Triangular-range-intensity profile spatial-correlation method for 3D super-resolution range-gated imaging Wang Xinwei,1,2 Li Youfu,2,* and Zhou Yan1 1
Optoelectronic System Laboratory, Institute of Semiconductors, CAS, Beijing 100083, China
2
Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China *Corresponding author: [email protected] Received 28 May 2013; revised 24 September 2013; accepted 26 September 2013; posted 1 October 2013 (Doc. ID 191226); published 18 October 2013
We present a triangular-range-intensity profile (RIP) spatial-correlation method for 3D range-gated imaging with a depth of super resolution. In this method, spatial sampling volumes with triangular-RIPs are established by matching laser-pulse width and sensor gate time, and then depth information collapsed in gate images can be reconstructed by spatial correlation of overlapped gate images corresponding to sampling volumes. Compared with super-resolution depth mapping under trapezoidal-RIPs, range accuracy and precision are improved, and a large range fluctuation due to noise disturbance is smoothed by noise suppression under triangular-RIPs. In this paper, a proof experiment is demonstrated with a range precision 2.5 times better than that obtained under trapezoidal-RIPs. © 2013 Optical Society of America OCIS codes: (110.0110) Imaging systems; (110.6880) Three-dimensional image acquisition; (100.6890) Three-dimensional image processing; (280.0280) Remote sensing and sensors. http://dx.doi.org/10.1364/AO.52.007399
1. Introduction
Three-dimensional range-gated imaging (3DRGI) first proposed in the 2000s is a relatively new technique in 3D imaging [1,2]. It has large potential in underwater imaging [3], target detection and identification [4], and spatial navigation [5] due to its long range detection, high spatial resolution, and effective work through obscurants, such as fog, camouflage, and turbid water. Up to now, 3DRGI has been mainly developed with three approaches by depth scanning [1–4], gain modulation [6,7], and super-resolution depth mapping [8,9]. For the depth scanning method, high-range accuracy needs shorter scanning step sizes and greater data-processing efforts, and thus high-range accuracy and large depth of field cannot be simultaneously realized. To solve the problem, the methods of gain modulation and super-resolution 1559-128X/13/307399-08$15.00/0 © 2013 Optical Society of America
depth mapping were developed. In the gain modulation method, a depth map can be obtained from two gate images, a gain-modulated image, and a gainconstant image, and its range accuracy is independent of laser pulse shapes. In the super-resolution depth mapping method, a 3D scene with depth resolution far beyond depth step size can be reconstructed by at least two gate images with trapezoidal rangeintensity profiles (RIPs) where the sensor gate time is twice as large as the laser pulse width. The two methods can both reduce raw data volume and simplify data processing with good real-time performance even under high-range accuracy. In the same conditions, the super-resolution depth mapping method has higher range accuracy than the gain modulation method, even though the former has lower signalto-noise ratio [10]. Furthermore, to enlarge the depth mapping range (or depth of field) while maintaining high-range accuracy, an image coding technique has been introduced into super-resolution depth mapping [11]. In other words, within the same depth of field, 20 October 2013 / Vol. 52, No. 30 / APPLIED OPTICS
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image coding of super-resolution depth mapping can obtain higher range accuracy than the gain modulation method, which is valuable in 3D imaging applications. However, energy of targets in the rising and falling ramps of trapezoidal RIPs is easily interfered by ambient noise [12], and thus big noise disturbance to the two ramps should be suppressed to resolve large range fluctuation in super-resolution depth mapping. In this paper, we present a triangular-RIP spatial-correlation method for 3D super-resolution range-gated imaging to solve large range fluctuation due to noise disturbance and further improve range accuracy and precision. 2. Triangular-Range-Intensity Profile Spatial-Correlation Method
In our method, the laser pulse width equals the sensor gate time, so that under the convolution of the laser pulse and the gate pulse the RIP of a spatial sampling volume formed by gate viewing is a triangle, which can be divided into two parts—a head part and a tail part as illustrated in Fig. 1(a)—different from three parts of a trapezoidal RIP. For gate image Si , its position Ri is determined by τi c∕2 where τi is the time
Fig. 1. Triangular RIP spatial-correlation method for 3DRGI. (a) Triangular RIP formed by gate viewing. (b) Two adjacent spatial-correlation sampling volumes. (c) 3D reconstruction by spatial correlation of triangular RIPs. 7400
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delay between the laser pulse and the gate pulse, c is the light speed in vacuum, and Si denotes the ith gate image with time delay τi. In this paper, the atmospheric refractive index is approximated as 1. The spatial interval of image Si is Rhead;i ; Rtail;i where Rhead;i and Rtail;i are the positions of the head and tail parts, Rhead;i Ri − tL c∕2 and Rtail;i Ri tL c∕2 where tL is the laser pulse width. By changing the time delay two adjacent spatial-correlation sampling volumes can be constructed in Fig. 1(b), where the tail part of image Si is overlapped by the head part of image Si1 where Si1 denotes the i 1th gate image with time delay τi1, and hereby a spatial-correlation zone is formed in Fig. 1(c). Apparently, targets in correlation zones will appear in successive gate images with a given delay step, which meets Δt tL tg where Δt is the delay step between images Si and Si1 and tg is the sensor gate time. The time sequence of the method is given in Fig. 2, where multiple exposures are adopted so that a frame of sampling gate image integrates many subframes formed by pulse pairs. A pulse pair consists of a laser pulse and a gate pulse to realize gate viewing (or range-gated imaging). The delay time τi1 of image Si1 meets τi1 τi Δt. To avoid the cross talk of target echo signal, the delay time should be less than the value of (1∕f L − tg ) where f L is the laser-pulse-repetition frequency. In our range-gated imaging system, the imaging sensor is a CCD with a gated intensifier. When the exposure time of one frame is much larger than the reading time of CCD, the number of subframes in one frame is f L ∕f CCD where f CCD is the frame rate of CCD. References [1] and [13] demonstrate that multiple exposures used in range-gated imaging can reduce the impact of ambient noise and improve image signal-to-noise ratio. It should be noted that depth information is lost in gate images, since in the process of imaging a spatial sampling volume is compacted and projected into a 2D image with depth information collapsed. Fortunately, the depth information can be reconstructed by the spatial correlation of overlapped gate images. Since the target gray level in gate images is proportional to irradiance of targets under laser illumination, to targets in the correlation zone the relationship between the pixel-to-pixel gray level ratio of the two successive gate images and target radiant energy ratio can be established as
Fig. 2. Time sequence of CCD, laser, and gate in triangular RIP spatial-correlation method.
