Singular value decomposition ghost imaging

Xue Zhang; Xiangfeng Meng; Xiulun Yang; Yurong Wang; Yongkai Yin; Xianye Li; Xiang Peng; Wenqi He; Guoyan Dong; Hongyi Chen

doi:10.1364/OE.26.012948

1. Introduction

Traditional ghost imaging (TGI) [1–9] experimental system consists of two optical arms. At object arm, the single-pixel detector (SPD) collects the total light intensity modulated by the object; at reference arm, the charge-coupled device (CCD) records the light distribution of each illumination pattern. TGI allows object to be imaged by correlating the two series of light intensities. Computational ghost imaging (GI) [10–22] is developed from TGI, and it abandons the beam splitter and CCD by using spatial light modulator (SLM) [23] that is capable of producing programmable illumination patterns. Whereas, improving the imaging quality of GI is a key project. Differential ghost imaging (DGI) [24,25] introduces another differential bucket signal, which is only sensitive to the object’s fluctuating part, so that it effectively enhances the signal-to-noise ratio of GI. Especially for high transparent objects, DGI works much better. Recently, pseudo-inverse ghost imaging (PGI) [26, 27] has been researched, where the object can be imaged from measurements lower than Nyquist limitation by computing the pseudo-inverse of the measurement matrix. However, the sensitivity to noise and reconstruction time wasting limits its usage [26]. Most recently, some advanced basis illumination patterns based ghost imaging methods have been proposed, including Hadamard basis based ghost imaging [18, 28, 29], Fourier basis based ghost imaging [30–33], discrete cosine basis based ghost imaging [34] and wavelet basis based ghost imaging [35, 36]. Because the basis patterns constitute a complete orthogonal set, and the natural image often can be sparsely representation in transform domain, these techniques can enhance the quality of GI and shorten the data acquisition time. However, these techniques have also some deficiencies [32], for example, the Hadamard basis based ghost imaging is only applicable for the original images of size H-by-H pixels, where, H, H / 12, or H / 20 has to be the power of 2. So it is urgent to find an alternative method that not only improves the imaging result with less measurements but also shortens the reconstruction time to fit practical applications.

To further improve the fidelity of GI and shorten its reconstruction time, in this paper, we construct the measurement matrix using the singular value decomposition method [37–39] which is termed as SVDGI. We construct an orthogonal matrix for the GI as a measurement matrix, but this orthogonal matrix is not constrained by the size of the target object. With the increase of the number of measurements, the measurement matrix tends to be a complete orthogonal set, so theoretically, when the number of measurements is equal to the number of pixels in the image, our method can perfectly reconstruct the object. In addition, because the measurement matrix contains both positive values and negative values, the differential measurements are required in the experiment, so that the background noise can be effectively suppressed, which is not achievable using the random matrix as a measurement matrix. In numerical simulation experiments, where the noise is not considered, our method performs as well as PGI on clarity and better than GI and DGI. But the optical experimental results show that, with the same number of measurements, SVDGI can always get clearer images than the other three methods, where the background noise and the detector dark current are inevitable. We firstly give the theoretical analysis, then provide its simulated and experimental results, and finally draw the conclusion.

2. Analysis of the fundamental principles

The experimental setup is shown in Fig. 1, which consists of projector (Panasonic 3LCD) to project the i-th known patterns I_i(x, y); the object O(x, y) located at the image surface of the lens; the SPD (Thorlabs PDA-100A) facing the object located at the lens plane to collect the total reflected light B_i ; the data acquisition card (DAQ card: NI USB-6341) to digitalize the B_i ; and the computer (Intel(R) Core(TM) i5-4590CPU @ 3.30GHz; RAM: 8.00GB; No GPU) is capable of generating the I_i (x, y) and restoring the object.

Fig. 1 Schematic diagram of the experimental setup. A cup represents the object which can be arbitrarily replaced by anything.

