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Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system
ARTICLE IN PRESS Optics & Laser Technology 42 (2010) 561–569
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Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system Hui Zhao a,n, Yingcai Li a, Huajun Feng b, Zhihai Xu b, Qi Li b a b
Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Science, Xi’an 710119, China Zhejiang University, State Key Laboratory of Modern Optical Instrumentation, Department of Optical Engineering, Hangzhou 310027, China
a r t i c l e in fo
abstract
Article history: Received 7 January 2009 Received in revised form 11 August 2009 Accepted 23 October 2009 Available online 22 November 2009
Wave-front coding is a well known technique used to extend the depth of field of incoherent imaging system. The core of this technique lies in the design of suitable phase masks, among which the most important one is the cubic phase mask suggested by Dowski and Cathey (1995) [1]. In this paper, we propose a new type called cubic sinusoidal phase mask which is generated by combing the cubic one and another component having the sinusoidal form. Numerical evaluations and real experimental results demonstrate that the composite phase mask is superior to the original cubic phase mask with parameters optimized and provides another choice to achieve the goal of depth extension. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Wave-front coding Sinusoidal component Composite phase mask
1. Introduction As is widely known, wave-front coding is a powerful design method suggested by Dowski and Cathey [1] to extend the depth of field of incoherent imaging system in 1995. By introducing one phase mask to the aperture plane, the optical transfer function (OTF) can be modified to be insensitive to focus errors and defocus related aberrations, and it is this modification that greatly extends the depth of field. The most important part of wave-front coding technique lies in the design of suitable phase masks. During the last decade, lots of researchers have devoted themselves to this topic and many kinds of phase masks have been proposed. Among all phase masks, the cubic type is the most popular one because of its simplicity and superior performance in the real application. Generally speaking, there are mainly two important metrics to judge the performance of wave-front coding system. One is the defocused MTF (modulation transfer function) and the other is Fisher information (FI). The goal of designing one phase mask lies in two aspects. On the one hand, the magnitude of defocused MTF should be large enough to make the digital post processing easier. On the other hand, FI values should be small enough to make the overall system less sensitive to defocus as much as possible. Unfortunately, the two aspects are contradictory in most cases according to the existing researches [1–5]. In other words, it is very hard to achieve the satisfactory results of big magnitude of MTF and small values of FI at the same time. However, several
n
Corresponding author. E-mail address: [email protected] (H. Zhao).
0030-3992/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2009.10.004
masks, such as modified cubic type [3], exponential type [4] and logarithmic type [5,8] and so on, are still designed to improve the performance of wave-front coding system to a certain extent. Researchers never stop looking for new masks and so do we. As known in the literatures, there are two ways to obtain the satisfactory phase mask. One is operated in spatial domain and the other is in frequency domain. In spatial domain, defocused point spread function (PSF) is derived using scalar diffraction theory [5,9] and the phase function can be obtained by solving a differential equation generated by making the derivative of defocused PSF with respect to defocus parameter be zero. In frequency domain, the operation is nearly the same except that the defocused MTF rather than the defocused PSF is used to do the computing [1,6]. For many years, the derivation procedure did not change too much, which meant that a standard computation routine had been established. So, it is not easy for us to create a new phase mask completely from mathematical point of view. Under these circumstances, we change our viewpoint. In this paper, we propose the concept of composite phase mask by adding additional component to the existing phase mask to try to improve the system performance. The simulation results and real experiments demonstrate the effectiveness of the new phase mask and the researches done in this paper are not only interesting, but also meaningful.
2. Proposal of composite phase mask The phase profiles corresponding to several popular phase masks are plotted in Fig. 1, where all the curves are generated
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Fig. 2. Phase profiles of the composite phase mask with different parameters. Fig. 1. Phase profiles of several popular phase masks with typical parameters.
with typical parameters. In Fig. 1, the symbols ‘Cub’, ‘Exp’, ‘Log1’, ‘Log2’, ‘Ho’, denote cubic type, exponential type, logarithmic type, improved logarithmic type and high-order type, respectively. By observing Fig. 1, we can make a bold hypothesis: any phase mask whose profile is similar to those of existing ones can probably be used to extend the depth of field. The underlying phase mask candidates can be easily obtained by superimposing additional component to the existing phase masks and these masks are named as composite phase masks. Compared with the rigorous mathematical derivation, this way of generating new phase masks is direct and easier to operate. In this paper, we only take the cubic phase mask for example to explain the concept of composite phase mask, because the cubic phase mask is the most classical one among all kinds of phase masks. Whether similar results can be obtained or not on the condition that other types of masks are used as the basis will be further studied in future. The idea of composite phase mask is interesting and our motivation comes from reference [7]. However, reference [7] just mentioned the possibility of doing the combination and did not have an implementation. The work done in this paper is the first time to change the possibility into real fact and is a good proof to the wild guess. The ‘additional component’ is crucial to the composite phase mask. Factually, there are lots of candidates, such as exponential function, logarithmic function, sinusoidal function and cosine function and so on. Finally, the component which has sinusoidal form is selected for investigation. There are two reasons to do so. One is that the new phase mask generated by superimposing sinusoidal component demonstrates more versatile variation in phase profile, as shown by Fig. 2. The other is that sine or cosine function is rare to be studied in the literature centering on the topic of depth extension effect. Besides those, considering that sinusoidal and cosine function can be transformed between each other, the composite phase mask discussed here can be described by Eq. (1): f ðx; yÞ ¼ aðx3 þ y3 Þ þ bðsinðoxÞ þ sinðoyÞÞ
ð1Þ
where a and b together control the phase deviation; o is the angular frequency. In Eq. (1), the first part denotes the original cubic phase mask and the second one denotes ‘additional component’. By rearranging Eq. (1), Eq. (2) can be obtained: f ðx; yÞ ¼ ðax3 þ b sinðoxÞÞ þ ðay3 þ b sinðoyÞÞ
ð2Þ
