- Email: [email protected]
Imaging resolution analysis in limited-view Laser Radar reflective tomography
Optics Communications 285 (2012) 2575–2579
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Imaging resolution analysis in limited-view Laser Radar reflective tomography Xiaofeng Jin, Jianfeng Sun ⁎, Yi Yan, Yu Zhou, Liren Liu Key Laboratory of Space Laser Communication and Testing Technology, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
a r t i c l e
i n f o
Article history: Received 25 September 2011 Accepted 30 January 2012 Available online 13 February 2012 Keywords: Reflective tomography Limited views Imaging resolution
a b s t r a c t In many applications of Laser radar reflective tomography, anisotropic resolution of images reconstructed from an incomplete set of reflective projections will degrade imaging quality. In this paper, we present the theoretical point spread function (PSF) of imaging system over full views for reflective tomography. The corresponding imaging resolution of the PSF was determined by Rayleigh's criterion, here the most common algorithm of filtered backprojection was used for image reconstruction. The theoretical imaging resolution was derived approximately from the range resolution resolved by the laser pulse. The simulated horizontal imaging resolutions of an ideal diffused single point target in different limited-views were compared with the theoretical value. Experimental reconstructed images from limit-view projections of letter “E” illustrated the effects of anisotropic resolution. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Reflective tomography is one of the most effective high-resolution imaging methods for Laser sensing and imaging technologies. In reflective tomography, Laser radar systems can be designed to provide remote sensing of range-resolved, Doppler-resolved and rangeDoppler resolved signals. Direct detection of rang-resolved reflective tomography techniques can be used to obtain an image of a target, which is angularly unresolved when the data are range resolved. These techniques are especially applicable for a target that is not rotating fast enough to have its Doppler spectrum resolved by the detector, and the projection data could be collected from direct detection or heterodyne detection system. The range-resolved Laser radar reflective tomography imaging was firstly introduced by Parker [1,2] and Knight [3,4] at the Massachusetts Institute of Technology (MIT). Several years later, Matson [5–9] in Air Force began exploring the technique of using the HI-CLASS coherent Laser radar system to obtain images reconstructed by carrying out a heterodyne system analysis, deriving and validating imaging signal-to-noise ratio expressions. One of the main issues in reflective tomography concentrates on how to estimate numerically tomographic imaging results, when sampling views are not theoretically sufficient for exact image reconstruction. Insufficient projection data problems occur quite frequently because of practical constraints due to the imaging hardware, scanning geometry or target rotation. For example, in actual Earthstabilized satellites imaging case, projections can be obtained at
most extent over approximately 130° [10]. The laser signal returned in reflective tomography is reflected off the illuminated outer surface of the target, and only information about the exterior can be obtained. The reflective projections at different views usually do not contain the same information about the target, and the projection data over full views would be difficult to infer from limited-view projections data. Artifacts and geometric distortion would be encountered in the imaging results of limited-view images reconstruction. Furthermore the images that are distinguishable when all projections are given may not be distinguishable when only a few projections are available. In this paper, the theoretical point spread function (PSF) associated with filtered back-projection algorithm in reflective tomography was derived, and shown to depend upon the cutoff frequency. Because the cutoff frequency was related to the range resolution resolved by the laser pulse [11], the corresponding theoretical imaging resolution of the PSF can be represented by the range resolution. For all images reconstructed from limited-views in this paper, the image pixel interval is equal to the range resolution. In this way, the theoretical imaging resolution determined by the PSF could be compared with the simulated horizontal imaging resolutions in different limited-views. Anisotropic resolution will degrade the quality of reconstructed image. Target would be difficult to distinguish due to the artifacts from the process of image reconstructed [22]. 2. Review of filtered backprojection algorithm Firstly, brief description of filtered backprojection [8,9] is provided for image reconstruction of reflective tomography. Let f(x, y) denote the image to be reconstructed, and Lr, ϕ denote the solid line r = x cos ϕ + y sin ϕ, see Fig. 1
⁎ Corresponding author. E-mail address: [email protected] (J. Sun). 0030-4018/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2012.01.075
pðr; ϕÞ ¼ ∫Lr;ϕ f ðx; yÞds;
ð1Þ
2576
X. Jin et al. / Optics Communications 285 (2012) 2575–2579
like artifacts, and then the image reconstruction gFB(x, y), can be given by g FB ðx; yÞ ¼
m X
F
−1
ðjvjF ðpðx cosϕi þ y sinϕi ; ϕÞÞÞΔϕ
ð3Þ
i¼1
where F and F − 1 represent the 1-dimensional Fourier transform and inverse Fourier transform operators, v represents frequency variable. 3. The derivation of the PSF over full views In the following, we give the PSF associated with filtered backprojection algorithm for parallel beams in reflective tomography. And then the resolution of the PSF is given by the range resolution using the Rayleigh's criterion. The expression of theoretical PSF is a very useful measurement of the overall performance of the imaging system, because it represents the maximum theoretical resolving capability of the imaging reconstruction technique. For simplification, it is assumed that the Laser pulse receiver is at the same location as the transmitter illustrated in Fig. 2. For a single scatter at (r, θ), the round-trip distance between the point target and the location of the transmitter and receiver at (R, ϕ) can be given by,
Fig. 1. The schematic diagram of reflective tomography.
