August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
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Refocusing criterion via sparsity measurements in digital holography Pasquale Memmolo,1,2,* Melania Paturzo,2 Bahram Javidi,3 Paolo A. Netti,1 and Pietro Ferraro2 1
Center for Advanced Biomaterials for Health Care@CRIB, Istituto Italiano di tecnologia, Largo Barsanti e Matteucci 53, 80125 Napoli, Italy 2 3
CNR-Istituto Nazionale di Ottica, via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy
ECE Department, University of Connecticut, U-157, Storrs, Connecticut 06269, USA *Corresponding author: [email protected] Received May 29, 2014; accepted June 26, 2014; posted July 9, 2014 (Doc. ID 213105); published August 6, 2014
Several automatic approaches have been proposed in the past to compute the refocus distance in digital holography (DH). However most of them are based on a maximization or minimization of a suitable amplitude image contrast measure, regarded as a function of the reconstruction distance parameter. Here we show that, by using the sparsity measure coefficient regarded as a refocusing criterion in the holographic reconstruction, it is possible to recover the focus plane and, at the same time, establish the degree of sparsity of digital holograms, when samples of the diffraction Fresnel propagation integral are used as a sparse signal representation. We employ a sparsity measurement coefficient known as Gini’s index thus showing for the first time, to the best of our knowledge, its application in DH, as an effective refocusing criterion. Demonstration is provided for different holographic configurations (i.e., lens and lensless apparatus) and for completely different objects (i.e., a thin pure phase microscopic object as an in vitro cell, and macroscopic puppets) preparation. © 2014 Optical Society of America OCIS codes: (090.1995) Digital holography; (100.2960) Image analysis; (100.3010) Image reconstruction techniques. http://dx.doi.org/10.1364/OL.39.004719
One of the most interesting features in digital holography (DH) is the possibility to numerically manage the focusing plane, by propagating a discrete implementation of the diffraction Fresnel propagation integral to perform a digital holographic reconstruction at a chosen distance [1]. This property makes DH an elective method for several applications, such as 3D tracking [2–4], 3D display [5,6], and quantitative imaging [7–9]. On the other hand, an automatic method that is able to retrieve the focus distance of an object of interest is challenging and several approaches have been proposed [10–20], addressing this issue in different experimental conditions. In fact, many strategies are suitable for digital holograms recorded in a lensless configuration [10–12] as well as in the microscopic scenario [13–20], but not all methods can be applied in both cases. In particular, it was established in [18] that for DH in microscopy, an amplitude (phase) object is refocused at a distance that minimizes (maximizes) an energy-based measure, calculated by its amplitude reconstructions. Basically, most autofocusing strategies are based on maximization/minimization of a suitable image contrast measure and each of them provides an accurate estimation of the focal plane of the object of interest. However, as demonstrated in [10], most of them present several local extrema points, causing difficulties in the search for the global extreme. On the other hand, the method proposed in [10], based on the Tamura coefficient (TC), has the property to have only one extreme in the whole focus range. The latter is a fundamental feature that any refocusing criterion should own. Here we propose a new metric called Gini’s index (GI) [21] as a refocusing criterion that is used, to the best of our knowledge, for the first time in DH. In particular, GI is proposed in [21] as a sparsity measure that significantly improves the performance in signal reconstruction from compressive samples [22,23]. However, a sparsity-based refocusing approach 0146-9592/14/164719-04$15.00/0
has been already proposed in [12], in which a sharpness metric, related to the sparsity of the hologram’s expansion in distance-dependent wavelet-like Fresnelet bases, is used. This metric is related to the signal energy, when summing up the Fresnelet coefficients that have the most part of the intensity. Although it represents a good choice as a refocusing criterion, it is not an effective measure of sparsity. In fact, it was established in [21] that an optimal sparsity measure should be a weighted sum of the coefficients of signal representation, i.e., based on the weight of all coefficients in the overall sparsity. Our idea is to use GI in DH for calculating the in-focus distance of holograms and, at the same time, the degree of sparsity in holographic reconstructions at different distances. In fact, it has already been shown that samples of the diffraction Fresnel propagation integral can be used as a sparse representation of digital holograms for denoising purposes [24]. When GI is used as a refocusing criterion, i.e., as a function of refocusing distance, we demonstrate that it is maximized