In order to study the characteristics of the new phase mask denoted by Eq. (2), how to choose values for three parameters (a, b and o) is the first question that should be answered. According to references [3,5], two criteria have to be satisfied. First, the value of a and b has to be set to the same level. This constraint is reasonable because the purpose of composite phase mask lies in the combination of different components. As Eq. (2) tells, a and b together control the phase deviation of the mask. If the value of one parameter is much bigger or smaller than another, the composite phase mask will degrade into a single phase mask again and this violates the purpose of our research. Second, with the sinusoidal part added, the odd-symmetry and monotonicity should not be broken, which allows o to change within a relatively small range. The sinusoidal part of the composite phase mask is periodic. When the value of o is big enough, the phase function will start to oscillate on the condition that the values of a and b stay at the same order and this breaks the monotonicity constraint. However, if the value of o is too small, the influence of the sinusoidal part will become less important and this is similar to the situation as discussed in the former paragraph. For clarity, the comparison among 1-D phase profiles with different parameters is shown in Fig. 2 to explain the two criteria. In Fig. 2, the black solid curve denotes the original cubic phase mask with a set to 90 for comparison. The blue dotted curve corresponds to the new mask with b and o set to a/20 and 1, respectively. It is easy to see that the new phase mask is nearly the same as the original one in this situation (corresponds to the first criterion). The red dashed and magenta dot-dashed line is generated on the condition that o is set to 5 and 1, respectively, while b equals a. As you see, the phase profile denoted by the red dashed curve violates the characteristic of monotonicity (corresponds to the second criterion) and only magenta dot-dashed curve satisfies the two criteria at the same time. It is not difficult to find out that if o equals pi/2, the monotone part of sinusoidal function will just lie within the range of [ 1, 1] that covers the entire pupil plane as displayed in Fig. 1 and it is impossible for the second criterion to be violated. If the relationship between b and a can be determined at the same time, the complete model of the composite phase mask discussed in this paper will be established. As mentioned above, the magnitude of b and a has to have the same order and thus the ratio between them should lie within a reasonable range. If the ratio is seen as a
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that a is set to same value. The simulation results corresponding to two situations will be shown below: (1) Fig. 4 is one example in which the maximum phase deviation of the two masks is same. As Fig. 4 tells, when a in CSPM equals 90, the corresponding one in CPM should be set to 135 to comply with the first kind of constraint. Then, with defocus parameter set to 0, p, 3p, 6p and 10p, respectively, defocused MTF curves can be obtained, as shown in Fig. 5. Clearly, CSPM cannot only be used to extend the depth of field, but also can increase the magnitude of defocused MTF. This is an appealing characteristic which is helpful to digital post processing. Besides defocused MTF curves, FI is used as well to judge the performance of CSPM and CPM. As Fig. 6 reveals, when the additional component is added to the cubic phase mask, the value of FI increases and this means that the CSPM is a little bit sensitive to focus errors than CPM is. This phenomenon can be easily and clearly explained. According to the literatures, to any phase mask, the bigger the deviation degree is, the smaller the magnitude of defocused MTF is, and vice versa. Fig. 4 shows that the central part Fig. 3. Comparison between phase profiles of CPM and CSPM with same parameters.
variable, the following discussions will become complicated because at least two parameters should be jointly considered. Here, we simply set the ratio to 2 without loss of generality and the corresponding investigation can be greatly simplified. Discussion to other cases with ratio set to different values can be done with the same procedure. More complicated cases, in which the ratio is seen as a variable, will be left for future study. So, the condition under which the new phase mask can work is given, as shown below:
b¼
a 2
;
o¼
p 2
With Eq. (3), Eq.(2) can be rewritten as a p a p f ðx; yÞ ¼ ax3 þ sin x þ ay3 þ sin y 2 2 2 2
ð3Þ
ð4Þ
The phase mask described by Eq. (4) is called CSPM which is the abbreviation for cubic-sinusoidal phase mask. With a set to different values, the 1-D phase profile corresponding to CSPM can be obtained as shown in Fig. 3, in which the phase profile of original cubic mask is also drawn for comparison. With the expression of CSPM, numerical evaluations can be carried out to show the performance of the composite phase mask.
Fig. 4. Phase profiles of CPM and CSPM with the constraint that the maximum phase deviation is same.
3. Characteristics of CSPM and corresponding discussions Researchers usually use defocused MTF and Fisher information to evaluate the performance of one phase mask. We do the similar thing in this section with two cases considered. In the first part, the evaluation is carried out with CSPM and CPM (cubic phase mask) not optimized and we do the numerical evaluating in the second part under the condition that CSPM and CPM are optimized. 3.1. Performance comparison between CSPM and CPM with optimization procedure not introduced Although the optimization procedure is not used, certain constraints should be satisfied to make sure that the following comparisons are meaningful. Two kinds of constraints are considered here. The first one is that the maximum phase deviation of the two masks remains same and the second one is
Fig. 5. Comparison between defocused MTF curves of CPM and CSPM corresponding to Fig. 4.
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Fig. 6. Comparison between Fisher information of CPM and CSPM corresponding to Fig. 4.
Fig. 7. Another example in which the constraint of same maximum phase deviation is satisfied. (The left part is the comparison of defocused MTF curves and the right part is the comparison of Fisher information curves.)
of the new phase mask gradually approaches a tilt and this tilt contributes less effect on extending the depth of field. This means that by adding the sinusoidal component, the overall deviation degree of the new phase mask becomes relatively small, which causes the increase of the magnitude of MTF and thus makes phase mask more sensitive to defocus. Another example in which a in CPM and CSPM equals 180 and 120 is also provided and similar results are shown in Fig. 7. (2) When a in the two phase masks is set to same value, the corresponding phase profiles can be drawn in Fig. 8. At the same time, defocused MTF and Fisher information can also be computed and the results are shown in Figs. 9 and 10. As Figs. 9 and 10 tell, besides the increase of the magnitude of defocused MTF, the increase of FI curves of CSPM can also bee seen. The reason leading to this phenomenon has been discussed above and is omitted here. In order to prove that the results are not occasional, another example where a is set to 150 is given
Fig. 8. Phase profiles of CPM and CSPM with the constraint that the values of a in both masks are same.