2ρ ¼ 2R−2r cosðϕ−θÞ:
ð4Þ
We consider the shape of a received pulse reflected from a point target in terms of the temporal impulse response of a linear system which consists of the transmitter, receiver, and an isolated diffuse point target [12–14]. Let h(t) denote the impulse response of the linear system, which is shown in Fig. 3. h(t) can be assumed invariant with respect to target position, and is only proportional to the target reflectivity. Under these linear and time-invariant conditions, for the transmitted pulse pt(t), the received pulse pr(t, ϕ) can be written as, 2ρ 2ρ pr t− ; ϕ ¼ pt ðt Þ⊗h t− c c
ð5Þ
where ⊗ represents convolution operation, in theoretical consideration, the Eq. (5) can be given by, Fig. 2. The schematic diagram of single scattering point model.
where s represents solid line along Lr, ϕ, p(r, ϕ) is the projection of the target f(x, y) at angle ϕ, and the variable r is the spatial variable along the integration path in the ϕ direction. Using back-projection algorithm, the estimated image, g(x, y), is given by g ðx; yÞ ¼
m X
pðx cosϕi þ y sinϕi ; ϕi ÞΔϕ
ð2Þ
i¼1
pt ðt Þ ¼ δðt Þ; hðt Þ ¼ δðt Þ;
ð6Þ
2ρ 2ρ 2ρ ¼ δ t− pr t− ; ϕ ¼ δðt Þ⊗δ t− c c c
ð7Þ
where ε = ct, the reflective projection data versus the corresponding distance can be expressed as, pr ðε−2ρ; ϕÞ ¼ δðε−2ρÞ:
ð8Þ
For a single scatter at (r0, θ0),
where ϕi is the angle of the ith projection, Δϕ is the sampling angular separation, and m is the total number of projections. Filtered backprojection was introduced to modify projections to reduce star-
pr ðε−2ρ; ϕÞ ¼ δðε−2ρÞ ¼ δðε−2R−2r0 cosðϕ−θ0 ÞÞ:
Fig. 3. The linear system consisting of a transmitter/receiver and an isolated diffuse point target.
ð9Þ
X. Jin et al. / Optics Communications 285 (2012) 2575–2579
2577
Here the filter function of Ramachandran and Lakshminarayanan (RL) [15] was used. The cutoff frequency was set at vc, and the filter function H(v) can be given by, H ðvÞ ¼
jvj jvjbvc : 0 elsewhere
ð13Þ
Using the filter function above, the PSF of the image reconstruction can be transformed to, v
c f ðr; θÞ ¼ 2π∫−v jvjJ 0 ð4πvrÞdv: c
ð14Þ
Since J0(•) is an even function, it is easy to simplify Eq. (16), v
f ðr; θÞ ¼ 4π∫0c vJ 0 ð4πvrÞdv:
ð15Þ
Based on the integral transform characteristics of Bessel function, Fig. 4. The slice of PSF distribution given by Eq. (21).
d n n x J n ðxÞ ¼ x J n−1 ðxÞ: dx
Using the filtered back-projection algorithm described in Section 2, the estimated image can be given by, −1 f ðr; θÞ ¼ ∫2π fHðvÞF ½pr ðε−2ρ; ϕÞgdϕ 0 F
¼
∞ ∫2π 0 dϕ∫−∞ H ðvÞ exp½j2πvðτ−2ρ0 Þdv
ð10Þ
where F and F − 1 denote Fourier transform and inverse Fourier transform respectively, variable τ = 2ρ. When the scattering center is located at the origin, using the Bessel identity, J 0 ðr Þ ¼
1 2π ∫ exp½jr cosðϕ−θÞdϕ: 2π 0
ð11Þ
The PSF of the image reconstruction can be further simplified, ∞
f ðr; θÞ ¼ 2π∫−∞ HðvÞJ 0 ð4πvr Þdv:
ð12Þ
ð16Þ
The final PSF of image reconstruction is given by, f ðr; θÞ ¼
vc J ð4πvc r Þ: r 1
ð17Þ
The pulse δ(t) is not practical as transmitted pulse. The form for the laser pulse outgoing radiation must be assumed. For simplicity, suppose that transmitted pulse pt(t) in Eq. (5) is a rectangle with time duration τ, which has an effective spatial extent cτ/2 due to the reflection off of the target. We defined vc as the location of the first zero of the Sinc function that is the Fourier transform of the transmitted pulse pt(t) [11]. Then the cutoff frequency vc can be written as, vc ¼
1 cτ ; ΔR1 ¼ ΔR1 2
Fig. 5. Image reconstruction of single point target in different limited views with 1° step: limited-views over (a) 15°, (b) 30°, (c) 45°, (d) 60°, (e) 90°, (f) 360°.