I tail;i I head;i1
Etail;i ri ; Ehead;i1 ri
(1)
where I tail;i and I head;i1 are, respectively, the gray level of target in the tail part of gate image Si and the head part of gate image Si1 , ri is the target range, and Etail;i ri and Ehead;i1 ri are, respectively, the radiant energy of target in the tail part of the sampling volume of gate image Si and the head part of the sampling volume of gate image Si1 . According to [14] and [15], the laser pulse and the gate pulse are both in rectangular form, and then Etail;i ri can be given as Etail;i ri ξ
exp−2αri EL ri − Ri ; t − 2 tL L c r2i
(2)
where the coefficient ξ Gηt ηr f L ∕f CCD ρ∕π At ∕AL Ar . G is the gain of the imaging system, ηr and ηt are, respectively, the transmission efficiencies of receiving and transmitting optical modules, ρ is the target reflection coefficient, At is the effective laser cross section, AL is the illuminated area, Ar is the receiver area, and α is the atmospheric total extinction coefficient. For a fixed imaging system and given targets, the coefficient ξ is a constant. Similarly, from [14] and [15] Ehead;i1 ri can be given as
Fig. 3. Comparison of super-resolution 3DRGI under different RIPs. (a) Under triangular RIPs. (b) Under trapezoidal RIPs.
Based on the rules of error propagation and considering time jitter and noise disturbance, from Eq. (4) the range precision under three-dimensional rangegated imaging based on triangular RIPs (triangular RIP 3DRGI) is " I 2head;i1 c 2 σ τi σ2 σ ri 2 I head;i1 I tail;i 2 tL
exp−2αri EL R − ri Ehead;i1 ri ξ tL − 2 i1 : (3) 2 tL c ri
Therefore, one can substitute Eqs. (1) and (2) into Eq. (3) and obtain the target range as ri
I head;i1 τi c tL c ; I head;i1 I tail;i 2 2
(4)
where ri is the target range. From Eq. (4) depth information collapsed in gate images can be picked up, and a 3D scene also can be further reconstructed by a camera model. As illustrated in Fig. 3, different from trapezoidal RIPs there is no available plateau to be directly used for 3D reconstruction under triangular RIPs. In fact, an equivalent plateau is constructed in Eq. (4) by the superposition of the tail part of the current gate image Si and the head part of the adjacent gate image Si1 . The denominator of the gray-level ratio of I head;i1 ∕I tail;i I head;i1 is the equivalent plateau, which will be further discussed in Section 3A. 3. Comparison with Super-Resolution Depth Mapping under Trapezoidal RIPs A.
Range Precision
Our method and super-resolution depth mapping under trapezoidal RIPs can reconstruct spatial information collapsed in gated images and realize 3D imaging with depth super-resolution. In this section, range precision is compared under the two methods.
t2L I 2tail;i I head;i1 I tail;i 4 t2L I 2head;i1 I head;i1 I tail;i
σ 2Ihead;i1
σ2 4 Itail;i
#1∕2 ;
(5)
where σ 2τi is the variance of the delay time τi , σ 2tL is the variance of the laser pulse width tL , and σ 2Ihead;i1 and σ 2Itail;i are, respectively, the variance of I head;i1 and I tail;i . For super-resolution depth mapping under trapezoidal RIPs [8], the gate time is twice as large as the laser pulse width, and target range r0i is r0i
τi c I 0head;i1 tL c 0 ; I body;i 2 2
(6)
where I 0body;i and I 0head;i1 are, respectively, the gray level of target in the body part of gate image Si and the head part of gate image Si1 as shown in Fig. 3(b). Here, the head, body and tail parts correspond to the rising ramp, plateau, and falling ramp in [8]. The range precision under 3D range-gated imaging based on trapezoidal RIPs (trapezoidal RIP 3DRGI) is σ r0i
t2L I 02 c 2 I 02 body;i 2 2 σ τi head;i1 σ σ I0 tL 04 head;i1 2 I 02 I body;i body;i !1∕2 t2 I 02 L head;i1 σ 2I0 ; body:i I 04 body;i
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(7)
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where σ 2I0
head;i1
and σ 2I0
I 0head;i1
are, respectively, the vari-
body:i
I 0tail;i .
and ance of In order to compare the range precision under the same depth step size, Eqs. (5) and (7) have the same time delay and laser-pulse width, and then their depth step sizes are both tL c∕2. Furthermore, laser power and other system parameters are all the same except for the gate time. Therefore Eq. (7) has the same σ 2τi and σ 2tL with Eq. (5). According to [14–16] and considering that target gray level in gate images is proportional to target radiant energy, I 0head;i1 and I 0tail;i can be expressed as I 0head;i1 κE0head;i1 ri κξ
exp−2αri EL Ri1 − ri t − 2 tL L c r2i
κEhead;i1 ri I head;i1 ;
I 0body;i κE0body;i ri κξ
(9)
where κ is a constant conversion coefficient between target gray level of I 0head;i1 and I 0tail;i and target radiant energy of E0head;i1 and E0tail;i . Similarly to Eq. (8), for the term of I head;i1 I tail;i in Eqs. (4) and (5), one can obtain I head;i1 I tail;i κEhead;i1 Etail;i κξ
exp−2αri EL I 0body;i : r2i
(10)
Apparently from Eqs. (8) and (10), I 0head;i1 equals I head;i1 , and I 0body;i equals the sum of I tail;i and I head;i1 in theory. Although in our method there is no plateau under triangular RIPs, the superposition of the tail part of gate image Si and the head part of gate image Si1 constructs a plateau. Therefore, Eq. (4) has the equivalent expression with Eq. (6) under Eq. (10). , and σ 2I0 , they are For σ 2Ihead;i1, σ 2Itail;i , σ 2I0 head;i1
body:i
mainly caused by noise disturbance from atmospheric optical turbulence, ambient background and sensor noise. They have relationships as σ 2Itail;i < σ 2I0
body:i
and σ 2Ihead;i1 < σ 2I0
due to noise suppression
head;i1
under triangular RIPs, which will be discussed in the next section. Therefore, in terms of Eqs. (8) and (10), by comparing Eq. (5) with Eq. (7) one can obtain σ ri < σ r0i ;
(11)
and apparently the range precision is improved under triangular RIPs under the same depth step size. It means that the two methods have the same number of gate images when scanning a given scene 7402
B. Noise Suppression
For 3D super-resolution range-gated imaging based on RIPs, the 3D reconstruction algorithm is a head-tail mode for triangular RIPs and a head-body or tail-body mode for trapezoidal RIPs as shown in Fig. 3. Since target energy in the head and tail parts of RIPs is gradually decreased and easily influenced by noise, to obtain high signal-to-noise ratio for the head and tail parts is crucial for high-accuracy 3D reconstruction. In range-gated imaging, the total noise can be expressed as Enoise EB ES EENE ;
(8)
exp−2αri EL ; r2i
with a large depth of field under the same depth step size, while triangular RIP 3DRGI has higher range precision than trapezoidal RIP 3DRGI.
