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Study on compressed sensing imaging based on intensity modulation in Fourier domain
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ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
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Study on compressed sensing imaging based on intensity modulation in Fourier domain Ruiqing He ∗ , Qian Chen, Weiji He Jiangsu Key Lab of Spectral Imaging & Intelligence Sense (SIIS), Nanjing University of Science and Technology, Nanjing 210094, PR China
a r t i c l e
i n f o
Article history: Received 20 July 2013 Accepted 15 January 2014 Available online xxx Keywords: Compressed sensing Intensity modulation Coherence
a b s t r a c t In 4F system, compressed sensing is usually implemented by using phase modulation in Fourier domain. In this paper, we present a type of 4F system based on intensity modulation in Fourier domain as the measurement system for compressed sensing. The feasibility of this system is demonstrated. At the point of coherence, the two modulation methods are compared and superiority of intensity modulation in Fourier domain was verified. Simulations are presented and the conclusion we presented is validated. Finally, we analyze the results. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Recently, compressed sensing (CS) theory was discussed widely [1–3]. Compared with the traditional sampling method, compressed sensing is based on the sparsity of signal. In the compressed sensing framework, we need to design a random matrix to measure signal and recover the signal by convex optimization. Actually, the measurement procedure can be understood as projecting the signal into a random vector of a measurement matrix. By using random measurement, CS can hugely reduce the sampling ratio which is lower than the Nyquist sampling ratio, as a result we save the time spent on the sampling and data size of the sampling value is smaller than before. Its appearance is very important to the signal process. CS has been found to have wide usage in many applications, such as optical imaging [4,5], biomedical imaging [6], high-speed wireless communication [7], etc. In the imaging field, the single pixel camera based on compressed sensing has been presented by Davenport M.A. [8]. The advantage of single pixel camera is its suitability for optical systems. In the compressed imaging field, Rebecca M. Willett’s group improved the image resolution by using compressed coded aperture [9]. Gonzalo R. Arceze presented the variable density compressed image sampling [10]. Justin Romberg proved that the measurement system consisted of the 4F system based on phase modulation in the Fourier domain can meet compressed sensing requirements [11].
Inspired by Justin Romberg, we present our purposed measurement system consisting of the 4F system based on intensity modulation in the Fourier domain and proves it to be fitted for compressed sensing. Another contribution of this paper is that compared to the measurement system using phase modulation in the Fourier domain [11], the superiority of our purposed system was verified at the point of coherence. This work is organized as follows: Section 2 introduces the principle of compressed sensing. Section 3 contains the main theoretical work of this paper and we construct the 4F system based on intensity modulation in the Fourier domain as the measurement system. Section 4 is the numerical simulation, in which the feasibility of our imaging system is validated. Finally, Section 5 concludes this paper. 2. Principle of compressed sensing The principle of compressed sensing can be expressed as follow. To recover the unknown signal x ∈ Rn , we need to measure it and the measurement results can be written as: y1 = x, 1 · · ·ym = x, m
k is a measurement vector whose size is same as signal x·yi is measurement value. We can use as the measurement system, where the k are stacked up as the rows in . The measurement process can be expressed as: y = ˚x = ˚˛ (x = ˛).
∗ Corresponding author. E-mail address: [email protected] (R. He).
(1)
(2)
In general, if one wants to recover signal x from y, we need m ≥ n, but when signal x is sparse in an orthogonal basis (i.e. there are relatively few nonzero entries in ˛ and the number of nonzero
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Please cite this article in press as: R. He, et al., Study on compressed sensing imaging based on intensity modulation in Fourier domain, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.123
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back focal plane of the second Fourier lens), the modulated image is sampled randomly. Finally, the original signal is reconstructed from the sampling result by using convex optimization. The measurement system in Fig. 1 is shown as follow: √ ˚ = nF ∗ PF, (6) Fig. 1. Compressed sensing based on intensity modulation in 4f system.
entries is K, called K-sparse), then it is possible to reconstruct signal x with m n. For general images, there always is a space spanned by a proper orthogonal basis where the image signal is sparse. Aurelien Bourquard believes that the importance of is its mere existence [12]. Based on the sparse property, if matrix = ˚ satisfies the RIP (restricted isometry property), the signal can be recovered accurately from the relatively fewer samples. The RIP is briefly reviewed later. Suppose T ⊂ {1, 2 . . . N} and |T| ≤ K, ˚T which is indexed by T, is a subset of . ık is the minimal constant which satisfies Eq. (4) and it is correlated to T and K. If there are that meets Eq. (3), then we say that satisfies RIP. From the point of RIP, the goal of the designing or optimizing measure system is making ık as minimal as possible. (1 − ık )||x||22 ≤ ||T x||22 ≤ (1 + ık )||x||22
(3)
RIP is a sufficient but not necessary condition, and it is hard to determine whether or not the measurement system meets RIP by algorithm. In fact, to implement compressed sensing, one always designs the measurement system which is incoherent or low coherent with . The coherence can be defined as follows. Suppose the measurement system is , and the orthogonal basis is . The coherence can then be written as:
= maxl,k l , k ,
(4)
where l is the lth √row in and k is the kth column in . In (4), satisfies 1 ≤ ≤ N. Based on conditions before, recovering signal x is a convex optimization problem as shown in Eq. (5)
min T x
1
s.t. y = ˚x.
