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Quantitative assessment of lateral resolution improvement in digital holography
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Optics Communications 281 (2008) 3454–3460 www.elsevier.com/locate/optcom
Quantitative assessment of lateral resolution improvement in digital holography Freddy Monroy a, Oscar Rincon a, Yaneth Marcela Torres a, Jorge Garcia-Sucerquia b,c,* b
a Department of Physics, Universidad Nacional de Colombia Sede Bogota´, A.A. 3840, Medellin, Colombia Physics School, Universidad Nacional de Colombia Sede Medellı´n, Ciudad Universitaria Bogota, DC, Colombia c COPL, Department of Physics, Engineering Physics and Optics, Laval University, Quebec, Canada G1K 7P4
Received 2 May 2007; received in revised form 4 March 2008; accepted 5 March 2008
Abstract Controllable experimental features such as wavelength, camera’s specifications and reconstruction distance, determine the theoretical limit for lateral resolution in digital holography; however, due to the speckle noise associated to any coherent imaging system it is not possible to reach this theoretical limit. In this paper, a quantitative study the lateral resolution for digital holography under the effect the speckle noise is carried out. It is shown that by reducing the contrast of the speckle noise the resolution capabilities of digital holography are improved; the signal-to-noise ratio of the reconstructed holograms is the metric use to quantitatively assess the reached resolution in the holographic experiments. Ó 2008 Elsevier B.V. All rights reserved. PACS: 42.40.i; 42.30.Kq; 42.40.My Keywords: Digital holography; Lateral resolution; Speckle noise
1. Introduction Numerical reconstruction of digitally recorded holograms, named as digital holography, has become a field of intense investigation; several cornerstone books have been published [1–3] that accomplish the general theory and applications of this field of research and development on optics. Since its beginnings, the number and kind of applications in many areas of science and engineering have been comparable to those of the optical holography. In digital holography, the wet-chemical process associated with the optical holography has been replaced with the power and versatility of numerical calculation of the recon-
*
Corresponding author. Address: COPL, Department of Physics, Engineering Physics and Optics, Laval University, Quebec, Canada G1K 7P4. E-mail address: [email protected] (J. Garcia-Sucerquia). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.03.011
structed optical field providing it with unique features as for instance the phase-contrast images [2]. As it is common to any coherent imaging system, digital holography suffers of the presence of high contrast speckle noise; some direct consequences of this fact are reconstructed images with smaller lateral resolution than theoretically predicted, low signal-to-noise ratio (SNR), and all the nuisance effects associated with it. Since in digital holography, as in other optics-digital system, the dynamic range for results visualization is smaller than in its optical counterpart, the ruining effects of the speckle noise are even worse. This fact has propelled the development of different ways to somehow reduce the speckle noise and partially alleviate their ruining effects [4–16]; the common denominator to all of these approaches is the use of the versatility and power that the digital component has given to holography. The first and immediate way to reduce the speckle noise is digital image processing [4–8]. By application of conventional and non-conventional filtering techniques some
F. Monroy et al. / Optics Communications 281 (2008) 3454–3460
important reduction of the speckle noise over the images of the reconstructed holograms have been achieve. Convolution filters like media and median [4,5] have shown a qualitative improvement of the reconstructed images, while more elaborated approaches that account for the statistics of the speckle phenomenon [6,7] and filtering methods via the Fourier domain [8] have proven greater efficiency and some degree of quantification of their power has been established. All of these approaches have the disadvantage of diminishing the image intensity and resolution by blurring up edges and details. While the speckle noise is originated by the coherent superposition of randomly scattered light on the rough surface of the object, another different approach to speckle noise reduction is to decrease the spatial coherence of light. This idea was originally proposed and applied by the inventors of holography [17,18] and its statistically formalisation was presented a bit later [19]. The general idea that searches for the reduction of the spatial coherence of light, introduces a rotating ground glass in the