APPLIED OPTICS / Vol. 52, No. 30 / 20 October 2013
(12)
where EB is ambient background noise, ES is atmospheric backscattering noise, and EENE is the equivalent noise energy of imaging system. For the total noise, the sum of ambient background noise and atmospheric backscattering noise is external noise, and the equivalent noise is internal noise. EB and ES can be obtained from [12] and [17], and EENE is a constant for a fixed imaging system. To maintain the same depth of field of 3D imaging, the two methods have the same laser parameters, including laser pulse width and laser power, and the gate time in our method is half of that in trapezoidalRIP 3DRGI. According to Eq. (12) in [17], the gate time determines the exposure time to ambient background noise, and thus larger exposure time leads to bigger ambient background noise. In terms of Eq. (3) in [12], both the laser pulse width and the gate time determine the depth of a gate image, which is the spatial integration interval of atmospheric backscattering noise. In our method, the depth of integration interval of backscattering noise is tL c, and the depth in trapezoidal-RIP 3DRGI is 3tL c∕2 as shown in Fig. 3. Therefore, for the i 1th gate images in the two methods one can obtain as 2EB;i1 E0B;i1 and ES;i1 < E0S;i1 where EB;i1 and ES;i1 are under a triangular RIP, and E0B;i1 and E0S;i1 are under a trapezoidal RIP. Apparently, under triangular RIP, ambient background noise and atmospheric backscattering noise are suppressed so that the external noise is suppressed, and thus based Eq. (12) the total noise under a triangular RIP is less than that under a trapezoidal RIP. For head parts of I head;i1 and I 0head;i1 used in Eqs. (4) and (6), the following relationship can be obtained as SNRhead;i1 >
Ehead;i1 EB;i1 ES;i1 EENE E0head;i1 SNR0head;i1 ; E0B;i1 E0S;i1 E0ENE (13)
where SNRhead;i1 and SNR0head;i1 are, respectively, signal-to-noise ratio under triangular and trapezoidal RIPs, Ehead;i1 E0head;i1 at the same target range, and EENE E0ENE for a fixed imaging system. Equation (13) demonstrates that the signal-tonoise ratio of a head part under a triangular RIP is higher than that under a trapezoidal RIP, so that targets with weak radiant energy can be better picked up from noise disturbance in our method. This result is beneficial to robustly reconstruct a high-resolution 3D image with noise suppression. 4. Experimental Result and Discussion
For experimental research, a range-gated imaging system is established by a pulsed laser illuminator, a gated camera, and a timing control unit as shown in Fig. 4(a). The laser illuminator is realized by a laser diode with a center wavelength of 808 nm, and its laser pulse width can be changed from 10 ns to several microseconds under the trigger of the timing control unit. For the gated camera, a gated GEN II intensifier is coupled to a CCD with 1392 × 1040 pixels and acts as a gate with a minimal gate time of 40 ns and a maximal repetition frequency of 100 kHz. The timing control unit realized by FPGA can provide desired time sequence for the pulsed laser and the gated camera. In the two following experiments, the laser illuminator has a pulse width of 200 ns and a peak power of 20 W at a pulserepetition frequency of 80 kHz, and CCD has a frame rate of 15 frames per second with an exposure time of 40 ms. For the gated image intensifier, it has the same repetition frequency with the pulsed laser. Therefore, there are 3200 pulse pairs formed in one frame of CCD so that a frame of gate image integrates by 3200 subframes, which is beneficial to output gate images with high SNR [1] and [13]. Figure 4(b) shows the experimental target of a concrete bridge pier over sea where the area encircled by the white-dashed line is the region of interest, which is imaged. The experiment under triangular RIPs is performed in Fig. 5. In this experiment, the laser pulse width and the gate time are both 200 ns, and two
Fig. 4. (a) Experimental setup. (b) Concrete bridge at sea.
Fig. 5. Experimental results of a bridge pier at sea under triangular RIP. (a) and (b) Two successive gate images. (c) Depth map. (d) Trace across lines A-B of (c).
time delays are set as 1440 and 1640 ns. Therefore, two gate images are captured in Figs. 5(a) and 5(b). According to Eq. (4), a depth map is deduced in Fig. 5(c). Figure 5(d) is the trace across line A-B corresponding to the 600th row in Fig. 5(c), which demonstrates that the range precision is about 0.8 m. From Fig. 5(c), the bridge deck, pier cap, and pier column can be easily distinguished by depth. Figure 6 is a comparison experiment under trapezoidal RIPs where the laser pulse width is still 200 ns, while the gate time is 400 ns. A depth map is obtained in Fig. 6(c) from Figs. 6(a) and 6(b) under the same time delay setting with the experiment of Fig. 5. Figure 6(d) is the trace across line A-B corresponding to the 600th row in Fig. 6(c) with range precision about 2 m. In Fig. 6(d), large range fluctuation exists due to noise disturbance, while in Fig. 5(d) range fluctuation is smoothed by noise suppression under triangular RIPs. This phenomenon is explained in Fig. 7. In Fig. 7(a) the four curves depict gray-level distributions from pixel column 900 to 1000 at the 600th row of gate images. In detail, the red solid curve represents I head;2, the magenta dashed–dotted curve represents the sum of I tail;1 and I head;2 , the blue-dashed curve is I 0head;2, and the green-dotted curve shows I 0body;1 where I tail;1 , I head;2 , I 0body;1 , and I 0head;2 correspond to the gray levels from pixel column 900 to 1000 at the 600th row of 20 October 2013 / Vol. 52, No. 30 / APPLIED OPTICS
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Fig. 6. Experimental results of a bridge pier at sea under trapezoidal RIP. (a) and (b) Two successive gate images. (c) Depth map. (d) Trace across line A-B of (c).