Fig. 9. Comparison between defocused MTF curves of CPM and CSPM corresponding to Fig. 8.
below, as shown in Fig. 11, and we can find out that the results are similar. Now we can make a simple conclusion to end the first part of this section. With the two constraints considered above, CSPM can improve the magnitude of defocused MTF, but becomes more sensitive to focus error. At the same time, it is not difficult to find out that the magnitude of MTF corresponding to CSPM mainly increases in low to mid frequencies, but decreases at higher frequencies. So, the observations above mean that CSPM can perform better than CPM within a reduced range of depth of field. In other words, the MTF curves corresponding to CSPM will become instable with increase of focus error. Just as mentioned in Section 1, the results correspond to the fact that larger magnitude of MTF and smaller FI values usually contradict each other. Besides that, a trend is obvious. If a of CSPM is bigger than that of CPM and gradually increases to a certain value, the MTF curves of CPM and CSPM will approach each other, so does the FI curves. To this point, an interesting question is proposed. In the case when the MTF of both phase masks is nearly same, which mask is
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mask is optimized in order to achieve two goals. One is to make the magnitude of defocused MTF as large as possible and the other is to make the defocused MTF curves stable enough within the extended depth of field. The similar description can be found in reference [5]. Here, we adopt the Fisher-information based optimization method [4] and the optimization procedure can be simply described as follows: min FIða; c0 Þ ¼ ( subjectto
Fig. 10. Comparison between Fisher information of CPM and CSPM corresponding to Fig. 8.
Fig. 11. Another example in which the values of a in both CPM and CSPM are same. (The left part is the comparison of defocused MTF curves and the right part is the comparison of Fisher information curves.)
much less sensitive to defocus, CPM or CSPM? The next part of this section will give a positive answer. 3.2. Performance comparison between CSPM and CPM with optimization procedure introduced In the first part of this section, numerical evaluations are carried out with CPM and CSPM not optimized. However, this is not the usual case. When we decide to use one type of phase mask, the first step is to optimize the mask to obtain optimum parameters. With suitable optimization procedure introduced, the optimum phase mask can be computed and it will perform the best in theoretical sense. Generally speaking, there are 3–4 methods that can be used to optimize one wave-front coded imaging system, such as Strehl– Ratio based optimization [3], Fisher-information based optimization [3,4], PSF or MTF stability based optimization [5] and so on. No matter which method is used, the purpose is same. One phase
R c0
c0
2 ! ! R@ Hðu; cÞ du dc @c
a a Z a0 R AðaÞ ¼ 9Hðu; 0Þ9du Z k
ð5Þ
where a is the mask parameter, c0 the maximum focus error; u is the normalized frequency; H(u,c) is the 1-D defocused optical transfer function; A(a) is the integrated area of the in-focus MTF curve and its minimum value is set to k; FI denotes the Fisher information and is the object function we would like to minimize under two constraints. The first constraint gives the lower bound of phase parameter denoted as a0 and the second one has to be satisfied in order to make the values of in-focus MTF as large as possible, which is the requirement for digital post processing. According to Eq. (5), the optimization of CPM and CSPM can be easily operated with the help of optimization tool box in Matlab. The related constants and final optimum parameters are provided in Table 1. There are five groups of optimum parameters in Table 1 and the performance comparison between CPM and CSPM can be carried out immediately. The optimization procedure adopted here will lead to a result that the defocused MTF curves of CSPM and CPM have similar shapes. As Figs. 12–16 tell, no matter which group of optimum parameters is used, the defocused MTF curves of the two masks are nearly the same. However, by zooming in the local part of the MTF curves, we can find that the MTF curves of two phase masks do not match exactly point to point. In some regions, the MTF value of CSPM is bigger than that of CPM; but in other regions, the MTF value of CSPM is relatively smaller instead. This phenomenon is decided by the principle of conservation of ambiguity. The difference mainly lies in the computation of corresponding Fisher information. It is obvious that FI values of CSPM are smaller than those of CPM in the extended depth of field, which means that CSPM is less sensitive to focus error and performs better when the parameters are optimized. This part just answers the question proposed at the end of the first part. When the value of k is same, the MTF curves of the two phase masks indeed become very similar to each other in a overall view and the results show that CSPM is superior to CPM in this case. A conclusion can be made: with magnitude of MTF improved at certain frequency regions, the CSPM also becomes less sensitive to defocus at the same time. We believe that this improvement is meaningful in the real application. In this section, we investigate the characteristics of CSPM with the help of defocused MTF and Fisher information curves in two situations. Simulations demonstrate the effectiveness of CSPM no matter which situation is considered. Besides the two merits, Table 1 Five groups of optimum parameters of CPM and CSPM.
c0 = 30, a0 =30
k CPM CSPM
0.2 246.9561 354.9509
0.24 169.8310 244.0747
0.28 123.6294 177.3919
0.32 93.6136 134.5060
0.36 73.2951 105.2304
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Fig. 12. Comparison of defocused MTF curves and Fisher information between optimized CPM and CSPM when k in Table 1 equals 0.2.
Fig. 13. Comparison of defocused MTF curves and Fisher information between optimized CPM and CSPM when k in Table 1 equals 0.24.
defocused PSF (point spread function) is another merit usually used to judge the performance of imaging system. To end this part, we display the defocused PSF of the two phase masks in Figs. 17 and 18, respectively, with a set to 90 and 135. As clearly seen in Figs. 17 and 18, the corresponding PSF remains quite stable with the defocus parameter set to pi, 3pi, 6pi and 10pi. At the same time, it is not difficult to find that the PSF of CSPM is smaller than that of CPM, which just corresponds to the higher MTF curve of CSPM in Fig. 5. When CSPM and CPM are optimized, their defocused MTF curves are similar and thus the corresponding PSF will become similar as well, but the results are omitted. 3.3. Experimental demonstration of CSPM’s performance The characteristic of CSPM has been proven with the help of numerical evaluations in the first and second part of this section. Although the simulations above are correct and persuasive, the
Fig. 14. Comparison of defocused MTF curves and Fisher information between optimized CPM and CSPM when k in Table 1 equals 0.28.
Fig. 15. Comparison of defocused MTF curves and Fisher information between optimized CPM and CSPM when k in Table 1 equals 0.32.
reader might still want to know the real effect when CSPM is incorporated into traditional imaging system. This is reasonable because real experimental results would be more persuasive. So in this part, we design a prototype imaging system having CSPM and carry out some interesting experiments. Before results are shown, first of all we would like to make necessary discussions and explanations to the configuration of imaging system. First, the parameter of CSPM is set to 177.3919, which corresponds to the case of k =0.28 in Table 1; and the size of CSPM is 15 mm 15 mm, which just matches the aperture of the original imaging system. In order to carry out the following experiments, the fabrication of phase mask is very crucial. As is known to all, the aspheric optical element is hard to fabricate and test, especially when the materials used are glass. Through the detailed investigations, from several candidate institutes, we ask Hong Kong Polytechnic University for research help at last. They use high-precision freedom surface fabrication machine which has five degree of
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Fig. 16. Comparison of defocused MTF curves and Fisher information between optimized CPM and CSPM when k in Table 1 equals 0.36. Fig. 18. Defocused PSF curves of CPM with defocus parameter set to pi, 3pi, 6pi and 10pi.