for holograms recorded in lensless configuration, while it is minimized in the case of a microscope scenario. The latter result is related to the amplitude property of phase objects, which present a quasi-constant amplitude reconstruction in their focus plane. We will show that GI presents the same feature of the TC in terms of convexity with respect to the focus distance and, in addition, it furnishes information about the sparsity of the considered holograms. Notice that, only empirical evidence confirms that sparsity metrics lead to effective focus measures. In fact, the value of GI, calculated at the estimated focus distance, measures a percentage of sparsity of the hologram’s reconstruction. The latter could be fundamental in the field of compressive holography [25–28] permitting the discovery of the sparsest reconstruction plane in which the recovery of digital holograms is better. © 2014 Optical Society of America
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The proposed metric is applied in two different experiments: (1) an astronaut puppet, recorded in lensless configuration; and (2) a biological sample in which a digital hologram of mouse cells is recorded in microscope configuration, to demonstrate the results of the proposed approach for all possible configurations in which DH is adopted. Moreover, a comparison with the TC [10] and energy-based [18] methods is performed to establish the accuracy of the proposed one. Object reconstruction is numerically obtained by propagating the corresponding holograms at a distance δ from the hologram plane. This calculation uses the wellknown Fresnel diffraction propagation integral, given by: 1 Cx; y; δ iλδ
ZZ Hξ; ηRξ; η
2πδ x − ξ2 y − η2 × exp i 1 dξdη; λ 2δ2 (1)
where λ is the recording wavelength, and Hξ; η and Rξ; η are the hologram and reference beams, respectively. Usually, it is assumed that Rξ; η 1, which means that the reference beam is a plane wavefront. Since we are interested in the refocusing capability, only the amplitude of Eq. (1) is considered. After simple mathematical operations, we calculate the amplitude reconstruction as iπ 2 ξ η2 (2) Aδ jCδj FT Hξ; η exp λδ
a pixel size of 6.7 μm × 6.7 μm. One is the hologram of an astronaut puppet, which was acquired in lensless configuration, while the other is the hologram of a biological sample, that is, a cells, recorded in microscope configuration. From the recorded holograms, we extract only the 1 diffraction order by Fourier filtering. We compare the proposed metric with the TC and energy-based methods, properly normalized, calculating their values for different reconstruction distances. The TC metric is defined as [σI∕hIi] where σI and hIi represent the image variance and mean of texture, respectively. In all experiments, the search range of the focus distance is chosen around the value of the nominal distance, measured in the recording step. First, we test the proposed method for the hologram of the astronaut puppet, where the nominal focus distance is 73 cm. We report these results in Fig. 1. In Fig. 1(a) we observe that the three metrics present extrema at distance values δGI 72.6 cm, δTC 72.7 cm, and δEn 72.6 cm, which are in agreement with the nominal focus distance. In fact, in Fig. 1(b) we show the amplitude reconstruction obtained at δGI 72.6 cm, while in Fig. 1(c) we show the amplitude of the hologram Fourier transform. On the other hand, the sparsity
in which FTf·g is the Fourier transform. We omitted, for simplicity, the spatial frequencies dependency and the constant terms. The GI of holographic reconstruction in Eq. (2) is given by: GIfcδg 1 − 2
N X ak δ N − k 0.5 ; N ‖cδ‖1 k1
(3)
where a vecfAg, c vecfCg, in which vecf·g is an operator that creates a column vector from a matrix, ‖ · ‖1 is the l1 norm, and ak for k 1; …; N are the sorted entries of vector a in ascending order. The GI is a quasi-convex function of c and it assumes values in the range [0,1]. In particular, GI 0 is for a vector with all the entries having an equal amount of energy, while GI 1 is for the most sparse vector which has all the energy concentrated in only one element [21]. Our analysis consists of the assessment of the GI for different reconstruction distances, finding its extrema. A special case is represented by the GI value when δ goes to infinity. In fact, in this case, Eq. (2) becomes jC ∞ j lim Aδ jFTfHξ; ηgj: δ→∞
(4)
Equation (4) states that an asymptote for the GI is the value obtained when the amplitude of the hologram Fourier transform is used in Eq. (3). To test the proposed method, two different digital holograms of 1024 × 1024 pixels were recorded at a wavelength of λ 532 nm by using a CCD camera with
Fig. 1. Comparing GI, TC, and energy metrics for the astronaut’s hologram. In (a) we show the values assumed by the three metrics for different values of the reconstruction distance. The TC and the energy metrics are properly normalized to be compared with GI. In (b) and (c) we present the retrieved amplitude reconstruction, calculated at a distance corresponding to the extreme point of the GI, and the amplitude of the Fourier transform, respectively (see the dotted black circles).