(5)
3. System scheme Romberg has proven that the 4F system using phase modulation in the Fourier domain can be incoherent with any orthogonal basis [11]. Based on Romberg’s research, we present a type of 4F system based on intensity modulation in the Fourier domain as the measurement system. Section 3.1 introduces the composition of our imaging system. Sections 3.2 and 3.3 verify feasibility and superiority of our purposed measurement system respectively in theory. 3.1. Composition of imaging system The basic composition of our purposed imaging system contains a measurement and sparse sampling system which are schematically presented in Fig. 1. The measurement system is composed of a pair of Fourier lenses and intensity SLM (spatial light modulator) and finishing modulation of input signal. The sparse sampling system randomly selects pixel locations and uses the pixel values of these positions as input signal for imaging system. The process is shown as follows. At first, laser beam is expanded and collimated. A transmissive object is located in the way of beam. Intensity distribution after the object is regarded as the input signal for this imaging system. In the Fourier domain (i.e. the back focal plane of the first Fourier lens), the intensity SLM modulates the frequency of the input signal. On the imaging plane (i.e. the
where F is a two-dimensional discrete Fourier matrix which represents a two-dimensional discrete Fourier transformation (as shown in (7)), F* is the conjugate matrix of F, and P (as shown in (8)) represents the intensity modulation in Fourier domain. The elements of P distribute uniformly and independently in the interval [0,1]. 1 F(u, v) = √ e−i2 (u−1)(v−1)/N N
⎡
⎢ ⎢ ⎢ ⎢ P=⎢ ⎢ ⎢ ⎢ ⎣
(7)
⎤
a11 ..
⎥ ⎥ ⎥ ⎥ ⎥ aii ∼uniform[0, 1] ⎥ ⎥ ⎥ ⎦
. aii ..
.
(8)
ann 3.2. Incoherence analysis Many works show that the key of implementing compressed sensing is designing a measurement system which is incoherent with any fixed orthogonal basis . According to the definition of coherence in Eq. (4) and the conclusion in [11], when coherence satisfies Eq. (9), the measurement system is nearly incoherent with the orthogonal basis. As a at the smallest sampling ratio. result, we can reconstruct a signal With the coherence smaller than 2 log(4n2 /ı), the probability of
P(maxl,k |˚l , k | < 2 log(4n2 /ı)) will exceed 1 − ı (as shown in (8)) and will approach 1 [11].
P(maxl,k |˚l , k | < 2
log(4n2 /ı)) > 1 − ı
(9)
Based on the analysis before, it is obvious that if the proposed measurement system can satisfy (8), then it will fit for compressed sensing. According to the expression of matrix P in (7), the coherence can be written as:
˚l , k =
n
e
i2 (l−1)(j−1) n
ˆ k (j) ajj
(10)
j=1
k (j) = Fk , e(i2 (l−1)(j−1))/n = i , Eq. (8) can be written In Eq. (9), as:
˚l , k =
n
ˆ k (j) ei ajj
(11)
j=1
In the complex plane, it is obvious that |ei | = | cos + i sin | = 1
(cos + i sin )ajj ˆ k (j) ≤ ajj ˆ k (j)
(12) (13)
Because of |ajj | ∈ [0, 1], (13) can be written as:
(cos + i sin )ajj ˆ k (j) ≤ ˆ k (j)
(14)
The complex Hoeffding inequality tells us that,
⎧ ⎫ ⎨ ⎬ 2 P ˚l , k > ≤ 4 exp − ⎩ 2 n ˆ (j)2 ⎭ k j=1
(15)
Please cite this article in press as: R. He, et al., Study on compressed sensing imaging based on intensity modulation in Fourier domain, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.123
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ˆ k (j) = 1, choosing = Since 2 written as: P(|˚l , k | > ) ≤
3
2 ln (4n2 /ı) (15) can be
ı n2
(16)
Applying Boole’s inequality, Eq. (16) can be written as:
P(maxl,k |˚l , k | > 2
log(4n2 /ı)) ≤ 1 − ı
(17)
According to Eqs. (7) and (15), we conclude that our purposed measurement system based on intensity modulation in the Fourier domain is incoherent with an arbitrary orthogonal basis, therefore this measurement system satisfies CS requirements. 3.3. Advantage of intensity modulation method In this section, we will show that at the point of incoherence the measurement system based on intensity modulation in Fourier domain is better than the one which uses phase modulation framework. Construct the phase modulation system [6] as follow:
⎡
⎢ ⎢ ⎢ ⎢ P =⎢ ⎢ ⎢ ⎢ ⎣
⎤
b11 ..