Fig. 19. The cone used in the imaging system. Fig. 17. Defocused PSF curves of CSPM with defocus parameter set to pi, 3pi, 6pi and 10pi.
freedom to do the job. According to the existing experiment results, the fabrication can be considered successful. Second, the main part of imaging system is one doublet lens with focal length of 50 mm and the ideal imaging distance is approximately 365 mm due to limitation of the fixed cone, as shown in Fig. 19. At the same time, Fig. 19 also demonstrates the location where the CSPM will be put. The detailed description of the system can be concluded as follows: The radius of the doublet equals 29.418, 22.731 and 68.491 mm, respectively. The front and rear principal plane lies in 7.0 mm and 8.2 mm. The object distance changes from 308.5 to 423.5 mm and the corresponding image distance is from 56.7 to 59.7 mm. So, according to the classical defocus computation equation, the range of defocus in the experiments denoted by W20 equals [ 3.9p, 5.4p], approximately.
As mentioned above, the system aperture is rectangular and its size equals 15 mm 15 mm as well. So, according to theory of Geometrical Optics, when the imaging system does not introduce CSPM, we can easily compute the depth of field which is nearly 15 mm. It is obvious that the original imaging system has very low depth of field; because of this, we can demonstrate the performance of CSPM better. Third, as we already know, the images captured by wave-front coding system are purposely blurred and have to be deblurred to obtain the final clear images. Generally speaking, any image restoration methods can be used such as constrained inverse filtering, wiener filtering, Richard-Lucy, EM and so on. However, because the degradation function of wave-front coding system can be known at the stage of system design, relatively simple algorithms such as inverse filtering or wiener filtering becomes the first choices. In our experiment, wiener filtering is used to restore the intermediate images and depth extension effect can be clearly proven. Besides these, there is one point that should be
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noted: the filter we adopt in the restoration process is complex because phase part is also important. Now, with the experimental system described above, the intermediate and restored images can be obtained. We have carried out two groups of experiments, as shown in Figs. 20 and 21, respectively. In Fig. 20, a planar plate is used as imaging target and we have to make it move along optical axis with a fixed interval to generate the defocus effect. The first row in Fig. 20 denotes images
captured by imaging system without CSPM; the second one and the third one give the intermediate and restored images when CSPM is introduced. It is obvious that the depth of field has been effectively extended. At the same time, the magnification effect can be seen as well because of the change of object distance. In Fig. 21, a tilted notebook is imaged. The characters that cannot be distinguished by the traditional imaging system are easy to recognize in the CSPM wave-front coding system. This could be a very good proof of the success of CSPM.
Fig. 20. Comparison between the traditional and wave-front coding imaging system; real images captured without CSPM (first row); intermediate images with CSPM (second row); restored images with wiener filtering (third row).
Fig. 21. Another example for demonstrating the effectiveness of CSPM; real images captured without CSPM (left); intermediate images with CSPM (middle); restored images with wiener filtering (right).
Fig. 22. Comparison between performance of CPM and CSPM.
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Fig. 23. Objective assessment of quality of restored images captured by two phase masks, respectively.
So, with the experiments carried out in this part, we are very sure that CSPM indeed can be used to extend the depth of field, which is served as a good complement to the theoretical analysis as well. In order to make a comparison with CPM, we also fabricate CPM which has the same physical size as CSPM and the results are shown in Fig. 22. In Fig. 22, the first two rows denote the intermediate images captured by CSPM and CPM, respectively; the second two rows denote the corresponding restored results. As Fig. 22 shows, it is hard to tell which kind of mask has larger depth of filed through visual comparisons. However, there is still one method that can be used to judge the performance of two phase masks objectively. The well-known focus measure operator is very easy, but powerful tool to assess the image clarity. Here, we adopt the easiest one, which is called global variance, to do the objective assessment. The result is shown in Fig. 23. Fig. 23 is generated by assessing two groups of restored images with the global variance operator. As Fig. 23 tells, the value of focus measure of images captured by CSPM is larger than that of CPM in more than 80% imaging space. We believe that this can prove that CSPM indeed performs better than CPM in extending the depth of field, because not only the depth extension effect is realized, but also the quality of restored images are improved as well. Although the desired effect has been obtained, we have to admit that the experiments are still at the initial stage. In the future, more experiments would be designed and the characteristics of CSPM would also be further investigated.
4. Conclusions Although lots of phase masks have been suggested during the last decade, researchers still try their best to find the underlying
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phase masks with better performance. In this paper, we show the effort we have done about this topic. Considering the difficulty in computing the new phase mask from mathematical viewpoint, we turn to other ways and propose the concept of composite phase mask. As shown in the paper, by adding one component having sinusoidal form to the cubic phase mask, a completely new phase mask called cubic sinusoidal phase mask (CSPM) is generated. We have studied the characteristics of CSPM in a numerical way and simulation results demonstrate the superior performance of CSPM in extending the depth of field when certain constraints are considered. If CSPM is not optimized, it can improve the magnitude of defocused MTF, but become a little bit sensitive to defocus; if CSPM is optimized, it is made much less sensitive to focus error when the magnitude of MTF in certain regions is also improved. Besides those, we also carry out two groups of interesting experiments and demonstrate the effectiveness of CSPM with real persuasive data, which are really promising. So, we believe that the optimized CSPM should be used in real application. However, only cubic phase mask is used as a basis to generate the composite phase mask in this paper and this is not enough. In future, other kinds of composite phase masks will be investigated as well.
Acknowledgements This work has been supported by: (1) Basic Research Development Program of China (973 Program) under contract No. 2009CB724006; (2) Basic Research Development Program of China (973 Program) under contract No. 2009CB724002; (3) National Natural Science Foundation of China No. 60977010. At the same time, the authors would also like to thank anonymous reviewers for their wonderful comments to improve the research quality.