Fig. 7. Comparison of gray-level curves and gray-level ratio from pixel column 900 to 1000 at the 600th row of gate images under triangular and trapezoidal RIPs. (a) Gray level curves in Figs. 5(a) and 5(b) as well as Figs. 6(a) and 6(b). (b) Gray-level ratio curves under triangular and trapezoidal RIPs. 7404
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Figs. 5(a), 5(b), 6(a), and 6(b), respectively. The Pearson correlation coefficient between I head;2 and I tail;1 I head;2 is 0.989, while the coefficient between I 0head;2 and I 0body;1 is 0.829. Apparently I head;2 and I tail;1 I head;2 has stronger linear dependence than I 0head;2 and I 0body;1 . Figure 7(b) also demonstrates this. In Fig. 7(b), the red solid curve is the gray level ratio of I head;2 ∕I tail;1 I head;2 under triangular RIPs, and the blue-dashed curve is the gray-level ratio of I 0head;2 ∕I 0body;1 under trapezoidal RIPs. It is clear that in Fig. 7(b) the blue-dashed curve has a large fluctuation, while for the red solid curve, the large fluctuation is smoothed with a better linear property. Lines A-B in Figs. 5(c) and 6(c) lie in the pier cap, and thus the four curves in Fig. 7(a) represent the reflective properties of the pier cap along lines A-B as well as noise disturbance caused by atmospheric optical turbulence, ambient background noise, and camera noise. Since the pier cap is a planar surface as shown in two-dimensional gate images of Figs. 5 and 6, the amplitude fluctuation of gray-level ratio curves in Fig. 7(b) is mainly caused by the noise disturbance. Therefore, the good linear property of the red solid curve with small amplitude fluctuation in Fig. 7(b) demonstrates effective noise suppression under triangular RIPs. This is also proved by Fig. 7(a) where the magenta dashed–dotted curve of I tail;1 I head;2 is below the green-dotted curve of I 0body;1 , which is more evident in Fig. 8. In Fig. 8, the four curves correspond to those four curves in Fig. 7(a), and they are gray-level curves from pixel column 900 to 1000 at the 600th row by, respectively, averaging six frames of gate images under triangular or trapezoidal RIPs. It is clear that I tail;1 I head;2 < I 0body;1 and I head;2 < I 0head;2 in Fig. 8. In terms of Eqs. (8) and (10), one should obtain I tail;1 I head;2 I 0body;1 and I head;2 I 0head;2 . However, in practical applications, pixel gray level in gate images is a sum of contributions from target reflection, atmospheric backscatter, ambient background, and camera noise different from the conditions of Eqs. (8) and (10) where the pixel gray level just reflects signal energy of target reflection. Therefore, the results of
Fig. 8. Gray-level curves from pixel column 900 to 1000 at the 600th row by averaging six frames of gate images under triangular and trapezoidal RIPs.
Figs. 7(a) and 8 demonstrate that noise under triangular RIPs is better suppressed than that under trapezoidal RIPs. To compare range accuracy and precision under the two methods, the experiments of Figs. 5 and 6 are performed another five times, respectively. Figure 9 gives the results of range accuracy and precision analysis based on these experiments. In Fig. 9(a), the red solid curve is the average of target range ri from six times of experiment Fig. 5 under triangular RIP 3DRGI, the blue-dotted curve is the average of target range r0i from six times of experiment Fig. 6 under trapezoidal RIP 3DRGI, and the green-dashed line is the real target range. Correspondingly, Fig. 9(b) shows root-mean-square errors (RMSE) under the two methods. In Fig. 9(b), the red solid curve is RMSE of ri, and the blue-dotted curve is RMSE of r0i. Apparently, target range accuracy under triangular RIP 3DRGI is improved by a factor of 3, which meets the results of Fig. 7(b). In Fig. 7(b),
the gray-level ratio under triangular RIPs is larger than that under trapezoidal RIPs, and the two gray-level ratios are just the coefficients of the second terms in Eqs. (4) and (6). Therefore, from the two equations, one can also conclude that the calculated target range in triangular RIP 3DRGI corresponds better to the real range. Figure 9(c) is the result of range precision analysis where the blue-dashed curve and the green-dotted curve are, respectively, σ ri and σ r0i predicted by Eqs. (5) and (7), and the magenta dashed–dotted curve and the red solid curve are, respectively, standard deviations of ri and r0i in experiment. In the calculation of σ ri and σ r0i , σ τi and σ tL are both 1 ns, and the other parameters are all calculated from gate images. Apparently, in Fig. 9(c) Eqs. (5) and (7) predict well the range precision compared with experimental results. According to Fig. 9(c), the range precisions under triangular RIP 3DRGI and trapezoidal RIP 3DRGI are, respectively, about 0.8 and 2 m, and the former range precision is 2.5 times better than the latter in the same depth step size of 30 m. 5. Conclusion
In summary, a triangular-range-intensity-profile spatial-correlation method for 3DRGI with superresolution depth is proposed. In the method, spatial sampling volumes with triangular RIPs are established by adjusting the laser pulse width and the gate time. Different from trapezoidal RIPs in [8] and [11], there is no available plateau in triangular RIPs for 3D reconstruction, and thus an equivalent plateau is constructed by the superposition of two adjacent gate images in our method. Our research demonstrates that the constructed plateau has better noise suppression. Finally, a depth map of a 3D scene can be obtained by spatial correlation of overlapped gate images corresponding to those sampling volumes. Compared with super-resolution depth mapping under trapezoidal RIPs, in the same depth step sizes the proposed method has higher range accuracy and precision with better noise suppression. Therefore, in image coding [11], the 3D reconstruction algorithm of triangular RIP spatial correlation can be used instead of the trapezoidal RIP algorithm to further improve range accuracy and precision, or the two algorithms can combine together to expand coding length and enlarge 3D imaging range. The authors acknowledge the financial funding of this work by the National Natural Science Foundation of China (NSFC) (grant 61205019) and the Hong Kong Scholars Program (grant XJ2012046). References
Fig. 9. Comparison of range accuracy and precision under triangular RIP and trapezoidal RIP 3D imaging. (a) Target range obtained from six times of experiments. (b) Range accuracy. (c) Range precision.