ð18Þ
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X. Jin et al. / Optics Communications 285 (2012) 2575–2579
where ΔR1 is the range resolution resolved by the laser pulse, the final expression of PSF associated with filtered back-projection algorithm for reflective tomography can be given by, f ðr; θÞ ¼
1 J 1 ð4πr=ΔR1 Þ : ΔR1 r
ð19Þ
Based on the Rayleigh criterion, the imaging resolution X0 of the f(r, θ) at the first zero can be given by, X 0 ¼ 0:3049ΔR1 :
ð20Þ
If r = NΔR2, ΔR2 is the image reconstruction pixel interval. The PSF of image reconstruction is shown by, 1 J 1 ð4πNΔR2 =ΔR1 Þ : ΔR1 ΔR2 N
ð21Þ
Fig. 7. The horizontal imaging resolution results using the Rayleigh criterion in limited views.
If ΔR2 = ΔR1/3, the slice of PSF distribution given by Eq. (21) was shown in Fig. 4. We see that the PSF does not fall off smoothly from their peak values, but have obvious oscillations which gradually fall off in amplitude with increasing distance away from the peak values. This oscillatory behavior in the PSF induces artifacts in the image reconstruction of complicated structures. This oscillatory behavior is essentially caused by the sharp cutoffs in Fourier space restricted by the RL filter [16]. Using the Rayleigh criterion, the theoretical imaging resolution of filtered back-projection algorithm for reflective tomography was given in Eq. (20).
and the corresponding horizontal slices through the center of images reconstructed were illustrated in Fig. 6. For all tomographic image reconstruction in this section, the adjacent pixel interval was equal to the range resolution resolved by the laser pulse, so the simulated imaging resolution in different limited views can be expressed by the range resolution. Because we do not have projection data for full 360°, tomographic striping artifacts and geometric distortion in the image reconstruction are obvious in Fig. 6. Especially in the limitedview extent over 5° and 15°, the single scattering point can not be distinguished from these imaging results. These artifacts would be eliminated if the views spread out evenly over full 360°. Using the Rayleigh criterion (taking the resolution limit as the distance for which the ratio of the value at the central dip in the intensity distribution to that at the maxima on either side is equal to 0.81) [16], the horizontal imaging resolutions of different limited views in 1° step were shown in Fig. 7. The abscissa denotes the limited-view extent, the ordinate denotes the corresponding imaging resolution. It is easy to find that the horizontal resolution of the filtered backprojection algorithm improves with increasing sampling views extent. And this increasing process is non-linear. Using Fourier-slice theorem for reflective tomography, the ideal reconstruction algorithm produces an estimate of the Fourier transform of the object for all frequencies within a disk [19,20]. This effect of the limited-view only gives partial frequencies in Fourier domain of the target. In this way, a limited number of views degrade the imaging resolution by low pass filtering the Fourier domain in certain directions. We also collect projection data with smaller sampling view interval in limited-view, and the transverse slices through the center of images reconstructed in 0.1° step were shown in Fig. 8. Compared
f ðNΔR2 ; θÞ ¼
4. Imaging resolution in limited-view The image reconstruction of single diffuse scatter point with limited projections data was simulated in this section. It is assumed that the diffused point target has the same reflective characteristics for all directions. Projections and back-projections light ray tracing required for forward simulation of reflected projections data and reconstruction are preformed using the Siddon algorithm [17]. It is assumed that the laser pulse receiver is at the same location as the transmitter. For all tomographic reconstructions in this section, a standard filtered backprojection algorithm is adopted to reconstruct the point target from limited-view domain. For the limited-view extent over angle φ, the sampling views of projection and backprojection operation range between − φ2 and φ2 . Limited data will introduce anisotropic resolution [18] in the final imaging results, and only the horizontal resolution was considered here. Fig. 5 shows the images reconstruction results of a point target in different limited views,
Fig. 6. The horizontal slice through the center of image reconstruction in different limited-views in 1° step.
Fig. 8. The horizontal slice through the center of image reconstruction in 0.1° step.