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Cubic sinusoidal phase mask: Another choice to extend the depth of field of incoherent imaging system Hui Zhao a,n, Yingcai Li a, Huajun Feng b, Zhihai Xu b, Qi Li b a b
Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Science, Xi’an 710119, China Zhejiang University, State Key Laboratory of Modern Optical Instrumentation, Department of Optical Engineering, Hangzhou 310027, China
a r t i c l e in fo
abstract
Article history: Received 7 January 2009 Received in revised form 11 August 2009 Accepted 23 October 2009 Available online 22 November 2009
Wave-front coding is a well known technique used to extend the depth of field of incoherent imaging system. The core of this technique lies in the design of suitable phase masks, among which the most important one is the cubic phase mask suggested by Dowski and Cathey (1995) [1]. In this paper, we propose a new type called cubic sinusoidal phase mask which is generated by combing the cubic one and another component having the sinusoidal form. Numerical evaluations and real experimental results demonstrate that the composite phase mask is superior to the original cubic phase mask with parameters optimized and provides another choice to achieve the goal of depth extension. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Wave-front coding Sinusoidal component Composite phase mask
1. Introduction As is widely known, wave-front coding is a powerful design method suggested by Dowski and Cathey [1] to extend the depth of field of incoherent imaging system in 1995. By introducing one phase mask to the aperture plane, the optical transfer function (OTF) can be modified to be insensitive to focus errors and defocus related aberrations, and it is this modification that greatly extends the depth of field. The most important part of wave-front coding technique lies in the design of suitable phase masks. During the last decade, lots of researchers have devoted themselves to this topic and many kinds of phase masks have been proposed. Among all phase masks, the cubic type is the most popular one because of its simplicity and superior performance in the real application. Generally speaking, there are mainly two important metrics to judge the performance of wave-front coding system. One is the defocused MTF (modulation transfer function) and the other is Fisher information (FI). The goal of designing one phase mask lies in two aspects. On the one hand, the magnitude of defocused MTF should be large enough to make the digital post processing easier. On the other hand, FI values should be small enough to make the overall system less sensitive to defocus as much as possible. Unfortunately, the two aspects are contradictory in most cases according to the existing researches [1–5]. In other words, it is very hard to achieve the satisfactory results of big magnitude of MTF and small values of FI at the same time. However, several
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Corresponding author. E-mail address: [email protected] (H. Zhao).
0030-3992/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2009.10.004
masks, such as modified cubic type [3], exponential type [4] and logarithmic type [5,8] and so on, are still designed to improve the performance of wave-front coding system to a certain extent. Researchers never stop looking for new masks and so do we. As known in the literatures, there are two ways to obtain the satisfactory phase mask. One is operated in spatial domain and the other is in frequency domain. In spatial domain, defocused point spread function (PSF) is derived using scalar diffraction theory [5,9] and the phase function can be obtained by solving a differential equation generated by making the derivative of defocused PSF with respect to defocus parameter be zero. In frequency domain, the operation is nearly the same except that the defocused MTF rather than the defocused PSF is used to do the computing [1,6]. For many years, the derivation procedure did not change too much, which meant that a standard computation routine had been established. So, it is not easy for us to create a new phase mask completely from mathematical point of view. Under these circumstances, we change our viewpoint. In this paper, we propose the concept of composite phase mask by adding additional component to the existing phase mask to try to improve the system performance. The simulation results and real experiments demonstrate the effectiveness of the new phase mask and the researches done in this paper are not only interesting, but also meaningful.
2. Proposal of composite phase mask The phase profiles corresponding to several popular phase masks are plotted in Fig. 1, where all the curves are generated
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Fig. 2. Phase profiles of the composite phase mask with different parameters. Fig. 1. Phase profiles of several popular phase masks with typical parameters.
with typical parameters. In Fig. 1, the symbols ‘Cub’, ‘Exp’, ‘Log1’, ‘Log2’, ‘Ho’, denote cubic type, exponential type, logarithmic type, improved logarithmic type and high-order type, respectively. By observing Fig. 1, we can make a bold hypothesis: any phase mask whose profile is similar to those of existing ones can probably be used to extend the depth of field. The underlying phase mask candidates can be easily obtained by superimposing additional component to the existing phase masks and these masks are named as composite phase masks. Compared with the rigorous mathematical derivation, this way of generating new phase masks is direct and easier to operate. In this paper, we only take the cubic phase mask for example to explain the concept of composite phase mask, because the cubic phase mask is the most classical one among all kinds of phase masks. Whether similar results can be obtained or not on the condition that other types of masks are used as the basis will be further studied in future. The idea of composite phase mask is interesting and our motivation comes from reference [7]. However, reference [7] just mentioned the possibility of doing the combination and did not have an implementation. The work done in this paper is the first time to change the possibility into real fact and is a good proof to the wild guess. The ‘additional component’ is crucial to the composite phase mask. Factually, there are lots of candidates, such as exponential function, logarithmic function, sinusoidal function and cosine function and so on. Finally, the component which has sinusoidal form is selected for investigation. There are two reasons to do so. One is that the new phase mask generated by superimposing sinusoidal component demonstrates more versatile variation in phase profile, as shown by Fig. 2. The other is that sine or cosine function is rare to be studied in the literature centering on the topic of depth extension effect. Besides those, considering that sinusoidal and cosine function can be transformed between each other, the composite phase mask discussed here can be described by Eq. (1): f ðx; yÞ ¼ aðx3 þ y3 Þ þ bðsinðoxÞ þ sinðoyÞÞ
ð1Þ
where a and b together control the phase deviation; o is the angular frequency. In Eq. (1), the first part denotes the original cubic phase mask and the second one denotes ‘additional component’. By rearranging Eq. (1), Eq. (2) can be obtained: f ðx; yÞ ¼ ðax3 þ b sinðoxÞÞ þ ðay3 þ b sinðoyÞÞ
ð2Þ