recording process so that the effective coherence patch is shrunk. The direct result is a speckle spot with controlled size, which allows one to tune its net effect over the reconstructed image. The same idea has been applied to digital holography [9,10] and important improvement has been reach without diminishing considerably neither resolution nor intensity of the reconstructed images. Studies of the influence of the hologram aperture on the generation of the speckle noise in the reconstructed holograms [11–14] have proven that by engineering the hologram aperture is it possible to reduce the speckle noise. The most effective make use of synthetic apertures generated by either fine positioning [11] of recording systems or by the use of multiple cameras [12]; digitally engineering aperture have as well shown a suitable performance [13,14]. Based on the ideas of the statistics of the speckle phenomenon [19] an additional approach to its reduction has been published on the specialised literature [15,16]. In this attempt, the uncorrelated superposition of reconstructed holograms of the same static scene, leads to diminishing the speckle contrast, increasing the image intensity and recovering the spatial resolution. To our best knowledge, the only quantification of the performance of this methodology has been focussed to the evaluation of the of digital holography interferometry [16], but none report has dealt with a quantitative evaluation of the recovering of the lateral spatial resolution, as stated elsewhere by qualitative inspection [4,7,11,12,15]. In this paper, a quantitative study of the influence of the speckle noise on numerically reconstructed holograms is reported. It is verified that the experimentally achievable lateral resolution in digital holography is always smaller than the predicted by the theory, owing to the presence of the speckle noise. It is demonstrated that the size of the grains of speckle equals the theoretically feasible lateral resolution; then the contrast of speckle noise is reduced by
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means of the superposition of multiple uncorrelated images [15] as a way of improving the experimentally attainable lateral resolution. The SNR of reconstructed holograms is used to quantify the reached lateral resolution; its metrics is used to show that the experimentally achievable lateral resolution of the digital holography can be improved by reducing the contrast of the speckle noise. 2. Speckle noise in digital holography Digital recording of holograms with numerical reconstruction is supported on the same foundations as of optical holography; therefore, it can be modelled into two diffraction steps. Firstly, an object located at the plane z = 0 (Fig. 1) is coherently illuminated. The transmitted or reflected object optical field interferes with a reference field at a distance z = d on the so-called hologram plane. The hologram reconstruction constitutes the second step and it is carried out by numerical evaluation of the diffraction process of a coherent optical field (reconstruction wave) illuminating the recorded hologram. In a similar fashion as of optical holography, the reconstruction wave must be similar to the reference field to recover the object optical field [20]. The recorded intensity at the hologram plane I(xh, yh) is the square module of the amplitude superposition of the object and the reference wave. If the complex amplitude of these waves at the hologram plane are represented by O(xh, yh) and R(xh, yh), respectively, I(xh, yh) is given by Iðxh ; y h Þ ¼ jRðxh ; y h Þ þ Oðxh ; y h Þj2 2
2
¼ jRðxh ; y h Þj þ jOðxh ; y h Þj
þ R ðxh ; y h ÞOðxh ; y h Þ þ Rðxh ; y h ÞO ðxh ; y h Þ
ð1Þ
where the subscript h denotes the hologram plane and * stands for the complex conjugate. The first two terms of Eq. (1) represent the intensity of the reference and the object wave, respectively. These do not provide spatial information about the object optical field and represent the so-called zeroth-order diffraction. The last two terms provide the spatial frequency of the
Fig. 1. Coordinate system in a digital holographic process.