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2.1. Computational ghost imaging

To make the reader understand our following work, we immediately introduce the notation and concept of GI. Assume that the reflective of object is a matrix O(x, y) of size p × p, and the random measurement matrix Ф of size M × N, where N = p × p:

Φ = [\begin{matrix} \begin{matrix} I_{1} (1, 1) \\ I_{2} (1, 1) \\ ⋮ \\ I_{M} (1, 1) \end{matrix} & \begin{matrix} I_{1} (1, 2) \\ I_{2} (1, 2) \\ ⋮ \\ I_{M} (1, 2) \end{matrix} & \begin{matrix} \dots \\ \dots \\ ⋱ \\ \dots \end{matrix} & \begin{matrix} I_{1} (p, p) \\ I_{2} (p, p) \\ ⋮ \\ I_{M} (p, p) \end{matrix} \end{matrix}],

before i-th projection, we should firstly rearrange i-th row of Ф into 2-D pattern, then the corresponding B_i can be obtained. Mathematically speaking, if we present O(x, y) as a column vector of size p² × 1, after M detections, it must have the following linear relationship with B:

[\begin{matrix} B_{1} \\ B_{2} \\ ⋮ \\ B_{M} \end{matrix}] = Φ [\begin{matrix} O (1, 1) \\ O (1, 2) \\ ⋮ \\ O (p, p) \end{matrix}] .

Briefly, we will get the estimation of O(x, y) by GI’s correlation algorithm that is the weighted average of each pattern:

\hat{O} (x, y) = \frac{1}{M} \sum_{i = 1}^{M} (B_{i} - 〈 B 〉) I_{i} (x, y),

where, the symbol ‘

\sum_{i = 1}^{M}

’ calculates the summation of first M terms, ‘

〈 〉

’ calculates the average value of a vector,

B_{i} = \iint I_{i} (x, y) O (x, y) d x d y

, and

\hat{O} (x, y)

is the estimation of O(x, y). In essence, Eq. (3) can be expressed as [26]:

\hat{O} (x, y) = \frac{1}{M} [\begin{matrix} \begin{matrix} I_{1} (1, 1) \\ I_{1} (1, 2) \\ ⋮ \\ I_{1} (p, p) \end{matrix} & \begin{matrix} I_{2} (1, 1) \\ I_{2} (1, 2) \\ ⋮ \\ I_{2} (p, p) \end{matrix} & \begin{matrix} \dots \\ \dots \\ ⋱ \\ \dots \end{matrix} & \begin{matrix} I_{M} (1, 1) \\ I_{M} (1, 2) \\ ⋮ \\ I_{M} (p, p) \end{matrix} \end{matrix}] [\begin{matrix} B_{1} \\ B_{2} \\ ⋮ \\ B_{M} \end{matrix}] - 〈 B 〉 [\begin{matrix} 〈 I (1, 1) 〉 \\ 〈 I (1, 2) 〉 \\ ⋮ \\ 〈 I (p, p) 〉 \end{matrix}],

then, extracting the first term on the right hand of Eq. (4) and substituting Eqs. (1) - (2) into it, we can get:

\hat{O} (x, y) = \frac{1}{M} Φ^{T} Φ [\begin{matrix} O (1, 1) \\ O (1, 2) \\ ⋮ \\ O (p, p) \end{matrix}],

where, the symbol ^T indicates the transposition operation. Obviously, the closer the Ф^T Ф is to unit matrix, the higher the quality of the restored image is. When Ф is an orthogonal matrix, the target object can be quickly and accurately reconstructed.

2.2. Singular value decomposition ghost imaging

It is very easy to orthogonalize Ф when M = N. But, can we construct a measurement matrix such that it still performs good orthogonal property even when M < N? The answer is yes, SVD [37–39] modified by us is competent for this work. As we all know, after taking SVD on random Ф of size M × N, one can get two orthogonal matrices, U and V, and one singular value matrix S: Ф = USV ^T, then let these singular values in S are all equal to 1.0, a man-made measurement matrix can be written as:

Φ_{s v d} = U_{M \times M} {[Λ_{M \times M} 0]}_{M \times N} V^{T}_{N \times N},

where, Ʌ is a unit matrix. Substituting Eq. (6) into Eq. (5) and offsetting U, we get:

\hat{O} (x, y) = \frac{1}{M} V {[\begin{matrix} \begin{matrix} Λ_{M \times M} \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix}]}_{N \times N} V^{T} O (x, y),

where, V still holds absolute orthogonal character, which demonstrates that, with the same number of measurements, replacing Ф by Ф_svd in Eq. (5) can dramatically enhance GI’s imaging quality. And as the number of measurements increases, the image will get clearer, until M is equal to N, the object is perfectly restored.