In order to study the characteristics of the new phase mask denoted by Eq. (2), how to choose values for three parameters (a, b and o) is the first question that should be answered. According to references [3,5], two criteria have to be satisfied. First, the value of a and b has to be set to the same level. This constraint is reasonable because the purpose of composite phase mask lies in the combination of different components. As Eq. (2) tells, a and b together control the phase deviation of the mask. If the value of one parameter is much bigger or smaller than another, the composite phase mask will degrade into a single phase mask again and this violates the purpose of our research. Second, with the sinusoidal part added, the odd-symmetry and monotonicity should not be broken, which allows o to change within a relatively small range. The sinusoidal part of the composite phase mask is periodic. When the value of o is big enough, the phase function will start to oscillate on the condition that the values of a and b stay at the same order and this breaks the monotonicity constraint. However, if the value of o is too small, the influence of the sinusoidal part will become less important and this is similar to the situation as discussed in the former paragraph. For clarity, the comparison among 1-D phase profiles with different parameters is shown in Fig. 2 to explain the two criteria. In Fig. 2, the black solid curve denotes the original cubic phase mask with a set to 90 for comparison. The blue dotted curve corresponds to the new mask with b and o set to a/20 and 1, respectively. It is easy to see that the new phase mask is nearly the same as the original one in this situation (corresponds to the first criterion). The red dashed and magenta dot-dashed line is generated on the condition that o is set to 5 and 1, respectively, while b equals a. As you see, the phase profile denoted by the red dashed curve violates the characteristic of monotonicity (corresponds to the second criterion) and only magenta dot-dashed curve satisfies the two criteria at the same time. It is not difficult to find out that if o equals pi/2, the monotone part of sinusoidal function will just lie within the range of [ 1, 1] that covers the entire pupil plane as displayed in Fig. 1 and it is impossible for the second criterion to be violated. If the relationship between b and a can be determined at the same time, the complete model of the composite phase mask discussed in this paper will be established. As mentioned above, the magnitude of b and a has to have the same order and thus the ratio between them should lie within a reasonable range. If the ratio is seen as a
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that a is set to same value. The simulation results corresponding to two situations will be shown below: (1) Fig. 4 is one example in which the maximum phase deviation of the two masks is same. As Fig. 4 tells, when a in CSPM equals 90, the corresponding one in CPM should be set to 135 to comply with the first kind of constraint. Then, with defocus parameter set to 0, p, 3p, 6p and 10p, respectively, defocused MTF curves can be obtained, as shown in Fig. 5. Clearly, CSPM cannot only be used to extend the depth of field, but also can increase the magnitude of defocused MTF. This is an appealing characteristic which is helpful to digital post processing. Besides defocused MTF curves, FI is used as well to judge the performance of CSPM and CPM. As Fig. 6 reveals, when the additional component is added to the cubic phase mask, the value of FI increases and this means that the CSPM is a little bit sensitive to focus errors than CPM is. This phenomenon can be easily and clearly explained. According to the literatures, to any phase mask, the bigger the deviation degree is, the smaller the magnitude of defocused MTF is, and vice versa. Fig. 4 shows that the central part Fig. 3. Comparison between phase profiles of CPM and CSPM with same parameters.
variable, the following discussions will become complicated because at least two parameters should be jointly considered. Here, we simply set the ratio to 2 without loss of generality and the corresponding investigation can be greatly simplified. Discussion to other cases with ratio set to different values can be done with the same procedure. More complicated cases, in which the ratio is seen as a variable, will be left for future study. So, the condition under which the new phase mask can work is given, as shown below:
b¼
a 2
;
o¼
p 2
With Eq. (3), Eq.(2) can be rewritten as a p a p f ðx; yÞ ¼ ax3 þ sin x þ ay3 þ sin y 2 2 2 2
ð3Þ
ð4Þ
The phase mask described by Eq. (4) is called CSPM which is the abbreviation for cubic-sinusoidal phase mask. With a set to different values, the 1-D phase profile corresponding to CSPM can be obtained as shown in Fig. 3, in which the phase profile of original cubic mask is also drawn for comparison. With the expression of CSPM, numerical evaluations can be carried out to show the performance of the composite phase mask.
Fig. 4. Phase profiles of CPM and CSPM with the constraint that the maximum phase deviation is same.
3. Characteristics of CSPM and corresponding discussions Researchers usually use defocused MTF and Fisher information to evaluate the performance of one phase mask. We do the similar thing in this section with two cases considered. In the first part, the evaluation is carried out with CSPM and CPM (cubic phase mask) not optimized and we do the numerical evaluating in the second part under the condition that CSPM and CPM are optimized. 3.1. Performance comparison between CSPM and CPM with optimization procedure not introduced Although the optimization procedure is not used, certain constraints should be satisfied to make sure that the following comparisons are meaningful. Two kinds of constraints are considered here. The first one is that the maximum phase deviation of the two masks remains same and the second one is
Fig. 5. Comparison between defocused MTF curves of CPM and CSPM corresponding to Fig. 4.
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Fig. 6. Comparison between Fisher information of CPM and CSPM corresponding to Fig. 4.
Fig. 7. Another example in which the constraint of same maximum phase deviation is satisfied. (The left part is the comparison of defocused MTF curves and the right part is the comparison of Fisher information curves.)
of the new phase mask gradually approaches a tilt and this tilt contributes less effect on extending the depth of field. This means that by adding the sinusoidal component, the overall deviation degree of the new phase mask becomes relatively small, which causes the increase of the magnitude of MTF and thus makes phase mask more sensitive to defocus. Another example in which a in CPM and CSPM equals 180 and 120 is also provided and similar results are shown in Fig. 7. (2) When a in the two phase masks is set to same value, the corresponding phase profiles can be drawn in Fig. 8. At the same time, defocused MTF and Fisher information can also be computed and the results are shown in Figs. 9 and 10. As Figs. 9 and 10 tell, besides the increase of the magnitude of defocused MTF, the increase of FI curves of CSPM can also bee seen. The reason leading to this phenomenon has been discussed above and is omitted here. In order to prove that the results are not occasional, another example where a is set to 150 is given
Fig. 8. Phase profiles of CPM and CSPM with the constraint that the values of a in both masks are same.