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
analysis reveals that in a lensless configuration, the sparsest reconstruction plane, i.e., the maximum value of the GI, is given by the focus plane, which has a sparsity percentage equal to 0.816. Instead, the sparsity in the Fourier basis is 0.722. This means that, in a compressive holography scenario, and for this application, the Fourier basis may not be a good candidate as a representative basis, though it may be considered a good universal estimator and may have been used in compressive sensing in many scenarios. In the microscope configuration scenario, we consider the digital hologram of a mouse’s cell. In this case the nominal focus distance is set to 0, which corresponds to the middle plane of the sample volume. Figure 2 shows the comparison of the three metrics and the amplitude and phase reconstructions. We observe that the GI is in agreement with the TC and energy, recovering the correct focus distance. In this case, the GI is at a minimum in the focus plane; therefore the corresponding sparsity percentage does not provide any useful information. Notice that the GI reaches the minimum because, for phase objects the focus plane is detected when the amplitude reconstruction is quasiconstant, as well known in the literature [3,18]. To study the sparsity properties of holograms recorded in microscope configuration, we calculate the GI in the range of large distances, in order to establish its maximum. This analysis is reported in Fig. 3 for the hologram of a mouse’s cell reported in Fig. 2(b). We observe that the local extremes of the three metrics give an estimation of the back focal plane (BFP) of the microscope objective. This represents an important feature, in fact several applications in DH such as multiplexing [29,30] and
Fig. 2. Comparing GI, TC, and energy metrics for the hologram of a mouse’s cell. In (a) we show the values given by the three metrics for different values of reconstruction distance. In (b) we show the amplitude reconstruction, calculated at a distance corresponding to the extreme point of GI (black circles), while in (c) the corresponding phase reconstruction is reported.
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Fig. 3. Comparing among GI, TC, and energy metrics for the hologram of a mouse’s cell in the case of long reconstruction distances. In (a) we show the values given by the three metrics. In (b) we show the amplitude reconstruction calculated at a distance retrieved by the extreme point of GI (black circles), which in this case corresponds to the BFP of microscope objective, while in (c) we show the Fourier transform amplitude.
compression methods [31] are made by exploiting the property of the BFP. However, the corresponding sparsity percentage is not the maximum value of GI. In fact, in Fig. 3(a), the recovered value in the local extreme of GI is 0.937 while results with higher value tend to be the GI calculated for the Fourier transform, which is equal to 0.974. This result appears to be in contradiction to what is shown in Figs. 3(b) and 3(c), from which the BFP [Fig. 3(b)] is more spatially concentrated with respect to the Fourier spectrum [Fig. 3(c)]. However, this mismatch reveals an interesting property of the BFP. Because the GI is based on a weighted sum of coefficients of the signal representation, its highest value in the Fourier plane reveals that the signal energy is better distributed in relation to the BFP. In other words, there are fewer energy coefficients of hologram representation in the Fourier plane as compared to the BFP representation. This consideration is also demonstrated by observing the trend of energy metric in Fig. 3(c), which presents an energy local minimum in the BFP. This means that, in a compressive holography scenario and for the microscope case, the sparsest representation is achieved in the Fourier plane. In conclusion, we have investigated an automatic focusing approach for a digital hologram that computes the refocus distance of an object of interest exploiting its sparsity properties. We have demonstrated the feasibility of the GI as a refocusing metric in the DH. The ability of the GI to recover the focus distance of digital holograms, recorded in different experimental conditions, is confirmed by a comparison with two other metrics, that