⎥ ⎥ ⎥ ⎥ ⎥ bk = ei k , k ∼uniform[0, 2 ) ⎥ ⎥ ⎥ ⎦
. bk ..
. bn
(18) where P represents phase modulation in Fourier domain. The k in each element bk of P distribute uniformly and independently in the interval [0,2). Based on the phase modulation method, the measurement sys√ tem is given by ˚ = nF ∗ P F which is orthogonal. According to Eq. (8), it is not hard to see:
˚ l , k =
n
e
i2 (l−1)(j−1) n
ˆ k (j) ei k
the image plane. Suppose the sampling matrix S is a logical matrix which contains 0 and 1. In the sampling matrix, 1 represents sampling at this location and 0 indicates the opposite operation. The percentage of 1s in sampling matrix S can be regarded as the sampling ratio. For example, if the input image is “Lena” whose size is 100 × 100 in numerical simulation, then we recover the image at a 50% sampling ratio. The simulation result is shown in Fig. 2. To denote the sampling procedure, define the operation between two matrices A {ai } and B = {b1 } with identical dimension as A B = {ai · bi }. The modulated and IFT transformed signal can be described as: y = S (˚x)
(22)
(19) 4.2. Comparative simulation
j=1
i2 (l−1)(j−1) i2 (l−1)(j−1) ˆ ˆ k (j) ≤ e n n ei k ei k e k (j) i2 (l−1)(j−1) n ei k = 1, Eq. (20) can be written as: For e i2 (l−1)(j−1) ˆ k (j) < ˆ k (j) n ei k e
Fig. 2. Compressed sensing simulation based on 4F system by using intensity modulation in Fourier domain. (a) Original image. (b) Modulated image. (c) Sampling on the imaging plane at 50% sampling ratio. (d) Reconstructed image.
(20)
(21)
of Comparing Eq. (13) with Eq. (21), in Eq. (13), the upper bound ˆ k (j), every term in coherence with intensity modulation is ajj ˆ k (j). but in Eq. (21), the upper bound in phase modulation is The key points of the compressed sensing are sparsity and incoherence. For the same signal, without considering sparsity, the quality of the compressed sensing system is dependent on the incoherence. With lower coherence, the measurement system ˚ is more incoherent with . As a result, we need a smaller sampling number ˆ k (j)| < | ˆ k (j)|, our to recover the original signal. Because of |ajj || proposed measurement method is better. 4. Numerical simulations 4.1. Simulations based on intensity modulation The feasibility of the proposed system is validated by a numerical experiment in this section. Based on the measurement mentioned in Section 3, the same sampling method [11,13] has been used—sampling randomly on
4.2.1. Recover the image in two frameworks We contrast the recovered images modulated by intensity and phase modulation in the Fourier domain. The results are shown in Fig. 3. When the sampling ratio is relative low, the quality of images recovered by using intensity modulation are better than the ones modulated by using phase modulation. For example, when the sampling ratio is 45%, the details in (a3) are clearer compared to (b3). But it is just subjective judgment. In the next section we will introduce the SNR (signal to noise ratio) to evaluate the two groups of Fig. 3. 4.2.2. Image quality assessment by SNR We introduce signal-to-noise to evaluate the recovered images. The SNR [10] can be written as:
M−1 N−1 x=0
y=0
SNR = 101 g M−1 N−1 x=0
y=0
[f (x, y) − f¯ ]
2
[I(x, y) − f (x, y)]
2
(23)
In Eq. (16), f is the original image, f¯ is average value of f and I is the reconstructed image. The calculated results are presented in Tables 1 and 2. The contrast line is shown in Fig. 4. It is necessary to declare that at the sampling rate 65%, the recovered images are so good by both modulation methods and their SNRs are infinite which cannot be drawn, thus we use 100 to replace the infinite value. In Fig. 4, the red lines represent the SNRs for recovered images by using intensity modulation in the Fourier domain, and
Please cite this article in press as: R. He, et al., Study on compressed sensing imaging based on intensity modulation in Fourier domain, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.123
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Fig. 3. Contrast simulations between two modulation methods. Images above are reconstructed by using intensity modulation in Fourier domain and their sampling ratios from left to right are 35%, 40%, 45%, 50%, 55%, 60%, respectively. Images below are reconstructed by using phase modulation in Fourier domain and their sampling ratios are from left to right 35%, 40%,45%, 50%, 55%, 60%, respectively.