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recorded hologram and are responsible for the twin-images [20]. A two-dimensional discrete detector (CCD or CMOS camera) located at the hologram plane records a sampled version of Eq. (1). If we consider a detector with Nx Ny pixels with their centers separated Dxh(Dyh) and pixel size aDxh(aDyh) (0 < a 6 1) along the x(y) coordinate, the sampled version of Eq. (1) is given by xh yh I S ðxh ; y h Þ ¼ Iðxh ; y h Þrect ; N x Dxh N y Dy h xh y h xh yh comb ; ; rect Dxh Dy h aDxh aDy h ð2Þ with the convolution symbol, rect and comb the rect and comb function, respectively [20]. The first rect function represents the window of the whole sensor, i.e. accounts for the finite size of the detector chip. The curl-bracketed functions are the responsible for the hologram sampling; the comb function represents the location of each pixel and the rect function regards to the finite size of each individual sensor element (pixel). The hologram reconstruction is carried out by numerical calculation of the diffraction of Eq. (2) when it is illuminated by the reference wave. In many practical applications the reference wave is a homogeneous plane wave impinging perpendicular to the hologram plane, then it can be represented as a constant field of amplitude E0. The used experimental arrangement was set such that can be correctly described by the Fresnel diffraction theory; the hologram reconstruction can be performed by calculating the Rayleigh–Sommerfeld diffraction formula in the Fresnel–Fraunhofer approximation [20]: iE0 ip 2 2 Eðxi ; y i ; zÞ ¼ exp ðxi þ y i Þ kz kz Z 1 Z 1 ip 2 2 I S ðxh ; y h Þ exp ðxh þ y h Þ kz 1 1 i2p exp ðxh xi þ y h y i Þ dxh dy h kz iE0 ip exp ðx2i þ y 2i Þ ¼ kz kz ip F I S ðxh ; y h Þ exp ðx2h þ y 2h Þ ð3Þ kz with z the propagation distance, k is the wavelength of the reconstruction field, the subscript i stands for the image plane, F{} represents a two-dimensional Fourier transform and E(xi, yi, z) is the complex amplitude of the reconstructed field at the distance z. Eq. (3) states that the reconstructed complex amplitude is the result of multiplying 2 the Fourier transform ip F I S ðxh ; y h Þ exp kz xh þ y 2h times Fresnel’s phase at
ip 2 and times a the reconstruction plane exp kz xi þ y 2i phase factor due to the propagation distance iEkz0 . Both
phase factors outside the Fourier transform are essential in the calculation of phase-contrast images, but can be neglected in the case of amplitude or intensity contrast images [2]. By introducing the sampled intensity IS(xh, yh) (Eq. (2)) into the expression for the reconstructed optical field (Eq. (3)) one obtains iE0 ip Eðxi ; y i ; zÞ ¼ exp ðx2i þ y 2i Þ kz kz ip F Iðxh ; y h Þ exp ðx2h þ y 2h Þ kz xi y N x Dxh N y Dy h sinc N x Dxh ; N y Dy h i kz kz n xi yi 2 Dxh Dy h comb Dxh ; Dy h a Dxh Dy h kz kz o xi y sinc aDxh ; aDy h i ð4Þ kz kz The terms in the first line are similar to
the explained in the ip x2h þ y 2h repreformer paragraph, provided that exp kz sents the Fresnel phase at the hologram plane. The sinc function on the second line blurs-up the reconstructed optical field. Its extension, first minimum of it, is given by kz N x Dxh kz dy ¼ N y Dy h dx ¼
ð5Þ
for the x and y coordinates, respectively, and have been recognized as the speckle size [3,21]. The curl-bracketed functions in Eq. (4) are the result of the sampling functions in Eq. (2). Their effect over the reconstructed amplitude is determined by the value of a as it follows: (i) if a = 1, the pixel size equals the distance between their centers and the zeroes of the sinc function coincides with the pitch of the comb function. In this situation, the curl-bracketed functions reduces to a delta function at the origin and the hologram reconstruction suffers minimum distortion; (ii) when there is a distance between the pixels of the sensor (0 < a < 1) the sinc produces a modulation of the reconstructed amplitude. It has an extension along x coordinate given by kz/Dxh (first minimum of the sinc function), and therefore to guarantee the smallest distortion of the reconstructed image, the latter amount must be greater than the largest lateral dimension of the image [21]; the same analysis applies to the y coordinate with kz/Dyh the constraining amount. The comb function has a pitch given by the extension of the complete reconstructed field along each image plane coordinate. The extension of this function must fit the size of the desired reconstruction dimension, in order the avoid nuisance effects such as aliasing. The practical implementation of hologram reconstruction via the calculation of the diffraction integral (Eq. (3)) relies upon the numerical implementation of it. Eq. (3) can be numerically calculated via its discrete version called discrete Fresnel transform (DFT) [1,3] given by
F. Monroy et al. / Optics Communications 281 (2008) 3454–3460
" !# iE0 m2 n2 exp ipkz Eðm; n; zÞ ¼ þ kz N 2x Dx2h N 2y Dy 2h Ny Nx X X ip 2 2 2 2 Iðk; lÞ exp ðk Dxh þ l Dy h Þ kz k¼1 l¼1 km ln exp i2p þ ð6Þ Nx Ny In the discrete version, the coordinates of sampled hologram (Eq. (2)) have been replaced for integer indices (k, l) and the hologram intensity I(k, l) spreads over an area of Nx Ny pixels. The indices m = 1, . . . , Nx and n = 1, . . . , Ny account the extension of the reconstructed field and the pixel size Dxi Dyi in the image (reconstructed hologram) is related to the camera pixel size Dxh Dyh by kz N x Dxh kz Dy i ¼ N y Dy h Dxi ¼
ð7Þ