X. Jin et al. / Optics Communications 285 (2012) 2575–2579
2579
Fig. 9. Images reconstructed by filtered back-projection in limited views: (a)45°; (c) 90°; (d) 360°.
with the slices in Fig. 6, the boundary of the imaging point becomes smooth, however the imaging resolution does not obtain substantial improvement. Experimental setup was used to measure range-resolved projections of a letter “E” [22]. Here simplified description and imaging results from limit-view projections were given. A letter “E” is placed on a plane with a tilt angle to the horizontal plane and rotated about the axis perpendicular to it, the target is illuminated by parallel light pulses. Range-resolved return signal is collected by a nonimaging optical system. Fig. 9(a)(b)(c) are the reconstructed images from 45°, 90° and 360° by filter back-projection algorithm. It can be seen from the images that with 360 degree view of projections a high-quality image can be acquired, when the view of projections decrease, noise and distortion shows up in the reconstructed images. The letter E could not be distinguished from 45°. 5. Conclusion We have derived the PSF associated with filtered backprojection algorithm over full views for reflective tomography in this paper. And the theoretical imaging resolution using the Rayleigh criterion of this imaging technology was represented by the range resolution resolve by the laser pulse. The simulated horizontal imaging resolutions in different limited-views were compared with the theoretical resolution. It is found that limited number of views degrades the imaging resolution by low pass filtering the Fourier domain in certain directions. Also the imaging resolutions do not obtain substantial improvement, even with smaller sampling angle interval in limited views. Images reconstructed of letter “E” were given from limitview projections, and varying noises and geometric distortion show up in different views. In real application of reflective tomography, it is difficult to determine the illuminated views extent of a point target on the object surfaces. Also the reflection from an elemental area on the object depends upon the incident and reflective orientation of that area. Shadowing effects, artifacts can lead to bad resolution in the image reconstructed even when no noise exists in the projections. For the image reconstruction of target surfaces with complicated reflective
characteristics [21], only diffuse model would not be enough to describe accurately, in this case, the imaging resolution in limited views needs further study. Acknowledgment The authors acknowledge the support of the Key Laboratory of Space Laser Communication and Testing Technology of Chinese Academy of Sciences and the National Nature Science Foundation of China (Grant No. 61108069). References [1] J.K. Parker, E.B. Cralg, D.I. Klick, et al., Applied Optics 27 (13) (1988) 2642. [2] R.M. Marino, R.N. Capes, J.K. Parker, et al., SPIE Laser Radar III 999 (1988) 248. [3] F.K. Knight, D. Klick, D.P. Ryan-Howard, et al., Applied Optics 28 (12) (1989) 2196. [4] F.K. Knight, D.I. Klick, D.P. Ryan-Howard, et al., Optical Engineering 30 (1) (1991) 55. [5] C.L. Matson, E.P. Magee, D.H. Stone, Proceedings of SPIE 2302 (1994) 73. [6] E.P. Magee, C.L. Matson, D.H. stone, SPIE Image Reconstruction and Restoration 2302 (1994) 95. [7] C.L. Matson, Proceeding of SPIE, SPIE Image Reconstruction and Restoration 2562 (1995) 184. [8] C.L. Matson, Optics Communications 137 (4-6) (1997) 343. [9] S.D. Ford, C.L. Matson, SPIE 3815 (1999) 189. [10] C.L. Matson, D.E. Mosley, Applied Optics 40 (14) (2001) 2290. [11] C.L. Matson, E.P. Magee, Donald E. Holland, Optical Engineering 34 (9) (1995) 2811. [12] A.C. Kak, M. Slaney, Principles of computerized tomographic imaging, IEEE Press, 1999. [13] S.J. Norton, Ultrasonic Imaging 1 (1979) 154. [14] S.J. Norton, IEEE Transactions on Biomedical Engineering BME-28 (1981) 220. [15] Y.S. Kwoh, I.S. Reed, IEEE Transaction on Nuclear Science NS-24 (5) (1977) 1990. [16] A.J. den Dekker, A. van den Bos, Journal of the Optical Society 14 (3) (1996) 547. [17] R.L. Siddon, Medical Physics 12 (1985) 252. [18] R.M. Rangayyan, R. Gordon, IEEE Transactions on Biomedical Engineering BME-30 (12) (1983) 806. [19] A.J. Devaney, IEEE Transactions on Geosciences and Remote Sensing GE-22 (1) (1984) 3. [20] A.J. Devaney, Inverse Problems 5 (1989) 501. [21] Xuemin Jin, Robert Y. Levine, Applied Optics 48 (21) (2009) 4191. [22] Yi Yan, Jianfeng Sun, Xiaofeng Jin, Liren Liu, Proceeding of SPIE 8162 (2011) 81620Y-1.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Imaging resolution analysis in limited-view Laser Radar reflective tomography Xiaofeng Jin, Jianfeng Sun ⁎, Yi Yan, Yu Zhou, Liren Liu Key Laboratory of Space Laser Communication and Testing Technology, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
a r t i c l e
i n f o
Article history: Received 25 September 2011 Accepted 30 January 2012 Available online 13 February 2012 Keywords: Reflective tomography Limited views Imaging resolution