Fig. 9. Comparison between defocused MTF curves of CPM and CSPM corresponding to Fig. 8.
below, as shown in Fig. 11, and we can find out that the results are similar. Now we can make a simple conclusion to end the first part of this section. With the two constraints considered above, CSPM can improve the magnitude of defocused MTF, but becomes more sensitive to focus error. At the same time, it is not difficult to find out that the magnitude of MTF corresponding to CSPM mainly increases in low to mid frequencies, but decreases at higher frequencies. So, the observations above mean that CSPM can perform better than CPM within a reduced range of depth of field. In other words, the MTF curves corresponding to CSPM will become instable with increase of focus error. Just as mentioned in Section 1, the results correspond to the fact that larger magnitude of MTF and smaller FI values usually contradict each other. Besides that, a trend is obvious. If a of CSPM is bigger than that of CPM and gradually increases to a certain value, the MTF curves of CPM and CSPM will approach each other, so does the FI curves. To this point, an interesting question is proposed. In the case when the MTF of both phase masks is nearly same, which mask is
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mask is optimized in order to achieve two goals. One is to make the magnitude of defocused MTF as large as possible and the other is to make the defocused MTF curves stable enough within the extended depth of field. The similar description can be found in reference [5]. Here, we adopt the Fisher-information based optimization method [4] and the optimization procedure can be simply described as follows: min FIða; c0 Þ ¼ ( subjectto
Fig. 10. Comparison between Fisher information of CPM and CSPM corresponding to Fig. 8.
Fig. 11. Another example in which the values of a in both CPM and CSPM are same. (The left part is the comparison of defocused MTF curves and the right part is the comparison of Fisher information curves.)
much less sensitive to defocus, CPM or CSPM? The next part of this section will give a positive answer. 3.2. Performance comparison between CSPM and CPM with optimization procedure introduced In the first part of this section, numerical evaluations are carried out with CPM and CSPM not optimized. However, this is not the usual case. When we decide to use one type of phase mask, the first step is to optimize the mask to obtain optimum parameters. With suitable optimization procedure introduced, the optimum phase mask can be computed and it will perform the best in theoretical sense. Generally speaking, there are 3–4 methods that can be used to optimize one wave-front coded imaging system, such as Strehl– Ratio based optimization [3], Fisher-information based optimization [3,4], PSF or MTF stability based optimization [5] and so on. No matter which method is used, the purpose is same. One phase
R c0
c0
2 ! ! R@ Hðu; cÞ du dc @c
a a Z a0 R AðaÞ ¼ 9Hðu; 0Þ9du Z k
ð5Þ
where a is the mask parameter, c0 the maximum focus error; u is the normalized frequency; H(u,c) is the 1-D defocused optical transfer function; A(a) is the integrated area of the in-focus MTF curve and its minimum value is set to k; FI denotes the Fisher information and is the object function we would like to minimize under two constraints. The first constraint gives the lower bound of phase parameter denoted as a0 and the second one has to be satisfied in order to make the values of in-focus MTF as large as possible, which is the requirement for digital post processing. According to Eq. (5), the optimization of CPM and CSPM can be easily operated with the help of optimization tool box in Matlab. The related constants and final optimum parameters are provided in Table 1. There are five groups of optimum parameters in Table 1 and the performance comparison between CPM and CSPM can be carried out immediately. The optimization procedure adopted here will lead to a result that the defocused MTF curves of CSPM and CPM have similar shapes. As Figs. 12–16 tell, no matter which group of optimum parameters is used, the defocused MTF curves of the two masks are nearly the same. However, by zooming in the local part of the MTF curves, we can find that the MTF curves of two phase masks do not match exactly point to point. In some regions, the MTF value of CSPM is bigger than that of CPM; but in other regions, the MTF value of CSPM is relatively smaller instead. This phenomenon is decided by the principle of conservation of ambiguity. The difference mainly lies in the computation of corresponding Fisher information. It is obvious that FI values of CSPM are smaller than those of CPM in the extended depth of field, which means that CSPM is less sensitive to focus error and performs better when the parameters are optimized. This part just answers the question proposed at the end of the first part. When the value of k is same, the MTF curves of the two phase masks indeed become very similar to each other in a overall view and the results show that CSPM is superior to CPM in this case. A conclusion can be made: with magnitude of MTF improved at certain frequency regions, the CSPM also becomes less sensitive to defocus at the same time. We believe that this improvement is meaningful in the real application. In this section, we investigate the characteristics of CSPM with the help of defocused MTF and Fisher information curves in two situations. Simulations demonstrate the effectiveness of CSPM no matter which situation is considered. Besides the two merits, Table 1 Five groups of optimum parameters of CPM and CSPM.
c0 = 30, a0 =30
k CPM CSPM
0.2 246.9561 354.9509
0.24 169.8310 244.0747
0.28 123.6294 177.3919
0.32 93.6136 134.5060
0.36 73.2951 105.2304
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Fig. 12. Comparison of defocused MTF curves and Fisher information between optimized CPM and CSPM when k in Table 1 equals 0.2.
Fig. 13. Comparison of defocused MTF curves and Fisher information between optimized CPM and CSPM when k in Table 1 equals 0.24.
defocused PSF (point spread function) is another merit usually used to judge the performance of imaging system. To end this part, we display the defocused PSF of the two phase masks in Figs. 17 and 18, respectively, with a set to 90 and 135. As clearly seen in Figs. 17 and 18, the corresponding PSF remains quite stable with the defocus parameter set to pi, 3pi, 6pi and 10pi. At the same time, it is not difficult to find that the PSF of CSPM is smaller than that of CPM, which just corresponds to the higher MTF curve of CSPM in Fig. 5. When CSPM and CPM are optimized, their defocused MTF curves are similar and thus the corresponding PSF will become similar as well, but the results are omitted. 3.3. Experimental demonstration of CSPM’s performance The characteristic of CSPM has been proven with the help of numerical evaluations in the first and second part of this section. Although the simulations above are correct and persuasive, the
Fig. 14. Comparison of defocused MTF curves and Fisher information between optimized CPM and CSPM when k in Table 1 equals 0.28.
Fig. 15. Comparison of defocused MTF curves and Fisher information between optimized CPM and CSPM when k in Table 1 equals 0.32.
reader might still want to know the real effect when CSPM is incorporated into traditional imaging system. This is reasonable because real experimental results would be more persuasive. So in this part, we design a prototype imaging system having CSPM and carry out some interesting experiments. Before results are shown, first of all we would like to make necessary discussions and explanations to the configuration of imaging system. First, the parameter of CSPM is set to 177.3919, which corresponds to the case of k =0.28 in Table 1; and the size of CSPM is 15 mm 15 mm, which just matches the aperture of the original imaging system. In order to carry out the following experiments, the fabrication of phase mask is very crucial. As is known to all, the aspheric optical element is hard to fabricate and test, especially when the materials used are glass. Through the detailed investigations, from several candidate institutes, we ask Hong Kong Polytechnic University for research help at last. They use high-precision freedom surface fabrication machine which has five degree of
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Fig. 16. Comparison of defocused MTF curves and Fisher information between optimized CPM and CSPM when k in Table 1 equals 0.36. Fig. 18. Defocused PSF curves of CPM with defocus parameter set to pi, 3pi, 6pi and 10pi.