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Optoelectronic System Laboratory, Institute of Semiconductors, CAS, Beijing 100083, China
2
Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China *Corresponding author: [email protected] Received 28 May 2013; revised 24 September 2013; accepted 26 September 2013; posted 1 October 2013 (Doc. ID 191226); published 18 October 2013
We present a triangular-range-intensity profile (RIP) spatial-correlation method for 3D range-gated imaging with a depth of super resolution. In this method, spatial sampling volumes with triangular-RIPs are established by matching laser-pulse width and sensor gate time, and then depth information collapsed in gate images can be reconstructed by spatial correlation of overlapped gate images corresponding to sampling volumes. Compared with super-resolution depth mapping under trapezoidal-RIPs, range accuracy and precision are improved, and a large range fluctuation due to noise disturbance is smoothed by noise suppression under triangular-RIPs. In this paper, a proof experiment is demonstrated with a range precision 2.5 times better than that obtained under trapezoidal-RIPs. © 2013 Optical Society of America OCIS codes: (110.0110) Imaging systems; (110.6880) Three-dimensional image acquisition; (100.6890) Three-dimensional image processing; (280.0280) Remote sensing and sensors. http://dx.doi.org/10.1364/AO.52.007399
1. Introduction
Three-dimensional range-gated imaging (3DRGI) first proposed in the 2000s is a relatively new technique in 3D imaging [1,2]. It has large potential in underwater imaging [3], target detection and identification [4], and spatial navigation [5] due to its long range detection, high spatial resolution, and effective work through obscurants, such as fog, camouflage, and turbid water. Up to now, 3DRGI has been mainly developed with three approaches by depth scanning [1–4], gain modulation [6,7], and super-resolution depth mapping [8,9]. For the depth scanning method, high-range accuracy needs shorter scanning step sizes and greater data-processing efforts, and thus high-range accuracy and large depth of field cannot be simultaneously realized. To solve the problem, the methods of gain modulation and super-resolution 1559-128X/13/307399-08$15.00/0 © 2013 Optical Society of America
depth mapping were developed. In the gain modulation method, a depth map can be obtained from two gate images, a gain-modulated image, and a gainconstant image, and its range accuracy is independent of laser pulse shapes. In the super-resolution depth mapping method, a 3D scene with depth resolution far beyond depth step size can be reconstructed by at least two gate images with trapezoidal rangeintensity profiles (RIPs) where the sensor gate time is twice as large as the laser pulse width. The two methods can both reduce raw data volume and simplify data processing with good real-time performance even under high-range accuracy. In the same conditions, the super-resolution depth mapping method has higher range accuracy than the gain modulation method, even though the former has lower signalto-noise ratio [10]. Furthermore, to enlarge the depth mapping range (or depth of field) while maintaining high-range accuracy, an image coding technique has been introduced into super-resolution depth mapping [11]. In other words, within the same depth of field, 20 October 2013 / Vol. 52, No. 30 / APPLIED OPTICS
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image coding of super-resolution depth mapping can obtain higher range accuracy than the gain modulation method, which is valuable in 3D imaging applications. However, energy of targets in the rising and falling ramps of trapezoidal RIPs is easily interfered by ambient noise [12], and thus big noise disturbance to the two ramps should be suppressed to resolve large range fluctuation in super-resolution depth mapping. In this paper, we present a triangular-RIP spatial-correlation method for 3D super-resolution range-gated imaging to solve large range fluctuation due to noise disturbance and further improve range accuracy and precision. 2. Triangular-Range-Intensity Profile Spatial-Correlation Method
In our method, the laser pulse width equals the sensor gate time, so that under the convolution of the laser pulse and the gate pulse the RIP of a spatial sampling volume formed by gate viewing is a triangle, which can be divided into two parts—a head part and a tail part as illustrated in Fig. 1(a)—different from three parts of a trapezoidal RIP. For gate image Si , its position Ri is determined by τi c∕2 where τi is the time
Fig. 1. Triangular RIP spatial-correlation method for 3DRGI. (a) Triangular RIP formed by gate viewing. (b) Two adjacent spatial-correlation sampling volumes. (c) 3D reconstruction by spatial correlation of triangular RIPs. 7400
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delay between the laser pulse and the gate pulse, c is the light speed in vacuum, and Si denotes the ith gate image with time delay τi. In this paper, the atmospheric refractive index is approximated as 1. The spatial interval of image Si is Rhead;i ; Rtail;i where Rhead;i and Rtail;i are the positions of the head and tail parts, Rhead;i Ri − tL c∕2 and Rtail;i Ri tL c∕2 where tL is the laser pulse width. By changing the time delay two adjacent spatial-correlation sampling volumes can be constructed in Fig. 1(b), where the tail part of image Si is overlapped by the head part of image Si1 where Si1 denotes the i 1th gate image with time delay τi1, and hereby a spatial-correlation zone is formed in Fig. 1(c). Apparently, targets in correlation zones will appear in successive gate images with a given delay step, which meets Δt tL tg where Δt is the delay step between images Si and Si1 and tg is the sensor gate time. The time sequence of the method is given in Fig. 2, where multiple exposures are adopted so that a frame of sampling gate image integrates many subframes formed by pulse pairs. A pulse pair consists of a laser pulse and a gate pulse to realize gate viewing (or range-gated imaging). The delay time τi1 of image Si1 meets τi1 τi Δt. To avoid the cross talk of target echo signal, the delay time should be less than the value of (1∕f L − tg ) where f L is the laser-pulse-repetition frequency. In our range-gated imaging system, the imaging sensor is a CCD with a gated intensifier. When the exposure time of one frame is much larger than the reading time of CCD, the number of subframes in one frame is f L ∕f CCD where f CCD is the frame rate of CCD. References [1] and [13] demonstrate that multiple exposures used in range-gated imaging can reduce the impact of ambient noise and improve image signal-to-noise ratio. It should be noted that depth information is lost in gate images, since in the process of imaging a spatial sampling volume is compacted and projected into a 2D image with depth information collapsed. Fortunately, the depth information can be reconstructed by the spatial correlation of overlapped gate images. Since the target gray level in gate images is proportional to irradiance of targets under laser illumination, to targets in the correlation zone the relationship between the pixel-to-pixel gray level ratio of the two successive gate images and target radiant energy ratio can be established as
Fig. 2. Time sequence of CCD, laser, and gate in triangular RIP spatial-correlation method.