2.3. Theoretical analysis for quality enhancement

To make the theoretical analysis process easy understanding, we repeat that the O(x, y) is a column vector representing the original object, and the random matrix Ф is the measurement matrix for GI, DGI and PGI, thus the total light intensity of GI and PGI is equivalent, which can be denoted by B_GP = Ф O, and the constructed matrix Ф_svd is the measurement matrix for SVDGI, then the corresponding total light intensity is termed as B_S = Ф_svd O.

According to the Eq. (5), we can restore two results: Ф^T B_GP for GI, and Ф^T_svd B_S for SVDGI, then let the

Φ^{T} Φ = E_{A S_{1}} = E_{O S_{1}},

and

Φ^{T}_{s v d} Φ_{S v d} = E_{A S_{2}} = E_{O S_{2}},

where, E_AS₁ and E_AS₂ are the diagonal axis elements of Ф^T Ф and Ф^T_svd Ф_svd, respectively, while E_OS₁ and E_OS₂ are the outside diagonal axis elements of Ф^T Ф and Ф^T_svd Ф_svd, respectively. Obviously, the more evenly the E_AS distributed, as well as the smaller the E_OS is, in other words, the closer the Ф^T Ф or Ф^T_svd Ф_svd is to unite matrix after normalization, the higher the fidelity of restored image is, which proves that the imaging quality of SVDGI is superior to the imaging quality of GI, because the former’s measurement matrix is partially orthogonal, while the latter is completely random and has no orthogonality.

Next, that the SVDGI is more noisy-resistant than PGI is deduced. Assuming that the F₁ and F₂ are two noise fluctuation for B_GP and B_S with M elements, respectively, the real detected total light intensities are Bˊ_GP = B_GP + F₁, and Bˊ_S = B_S + F₂, respectively. Then, the signal-to-noise ratio (SNR) of the two total light intensities can be expressed as:

{SNR}_{B'_{G P}} = \frac{\sum_{i = 1}^{M} {| B_{G P} |}^{2}}{\sum_{i = 1}^{M} {| F_{1} |}^{2}} = \frac{O^{T} Φ^{T} Φ O}{F_{1}^{T} F_{1}},

and

{SNR}_{B'_{S}} = \frac{\sum_{i = 1}^{M} {| B_{S} |}^{2}}{\sum_{i = 1}^{M} {| F_{2} |}^{2}} = \frac{O^{T} Φ^{T}_{s v d} Φ_{s v d} O}{F_{2}^{T} F_{2}},

because the Ф_svd is partially orthogonal, while Ф is completely random matrix and does not have orthogonality, O^T Ф^T Ф O >> O^T Ф^T_svd Ф_svd O. Then, under the circumstances of SNR_BˊGP = SNR_BˊS, we will get:

F_{1}^{T} F_{1} > > F_{2}^{T} F_{2} = \frac{O^{T} Φ_{s v d}^{T} Φ_{s v d} O F_{1}^{T} F_{1}}{O^{T} Φ^{T} Φ O},

substituting the Eq. (6) into the upper equation, we will get:

F_{2}^{T} F_{2} = \frac{O^{T} V [\begin{matrix} Λ & 0 \\ 0 & 0 \end{matrix}] V^{T} O F_{1}^{T} F_{1}}{O^{T} V [\begin{matrix} λ_{1}^{2} \\ ⋱ & 0 \\ λ_{m}^{2} \\ 0 & 0 \end{matrix}] V^{T} O},

where λ₁, λ₂,… λ_m denote the singular values of Ф.