Table 1 SNRs of reconstructed image based on intensity modulation (the first row represents sampling ratio and the first column represents experiment number).
1 2 3 4 5 6 7 8 9 10
35%
40%
45%
50%
55%
60%
65%
5.444546 8.458867 4.552902 5.682878 3.301818 4.06087 2.659736 5.807927 2.199507 8.949655
3.98903 5.990769 3.359868 3.946259 2.65929 3.018195 2.124349 4.206199 1.886059 6.424282
3.371566 4.908328 3.22066 3.51354 2.891225 3.108656 2.547626 3.590094 2.628427 4.680419
5.562787 6.416412 5.350706 5.360577 5.884283 5.800227 5.644601 5.414417 5.391698 6.79061
12.56286 13.34688 13.70908 11.43096 13.83721 12.00055 12.21455 11.995 15.29176 13.24459
20.36159 20.15585 23.77348 22.98276 26.35 21.06794 31.69163 20.29684 24.65212 27.30429
100 100 100 100 100 100 100 100 100 100
Table 2 SNRs of reconstructed image based on phase modulation (the first row represents sampling ratio and the first column represents experiment number).
1 2 3 4 5 6 7 8 9 10
35%
40%
45%
50%
55%
60%
65%
2.968621 2.604978 2.717369 2.6894 2.936318 2.836004 3.12834 2.6881 2.772514 2.997349
2.455048 2.044107 2.280039 2.212023 2.188175 2.28509 2.419455 2.300719 2.421862 2.531381
2.528757 2.365704 2.288243 2.801506 2.401609 2.678842 2.526866 2.752549 2.675432 2.92536
5.180993 4.489718 5.269328 5.861559 4.759121 5.163497 5.283897 5.500271 5.423828 5.423828
12.72266 10.34456 12.18256 10.76361 10.89408 11.47429 10.53019 12.27669 11.96032 10.77584
18.82 19.99719 22.35 24.29627 26.74 20.24783 21.2 22.87 23.4665 23.4665
100 100 100 100 100 100 100 100 100 100
recovered more accurately based on intensity modulation framework. 5. Conclusions
Fig. 4. SNR contrast simulations between two modulation systems. Red lines represent images SNRs of intensity modulation. Blue lines represent images’ SNRs of phase modulation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
the blue lines represent the ones recovered by using phase modulation in Fourier domain. It is easy to see that when the sampling ratio is low, the SNRs by intensity modulation are higher than the SNRs by phase modulation in the mass, therefore signal can be
We demonstrated that a 4F system based on intensity modulation in the Fourier domain is fit for compressed sensing. Then we verified the superiority of our proposed system by comparing it with the system of phase modulation in the Fourier domain. Existing problems still remain for future work. Although the two measurement systems are similar, the phase modulation measurement system is orthogonal but the intensity one is not. From the results of simulations, as can be seen, the SNRs from intensity modulation are higher than those from phase modulation which however are not stable. The reasons and how to overcome these disadvantages will be explained and carried out in the future. Acknowledgments This work is supported by the National Natural Science Foundation of China (NSFC) (Grant 61101196), the specialized research fund of Ministry of Education of China for the doctoral program of colleges and universities (Grant 20103219120016), National
Please cite this article in press as: R. He, et al., Study on compressed sensing imaging based on intensity modulation in Fourier domain, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.123
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Postdoctoral Foundation (Grant 2012M521085), National Natural Science Foundation of China (NSFC) (Grant 61271332). Natural Science Foundation of Jiangsu Province of China (Grant BK20131354). References [1] E. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory 52 (2006) 489–509. [2] E. Candès, T. Tao, Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inform. Theory 52 (2006) 5406–5425. [3] D. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (2006) 1289–1306. [4] Y. Rivenson, A. Stern, B. Javidi, Single exposure super-resolution compressive imaging by double phase encoding, Opt. Express 18 (14) (2010) 15094–15103. [5] Z. Wang, G.R. Arce, Variable density compressed image sampling, IEEE Trans. Image Process. 19 (1) (2010) 264–270.