I tail;i I head;i1
Etail;i ri ; Ehead;i1 ri
(1)
where I tail;i and I head;i1 are, respectively, the gray level of target in the tail part of gate image Si and the head part of gate image Si1 , ri is the target range, and Etail;i ri and Ehead;i1 ri are, respectively, the radiant energy of target in the tail part of the sampling volume of gate image Si and the head part of the sampling volume of gate image Si1 . According to [14] and [15], the laser pulse and the gate pulse are both in rectangular form, and then Etail;i ri can be given as Etail;i ri ξ
exp−2αri EL ri − Ri ; t − 2 tL L c r2i
(2)
where the coefficient ξ Gηt ηr f L ∕f CCD ρ∕π At ∕AL Ar . G is the gain of the imaging system, ηr and ηt are, respectively, the transmission efficiencies of receiving and transmitting optical modules, ρ is the target reflection coefficient, At is the effective laser cross section, AL is the illuminated area, Ar is the receiver area, and α is the atmospheric total extinction coefficient. For a fixed imaging system and given targets, the coefficient ξ is a constant. Similarly, from [14] and [15] Ehead;i1 ri can be given as
Fig. 3. Comparison of super-resolution 3DRGI under different RIPs. (a) Under triangular RIPs. (b) Under trapezoidal RIPs.
Based on the rules of error propagation and considering time jitter and noise disturbance, from Eq. (4) the range precision under three-dimensional rangegated imaging based on triangular RIPs (triangular RIP 3DRGI) is " I 2head;i1 c 2 σ τi σ2 σ ri 2 I head;i1 I tail;i 2 tL
exp−2αri EL R − ri Ehead;i1 ri ξ tL − 2 i1 : (3) 2 tL c ri
Therefore, one can substitute Eqs. (1) and (2) into Eq. (3) and obtain the target range as ri
I head;i1 τi c tL c ; I head;i1 I tail;i 2 2
(4)
where ri is the target range. From Eq. (4) depth information collapsed in gate images can be picked up, and a 3D scene also can be further reconstructed by a camera model. As illustrated in Fig. 3, different from trapezoidal RIPs there is no available plateau to be directly used for 3D reconstruction under triangular RIPs. In fact, an equivalent plateau is constructed in Eq. (4) by the superposition of the tail part of the current gate image Si and the head part of the adjacent gate image Si1 . The denominator of the gray-level ratio of I head;i1 ∕I tail;i I head;i1 is the equivalent plateau, which will be further discussed in Section 3A. 3. Comparison with Super-Resolution Depth Mapping under Trapezoidal RIPs A.
Range Precision
Our method and super-resolution depth mapping under trapezoidal RIPs can reconstruct spatial information collapsed in gated images and realize 3D imaging with depth super-resolution. In this section, range precision is compared under the two methods.
t2L I 2tail;i I head;i1 I tail;i 4 t2L I 2head;i1 I head;i1 I tail;i
σ 2Ihead;i1
σ2 4 Itail;i
#1∕2 ;
(5)
where σ 2τi is the variance of the delay time τi , σ 2tL is the variance of the laser pulse width tL , and σ 2Ihead;i1 and σ 2Itail;i are, respectively, the variance of I head;i1 and I tail;i . For super-resolution depth mapping under trapezoidal RIPs [8], the gate time is twice as large as the laser pulse width, and target range r0i is r0i
τi c I 0head;i1 tL c 0 ; I body;i 2 2
(6)
where I 0body;i and I 0head;i1 are, respectively, the gray level of target in the body part of gate image Si and the head part of gate image Si1 as shown in Fig. 3(b). Here, the head, body and tail parts correspond to the rising ramp, plateau, and falling ramp in [8]. The range precision under 3D range-gated imaging based on trapezoidal RIPs (trapezoidal RIP 3DRGI) is σ r0i
t2L I 02 c 2 I 02 body;i 2 2 σ τi head;i1 σ σ I0 tL 04 head;i1 2 I 02 I body;i body;i !1∕2 t2 I 02 L head;i1 σ 2I0 ; body:i I 04 body;i
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(7)
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where σ 2I0
head;i1
and σ 2I0
I 0head;i1
are, respectively, the vari-
body:i
I 0tail;i .
and ance of In order to compare the range precision under the same depth step size, Eqs. (5) and (7) have the same time delay and laser-pulse width, and then their depth step sizes are both tL c∕2. Furthermore, laser power and other system parameters are all the same except for the gate time. Therefore Eq. (7) has the same σ 2τi and σ 2tL with Eq. (5). According to [14–16] and considering that target gray level in gate images is proportional to target radiant energy, I 0head;i1 and I 0tail;i can be expressed as I 0head;i1 κE0head;i1 ri κξ
exp−2αri EL Ri1 − ri t − 2 tL L c r2i
κEhead;i1 ri I head;i1 ;
I 0body;i κE0body;i ri κξ
(9)
where κ is a constant conversion coefficient between target gray level of I 0head;i1 and I 0tail;i and target radiant energy of E0head;i1 and E0tail;i . Similarly to Eq. (8), for the term of I head;i1 I tail;i in Eqs. (4) and (5), one can obtain I head;i1 I tail;i κEhead;i1 Etail;i κξ
exp−2αri EL I 0body;i : r2i
(10)
Apparently from Eqs. (8) and (10), I 0head;i1 equals I head;i1 , and I 0body;i equals the sum of I tail;i and I head;i1 in theory. Although in our method there is no plateau under triangular RIPs, the superposition of the tail part of gate image Si and the head part of gate image Si1 constructs a plateau. Therefore, Eq. (4) has the equivalent expression with Eq. (6) under Eq. (10). , and σ 2I0 , they are For σ 2Ihead;i1, σ 2Itail;i , σ 2I0 head;i1
body:i
mainly caused by noise disturbance from atmospheric optical turbulence, ambient background and sensor noise. They have relationships as σ 2Itail;i < σ 2I0
body:i
and σ 2Ihead;i1 < σ 2I0
due to noise suppression
head;i1
under triangular RIPs, which will be discussed in the next section. Therefore, in terms of Eqs. (8) and (10), by comparing Eq. (5) with Eq. (7) one can obtain σ ri < σ r0i ;
(11)
and apparently the range precision is improved under triangular RIPs under the same depth step size. It means that the two methods have the same number of gate images when scanning a given scene 7402
B. Noise Suppression
For 3D super-resolution range-gated imaging based on RIPs, the 3D reconstruction algorithm is a head-tail mode for triangular RIPs and a head-body or tail-body mode for trapezoidal RIPs as shown in Fig. 3. Since target energy in the head and tail parts of RIPs is gradually decreased and easily influenced by noise, to obtain high signal-to-noise ratio for the head and tail parts is crucial for high-accuracy 3D reconstruction. In range-gated imaging, the total noise can be expressed as Enoise EB ES EENE ;
(8)
exp−2αri EL ; r2i
with a large depth of field under the same depth step size, while triangular RIP 3DRGI has higher range precision than trapezoidal RIP 3DRGI.