a b s t r a c t In many applications of Laser radar reflective tomography, anisotropic resolution of images reconstructed from an incomplete set of reflective projections will degrade imaging quality. In this paper, we present the theoretical point spread function (PSF) of imaging system over full views for reflective tomography. The corresponding imaging resolution of the PSF was determined by Rayleigh's criterion, here the most common algorithm of filtered backprojection was used for image reconstruction. The theoretical imaging resolution was derived approximately from the range resolution resolved by the laser pulse. The simulated horizontal imaging resolutions of an ideal diffused single point target in different limited-views were compared with the theoretical value. Experimental reconstructed images from limit-view projections of letter “E” illustrated the effects of anisotropic resolution. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Reflective tomography is one of the most effective high-resolution imaging methods for Laser sensing and imaging technologies. In reflective tomography, Laser radar systems can be designed to provide remote sensing of range-resolved, Doppler-resolved and rangeDoppler resolved signals. Direct detection of rang-resolved reflective tomography techniques can be used to obtain an image of a target, which is angularly unresolved when the data are range resolved. These techniques are especially applicable for a target that is not rotating fast enough to have its Doppler spectrum resolved by the detector, and the projection data could be collected from direct detection or heterodyne detection system. The range-resolved Laser radar reflective tomography imaging was firstly introduced by Parker [1,2] and Knight [3,4] at the Massachusetts Institute of Technology (MIT). Several years later, Matson [5–9] in Air Force began exploring the technique of using the HI-CLASS coherent Laser radar system to obtain images reconstructed by carrying out a heterodyne system analysis, deriving and validating imaging signal-to-noise ratio expressions. One of the main issues in reflective tomography concentrates on how to estimate numerically tomographic imaging results, when sampling views are not theoretically sufficient for exact image reconstruction. Insufficient projection data problems occur quite frequently because of practical constraints due to the imaging hardware, scanning geometry or target rotation. For example, in actual Earthstabilized satellites imaging case, projections can be obtained at
most extent over approximately 130° [10]. The laser signal returned in reflective tomography is reflected off the illuminated outer surface of the target, and only information about the exterior can be obtained. The reflective projections at different views usually do not contain the same information about the target, and the projection data over full views would be difficult to infer from limited-view projections data. Artifacts and geometric distortion would be encountered in the imaging results of limited-view images reconstruction. Furthermore the images that are distinguishable when all projections are given may not be distinguishable when only a few projections are available. In this paper, the theoretical point spread function (PSF) associated with filtered back-projection algorithm in reflective tomography was derived, and shown to depend upon the cutoff frequency. Because the cutoff frequency was related to the range resolution resolved by the laser pulse [11], the corresponding theoretical imaging resolution of the PSF can be represented by the range resolution. For all images reconstructed from limited-views in this paper, the image pixel interval is equal to the range resolution. In this way, the theoretical imaging resolution determined by the PSF could be compared with the simulated horizontal imaging resolutions in different limited-views. Anisotropic resolution will degrade the quality of reconstructed image. Target would be difficult to distinguish due to the artifacts from the process of image reconstructed [22]. 2. Review of filtered backprojection algorithm Firstly, brief description of filtered backprojection [8,9] is provided for image reconstruction of reflective tomography. Let f(x, y) denote the image to be reconstructed, and Lr, ϕ denote the solid line r = x cos ϕ + y sin ϕ, see Fig. 1
⁎ Corresponding author. E-mail address: [email protected] (J. Sun). 0030-4018/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2012.01.075
pðr; ϕÞ ¼ ∫Lr;ϕ f ðx; yÞds;
ð1Þ
2576
X. Jin et al. / Optics Communications 285 (2012) 2575–2579
like artifacts, and then the image reconstruction gFB(x, y), can be given by g FB ðx; yÞ ¼
m X
F
−1
ðjvjF ðpðx cosϕi þ y sinϕi ; ϕÞÞÞΔϕ
ð3Þ
i¼1
where F and F − 1 represent the 1-dimensional Fourier transform and inverse Fourier transform operators, v represents frequency variable. 3. The derivation of the PSF over full views In the following, we give the PSF associated with filtered backprojection algorithm for parallel beams in reflective tomography. And then the resolution of the PSF is given by the range resolution using the Rayleigh's criterion. The expression of theoretical PSF is a very useful measurement of the overall performance of the imaging system, because it represents the maximum theoretical resolving capability of the imaging reconstruction technique. For simplification, it is assumed that the Laser pulse receiver is at the same location as the transmitter illustrated in Fig. 2. For a single scatter at (r, θ), the round-trip distance between the point target and the location of the transmitter and receiver at (R, ϕ) can be given by,
Fig. 1. The schematic diagram of reflective tomography.