Fig. 19. The cone used in the imaging system. Fig. 17. Defocused PSF curves of CSPM with defocus parameter set to pi, 3pi, 6pi and 10pi.
freedom to do the job. According to the existing experiment results, the fabrication can be considered successful. Second, the main part of imaging system is one doublet lens with focal length of 50 mm and the ideal imaging distance is approximately 365 mm due to limitation of the fixed cone, as shown in Fig. 19. At the same time, Fig. 19 also demonstrates the location where the CSPM will be put. The detailed description of the system can be concluded as follows: The radius of the doublet equals 29.418, 22.731 and 68.491 mm, respectively. The front and rear principal plane lies in 7.0 mm and 8.2 mm. The object distance changes from 308.5 to 423.5 mm and the corresponding image distance is from 56.7 to 59.7 mm. So, according to the classical defocus computation equation, the range of defocus in the experiments denoted by W20 equals [ 3.9p, 5.4p], approximately.
As mentioned above, the system aperture is rectangular and its size equals 15 mm 15 mm as well. So, according to theory of Geometrical Optics, when the imaging system does not introduce CSPM, we can easily compute the depth of field which is nearly 15 mm. It is obvious that the original imaging system has very low depth of field; because of this, we can demonstrate the performance of CSPM better. Third, as we already know, the images captured by wave-front coding system are purposely blurred and have to be deblurred to obtain the final clear images. Generally speaking, any image restoration methods can be used such as constrained inverse filtering, wiener filtering, Richard-Lucy, EM and so on. However, because the degradation function of wave-front coding system can be known at the stage of system design, relatively simple algorithms such as inverse filtering or wiener filtering becomes the first choices. In our experiment, wiener filtering is used to restore the intermediate images and depth extension effect can be clearly proven. Besides these, there is one point that should be
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noted: the filter we adopt in the restoration process is complex because phase part is also important. Now, with the experimental system described above, the intermediate and restored images can be obtained. We have carried out two groups of experiments, as shown in Figs. 20 and 21, respectively. In Fig. 20, a planar plate is used as imaging target and we have to make it move along optical axis with a fixed interval to generate the defocus effect. The first row in Fig. 20 denotes images
captured by imaging system without CSPM; the second one and the third one give the intermediate and restored images when CSPM is introduced. It is obvious that the depth of field has been effectively extended. At the same time, the magnification effect can be seen as well because of the change of object distance. In Fig. 21, a tilted notebook is imaged. The characters that cannot be distinguished by the traditional imaging system are easy to recognize in the CSPM wave-front coding system. This could be a very good proof of the success of CSPM.
Fig. 20. Comparison between the traditional and wave-front coding imaging system; real images captured without CSPM (first row); intermediate images with CSPM (second row); restored images with wiener filtering (third row).
Fig. 21. Another example for demonstrating the effectiveness of CSPM; real images captured without CSPM (left); intermediate images with CSPM (middle); restored images with wiener filtering (right).
Fig. 22. Comparison between performance of CPM and CSPM.
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Fig. 23. Objective assessment of quality of restored images captured by two phase masks, respectively.
So, with the experiments carried out in this part, we are very sure that CSPM indeed can be used to extend the depth of field, which is served as a good complement to the theoretical analysis as well. In order to make a comparison with CPM, we also fabricate CPM which has the same physical size as CSPM and the results are shown in Fig. 22. In Fig. 22, the first two rows denote the intermediate images captured by CSPM and CPM, respectively; the second two rows denote the corresponding restored results. As Fig. 22 shows, it is hard to tell which kind of mask has larger depth of filed through visual comparisons. However, there is still one method that can be used to judge the performance of two phase masks objectively. The well-known focus measure operator is very easy, but powerful tool to assess the image clarity. Here, we adopt the easiest one, which is called global variance, to do the objective assessment. The result is shown in Fig. 23. Fig. 23 is generated by assessing two groups of restored images with the global variance operator. As Fig. 23 tells, the value of focus measure of images captured by CSPM is larger than that of CPM in more than 80% imaging space. We believe that this can prove that CSPM indeed performs better than CPM in extending the depth of field, because not only the depth extension effect is realized, but also the quality of restored images are improved as well. Although the desired effect has been obtained, we have to admit that the experiments are still at the initial stage. In the future, more experiments would be designed and the characteristics of CSPM would also be further investigated.
4. Conclusions Although lots of phase masks have been suggested during the last decade, researchers still try their best to find the underlying
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phase masks with better performance. In this paper, we show the effort we have done about this topic. Considering the difficulty in computing the new phase mask from mathematical viewpoint, we turn to other ways and propose the concept of composite phase mask. As shown in the paper, by adding one component having sinusoidal form to the cubic phase mask, a completely new phase mask called cubic sinusoidal phase mask (CSPM) is generated. We have studied the characteristics of CSPM in a numerical way and simulation results demonstrate the superior performance of CSPM in extending the depth of field when certain constraints are considered. If CSPM is not optimized, it can improve the magnitude of defocused MTF, but become a little bit sensitive to defocus; if CSPM is optimized, it is made much less sensitive to focus error when the magnitude of MTF in certain regions is also improved. Besides those, we also carry out two groups of interesting experiments and demonstrate the effectiveness of CSPM with real persuasive data, which are really promising. So, we believe that the optimized CSPM should be used in real application. However, only cubic phase mask is used as a basis to generate the composite phase mask in this paper and this is not enough. In future, other kinds of composite phase masks will be investigated as well.
Acknowledgements This work has been supported by: (1) Basic Research Development Program of China (973 Program) under contract No. 2009CB724006; (2) Basic Research Development Program of China (973 Program) under contract No. 2009CB724002; (3) National Natural Science Foundation of China No. 60977010. At the same time, the authors would also like to thank anonymous reviewers for their wonderful comments to improve the research quality.
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