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is, the TC and energy metric. In particular, for the phase object’s holograms, recorded in microscope configuration, we observe that the BFP can be detected. Because of the original definition of the GI as a measure of sparsity, we have investigated the sparsity properties of the Fresnel diffraction propagation integral varying the reconstruction distance. In particular, we observe that, for digital holograms recorded in lensless configuration, the sparsest reconstruction plane corresponds to the infocus plane, while in the microscope case, the sparsest representation is achieved in the Fourier plane. Our results improve the reconstruction recovery performance from compressive holographic samples. We are very grateful to Dr. Andrea Finizio for acquiring the holograms of the mouse’s cell and the astronaut puppet. This work was supported by Progetto Bandiera “La Fabbrica del Futuro” in the framework of the funded project “Plastic lab-on-chips for the optical manipulation of single cells” (PLUS). References 1. M. U. Schnars and W. Juptner, Appl. Opt. 33, 179 (1994). 2. L. Miccio, P. Memmolo, F. Merola, S. Fusco, V. Embrione, A. Paciello, M. Ventre, P. A. Netti, and P. Ferraro, Lab Chip 14, 1129 (2014). 3. P. Memmolo, M. Iannone, M. Ventre, P. A. Netti, A. Finizio, M. Paturzo, and P. Ferraro, Opt. Express 20, 28485 (2012). 4. T. W. Sua, L. Xuea, and A. Ozcan, Proc. Natl. Acad. Sci. 109, 16018 (2012). 5. M. Paturzo, P. Memmolo, A. Finizio, R. Nsnen, T. J. Naughton, and P. Ferraro, Opt. Express 18, 8806 (2010). 6. Y. S. Kim, T. Kim, S. S. Woo, H. Kang, T. C. Poon, and C. Zhou, Opt. Express 21, 8183 (2013). 7. M. Paturzo, A. Finizio, P. Memmolo, R. Puglisi, D. Balduzzi, A. Galli, and P. Ferraro, Lab Chip 12, 3073 (2012). 8. F. Merola, L. Miccio, P. Memmolo, G. Di Caprio, A. Galli, R. Puglisi, D. Balduzzi, G. Coppola, P. A. Netti, and P. Ferraro, Lab Chip 13, 4512 (2013). 9. Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
10. P. Memmolo, C. Distante, M. Paturzo, A. Finizio, P. Ferraro, and B. Javidi, Opt. Lett. 36, 1945 (2011). 11. T. Kim and T. C. Poon, Appl. Opt. 48, H153 (2009). 12. M. Liebling and M. Unser, J. Opt. Soc. Am. A 21, 2424 (2004). 13. C. A. Trujillo and J. Garcia-Sucerquia, Opt. Lett. 39, 2569 (2014). 14. M. F. Toy, J. Kühn, S. Richard, J. Parent, M. Egli, and C. Depeursinge, Opt. Lett. 37, 5094 (2012). 15. P. Gao, B. Yao, R. Rupp, J. Min, R. Guo, B. Ma, J. Zheng, M. Lei, S. Yan, D. Dan, and T. Ye, Opt. Lett. 37, 1172 (2012). 16. P. Gao, B. Yao, J. Min, R. Guo, B. Ma, J. Zheng, M. Lei, S. Yan, D. Dan, and T. Ye, Opt. Lett. 37, 3630 (2012). 17. P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, Appl. Opt. 47, D176 (2008). 18. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, Opt. Express 14, 5895 (2006). 19. A. El Mallahi and F. Dubois, Opt. Express 19, 6684 (2011). 20. W. Li, N. C. Loomis, Q. Hu, and C. S. Davis, J. Opt. Soc. Am. A 24, 3054 (2007). 21. D. Zonoobi, A. A. Kassim, and Y. V. Venkatesh, IEEE J. Sel. Topics Signal Process. 5, 927 (2011). 22. D. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006). 23. E. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 1289 (2006). 24. P. Memmolo, I. Esnaola, A. Finizio, M. Paturzo, P. Ferraro, and A. M. Tulino, Opt. Express 20, 17250 (2012). 25. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, Opt. Express 17, 13040 (2009). 26. Y. Rivenson, A. Stern, and B. Javidi, J. Disp. Technol. 6, 506 (2010). 27. Y. Rivenson, A. Stern, and B. Javidi, Appl. Opt. 52, A423 (2013). 28. Y. Rivenson, A. Stern, J. Rosen, and B. Javidi, eds. MultiDimensional Imaging (Wiley, 2014). 29. M. Paturzo, P. Memmolo, L. Miccio, A. Finizio, P. Ferraro, A. Tulino, and B. Javidi, Opt. Lett. 33, 2629 (2008). 30. M. Paturzo, P. Memmolo, A. Tulino, A. Finizio, and P. Ferraro, Opt. Express 17, 8709 (2009). 31. P. Memmolo, M. Paturzo, A. Pelagotti, A. Finizio, P. Ferraro, and B. Javidi, Opt. Lett. 35, 3883 (2010).