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[6] M. Lustig, D. Donoho, J.M. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magn. Reson. Med. 58 (6) (2007) 1182–1195. [7] W.U. Bajwa, A. Sayeed, R. Nowak, Compressed sensing of wireless channels in time, frequency, and space, Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, CA, October, 2008. [8] M.A. Davenport, D. Takhar, J.N. Laska, T. Sun, K.F. Kelly, G. Baraniuk, Singlepixel imaging via compressive sampling, Sig. Process. Mag. IEEE 25 (2) (2008) 83–91. [9] R.F. Marcia, Z.T. Harmany, R.M. Willet, Compressive coded aperture imaging, in: Proc. SPIE Conference on Computational Imaging VII, January, 2009. [10] Z. Wang, G.R. Arce, Variable density compressed image sampling, IEEE Trans. Image Process. 19 (1) (2010) 264–270. [11] J. Romberg, Sensing by random convolution, in: IEEE Int. Work. on Comp. Adv. Multi-Sensor Adaptive Proc. CAMPSAP, 2007, pp. 137–140. [12] A.Ú. Bourquard, F. Aguet, M. Unser, Optical imaging using binary sensors, Opt. Express 18 (5) (2010) 4876–4888. [13] J. Liu, J. Zhu, F. Yan, Z. Zhang, Random phase modulation and sparse samplingbased optical imaging system, Opt. Eng. 49 (4) (2010), 047002-047002-5.
Please cite this article in press as: R. He, et al., Study on compressed sensing imaging based on intensity modulation in Fourier domain, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.123
ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Study on compressed sensing imaging based on intensity modulation in Fourier domain Ruiqing He ∗ , Qian Chen, Weiji He Jiangsu Key Lab of Spectral Imaging & Intelligence Sense (SIIS), Nanjing University of Science and Technology, Nanjing 210094, PR China
a r t i c l e
i n f o
Article history: Received 20 July 2013 Accepted 15 January 2014 Available online xxx Keywords: Compressed sensing Intensity modulation Coherence
a b s t r a c t In 4F system, compressed sensing is usually implemented by using phase modulation in Fourier domain. In this paper, we present a type of 4F system based on intensity modulation in Fourier domain as the measurement system for compressed sensing. The feasibility of this system is demonstrated. At the point of coherence, the two modulation methods are compared and superiority of intensity modulation in Fourier domain was verified. Simulations are presented and the conclusion we presented is validated. Finally, we analyze the results. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Recently, compressed sensing (CS) theory was discussed widely [1–3]. Compared with the traditional sampling method, compressed sensing is based on the sparsity of signal. In the compressed sensing framework, we need to design a random matrix to measure signal and recover the signal by convex optimization. Actually, the measurement procedure can be understood as projecting the signal into a random vector of a measurement matrix. By using random measurement, CS can hugely reduce the sampling ratio which is lower than the Nyquist sampling ratio, as a result we save the time spent on the sampling and data size of the sampling value is smaller than before. Its appearance is very important to the signal process. CS has been found to have wide usage in many applications, such as optical imaging [4,5], biomedical imaging [6], high-speed wireless communication [7], etc. In the imaging field, the single pixel camera based on compressed sensing has been presented by Davenport M.A. [8]. The advantage of single pixel camera is its suitability for optical systems. In the compressed imaging field, Rebecca M. Willett’s group improved the image resolution by using compressed coded aperture [9]. Gonzalo R. Arceze presented the variable density compressed image sampling [10]. Justin Romberg proved that the measurement system consisted of the 4F system based on phase modulation in the Fourier domain can meet compressed sensing requirements [11].
Inspired by Justin Romberg, we present our purposed measurement system consisting of the 4F system based on intensity modulation in the Fourier domain and proves it to be fitted for compressed sensing. Another contribution of this paper is that compared to the measurement system using phase modulation in the Fourier domain [11], the superiority of our purposed system was verified at the point of coherence. This work is organized as follows: Section 2 introduces the principle of compressed sensing. Section 3 contains the main theoretical work of this paper and we construct the 4F system based on intensity modulation in the Fourier domain as the measurement system. Section 4 is the numerical simulation, in which the feasibility of our imaging system is validated. Finally, Section 5 concludes this paper. 2. Principle of compressed sensing The principle of compressed sensing can be expressed as follow. To recover the unknown signal x ∈ Rn , we need to measure it and the measurement results can be written as: y1 = x, 1 · · ·ym = x, m
k is a measurement vector whose size is same as signal x·yi is measurement value. We can use as the measurement system, where the k are stacked up as the rows in . The measurement process can be expressed as: y = ˚x = ˚˛ (x = ˛).
∗ Corresponding author. E-mail address: [email protected] (R. He).
