equivalent to 11.5% of the light speed and a $k$-calculus equal to 1.12. The feed speed was adjusted to the same speed with an accuracy of less than 1%. The maximum difference between the experimental temporal measurements and the theoretical temporal values was in the order of 0.15%. After having validated the measurements, the different comparisons between both movements concerning the frequency of the object position in the scene (${f_{\textit{Backward}}},\;{f_{\textit{Forward}}}$), the round-trip frequency of laser pulses (${F_{\textit{back}}},\;{F_{\textit{for}}}$), the period of pulse returns at the observer place ($Perio{d_{\textit{back}}},\;Perio{d_{\textit{for}}}$) and the visualization times of the object ($\Delta {t_{back,\:}}\Delta {t_{\textit{for}}}$) were conducted. Starting with the frequency of the object position in the scene, the position of the mirrors has been adapted to obtain the same speed in both directions. Therefore, the gap between the mirrors in the scene was of 2.5 cm for the backward movement and of 2 cm for the forward movement. By expressing these values in frequency, ${f_{\textit{Forward}}}$ of 50 Hz is well higher than ${f_{\textit{Backward}}}$ of 40 Hz as shown in Fig. 1. Concerning the round-trip period of laser pulses, the obtained values of ${T_{\textit{back}}}$ gave 4.8, 4.97, and 5.14 ns, and the obtained values of ${T_{\textit{for}}}$ gave 4.8, 4.66, and 4.54 ns. By expressing these values in frequency, ${F_{\textit{back}}}$ corresponds to 208, 201, and 194 MHz, and ${F_{\textit{for}}}$ corresponds to 208, 214, and 220 MHz. The backward frequency ${F_{\textit{back}}}$ decreases with the period number or more exactly with the used wavelength number and that the forward frequency ${F_{\textit{for}}}$ increases with the period number or more exactly with the used wavelength number. This observation reveals well the Doppler effect.
Fig. 9. Behaviors appearing at relativistic speeds with (a) a backward moving object, with (b) a forward moving object and when the speed of the forward moving object is (c) equal to or (d) faster than that of light.
Concerning the period of pulse returns at the observer place, with the illumination period equal to 0.64 ns, the period of pulse returns for a backward moving object $Perio{d_{\textit{back}}}$ gave 0.81 ns and for a forward moving object $Perio{d_{\textit{for}}}$ gave 0.51 ns. By comparing these values, we find the same observations as the previous theoretical approach [Eqs. (35) and (41)] like the period $Perio{d_{\textit{back}}}$ longer than the period $Perio{d_{\textit{for}}}$. Concerning the visualization times of the object, because it was possible to make the connection with the period of pulse returns, we did not try to divide temporally the laser pulse to differentiate the beginning with the end of the pulse according to the mirrors of the scene. Therefore, the visualization time $\Delta {t_{\textit{back}}}$ is greater than the visualization time $\Delta {t_{\textit{for}}}$. Consequently, all these measures and these comparisons confirmed that the multiple-wavelength range-gated active imaging in superimposed style was subject to the Doppler effect when the observed object moved in direction of the line-of-sight. It is important to take into account this effect to know the exact moment of each recorded view represented by a different wavelength. Extreme behaviors also appear with relativistic speeds. In the case of a backward moving object with a speed increasing more and more towards that of the light, the views represented by different wavelengths will be recorded more and more spaced in time [Fig. 9(a)]. Likewise, the visualization time of each view will tend to increase considerably, superimposing during this time the different attitudes of the object creating a blur in the image. In the case of a forward moving object with a speed increasing more and more towards that of the light, the views represented by different wavelengths will be recorded more and more rapidly [Fig. 9(b)]. But also, the visualization time will tend to decrease quickly, reducing at the same time the illuminance until the object is invisible in the image. In case the camera was sensitive enough to be able to render the object image and if the object speed corresponded to that of light, the different moments of the object route would arrive at the same time at the observer place [Fig. 9(c)]. Thereby, the camera would record in one instant the entire life of the object during its route. Likewise, using this type of graph [Fig. 9(d)] and by assuming that the object speed would be greater than that of light, we see that the different moments of the object route would arrive in reverse direction at the observer place. Thereby, the camera would record the objet going back in time of its route.