As described in Ref. 26,

Φ = U [\begin{matrix} λ_{1} \\ ⋱ & 0 \\ λ_{m} \end{matrix}] V^{T},

Φ^{†} = V [\begin{matrix} λ_{1}^{- 1} \\ ⋱ \\ λ_{m}^{- 1} \\ 0 \end{matrix}] U^{T},

where, Ф^† denotes the pseudo inverse of Ф. Therefore, the SNRs of the restored results for PGI and SVDGI are:

{SNR}_{PGI} = \frac{\sum_{i = 1}^{M} {| Φ^{†} B_{G P} |}^{2}}{\sum_{i = 1}^{M} {| Φ^{†} F_{1} |}^{2}} = \frac{{(Φ^{†} Φ O)}^{T} Φ^{†} Φ O}{{(Φ^{†} F_{1})}^{T} Φ^{†} F_{1}} = \frac{O^{T} V [\begin{matrix} Λ & 0 \\ 0 & 0 \end{matrix}] V^{T} O}{F_{1}^{T} U [\begin{matrix} λ_{1}^{- 2} \\ ⋱ \\ λ_{m}^{- 2} \end{matrix}] U^{T} F_{1}},

{SNR}_{SVDGI} = \frac{\sum_{i = 1}^{M} {| Φ^{T}_{s v d} B_{S} |}^{2}}{\sum_{i = 1}^{M} {| Φ^{T}_{s v d} F_{2} |}^{2}} = \frac{{(Φ^{T}_{s v d} Φ_{s v d} O)}^{T} Φ^{T}_{s v d} Φ_{s v d} O}{{(Φ^{T}_{s v d} F_{2})}^{T} Φ^{T}_{s v d} F_{2}} = \frac{O^{T} V [\begin{matrix} Λ & 0 \\ 0 & 0 \end{matrix}] V^{T} O}{F_{2}^{T} F_{2}},

it is obvious that the power of signal of PGI and SVDGI is equivalent, thus, the ratio of the two SNRs that is termed as r, can be written as:

r = \frac{{SNR}_{PGI}}{{SNR}_{SVDGI}} = \frac{F_{2}^{T} F_{2}}{F_{1}^{T} U [\begin{matrix} λ_{1}^{- 2} \\ ⋱ \\ λ_{m}^{- 2} \end{matrix}] U^{T} F_{1}},

substituting Eq. (13) into Eq. (18), and let the Ψ _N _{× 1} = V ^T O, Ω _M _{× 1} = U ^T F₁, then we can simplify it into:

r = \frac{\sum_{i = 1}^{M} \sum_{j = 1}^{M} Ψ_{i}^{2} Ω_{j}^{2}}{\sum_{i = 1}^{M} \sum_{j = 1}^{M} {(\frac{λ_{i}}{λ_{j}})}^{2} Ψ_{i}^{2} Ω_{j}^{2}},

as λ₁ > λ₂ > . . . > λ _M > 0 [37], and the square of a positive number that is smaller than 1.0 is smaller than itself, while the square of a positive number that is greater than 1.0 is greater than itself. Therefore, the Eq. (19) can be written as:

r \approx \frac{\sum_{i = 1}^{M} \sum_{j \geq i}^{M} Ψ_{i}^{2} Ω_{j}^{2}}{\sum_{i = 1}^{M} \sum_{j \geq i}^{M} {(\frac{λ_{i}}{λ_{j}})}^{2} Ψ_{i}^{2} Ω_{j}^{2}} < 1.0.

Therefore, it is theoretically deduced that the SVDGI will get a higher fidelity image than PGI in the noisy environment.

Finally, we define the notation η as the compression ratio denoting the relationship between M and N:

η = \frac{M}{N},

and we quote the correlated coefficients (CC) [37] to reflects the similarity between restored result and the ground truth image, the reconstruction time (t) is used to illustrate the speed of algorithm.

3. Simulated and optical experimental results

In the following, we verify the feasibility and validity of our method by the numerical simulation and the optical experiments.

3.1. Computer simulations

To begin with, we compute and display the products of Ф^T Ф and Ф^T_svd Ф_svd to reflect which one is closer to unit matrix. Since the paper size limitation, we adopt a random Ф of size 64 × 128, Ф_svd is constructed by our method. In Fig. 2, we normalized the two products into 0 ~255 grayscale and vividly display them using 1D curve, 2D matrix and 3D map forms. The illustrations at the top left of Figs. 2(a) and 2(b) are the matrix presentation of the two products, where x and y represent the row and column coordinates, respectively. Figure 2(c) shows two sectional views, the blue and red lines indicate 64-th row of Ф^T Ф and Ф^T_svd Ф_svd, respectively.