(1)
(2)
In general, if one wants to recover signal x from y, we need m ≥ n, but when signal x is sparse in an orthogonal basis (i.e. there are relatively few nonzero entries in ˛ and the number of nonzero
http://dx.doi.org/10.1016/j.ijleo.2014.01.123 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
Please cite this article in press as: R. He, et al., Study on compressed sensing imaging based on intensity modulation in Fourier domain, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.123
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2
back focal plane of the second Fourier lens), the modulated image is sampled randomly. Finally, the original signal is reconstructed from the sampling result by using convex optimization. The measurement system in Fig. 1 is shown as follow: √ ˚ = nF ∗ PF, (6) Fig. 1. Compressed sensing based on intensity modulation in 4f system.
entries is K, called K-sparse), then it is possible to reconstruct signal x with m n. For general images, there always is a space spanned by a proper orthogonal basis where the image signal is sparse. Aurelien Bourquard believes that the importance of is its mere existence [12]. Based on the sparse property, if matrix = ˚ satisfies the RIP (restricted isometry property), the signal can be recovered accurately from the relatively fewer samples. The RIP is briefly reviewed later. Suppose T ⊂ {1, 2 . . . N} and |T| ≤ K, ˚T which is indexed by T, is a subset of . ık is the minimal constant which satisfies Eq. (4) and it is correlated to T and K. If there are that meets Eq. (3), then we say that satisfies RIP. From the point of RIP, the goal of the designing or optimizing measure system is making ık as minimal as possible. (1 − ık )||x||22 ≤ ||T x||22 ≤ (1 + ık )||x||22
(3)
RIP is a sufficient but not necessary condition, and it is hard to determine whether or not the measurement system meets RIP by algorithm. In fact, to implement compressed sensing, one always designs the measurement system which is incoherent or low coherent with . The coherence can be defined as follows. Suppose the measurement system is , and the orthogonal basis is . The coherence can then be written as:
= maxl,k l , k ,
(4)
where l is the lth √row in and k is the kth column in . In (4), satisfies 1 ≤ ≤ N. Based on conditions before, recovering signal x is a convex optimization problem as shown in Eq. (5)
min T x
1
s.t. y = ˚x.
(5)
3. System scheme Romberg has proven that the 4F system using phase modulation in the Fourier domain can be incoherent with any orthogonal basis [11]. Based on Romberg’s research, we present a type of 4F system based on intensity modulation in the Fourier domain as the measurement system. Section 3.1 introduces the composition of our imaging system. Sections 3.2 and 3.3 verify feasibility and superiority of our purposed measurement system respectively in theory. 3.1. Composition of imaging system The basic composition of our purposed imaging system contains a measurement and sparse sampling system which are schematically presented in Fig. 1. The measurement system is composed of a pair of Fourier lenses and intensity SLM (spatial light modulator) and finishing modulation of input signal. The sparse sampling system randomly selects pixel locations and uses the pixel values of these positions as input signal for imaging system. The process is shown as follows. At first, laser beam is expanded and collimated. A transmissive object is located in the way of beam. Intensity distribution after the object is regarded as the input signal for this imaging system. In the Fourier domain (i.e. the back focal plane of the first Fourier lens), the intensity SLM modulates the frequency of the input signal. On the imaging plane (i.e. the
where F is a two-dimensional discrete Fourier matrix which represents a two-dimensional discrete Fourier transformation (as shown in (7)), F* is the conjugate matrix of F, and P (as shown in (8)) represents the intensity modulation in Fourier domain. The elements of P distribute uniformly and independently in the interval [0,1]. 1 F(u, v) = √ e−i2 (u−1)(v−1)/N N
⎡
⎢ ⎢ ⎢ ⎢ P=⎢ ⎢ ⎢ ⎢ ⎣
(7)
⎤
a11 ..
⎥ ⎥ ⎥ ⎥ ⎥ aii ∼uniform[0, 1] ⎥ ⎥ ⎥ ⎦
. aii ..
.
(8)
ann 3.2. Incoherence analysis Many works show that the key of implementing compressed sensing is designing a measurement system which is incoherent with any fixed orthogonal basis . According to the definition of coherence in Eq. (4) and the conclusion in [11], when coherence satisfies Eq. (9), the measurement system is nearly incoherent with the orthogonal basis. As a at the smallest sampling ratio. result, we can reconstruct a signal With the coherence smaller than 2 log(4n2 /ı), the probability of
P(maxl,k |˚l , k | < 2 log(4n2 /ı)) will exceed 1 − ı (as shown in (8)) and will approach 1 [11].