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Refocusing criterion via sparsity measurements in digital holography Pasquale Memmolo,1,2,* Melania Paturzo,2 Bahram Javidi,3 Paolo A. Netti,1 and Pietro Ferraro2 1
Center for Advanced Biomaterials for Health Care@CRIB, Istituto Italiano di tecnologia, Largo Barsanti e Matteucci 53, 80125 Napoli, Italy 2 3
CNR-Istituto Nazionale di Ottica, via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy
ECE Department, University of Connecticut, U-157, Storrs, Connecticut 06269, USA *Corresponding author: [email protected] Received May 29, 2014; accepted June 26, 2014; posted July 9, 2014 (Doc. ID 213105); published August 6, 2014
Several automatic approaches have been proposed in the past to compute the refocus distance in digital holography (DH). However most of them are based on a maximization or minimization of a suitable amplitude image contrast measure, regarded as a function of the reconstruction distance parameter. Here we show that, by using the sparsity measure coefficient regarded as a refocusing criterion in the holographic reconstruction, it is possible to recover the focus plane and, at the same time, establish the degree of sparsity of digital holograms, when samples of the diffraction Fresnel propagation integral are used as a sparse signal representation. We employ a sparsity measurement coefficient known as Gini’s index thus showing for the first time, to the best of our knowledge, its application in DH, as an effective refocusing criterion. Demonstration is provided for different holographic configurations (i.e., lens and lensless apparatus) and for completely different objects (i.e., a thin pure phase microscopic object as an in vitro cell, and macroscopic puppets) preparation. © 2014 Optical Society of America OCIS codes: (090.1995) Digital holography; (100.2960) Image analysis; (100.3010) Image reconstruction techniques. http://dx.doi.org/10.1364/OL.39.004719
One of the most interesting features in digital holography (DH) is the possibility to numerically manage the focusing plane, by propagating a discrete implementation of the diffraction Fresnel propagation integral to perform a digital holographic reconstruction at a chosen distance [1]. This property makes DH an elective method for several applications, such as 3D tracking [2–4], 3D display [5,6], and quantitative imaging [7–9]. On the other hand, an automatic method that is able to retrieve the focus distance of an object of interest is challenging and several approaches have been proposed [10–20], addressing this issue in different experimental conditions. In fact, many strategies are suitable for digital holograms recorded in a lensless configuration [10–12] as well as in the microscopic scenario [13–20], but not all methods can be applied in both cases. In particular, it was established in [18] that for DH in microscopy, an amplitude (phase) object is refocused at a distance that minimizes (maximizes) an energy-based measure, calculated by its amplitude reconstructions. Basically, most autofocusing strategies are based on maximization/minimization of a suitable image contrast measure and each of them provides an accurate estimation of the focal plane of the object of interest. However, as demonstrated in [10], most of them present several local extrema points, causing difficulties in the search for the global extreme. On the other hand, the method proposed in [10], based on the Tamura coefficient (TC), has the property to have only one extreme in the whole focus range. The latter is a fundamental feature that any refocusing criterion should own. Here we propose a new metric called Gini’s index (GI) [21] as a refocusing criterion that is used, to the best of our knowledge, for the first time in DH. In particular, GI is proposed in [21] as a sparsity measure that significantly improves the performance in signal reconstruction from compressive samples [22,23]. However, a sparsity-based refocusing approach 0146-9592/14/164719-04$15.00/0
has been already proposed in [12], in which a sharpness metric, related to the sparsity of the hologram’s expansion in distance-dependent wavelet-like Fresnelet bases, is used. This metric is related to the signal energy, when summing up the Fresnelet coefficients that have the most part of the intensity. Although it represents a good choice as a refocusing criterion, it is not an effective measure of sparsity. In fact, it was established in [21] that an optimal sparsity measure should be a weighted sum of the coefficients of signal representation, i.e., based on the weight of all coefficients in the overall sparsity. Our idea is to use GI in DH for calculating the in-focus distance of holograms and, at the same time, the degree of sparsity in holographic reconstructions at different distances. In fact, it has already been shown that samples of the diffraction Fresnel propagation integral can be used as a sparse representation of digital holograms for denoising purposes [24]. When GI is used as a refocusing criterion, i.e., as a function of refocusing distance, we demonstrate that it is maximized for holograms recorded in