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Fig. 2. One-dimensional optical signal and its two-dimensional phase space. (a) The optical signal arriving at an angle 6 and collected by a lens. (b) The phase space obtained from the Wigner transform is a two-dimensional representation as a func- tion of @ (in radians) and its reciprocal {spatial angle frequency {in cycles per radian (cpr)| a. If the FOV of the lens is Aé and the angular resolution of each ray is Aa/2 cpr, then the accep- tance range is a rectangle in the phase space. The area of the rectangle is AéAa and gives the SWP.

Figure 2 One-dimensional optical signal and its two-dimensional phase space. (a) The optical signal arriving at an angle 6 and collected by a lens. (b) The phase space obtained from the Wigner transform is a two-dimensional representation as a func- tion of @ (in radians) and its reciprocal {spatial angle frequency {in cycles per radian (cpr)| a. If the FOV of the lens is Aé and the angular resolution of each ray is Aa/2 cpr, then the accep- tance range is a rectangle in the phase space. The area of the rectangle is AéAa and gives the SWP.

Related Figures (15)

Fig. 3. FOV A@, is determined by the condition of nonoverlap- ping elemental images. where OTF ,.(a) denotes the OTF of the lenslet or aperture and OTF,,,(@) denotes the OTF of the recording device. The image can be recorded by a photographic film or by electronic means such as a charge-coupled device (CCD).
Fig. 3. FOV A@, is determined by the condition of nonoverlap- ping elemental images. where OTF ,.(a) denotes the OTF of the lenslet or aperture and OTF,,,(@) denotes the OTF of the recording device. The image can be recorded by a photographic film or by electronic means such as a charge-coupled device (CCD).
For a pixelated sensor such as a CCD, it is not possible to define an OTF in a regular way because the sampling process by the arrays of pixels is a space-variant process. However to preserve the convenience of the transfer func- tion approach, it is useful to define a spatially averaged OTF (AOTF)**?° by assuming that the sampled image is randomly positioned with respect to the sampling grid lo- cation. The AOTF of the capturing system is derived in Appendix A: where p,px is the pixel sensor size, FF, is the fill factor, M is lateral magnification of the pickup optics, and g, is the
For a pixelated sensor such as a CCD, it is not possible to define an OTF in a regular way because the sampling process by the arrays of pixels is a space-variant process. However to preserve the convenience of the transfer func- tion approach, it is useful to define a spatially averaged OTF (AOTF)**?° by assuming that the sampled image is randomly positioned with respect to the sampling grid lo- cation. The AOTF of the capturing system is derived in Appendix A: where p,px is the pixel sensor size, FF, is the fill factor, M is lateral magnification of the pickup optics, and g, is the
Fig. 4. Aperture-plate II pickup system with a pixelated sensor. The optical field passing through an array of pinholes (aperture plate) is collected by pickup optics (here illustrated by a lens) and captured by a digital sensor with pixels of size p.,,., and fill factor FF, . gap between the lenslet array and the pickup-optics im aging plane. Table 1. Parameters of the Capturing System
Fig. 4. Aperture-plate II pickup system with a pixelated sensor. The optical field passing through an array of pinholes (aperture plate) is collected by pickup optics (here illustrated by a lens) and captured by a digital sensor with pixels of size p.,,., and fill factor FF, . gap between the lenslet array and the pickup-optics im aging plane. Table 1. Parameters of the Capturing System
Fig. 5. Dependence of the Shannon number on the optimal gap g [expression (13)] for the setup in Table 1 and M = 1/2, X = 0.55 wm.
Fig. 5. Dependence of the Shannon number on the optimal gap g [expression (13)] for the setup in Table 1 and M = 1/2, X = 0.55 wm.
Fig. 6. Lenslet-array II capturing system with a pixelated sensor. The expression for an OTF with a circular aperture can be found in Ref. 7. In Eq. (14), the first term indicates the diffraction limit that is due to the lens width, and the sec- ond term sinc(-) indicates the effect of the focusing error, which is caused by a mismatch between the lenslet image position and the pickup-optics focusing position. The fo- cusing error can be minimized by appropriate focusing of the pickup lens. The AMTF of an object that is severely out of focus is shown in Fig. 8. The object in this example is assumed to AOTF,,; is dependent on the object location because of the dependence of OTF,; on z. Examples of the overall aver- age AMTF..,, of the capturing system in Fig. 6 are shown in Figs. 7 and 8. Table 2 shows the parameters of the capturing system designed to image an object located at 0.5 m from the lenslet array. The MTF for an object lo- cated in focus (0.5 m from the lenslet array) is shown in Fig. 7. Since the object is in focus, the optics MTF [Eq. (14)] is only diffraction limited. However, the pixelated sensor induces a much more severe limitation. The pix- elated sensor constraint is the main limiting source on the Shannon number unless the object is moved severely out of focus.
Fig. 6. Lenslet-array II capturing system with a pixelated sensor. The expression for an OTF with a circular aperture can be found in Ref. 7. In Eq. (14), the first term indicates the diffraction limit that is due to the lens width, and the sec- ond term sinc(-) indicates the effect of the focusing error, which is caused by a mismatch between the lenslet image position and the pickup-optics focusing position. The fo- cusing error can be minimized by appropriate focusing of the pickup lens. The AMTF of an object that is severely out of focus is shown in Fig. 8. The object in this example is assumed to AOTF,,; is dependent on the object location because of the dependence of OTF,; on z. Examples of the overall aver- age AMTF..,, of the capturing system in Fig. 6 are shown in Figs. 7 and 8. Table 2 shows the parameters of the capturing system designed to image an object located at 0.5 m from the lenslet array. The MTF for an object lo- cated in focus (0.5 m from the lenslet array) is shown in Fig. 7. Since the object is in focus, the optics MTF [Eq. (14)] is only diffraction limited. However, the pixelated sensor induces a much more severe limitation. The pix- elated sensor constraint is the main limiting source on the Shannon number unless the object is moved severely out of focus.