Eq. (7) determines the size of the smallest square detail that can be reconstructed, hence it has been regarded as the lateral resolution of the digital holographic process [1–3]. The calculated object field (Eq. (6)) is a complex function of the discrete reconstruction coordinates (m, n) for a particular distance z. Then, it is possible to numeri2 cally evaluate intensity, Iðm; n; zÞ ¼ jEðm; n; zÞj ¼ 2 2 Re½Eðm; n; zÞ þ Im½Eðm; n; zÞ as well as phase
/ðm; n; zÞ ¼ arctan 2
Im½Eðm;n;zÞ Re½Eðm;n;zÞ
of the reconstructed opti-
cal field; here, Re and Im stand for the real and imaginary parts of complex field in that order; and the function arctan2 accounts for the signs of the Im½Eðm; n; zÞ and
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Re½Eðm; n; zÞ in the computation of the inverse tangent function. Eqs. (5) and (7) equal, this means that the speckle size as well as the achievable lateral resolution are determined by the same experimental parameters. The smaller the speckle size the finer the details are possible to reconstruct. In digital holography, the presence of speckle noise ruins the reconstruction quality. It leads to images with low signal-to-noise ratios (SNR) and the theoretical lateral resolution limit dictated by Eq. (7) is not experimentally reachable. In order to decrease the effects of speckle noise in digital holography several procedures have been proposed and demonstrated [4–16]. In this work, the reducing of the contrast of the speckle noise has been adopted as a method to improve the SNR in digital reconstructed holograms. This procedure is based upon the superposition of N decorrelated reconstructed holograms as it is described in detail in ref [15]. By imaging a resolution test targets and by evaluating the SNR of the reconstructed holograms it is verified that the experimentally achievable resolution is always below the limit stated by Eq. (7). Then, it is demonstrated that the actual experimental resolution in digital holography can be improved by reducing the contrast of the speckle noise. 3. Experimental method and results To study the influence of the speckle noise on the resolution of digitally recorded and numerically reconstructed holograms, it has been implemented the experimental setup illustrated on Fig. 2. A 22.5 mW He–Ne laser (k = 632.8 nm) is split into the reference (R) and the object (O) beams. The variable beam (VBS) splitter allows one to set the proper reference-to-object intensity ratio,
Fig. 2. Experimental setup. (VBS) variable beam-splitter (CSP) Collimator and spatial filter, (RGG) rotating ground glass, (M) mirrors, (BSP) beamsplitter cube, (PC) personal computer (O) object beam; (R) reference beam.
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such that maximum contrasted holograms are recorded. Two collimating and spatial filtering systems (CSP) produce clean plane waves for R and O beams. These beams are directed by the beam-splitter cube (BSP) towards a CCD camera HitachiÒ KPM-32. The CCD camera records the intensity generated by the interference of R and O. This intensity is the hologram and its reconstruction is carried by implementing Eq. (6) in a PC. The object beam passes through a rotating ground glass (RGG), which allows for the implementation of the speckle reduction technique [15]. The core of this technique is the superposition of reconstructed holograms of the same static scene with uncorrelated speckle patterns, into one unique resulting image. This superposition produces a reduction of the contrast pffiffiffiof ffi the speckle noise in the final image that follows the 1= m law, with m the number of superimposed images. The reconstructed images with the uncorrelated speckle are obtained from the reconstruction of uncorrelated holograms; these holograms are generated by keeping still the RGG during the recording of each of the m individual holograms while by rotating it between the consecutive registers. In this experiment, 61 decorrelated holograms have been acquired and labelled h1,h2,h3, . . . , h61. Consecutive pairs (h1 h2, h2 h3, . . . , h50 h61) are pixel-wise subtracted to fully eliminate the zeroth-order diffraction and the resulting 60 holograms are individually reconstructed. The superposition of up to 60 reconstructed holograms leads to one unique image with reduced speckle noise, and therefore improved SNR. As object, a matte star test target with smallest (highest) spatial frequency 1.27 (10) lp/mm has been used. The target has been split into regions by means of circles of different radii, the smaller the radius the higher the spatial frequency. The largest radius is 15 mm and the target was set at a distance of 47 cm from the CCD. Theoretically, such a setup, allows one to reconstruct objects with a maximum spatial frequency of 8.7 lp/mm and for the used configuration, the maximum recordable lateral dimension is 35 mm [21].