Fig. 2 Product comparison. (a) Ф^T Ф; (b) Ф^T_svd Ф_svd ; (c) two sectional views.

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From the 3D maps of Figs. 2(a) and 2(b), we can see that the diagonal elements of the two products are almost equal and larger than any off-diagonal value. It is also apparent that the off-diagonal values of Ф^T Ф are much larger than those in Ф^T_svd Ф_svd and fluctuate violently, which is more intuitive in Fig. 2(c). This means that SVDGI brings much less additive noise into restored results than GI according to Eqs. (8) and (9), thus it gets the clearer images, besides, the illustrations can also demonstrate that Ф^T_svd Ф_svd is closer to unit matrix than Ф^T Ф. Therefore, our method is theoretically feasible.

Next, we verify SVDGI’s validity. Taking 8000 measurements for instance, we adopt binary ‘SDU’ and grayscale ‘Peppers’ of size 100 × 100 pixels as original object images, the random matrix Ф of size 8000 × 10000 as measurements matrix for GI, DGI and PGI, constructed Ф_svd by modified SVD method as measurement matrix of SVDGI. Reconstruction algorithm for GI is Eq. (3), DGI refers to [24] and PGI is in [26]. All the results, the respective CC as well as the reconstruction time t are arranged in Fig. 3.

Fig. 3 Simulated results comparison when η = 0.8.

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As shown in Fig. 3, to begin with, in aspect of visually observation, for highly absorbing binary image, GI and DGI perform quite similarity, but SVDGI makes a further progress on clarity; for highly transparent grayscale image, DGI works better than GI, but SVDGI is better than DGI. Besides, in aspect of numerical qualitative analysis, compared with GI, the CCs of SVDGI increased by 37% for binary image and increased by 137% for grayscale one, then, comparing with DGI, the CCs of SVDGI increased by 33% for binary image and increased by 37% for grayscale one, respectively. Last but not the least, SVDGI is the fastest one to accomplish reconstruction, because it needs only one step transposition operation, however, GI and DGI need 8000 iterations, respectively, and PGI needs to compute pseudo inverse of Ф.

As described above, with 8000 measurements, SVDGI indeed increases GI’s CC, but what happens to SVDGI when η changes? Taking grayscale ‘Pepper’ for instance, we plot four CC curves versus η in Fig. 4. To be intuitive, we give four results when η is 0.6. As expected, as η increases, all the CCs rise. Especially, the SVDGI’s CC is larger than GI and DGI all the time and more and more close to 1.0, which is related to Ф^T_svd Ф_svd getting closer to the unit matrix.

Fig. 4 Relationship between CC and the compression ratio η. Four results are restored by (a): SVDGI; (b): PGI; (c): DGI; (d): GI.

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However, the environmental noise is inevitable. We analyze the influence of noise on SVDGI: In Fig. 5, we add Additive Gaussian White noise to vector B and plot four CCs versus different signal-to-ratio (SNR) of B when η is 0.8. In addition, there are four results to show visual difference.

Fig. 5 The relationship between CC and the SNR of B. Four results are restored by (a): SVDGI; (b): PGI; (c): DGI; (d): GI.

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As shown in Fig. 5, with 8000 measurements, the CCs of four methods reach to their respective maximum value and then remain steady along with the growth of SNR of B, because the larger the SNR of B is, the smaller the noise intensity is. Apparently, the CC of SVDGI is always higher than GI and DGI, but PGI achieves this point only when the SNR of B is larger than 60 dB, and only when the SNR of B is greater than 90dB, the PGI matches the performance of SVDGI, where, the power of noise is so small that it can be neglected. Another important information is that when the SNR of B is greater than 30 dB, the CCs of SVDGI reach to its maximum prior to the others and converge to 0.9. Under the circumstance when the SNR of B is equal to 30 dB, the CCs of GI, DGI, and PGI are all less than 0.2, i.e., the reconstructed results are completely submerged by noise, so that no useful information can be distinguished, however, the clarity of the result reconstructed by SVDGI is much higher than the other three methods, which shows SVDGI performs much stronger noise resistance so that it is more suitable for the practical environment.