4. COMBINED WRAI PRINCIPLE ACCORDING TO DOPPLER EFFECT
After having studied the Doppler effect in the superimposed style of the WRAI principle, it was important to show the existence of this effect in the combined WRAI (the superimposed style with the juxtaposed style) [9]. From the beginning of this paper on, the light information return remained voluntarily free for the demonstration. Therefore, the temporal indication given by the wavelengths was lost in the recorded image. Each wavelength in the image represented well a time but with the Doppler effect its value was shifted. For the combined method, the juxtaposed style giving the object range according to the wavelength had to be added. These needs imposed some conditions [Fig. 10]. Thus, the shutter aperture period remained constant, and its aperture time remained shorter than the emission period of laser pulses dedicated to the range (juxtaposed style). This aperture time has been reduced to take into account a single wavelength of the juxtaposed style. This time is, so to speak, proportional to the radial velocity of the object. The laser pulse width of the superimposed style, with the wavelengths giving the different moments of the object trajectory, integrated all the laser pulses of the juxtaposed style. Consequently, according to the position of the moving object, its range is determined by one of the wavelengths of the juxtaposed style, and the time when the object is at this range is determined by one of the wavelengths of the superimposed style.
5. CONCLUSION
The combined WRAI principle allows the recording of a moving object in a four-dimensional space represented by a single image. This combination is composed of juxtaposed style and superimposed style. The juxtaposed style allows to restore the 3D of a scene where each wavelength corresponds to a zone visualized at a different distance. In superimposed style, each wavelength is superimposed in the scene at a precise moment in time. By combining these two styles, it is possible to know in a single image the trajectory and the direction of the object in 3D space and to determine its speed between two consecutive positions. By emitting a series of laser pulses at a certain frequency, it was necessary to know if a Doppler effect was present in relation to the radial velocity of the moving object and to study its behavior up to the relativistic speeds. A theoretical approach was carried out from the version of the superimposed style of the WRAI principle with the camera shutter opening after the return of all the emitted wavelengths. This approach showed some temporal behaviors leading to the Doppler effect. The version of the superimposed style where the camera shutter opens after each emitted wavelength return has not been fully investigated because, except for the way of recording information, the equations of the previous version were applicable. For the experimental part, a setup simulating relativistic speeds offered the possibility of adjusting the position that the object should have if it had moved at these speeds. Mirrors facing the observer were placed to materialize these positions in the scene. Concerning the laser pulses emission, the adjustment of the period was performed by delaying their different paths beforehand. Finally, in accordance with the theoretical approach, the set of experimental results confirmed that the Doppler effect appeared in the superimposed style of the WRAI. Extreme behaviors in the case of relativistic speeds disturbing the recorded image have also been shown. Consequently, when using the combined WRAI principle, it was essential to take into account this Doppler effect in the case of a fast moving object along the line-of-sight. For that, some conditions must be respected such as the constant period of the shutter aperture and its aperture time shorter than the emission period of the laser pulses of the juxtaposed style. So even with the Doppler effect, the range of the moving object in the scene and the corresponding time are well identified according to the wavelengths of the juxtaposed style and the superimposed style in the combined WRAI principle. From an application point of view, the advantage of this principle lies in the study of very fast or ultra-fast phenomena. For example, in a scene where several objects (swarm) move with different and unknown trajectories, speeds and accelerations, these physical quantities can be determined in the three dimensions of the space from the first image and without being hindered by the video frequency. From there, it is possible to anticipate the movements of objects or even to draw the cartography of movements as in PIV (particle image velocimetry).
Funding
Institut Franco-Allemand de Recherches de Saint-Louis.
Disclosures
The author declares no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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