The similar approximation in expression (10) holds for AOTF,,, in Eq. (16) if the OTF,, has a large cutoff angular spatial frequency. The AMTF.,, is given by the modulus of the AOTF.,,. Note that, unlike AOTF.,, in Eq. (9), The AOTF of the system is similar to Eq. (9) but with the lenslet OTF [Eq. (14)] replacing the aperture OTF:
The similar approximation in expression (10) holds for AOTF,,, in Eq. (16) if the OTF,, has a large cutoff angular spatial frequency. The AMTF.,, is given by the modulus of the AOTF.,,. Note that, unlike AOTF.,, in Eq. (9), The AOTF of the system is similar to Eq. (9) but with the lenslet OTF [Eq. (14)] replacing the aperture OTF:
Table 2. Parameters of the Capturing System
Table 2. Parameters of the Capturing System
Fig. 7. AMTF of an in-focus pickup system with parameters shown in Table 2. The dotted curve is the lenslet MTF [Eq. (14)], the dashed curve denotes the approximated pure sampling AMTF in expression (10), and the solid curve is the total system AMTF [Eq. (16)]. The solid and the dashed curves almost coin- cide. The cutoff frequencies due to diffraction, sampling, and out-of-focus are agi = 1818 cpr, a,, = 126 cpr, and ayer = ©, respectively. It can be seen that the effect of sampling is dominant.
Fig. 7. AMTF of an in-focus pickup system with parameters shown in Table 2. The dotted curve is the lenslet MTF [Eq. (14)], the dashed curve denotes the approximated pure sampling AMTF in expression (10), and the solid curve is the total system AMTF [Eq. (16)]. The solid and the dashed curves almost coin- cide. The cutoff frequencies due to diffraction, sampling, and out-of-focus are agi = 1818 cpr, a,, = 126 cpr, and ayer = ©, respectively. It can be seen that the effect of sampling is dominant.
Fig. 8. AMTF of a severely out-of-focus pickup system. The system is focusing at 0.5 m from the lenslet array while the ob- ject is located at 0.05 m from the lenslet array. The dotted curve is the MTF according to Eq. (14), the dashed curve denotes the approximated pure sampling AMTF in expression (10), and the solid curve is the total system AMTF [Eq. (16)]. It can be seen that the out-of-focus effect is dominant.
Fig. 8. AMTF of a severely out-of-focus pickup system. The system is focusing at 0.5 m from the lenslet array while the ob- ject is located at 0.05 m from the lenslet array. The dotted curve is the MTF according to Eq. (14), the dashed curve denotes the approximated pure sampling AMTF in expression (10), and the solid curve is the total system AMTF [Eq. (16)]. It can be seen that the out-of-focus effect is dominant.
Fig. 9. Optical display subsystem. If digital recording and display are used, then the sam- pling effects need to be considered, and the AOTF and AMTF can be defined. In such a case the sampled optical signal is typically interpolated by the capturing system, sampled again by the digital display device, and propa- gated by the output aperture plate or lenslet array. Fol- lowing the AMTF approach in Section 3 and Appendix A, we give the AOTF at the output of the digital display de-
Fig. 9. Optical display subsystem. If digital recording and display are used, then the sam- pling effects need to be considered, and the AOTF and AMTF can be defined. In such a case the sampled optical signal is typically interpolated by the capturing system, sampled again by the digital display device, and propa- gated by the output aperture plate or lenslet array. Fol- lowing the AMTF approach in Section 3 and Appendix A, we give the AOTF at the output of the digital display de-
vice by substituting into Eq. (A7) the capture AOTF [Eqs. (9) and (16)] multiplied by the interpolation filter OTF H int‘ Typically, the interpolation filter H;,; is a low-pass filter with a cutoff frequency that is the same as the Nyquist frequency a; = Mg,/2p.px of the capturing device.
vice by substituting into Eq. (A7) the capture AOTF [Eqs. (9) and (16)] multiplied by the interpolation filter OTF H int‘ Typically, the interpolation filter H;,; is a low-pass filter with a cutoff frequency that is the same as the Nyquist frequency a; = Mg,/2p.px of the capturing device.
Fig. 10. II imaging system with a pixelated sensor. The instantaneous angular FOV of each pixel is FF .p.p.1/Mg,.
Fig. 10. II imaging system with a pixelated sensor. The instantaneous angular FOV of each pixel is FF .p.p.1/Mg,.
where H(q) is the product of the lens-array or aperture- plane OTF with the pixel OTF in Eq. (A2). Another way to obtain the AOTF of the system is by first averaging the PSF in Eq. (A4) and then taking the Fourier transform, which yields To relate Eq. (A5) to the system in Fig. 10, we define the pickup-optics focusing plane to be the canonical plane so the sampling interval is A = p,,,/M and the angular parameter is 9 = tan \(x/g). Under paralaxial approxi- mation, the AOTF of the system in Fig. 10 is then
where H(q) is the product of the lens-array or aperture- plane OTF with the pixel OTF in Eq. (A2). Another way to obtain the AOTF of the system is by first averaging the PSF in Eq. (A4) and then taking the Fourier transform, which yields To relate Eq. (A5) to the system in Fig. 10, we define the pickup-optics focusing plane to be the canonical plane so the sampling interval is A = p,,,/M and the angular parameter is 9 = tan \(x/g). Under paralaxial approxi- mation, the AOTF of the system in Fig. 10 is then
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