Finally, we analyze the reconstruction time of SVDGI and compare it with other methods. Because PGI needs too much time to observe the fluctuation of three other methods at the same coordinate system, we insert the relationship curves excepting PGI to the top left corner of the relationship curves including PGI. As shown in Fig. 6, the proposed SVDGI performs the least time-wasting with the growth of η, while the PGI is just opposite. For example, when η is 0.8, the SVDGI is 7000 times faster than PGI, and 19 times faster than GI and DGI.

Fig. 6 Reconstruction time comparison.

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3.2. Optical experimental results

The optical experiment is the most direct and effective way to test the feasibility of SVDGI. To get a 100 × 100 pixels’ image, we construct Ф_svd of size M × 100² for SVDGI, where M represents the number of measurements, and the same size measurement matrix Ф random filled with ± 1 for GI, DGI and PGI. The optical experimental setup is shown in Fig. 1, to manifest that the SVDGI is applicable to enhance the quality of GI for binary and grayscale objects, two target objects: binary letter ‘SVDGI’ printed on A4 paper and the life-sized hand, are respectively positioned at about 1 m from the lens so that it fits within the projected pattern.

As shown in Fig. 7, original target objects and the results restored by the four methods with different η are given. In general, the experimental results are as our expected, on one hand, for both binary and grayscale objects, the restored images become clearer with the increase of η, but even when η is 1.0, the restored results by GI and DGI are still contaminated by noise while the SVDGI makes further progress. On the other hand, SVDGI always generates the clearer image than the other three methods under the same η, especially when η is 1.0, the restored images by SVDGI are nearly close to perfect reconstruction, while the other results are more or less polluted by noise. In our opinion, there are two factors contributing to this phenomenon: in the first place, Ф^T_svd Ф_svd is closer to the unit matrix than Ф^T Ф, thus the estimated images by SVDGI are closer to the ground truth objects than GI and DGI; Furthermore, the noise is objective reality in our environment, such as detector dark current [40] and other forms of detection noise, as demonstrated in the simulation, SVDGI has stronger noise resistance capacity than PGI, thus the former gets the clearer image than the latter. Besides, there is one point deserves to be discussed: We can see from Fig. 7 that the clarity of images restored by SVDGI is improved sharply with the growth of η, and the details can be distinguished when η is 0.4 for binary object and 0.8 for grayscale one. However, the visual enhancement is not obvious in GI column, especially for grayscale target objects. For example, the restored hand using GI with 4000 measurements is almost the same as the results when η is 1.0, which implies SVDGI is a good alternative for GI. Finally, the reconstruction time t increases along with the rise of η, and SVDGI is several orders of magnitude faster than PGI, which shows SVDGI accomplishes the better results while using the less computing time. Therefore, the SVDGI technique has a promising prospect in real applications.

Fig. 7 Comparison of the experimental results for binary and 3D objects.

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4. Conclusion

We have proposed a SVDGI method that can significantly enhance the fidelity of GI method even when the measurements far less than the Nyquist limitation. SVDGI is characteristic of strong noise resistance, and the experimental results demonstrate that, with the same number of measurements, it always accomplishes better work using the shortest time than GI, DGI and PGI. We believe that this new technique will pave the way to the use of GI, such as remote sensing, optical security and imaging in optically noise environment.

Funding

National Natural Science Foundation of China (NSFC) (61775121, 61307003, 61405122, 11574311); Key R&D Program of Shandong Province (2018GGX101002); Natural Science Foundation of Shandong province (ZR2016FM03), and Fundamental Research Funds of Shandong University (2015GN031).

Acknowledgements

We thank the reviewers for some useful suggestions.

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Singular value decomposition ghost imaging

Abstract

1. Introduction

2. Analysis of the fundamental principles

2.1. Computational ghost imaging

2.2. Singular value decomposition ghost imaging

2.3. Theoretical analysis for quality enhancement

3. Simulated and optical experimental results

3.1. Computer simulations

3.2. Optical experimental results

4. Conclusion

Funding

Acknowledgements

References and links

Cited By

Figures (7)

Equations (21)

Optics Express