2ρ ¼ 2R−2r cosðϕ−θÞ:
ð4Þ
We consider the shape of a received pulse reflected from a point target in terms of the temporal impulse response of a linear system which consists of the transmitter, receiver, and an isolated diffuse point target [12–14]. Let h(t) denote the impulse response of the linear system, which is shown in Fig. 3. h(t) can be assumed invariant with respect to target position, and is only proportional to the target reflectivity. Under these linear and time-invariant conditions, for the transmitted pulse pt(t), the received pulse pr(t, ϕ) can be written as, 2ρ 2ρ pr t− ; ϕ ¼ pt ðt Þ⊗h t− c c
ð5Þ
where ⊗ represents convolution operation, in theoretical consideration, the Eq. (5) can be given by, Fig. 2. The schematic diagram of single scattering point model.
where s represents solid line along Lr, ϕ, p(r, ϕ) is the projection of the target f(x, y) at angle ϕ, and the variable r is the spatial variable along the integration path in the ϕ direction. Using back-projection algorithm, the estimated image, g(x, y), is given by g ðx; yÞ ¼
m X
pðx cosϕi þ y sinϕi ; ϕi ÞΔϕ
ð2Þ
i¼1
pt ðt Þ ¼ δðt Þ; hðt Þ ¼ δðt Þ;
ð6Þ
2ρ 2ρ 2ρ ¼ δ t− pr t− ; ϕ ¼ δðt Þ⊗δ t− c c c
ð7Þ
where ε = ct, the reflective projection data versus the corresponding distance can be expressed as, pr ðε−2ρ; ϕÞ ¼ δðε−2ρÞ:
ð8Þ
For a single scatter at (r0, θ0),
where ϕi is the angle of the ith projection, Δϕ is the sampling angular separation, and m is the total number of projections. Filtered backprojection was introduced to modify projections to reduce star-
pr ðε−2ρ; ϕÞ ¼ δðε−2ρÞ ¼ δðε−2R−2r0 cosðϕ−θ0 ÞÞ:
Fig. 3. The linear system consisting of a transmitter/receiver and an isolated diffuse point target.
ð9Þ
X. Jin et al. / Optics Communications 285 (2012) 2575–2579
2577
Here the filter function of Ramachandran and Lakshminarayanan (RL) [15] was used. The cutoff frequency was set at vc, and the filter function H(v) can be given by, H ðvÞ ¼
jvj jvjbvc : 0 elsewhere
ð13Þ
Using the filter function above, the PSF of the image reconstruction can be transformed to, v
c f ðr; θÞ ¼ 2π∫−v jvjJ 0 ð4πvrÞdv: c
ð14Þ
Since J0(•) is an even function, it is easy to simplify Eq. (16), v
f ðr; θÞ ¼ 4π∫0c vJ 0 ð4πvrÞdv:
ð15Þ
Based on the integral transform characteristics of Bessel function, Fig. 4. The slice of PSF distribution given by Eq. (21).
d n n x J n ðxÞ ¼ x J n−1 ðxÞ: dx
Using the filtered back-projection algorithm described in Section 2, the estimated image can be given by, −1 f ðr; θÞ ¼ ∫2π fHðvÞF ½pr ðε−2ρ; ϕÞgdϕ 0 F
¼
∞ ∫2π 0 dϕ∫−∞ H ðvÞ exp½j2πvðτ−2ρ0 Þdv
ð10Þ
where F and F − 1 denote Fourier transform and inverse Fourier transform respectively, variable τ = 2ρ. When the scattering center is located at the origin, using the Bessel identity, J 0 ðr Þ ¼
1 2π ∫ exp½jr cosðϕ−θÞdϕ: 2π 0
ð11Þ
The PSF of the image reconstruction can be further simplified, ∞
f ðr; θÞ ¼ 2π∫−∞ HðvÞJ 0 ð4πvr Þdv:
ð12Þ
ð16Þ
The final PSF of image reconstruction is given by, f ðr; θÞ ¼
vc J ð4πvc r Þ: r 1
ð17Þ
The pulse δ(t) is not practical as transmitted pulse. The form for the laser pulse outgoing radiation must be assumed. For simplicity, suppose that transmitted pulse pt(t) in Eq. (5) is a rectangle with time duration τ, which has an effective spatial extent cτ/2 due to the reflection off of the target. We defined vc as the location of the first zero of the Sinc function that is the Fourier transform of the transmitted pulse pt(t) [11]. Then the cutoff frequency vc can be written as, vc ¼
1 cτ ; ΔR1 ¼ ΔR1 2
Fig. 5. Image reconstruction of single point target in different limited views with 1° step: limited-views over (a) 15°, (b) 30°, (c) 45°, (d) 60°, (e) 90°, (f) 360°.