lensless configuration, while it is minimized in the case of a microscope scenario. The latter result is related to the amplitude property of phase objects, which present a quasi-constant amplitude reconstruction in their focus plane. We will show that GI presents the same feature of the TC in terms of convexity with respect to the focus distance and, in addition, it furnishes information about the sparsity of the considered holograms. Notice that, only empirical evidence confirms that sparsity metrics lead to effective focus measures. In fact, the value of GI, calculated at the estimated focus distance, measures a percentage of sparsity of the hologram’s reconstruction. The latter could be fundamental in the field of compressive holography [25–28] permitting the discovery of the sparsest reconstruction plane in which the recovery of digital holograms is better. © 2014 Optical Society of America
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The proposed metric is applied in two different experiments: (1) an astronaut puppet, recorded in lensless configuration; and (2) a biological sample in which a digital hologram of mouse cells is recorded in microscope configuration, to demonstrate the results of the proposed approach for all possible configurations in which DH is adopted. Moreover, a comparison with the TC [10] and energy-based [18] methods is performed to establish the accuracy of the proposed one. Object reconstruction is numerically obtained by propagating the corresponding holograms at a distance δ from the hologram plane. This calculation uses the wellknown Fresnel diffraction propagation integral, given by: 1 Cx; y; δ iλδ
ZZ Hξ; ηRξ; η
2πδ x − ξ2 y − η2 × exp i 1 dξdη; λ 2δ2 (1)
where λ is the recording wavelength, and Hξ; η and Rξ; η are the hologram and reference beams, respectively. Usually, it is assumed that Rξ; η 1, which means that the reference beam is a plane wavefront. Since we are interested in the refocusing capability, only the amplitude of Eq. (1) is considered. After simple mathematical operations, we calculate the amplitude reconstruction as iπ 2 ξ η2 (2) Aδ jCδj FT Hξ; η exp λδ
a pixel size of 6.7 μm × 6.7 μm. One is the hologram of an astronaut puppet, which was acquired in lensless configuration, while the other is the hologram of a biological sample, that is, a cells, recorded in microscope configuration. From the recorded holograms, we extract only the 1 diffraction order by Fourier filtering. We compare the proposed metric with the TC and energy-based methods, properly normalized, calculating their values for different reconstruction distances. The TC metric is defined as [σI∕hIi] where σI and hIi represent the image variance and mean of texture, respectively. In all experiments, the search range of the focus distance is chosen around the value of the nominal distance, measured in the recording step. First, we test the proposed method for the hologram of the astronaut puppet, where the nominal focus distance is 73 cm. We report these results in Fig. 1. In Fig. 1(a) we observe that the three metrics present extrema at distance values δGI 72.6 cm, δTC 72.7 cm, and δEn 72.6 cm, which are in agreement with the nominal focus distance. In fact, in Fig. 1(b) we show the amplitude reconstruction obtained at δGI 72.6 cm, while in Fig. 1(c) we show the amplitude of the hologram Fourier transform. On the other hand, the sparsity
in which FTf·g is the Fourier transform. We omitted, for simplicity, the spatial frequencies dependency and the constant terms. The GI of holographic reconstruction in Eq. (2) is given by: GIfcδg 1 − 2
N X ak δ N − k 0.5 ; N ‖cδ‖1 k1
(3)
where a vecfAg, c vecfCg, in which vecf·g is an operator that creates a column vector from a matrix, ‖ · ‖1 is the l1 norm, and ak for k 1; …; N are the sorted entries of vector a in ascending order. The GI is a quasi-convex function of c and it assumes values in the range [0,1]. In particular, GI 0 is for a vector with all the entries having an equal amount of energy, while GI 1 is for the most sparse vector which has all the energy concentrated in only one element [21]. Our analysis consists of the assessment of the GI for different reconstruction distances, finding its extrema. A special case is represented by the GI value when δ goes to infinity. In fact, in this case, Eq. (2) becomes jC ∞ j lim Aδ jFTfHξ; ηgj: δ→∞
(4)
Equation (4) states that an asymptote for the GI is the value obtained when the amplitude of the hologram Fourier transform is used in Eq. (3). To test the proposed method, two different digital holograms of 1024 × 1024 pixels were recorded at a wavelength of λ 532 nm by using a CCD camera with
Fig. 1. Comparing GI, TC, and energy metrics for the astronaut’s hologram. In (a) we show the values assumed by the three metrics for different values of the reconstruction distance. The TC and the energy metrics are properly normalized to be compared with GI. In (b) and (c) we present the retrieved amplitude reconstruction, calculated at a distance corresponding to the extreme point of the GI, and the amplitude of the Fourier transform, respectively (see the dotted black circles).