APPLIED OPTICS / Vol. 52, No. 30 / 20 October 2013
(12)
where EB is ambient background noise, ES is atmospheric backscattering noise, and EENE is the equivalent noise energy of imaging system. For the total noise, the sum of ambient background noise and atmospheric backscattering noise is external noise, and the equivalent noise is internal noise. EB and ES can be obtained from [12] and [17], and EENE is a constant for a fixed imaging system. To maintain the same depth of field of 3D imaging, the two methods have the same laser parameters, including laser pulse width and laser power, and the gate time in our method is half of that in trapezoidalRIP 3DRGI. According to Eq. (12) in [17], the gate time determines the exposure time to ambient background noise, and thus larger exposure time leads to bigger ambient background noise. In terms of Eq. (3) in [12], both the laser pulse width and the gate time determine the depth of a gate image, which is the spatial integration interval of atmospheric backscattering noise. In our method, the depth of integration interval of backscattering noise is tL c, and the depth in trapezoidal-RIP 3DRGI is 3tL c∕2 as shown in Fig. 3. Therefore, for the i 1th gate images in the two methods one can obtain as 2EB;i1 E0B;i1 and ES;i1 < E0S;i1 where EB;i1 and ES;i1 are under a triangular RIP, and E0B;i1 and E0S;i1 are under a trapezoidal RIP. Apparently, under triangular RIP, ambient background noise and atmospheric backscattering noise are suppressed so that the external noise is suppressed, and thus based Eq. (12) the total noise under a triangular RIP is less than that under a trapezoidal RIP. For head parts of I head;i1 and I 0head;i1 used in Eqs. (4) and (6), the following relationship can be obtained as SNRhead;i1 >
Ehead;i1 EB;i1 ES;i1 EENE E0head;i1 SNR0head;i1 ; E0B;i1 E0S;i1 E0ENE (13)
where SNRhead;i1 and SNR0head;i1 are, respectively, signal-to-noise ratio under triangular and trapezoidal RIPs, Ehead;i1 E0head;i1 at the same target range, and EENE E0ENE for a fixed imaging system. Equation (13) demonstrates that the signal-tonoise ratio of a head part under a triangular RIP is higher than that under a trapezoidal RIP, so that targets with weak radiant energy can be better picked up from noise disturbance in our method. This result is beneficial to robustly reconstruct a high-resolution 3D image with noise suppression. 4. Experimental Result and Discussion
For experimental research, a range-gated imaging system is established by a pulsed laser illuminator, a gated camera, and a timing control unit as shown in Fig. 4(a). The laser illuminator is realized by a laser diode with a center wavelength of 808 nm, and its laser pulse width can be changed from 10 ns to several microseconds under the trigger of the timing control unit. For the gated camera, a gated GEN II intensifier is coupled to a CCD with 1392 × 1040 pixels and acts as a gate with a minimal gate time of 40 ns and a maximal repetition frequency of 100 kHz. The timing control unit realized by FPGA can provide desired time sequence for the pulsed laser and the gated camera. In the two following experiments, the laser illuminator has a pulse width of 200 ns and a peak power of 20 W at a pulserepetition frequency of 80 kHz, and CCD has a frame rate of 15 frames per second with an exposure time of 40 ms. For the gated image intensifier, it has the same repetition frequency with the pulsed laser. Therefore, there are 3200 pulse pairs formed in one frame of CCD so that a frame of gate image integrates by 3200 subframes, which is beneficial to output gate images with high SNR [1] and [13]. Figure 4(b) shows the experimental target of a concrete bridge pier over sea where the area encircled by the white-dashed line is the region of interest, which is imaged. The experiment under triangular RIPs is performed in Fig. 5. In this experiment, the laser pulse width and the gate time are both 200 ns, and two
Fig. 4. (a) Experimental setup. (b) Concrete bridge at sea.
Fig. 5. Experimental results of a bridge pier at sea under triangular RIP. (a) and (b) Two successive gate images. (c) Depth map. (d) Trace across lines A-B of (c).
time delays are set as 1440 and 1640 ns. Therefore, two gate images are captured in Figs. 5(a) and 5(b). According to Eq. (4), a depth map is deduced in Fig. 5(c). Figure 5(d) is the trace across line A-B corresponding to the 600th row in Fig. 5(c), which demonstrates that the range precision is about 0.8 m. From Fig. 5(c), the bridge deck, pier cap, and pier column can be easily distinguished by depth. Figure 6 is a comparison experiment under trapezoidal RIPs where the laser pulse width is still 200 ns, while the gate time is 400 ns. A depth map is obtained in Fig. 6(c) from Figs. 6(a) and 6(b) under the same time delay setting with the experiment of Fig. 5. Figure 6(d) is the trace across line A-B corresponding to the 600th row in Fig. 6(c) with range precision about 2 m. In Fig. 6(d), large range fluctuation exists due to noise disturbance, while in Fig. 5(d) range fluctuation is smoothed by noise suppression under triangular RIPs. This phenomenon is explained in Fig. 7. In Fig. 7(a) the four curves depict gray-level distributions from pixel column 900 to 1000 at the 600th row of gate images. In detail, the red solid curve represents I head;2, the magenta dashed–dotted curve represents the sum of I tail;1 and I head;2 , the blue-dashed curve is I 0head;2, and the green-dotted curve shows I 0body;1 where I tail;1 , I head;2 , I 0body;1 , and I 0head;2 correspond to the gray levels from pixel column 900 to 1000 at the 600th row of 20 October 2013 / Vol. 52, No. 30 / APPLIED OPTICS
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Fig. 6. Experimental results of a bridge pier at sea under trapezoidal RIP. (a) and (b) Two successive gate images. (c) Depth map. (d) Trace across line A-B of (c).