ð18Þ
2578
X. Jin et al. / Optics Communications 285 (2012) 2575–2579
where ΔR1 is the range resolution resolved by the laser pulse, the final expression of PSF associated with filtered back-projection algorithm for reflective tomography can be given by, f ðr; θÞ ¼
1 J 1 ð4πr=ΔR1 Þ : ΔR1 r
ð19Þ
Based on the Rayleigh criterion, the imaging resolution X0 of the f(r, θ) at the first zero can be given by, X 0 ¼ 0:3049ΔR1 :
ð20Þ
If r = NΔR2, ΔR2 is the image reconstruction pixel interval. The PSF of image reconstruction is shown by, 1 J 1 ð4πNΔR2 =ΔR1 Þ : ΔR1 ΔR2 N
ð21Þ
Fig. 7. The horizontal imaging resolution results using the Rayleigh criterion in limited views.
If ΔR2 = ΔR1/3, the slice of PSF distribution given by Eq. (21) was shown in Fig. 4. We see that the PSF does not fall off smoothly from their peak values, but have obvious oscillations which gradually fall off in amplitude with increasing distance away from the peak values. This oscillatory behavior in the PSF induces artifacts in the image reconstruction of complicated structures. This oscillatory behavior is essentially caused by the sharp cutoffs in Fourier space restricted by the RL filter [16]. Using the Rayleigh criterion, the theoretical imaging resolution of filtered back-projection algorithm for reflective tomography was given in Eq. (20).
and the corresponding horizontal slices through the center of images reconstructed were illustrated in Fig. 6. For all tomographic image reconstruction in this section, the adjacent pixel interval was equal to the range resolution resolved by the laser pulse, so the simulated imaging resolution in different limited views can be expressed by the range resolution. Because we do not have projection data for full 360°, tomographic striping artifacts and geometric distortion in the image reconstruction are obvious in Fig. 6. Especially in the limitedview extent over 5° and 15°, the single scattering point can not be distinguished from these imaging results. These artifacts would be eliminated if the views spread out evenly over full 360°. Using the Rayleigh criterion (taking the resolution limit as the distance for which the ratio of the value at the central dip in the intensity distribution to that at the maxima on either side is equal to 0.81) [16], the horizontal imaging resolutions of different limited views in 1° step were shown in Fig. 7. The abscissa denotes the limited-view extent, the ordinate denotes the corresponding imaging resolution. It is easy to find that the horizontal resolution of the filtered backprojection algorithm improves with increasing sampling views extent. And this increasing process is non-linear. Using Fourier-slice theorem for reflective tomography, the ideal reconstruction algorithm produces an estimate of the Fourier transform of the object for all frequencies within a disk [19,20]. This effect of the limited-view only gives partial frequencies in Fourier domain of the target. In this way, a limited number of views degrade the imaging resolution by low pass filtering the Fourier domain in certain directions. We also collect projection data with smaller sampling view interval in limited-view, and the transverse slices through the center of images reconstructed in 0.1° step were shown in Fig. 8. Compared
f ðNΔR2 ; θÞ ¼
4. Imaging resolution in limited-view The image reconstruction of single diffuse scatter point with limited projections data was simulated in this section. It is assumed that the diffused point target has the same reflective characteristics for all directions. Projections and back-projections light ray tracing required for forward simulation of reflected projections data and reconstruction are preformed using the Siddon algorithm [17]. It is assumed that the laser pulse receiver is at the same location as the transmitter. For all tomographic reconstructions in this section, a standard filtered backprojection algorithm is adopted to reconstruct the point target from limited-view domain. For the limited-view extent over angle φ, the sampling views of projection and backprojection operation range between − φ2 and φ2 . Limited data will introduce anisotropic resolution [18] in the final imaging results, and only the horizontal resolution was considered here. Fig. 5 shows the images reconstruction results of a point target in different limited views,
Fig. 6. The horizontal slice through the center of image reconstruction in different limited-views in 1° step.
Fig. 8. The horizontal slice through the center of image reconstruction in 0.1° step.