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analysis reveals that in a lensless configuration, the sparsest reconstruction plane, i.e., the maximum value of the GI, is given by the focus plane, which has a sparsity percentage equal to 0.816. Instead, the sparsity in the Fourier basis is 0.722. This means that, in a compressive holography scenario, and for this application, the Fourier basis may not be a good candidate as a representative basis, though it may be considered a good universal estimator and may have been used in compressive sensing in many scenarios. In the microscope configuration scenario, we consider the digital hologram of a mouse’s cell. In this case the nominal focus distance is set to 0, which corresponds to the middle plane of the sample volume. Figure 2 shows the comparison of the three metrics and the amplitude and phase reconstructions. We observe that the GI is in agreement with the TC and energy, recovering the correct focus distance. In this case, the GI is at a minimum in the focus plane; therefore the corresponding sparsity percentage does not provide any useful information. Notice that the GI reaches the minimum because, for phase objects the focus plane is detected when the amplitude reconstruction is quasiconstant, as well known in the literature [3,18]. To study the sparsity properties of holograms recorded in microscope configuration, we calculate the GI in the range of large distances, in order to establish its maximum. This analysis is reported in Fig. 3 for the hologram of a mouse’s cell reported in Fig. 2(b). We observe that the local extremes of the three metrics give an estimation of the back focal plane (BFP) of the microscope objective. This represents an important feature, in fact several applications in DH such as multiplexing [29,30] and
Fig. 2. Comparing GI, TC, and energy metrics for the hologram of a mouse’s cell. In (a) we show the values given by the three metrics for different values of reconstruction distance. In (b) we show the amplitude reconstruction, calculated at a distance corresponding to the extreme point of GI (black circles), while in (c) the corresponding phase reconstruction is reported.
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Fig. 3. Comparing among GI, TC, and energy metrics for the hologram of a mouse’s cell in the case of long reconstruction distances. In (a) we show the values given by the three metrics. In (b) we show the amplitude reconstruction calculated at a distance retrieved by the extreme point of GI (black circles), which in this case corresponds to the BFP of microscope objective, while in (c) we show the Fourier transform amplitude.
compression methods [31] are made by exploiting the property of the BFP. However, the corresponding sparsity percentage is not the maximum value of GI. In fact, in Fig. 3(a), the recovered value in the local extreme of GI is 0.937 while results with higher value tend to be the GI calculated for the Fourier transform, which is equal to 0.974. This result appears to be in contradiction to what is shown in Figs. 3(b) and 3(c), from which the BFP [Fig. 3(b)] is more spatially concentrated with respect to the Fourier spectrum [Fig. 3(c)]. However, this mismatch reveals an interesting property of the BFP. Because the GI is based on a weighted sum of coefficients of the signal representation, its highest value in the Fourier plane reveals that the signal energy is better distributed in relation to the BFP. In other words, there are fewer energy coefficients of hologram representation in the Fourier plane as compared to the BFP representation. This consideration is also demonstrated by observing the trend of energy metric in Fig. 3(c), which presents an energy local minimum in the BFP. This means that, in a compressive holography scenario and for the microscope case, the sparsest representation is achieved in the Fourier plane. In conclusion, we have investigated an automatic focusing approach for a digital hologram that computes the refocus distance of an object of interest exploiting its sparsity properties. We have demonstrated the feasibility of the GI as a refocusing metric in the DH. The ability of the GI to recover the focus distance of digital holograms, recorded in different experimental conditions, is confirmed by a comparison with two other metrics, that
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