P(maxl,k |˚l , k | < 2
log(4n2 /ı)) > 1 − ı
(9)
Based on the analysis before, it is obvious that if the proposed measurement system can satisfy (8), then it will fit for compressed sensing. According to the expression of matrix P in (7), the coherence can be written as:
˚l , k =
n
e
i2 (l−1)(j−1) n
ˆ k (j) ajj
(10)
j=1
k (j) = Fk , e(i2 (l−1)(j−1))/n = i , Eq. (8) can be written In Eq. (9), as:
˚l , k =
n
ˆ k (j) ei ajj
(11)
j=1
In the complex plane, it is obvious that |ei | = | cos + i sin | = 1
(cos + i sin )ajj ˆ k (j) ≤ ajj ˆ k (j)
(12) (13)
Because of |ajj | ∈ [0, 1], (13) can be written as:
(cos + i sin )ajj ˆ k (j) ≤ ˆ k (j)
(14)
The complex Hoeffding inequality tells us that,
⎧ ⎫ ⎨ ⎬ 2 P ˚l , k > ≤ 4 exp − ⎩ 2 n ˆ (j)2 ⎭ k j=1
(15)
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ˆ k (j) = 1, choosing = Since 2 written as: P(|˚l , k | > ) ≤
3
2 ln (4n2 /ı) (15) can be
ı n2
(16)
Applying Boole’s inequality, Eq. (16) can be written as:
P(maxl,k |˚l , k | > 2
log(4n2 /ı)) ≤ 1 − ı
(17)
According to Eqs. (7) and (15), we conclude that our purposed measurement system based on intensity modulation in the Fourier domain is incoherent with an arbitrary orthogonal basis, therefore this measurement system satisfies CS requirements. 3.3. Advantage of intensity modulation method In this section, we will show that at the point of incoherence the measurement system based on intensity modulation in Fourier domain is better than the one which uses phase modulation framework. Construct the phase modulation system [6] as follow:
⎡
⎢ ⎢ ⎢ ⎢ P =⎢ ⎢ ⎢ ⎢ ⎣
⎤
b11 ..
⎥ ⎥ ⎥ ⎥ ⎥ bk = ei k , k ∼uniform[0, 2 ) ⎥ ⎥ ⎥ ⎦
. bk ..
. bn
(18) where P represents phase modulation in Fourier domain. The k in each element bk of P distribute uniformly and independently in the interval [0,2). Based on the phase modulation method, the measurement sys√ tem is given by ˚ = nF ∗ P F which is orthogonal. According to Eq. (8), it is not hard to see:
˚ l , k =
n
e
i2 (l−1)(j−1) n
ˆ k (j) ei k
the image plane. Suppose the sampling matrix S is a logical matrix which contains 0 and 1. In the sampling matrix, 1 represents sampling at this location and 0 indicates the opposite operation. The percentage of 1s in sampling matrix S can be regarded as the sampling ratio. For example, if the input image is “Lena” whose size is 100 × 100 in numerical simulation, then we recover the image at a 50% sampling ratio. The simulation result is shown in Fig. 2. To denote the sampling procedure, define the operation between two matrices A {ai } and B = {b1 } with identical dimension as A B = {ai · bi }. The modulated and IFT transformed signal can be described as: y = S (˚x)
(22)
(19) 4.2. Comparative simulation
j=1
i2 (l−1)(j−1) i2 (l−1)(j−1) ˆ ˆ k (j) ≤ e n n ei k ei k e k (j) i2 (l−1)(j−1) n ei k = 1, Eq. (20) can be written as: For e i2 (l−1)(j−1) ˆ k (j) < ˆ k (j) n ei k e
Fig. 2. Compressed sensing simulation based on 4F system by using intensity modulation in Fourier domain. (a) Original image. (b) Modulated image. (c) Sampling on the imaging plane at 50% sampling ratio. (d) Reconstructed image.
(20)
(21)
of Comparing Eq. (13) with Eq. (21), in Eq. (13), the upper bound ˆ k (j), every term in coherence with intensity modulation is ajj ˆ k (j). but in Eq. (21), the upper bound in phase modulation is The key points of the compressed sensing are sparsity and incoherence. For the same signal, without considering sparsity, the quality of the compressed sensing system is dependent on the incoherence. With lower coherence, the measurement system ˚ is more incoherent with . As a result, we need a smaller sampling number ˆ k (j)| < | ˆ k (j)|, our to recover the original signal. Because of |ajj || proposed measurement method is better. 4. Numerical simulations 4.1. Simulations based on intensity modulation The feasibility of the proposed system is validated by a numerical experiment in this section. Based on the measurement mentioned in Section 3, the same sampling method [11,13] has been used—sampling randomly on
4.2.1. Recover the image in two frameworks We contrast the recovered images modulated by intensity and phase modulation in the Fourier domain. The results are shown in Fig. 3. When the sampling ratio is relative low, the quality of images recovered by using intensity modulation are better than the ones modulated by using phase modulation. For example, when the sampling ratio is 45%, the details in (a3) are clearer compared to (b3). But it is just subjective judgment. In the next section we will introduce the SNR (signal to noise ratio) to evaluate the two groups of Fig. 3. 4.2.2. Image quality assessment by SNR We introduce signal-to-noise to evaluate the recovered images. The SNR [10] can be written as:
M−1 N−1 x=0
y=0
SNR = 101 g M−1 N−1 x=0
y=0
[f (x, y) − f¯ ]
2
[I(x, y) − f (x, y)]
2
(23)
In Eq. (16), f is the original image, f¯ is average value of f and I is the reconstructed image. The calculated results are presented in Tables 1 and 2. The contrast line is shown in Fig. 4. It is necessary to declare that at the sampling rate 65%, the recovered images are so good by both modulation methods and their SNRs are infinite which cannot be drawn, thus we use 100 to replace the infinite value. In Fig. 4, the red lines represent the SNRs for recovered images by using intensity modulation in the Fourier domain, and
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Fig. 3. Contrast simulations between two modulation methods. Images above are reconstructed by using intensity modulation in Fourier domain and their sampling ratios from left to right are 35%, 40%, 45%, 50%, 55%, 60%, respectively. Images below are reconstructed by using phase modulation in Fourier domain and their sampling ratios are from left to right 35%, 40%,45%, 50%, 55%, 60%, respectively.