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Fig. 9. Images reconstructed by filtered back-projection in limited views: (a)45°; (c) 90°; (d) 360°.
with the slices in Fig. 6, the boundary of the imaging point becomes smooth, however the imaging resolution does not obtain substantial improvement. Experimental setup was used to measure range-resolved projections of a letter “E” [22]. Here simplified description and imaging results from limit-view projections were given. A letter “E” is placed on a plane with a tilt angle to the horizontal plane and rotated about the axis perpendicular to it, the target is illuminated by parallel light pulses. Range-resolved return signal is collected by a nonimaging optical system. Fig. 9(a)(b)(c) are the reconstructed images from 45°, 90° and 360° by filter back-projection algorithm. It can be seen from the images that with 360 degree view of projections a high-quality image can be acquired, when the view of projections decrease, noise and distortion shows up in the reconstructed images. The letter E could not be distinguished from 45°. 5. Conclusion We have derived the PSF associated with filtered backprojection algorithm over full views for reflective tomography in this paper. And the theoretical imaging resolution using the Rayleigh criterion of this imaging technology was represented by the range resolution resolve by the laser pulse. The simulated horizontal imaging resolutions in different limited-views were compared with the theoretical resolution. It is found that limited number of views degrades the imaging resolution by low pass filtering the Fourier domain in certain directions. Also the imaging resolutions do not obtain substantial improvement, even with smaller sampling angle interval in limited views. Images reconstructed of letter “E” were given from limitview projections, and varying noises and geometric distortion show up in different views. In real application of reflective tomography, it is difficult to determine the illuminated views extent of a point target on the object surfaces. Also the reflection from an elemental area on the object depends upon the incident and reflective orientation of that area. Shadowing effects, artifacts can lead to bad resolution in the image reconstructed even when no noise exists in the projections. For the image reconstruction of target surfaces with complicated reflective
characteristics [21], only diffuse model would not be enough to describe accurately, in this case, the imaging resolution in limited views needs further study. Acknowledgment The authors acknowledge the support of the Key Laboratory of Space Laser Communication and Testing Technology of Chinese Academy of Sciences and the National Nature Science Foundation of China (Grant No. 61108069). References [1] J.K. Parker, E.B. Cralg, D.I. Klick, et al., Applied Optics 27 (13) (1988) 2642. [2] R.M. Marino, R.N. Capes, J.K. Parker, et al., SPIE Laser Radar III 999 (1988) 248. [3] F.K. Knight, D. Klick, D.P. Ryan-Howard, et al., Applied Optics 28 (12) (1989) 2196. [4] F.K. Knight, D.I. Klick, D.P. Ryan-Howard, et al., Optical Engineering 30 (1) (1991) 55. [5] C.L. Matson, E.P. Magee, D.H. Stone, Proceedings of SPIE 2302 (1994) 73. [6] E.P. Magee, C.L. Matson, D.H. stone, SPIE Image Reconstruction and Restoration 2302 (1994) 95. [7] C.L. Matson, Proceeding of SPIE, SPIE Image Reconstruction and Restoration 2562 (1995) 184. [8] C.L. Matson, Optics Communications 137 (4-6) (1997) 343. [9] S.D. Ford, C.L. Matson, SPIE 3815 (1999) 189. [10] C.L. Matson, D.E. Mosley, Applied Optics 40 (14) (2001) 2290. [11] C.L. Matson, E.P. Magee, Donald E. Holland, Optical Engineering 34 (9) (1995) 2811. [12] A.C. Kak, M. Slaney, Principles of computerized tomographic imaging, IEEE Press, 1999. [13] S.J. Norton, Ultrasonic Imaging 1 (1979) 154. [14] S.J. Norton, IEEE Transactions on Biomedical Engineering BME-28 (1981) 220. [15] Y.S. Kwoh, I.S. Reed, IEEE Transaction on Nuclear Science NS-24 (5) (1977) 1990. [16] A.J. den Dekker, A. van den Bos, Journal of the Optical Society 14 (3) (1996) 547. [17] R.L. Siddon, Medical Physics 12 (1985) 252. [18] R.M. Rangayyan, R. Gordon, IEEE Transactions on Biomedical Engineering BME-30 (12) (1983) 806. [19] A.J. Devaney, IEEE Transactions on Geosciences and Remote Sensing GE-22 (1) (1984) 3. [20] A.J. Devaney, Inverse Problems 5 (1989) 501. [21] Xuemin Jin, Robert Y. Levine, Applied Optics 48 (21) (2009) 4191. [22] Yi Yan, Jianfeng Sun, Xiaofeng Jin, Liren Liu, Proceeding of SPIE 8162 (2011) 81620Y-1.