Fig. 7. Comparison of gray-level curves and gray-level ratio from pixel column 900 to 1000 at the 600th row of gate images under triangular and trapezoidal RIPs. (a) Gray level curves in Figs. 5(a) and 5(b) as well as Figs. 6(a) and 6(b). (b) Gray-level ratio curves under triangular and trapezoidal RIPs. 7404
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Figs. 5(a), 5(b), 6(a), and 6(b), respectively. The Pearson correlation coefficient between I head;2 and I tail;1 I head;2 is 0.989, while the coefficient between I 0head;2 and I 0body;1 is 0.829. Apparently I head;2 and I tail;1 I head;2 has stronger linear dependence than I 0head;2 and I 0body;1 . Figure 7(b) also demonstrates this. In Fig. 7(b), the red solid curve is the gray level ratio of I head;2 ∕I tail;1 I head;2 under triangular RIPs, and the blue-dashed curve is the gray-level ratio of I 0head;2 ∕I 0body;1 under trapezoidal RIPs. It is clear that in Fig. 7(b) the blue-dashed curve has a large fluctuation, while for the red solid curve, the large fluctuation is smoothed with a better linear property. Lines A-B in Figs. 5(c) and 6(c) lie in the pier cap, and thus the four curves in Fig. 7(a) represent the reflective properties of the pier cap along lines A-B as well as noise disturbance caused by atmospheric optical turbulence, ambient background noise, and camera noise. Since the pier cap is a planar surface as shown in two-dimensional gate images of Figs. 5 and 6, the amplitude fluctuation of gray-level ratio curves in Fig. 7(b) is mainly caused by the noise disturbance. Therefore, the good linear property of the red solid curve with small amplitude fluctuation in Fig. 7(b) demonstrates effective noise suppression under triangular RIPs. This is also proved by Fig. 7(a) where the magenta dashed–dotted curve of I tail;1 I head;2 is below the green-dotted curve of I 0body;1 , which is more evident in Fig. 8. In Fig. 8, the four curves correspond to those four curves in Fig. 7(a), and they are gray-level curves from pixel column 900 to 1000 at the 600th row by, respectively, averaging six frames of gate images under triangular or trapezoidal RIPs. It is clear that I tail;1 I head;2 < I 0body;1 and I head;2 < I 0head;2 in Fig. 8. In terms of Eqs. (8) and (10), one should obtain I tail;1 I head;2 I 0body;1 and I head;2 I 0head;2 . However, in practical applications, pixel gray level in gate images is a sum of contributions from target reflection, atmospheric backscatter, ambient background, and camera noise different from the conditions of Eqs. (8) and (10) where the pixel gray level just reflects signal energy of target reflection. Therefore, the results of
Fig. 8. Gray-level curves from pixel column 900 to 1000 at the 600th row by averaging six frames of gate images under triangular and trapezoidal RIPs.
Figs. 7(a) and 8 demonstrate that noise under triangular RIPs is better suppressed than that under trapezoidal RIPs. To compare range accuracy and precision under the two methods, the experiments of Figs. 5 and 6 are performed another five times, respectively. Figure 9 gives the results of range accuracy and precision analysis based on these experiments. In Fig. 9(a), the red solid curve is the average of target range ri from six times of experiment Fig. 5 under triangular RIP 3DRGI, the blue-dotted curve is the average of target range r0i from six times of experiment Fig. 6 under trapezoidal RIP 3DRGI, and the green-dashed line is the real target range. Correspondingly, Fig. 9(b) shows root-mean-square errors (RMSE) under the two methods. In Fig. 9(b), the red solid curve is RMSE of ri, and the blue-dotted curve is RMSE of r0i. Apparently, target range accuracy under triangular RIP 3DRGI is improved by a factor of 3, which meets the results of Fig. 7(b). In Fig. 7(b),
the gray-level ratio under triangular RIPs is larger than that under trapezoidal RIPs, and the two gray-level ratios are just the coefficients of the second terms in Eqs. (4) and (6). Therefore, from the two equations, one can also conclude that the calculated target range in triangular RIP 3DRGI corresponds better to the real range. Figure 9(c) is the result of range precision analysis where the blue-dashed curve and the green-dotted curve are, respectively, σ ri and σ r0i predicted by Eqs. (5) and (7), and the magenta dashed–dotted curve and the red solid curve are, respectively, standard deviations of ri and r0i in experiment. In the calculation of σ ri and σ r0i , σ τi and σ tL are both 1 ns, and the other parameters are all calculated from gate images. Apparently, in Fig. 9(c) Eqs. (5) and (7) predict well the range precision compared with experimental results. According to Fig. 9(c), the range precisions under triangular RIP 3DRGI and trapezoidal RIP 3DRGI are, respectively, about 0.8 and 2 m, and the former range precision is 2.5 times better than the latter in the same depth step size of 30 m. 5. Conclusion
In summary, a triangular-range-intensity-profile spatial-correlation method for 3DRGI with superresolution depth is proposed. In the method, spatial sampling volumes with triangular RIPs are established by adjusting the laser pulse width and the gate time. Different from trapezoidal RIPs in [8] and [11], there is no available plateau in triangular RIPs for 3D reconstruction, and thus an equivalent plateau is constructed by the superposition of two adjacent gate images in our method. Our research demonstrates that the constructed plateau has better noise suppression. Finally, a depth map of a 3D scene can be obtained by spatial correlation of overlapped gate images corresponding to those sampling volumes. Compared with super-resolution depth mapping under trapezoidal RIPs, in the same depth step sizes the proposed method has higher range accuracy and precision with better noise suppression. Therefore, in image coding [11], the 3D reconstruction algorithm of triangular RIP spatial correlation can be used instead of the trapezoidal RIP algorithm to further improve range accuracy and precision, or the two algorithms can combine together to expand coding length and enlarge 3D imaging range. The authors acknowledge the financial funding of this work by the National Natural Science Foundation of China (NSFC) (grant 61205019) and the Hong Kong Scholars Program (grant XJ2012046). References
Fig. 9. Comparison of range accuracy and precision under triangular RIP and trapezoidal RIP 3D imaging. (a) Target range obtained from six times of experiments. (b) Range accuracy. (c) Range precision.
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