Table 1 SNRs of reconstructed image based on intensity modulation (the first row represents sampling ratio and the first column represents experiment number).
1 2 3 4 5 6 7 8 9 10
35%
40%
45%
50%
55%
60%
65%
5.444546 8.458867 4.552902 5.682878 3.301818 4.06087 2.659736 5.807927 2.199507 8.949655
3.98903 5.990769 3.359868 3.946259 2.65929 3.018195 2.124349 4.206199 1.886059 6.424282
3.371566 4.908328 3.22066 3.51354 2.891225 3.108656 2.547626 3.590094 2.628427 4.680419
5.562787 6.416412 5.350706 5.360577 5.884283 5.800227 5.644601 5.414417 5.391698 6.79061
12.56286 13.34688 13.70908 11.43096 13.83721 12.00055 12.21455 11.995 15.29176 13.24459
20.36159 20.15585 23.77348 22.98276 26.35 21.06794 31.69163 20.29684 24.65212 27.30429
100 100 100 100 100 100 100 100 100 100
Table 2 SNRs of reconstructed image based on phase modulation (the first row represents sampling ratio and the first column represents experiment number).
1 2 3 4 5 6 7 8 9 10
35%
40%
45%
50%
55%
60%
65%
2.968621 2.604978 2.717369 2.6894 2.936318 2.836004 3.12834 2.6881 2.772514 2.997349
2.455048 2.044107 2.280039 2.212023 2.188175 2.28509 2.419455 2.300719 2.421862 2.531381
2.528757 2.365704 2.288243 2.801506 2.401609 2.678842 2.526866 2.752549 2.675432 2.92536
5.180993 4.489718 5.269328 5.861559 4.759121 5.163497 5.283897 5.500271 5.423828 5.423828
12.72266 10.34456 12.18256 10.76361 10.89408 11.47429 10.53019 12.27669 11.96032 10.77584
18.82 19.99719 22.35 24.29627 26.74 20.24783 21.2 22.87 23.4665 23.4665
100 100 100 100 100 100 100 100 100 100
recovered more accurately based on intensity modulation framework. 5. Conclusions
Fig. 4. SNR contrast simulations between two modulation systems. Red lines represent images SNRs of intensity modulation. Blue lines represent images’ SNRs of phase modulation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
the blue lines represent the ones recovered by using phase modulation in Fourier domain. It is easy to see that when the sampling ratio is low, the SNRs by intensity modulation are higher than the SNRs by phase modulation in the mass, therefore signal can be
We demonstrated that a 4F system based on intensity modulation in the Fourier domain is fit for compressed sensing. Then we verified the superiority of our proposed system by comparing it with the system of phase modulation in the Fourier domain. Existing problems still remain for future work. Although the two measurement systems are similar, the phase modulation measurement system is orthogonal but the intensity one is not. From the results of simulations, as can be seen, the SNRs from intensity modulation are higher than those from phase modulation which however are not stable. The reasons and how to overcome these disadvantages will be explained and carried out in the future. Acknowledgments This work is supported by the National Natural Science Foundation of China (NSFC) (Grant 61101196), the specialized research fund of Ministry of Education of China for the doctoral program of colleges and universities (Grant 20103219120016), National
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Postdoctoral Foundation (Grant 2012M521085), National Natural Science Foundation of China (NSFC) (Grant 61271332). Natural Science Foundation of Jiangsu Province of China (Grant BK20131354). References [1] E. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory 52 (2006) 489–509. [2] E. Candès, T. Tao, Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inform. Theory 52 (2006) 5406–5425. [3] D. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (2006) 1289–1306. [4] Y. Rivenson, A. Stern, B. Javidi, Single exposure super-resolution compressive imaging by double phase encoding, Opt. Express 18 (14) (2010) 15094–15103. [5] Z. Wang, G.R. Arce, Variable density compressed image sampling, IEEE Trans. Image Process. 19 (1) (2010) 264–270.
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