The improvement the lateral resolution by means of reducing the contrast of the speckle noise is quantify by means of the SNR; it measures the randomness of the quantity of interest about its mean value. In the reconstructed hologram the intensity distribution is the random quantity x and it can be assumed wide sense stationary and consequently its mean equals x. The variance is defined as customary to end up with a definition of SNR given by [22] x ffi SNR ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8Þ P N 2 1 ðx x Þ i i¼1 N PN 1 with x ¼ N i¼1 xi . To determine the value of the SNR of the image of the start test target at different spatial frequencies, donut-like areas are delimited by concentric circles of different radii. The calculation of Eq. (8) is carried out within areas like the one shown in panel A of Fig. 3 between the two white circles; the spatial frequency is given by the radius r of circle halfway between the smallest and the largest circle through 1 the relationship lp=mm ¼ 120 . To test the algorithm, the 2p r SNR of start test target shown in Fig. 3 panel A has been calculated with the results shown in panel B. By contrasting the measurement of the SNR at any given spatial frequency with the corresponding profile of the image for the same spatial frequency, we conclude that a particular spatial frequency is fully resolved if the SNR is equal or greater than 0.8. As it could be expected, the maximum SNR for the start test target is one. The SNR value drops to 0.8 when the spatial frequency of the image is of the order of 11 lp/mm, i.e. this is the maximum number of lp/mm that one can expect to image from this star test target. The maximum (minimum) spatial frequency one can obtain from the image in panel A of Fig. 3 corresponds to circle of radius 1.7 mm (15 mm). Fig. 4 shows the results of superimposing different number of decorrelated reconstructions. For panel A, B and C we have superimposed 2, 30 and 60 images, respectively. To test the achievable lateral resolution, the SNR has been calculated according to the former description. The circle for which the value of the SNR drops to 0.8 has been drawn in
Fig. 3. Calculation of the SNR value over an image of a star test target. Panel A is the image of the star test target with a sample of the area over which the calculation is done. In panel B, it is shown how the SNR changes for different values of spatial frequency for the image shown in panel A.
F. Monroy et al. / Optics Communications 281 (2008) 3454–3460
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Fig. 4. Superposition of hologram reconstructions of a star test target. The number of superimposed holograms is 2, 30 and 60 for panels A, B and C, respectively. The white circle has the radius r at which the SNR drops to 0.8, i.e. it gives the spatial frequency that is fully resolved.
white denoting the maximum spatial frequency of the corresponding image. For panel A this value corresponds to 1.6 lp/mm. As the number of superimposed reconstructed images raises the maximum achievable spatial resolution increases as well, for Panel B it corresponds to 2.6 lp/mm and for panel C to 3.6 lp/mm. To quantitatively study the behaviour of the improvement of the lateral resolution by this method, one can make a plot of the SNR vs. the number of superimposed reconstructed holograms for different spatial frequencies. Fig. 5 shows how this improvement grows up rapidly up to an almost stationary value; the plot is shown for 1.3, 1.6, 1.9, 2.6 and 3.7 lp/mm. Since the improving method of lateral resolution is based upon the reduction p offfiffifficontrast of speckle ffi noise, the ideal curve must follow a m shape [15,19,22], with m the number of superimposed reconstructions. The ideal behaviour is not completely followed due to the
Fig. 5. SNR vs. number of superimposed reconstructed holograms for different spatial frequencies.
slightly correlation existent among the different holograms, a situation that has been already recognized before [22]. According to Eq. (7) and the experimental setup that was used one should be able to reconstruct images with spatial frequencies up to 8.1 lp/mm. From Fig. 5, one can see that images with just 1.3 lp/mm can be resolved if at least two reconstructed images are superimposed. This value corresponds to the 16% of the theoretically expected value. This percentage grows up to a maximum value of approximately 46% (3.7 lp/mm) when 60 reconstructions are superimposed. Besides there is an important improvement of the lateral resolution with the superposition of non-correlated reconstructions, its maximum achievable value still diverges a 54% of the expected value. According to this result, one then must be sure that the experimental setup has resolution capabilities (Eq. (7)), that exceed almost in a 60% the maximum spatial frequency of the object to be imaged with digital holography. As another example of the technique of superimposing decorrelated reconstructions to improve lateral resolution in digital holography, in Fig. 6 are shown the results over reconstructed images of an USAF test target 1951. In Panel A only the element 3 of group 1 (2.52 lp/mm) can be resolved, whereas with the superposition of 30 images (panel B) the element 1 of group 2 (4 lp/mm) is fully resolved. The ultimate resolution is achieved when 60 holograms are superimposed as it shown in panel C; there a frequency of 4.49 lp/mm is fully resolved (element 2 of group 2). According to Eq. (7) the lateral resolution for the experiment from which those images were taken was 14.7 lp/ mm; these results confirm the previous criterion that in order to experimentally achieve a given lateral resolution, the experiment must be setup up to surpass at least in an 60% the expected resolution. Here it is observed the same behaviour of a rapid improvement up to an almost stationary value of the resolution as it is exhibited in Fig. 4. This example emphasizes the fact that with just the superposition of uncorrelated reconstructions of the same holograms one can improve the lateral resolution of the numerically reconstructed images from digitally recorded holograms.
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Fig. 6. The same technique of superimposing uncorrelated reconstructions to improve lateral resolution applied to a USAF test target 1951. Panel A, B and C 2, 30 and 60 holograms superimposed, respectively.
4. Conclusions
References
The recently proposed method of superimposing decorrelated reconstructions of a unique hologram has been successfully applied to study the improvement of the lateral resolution of the numerical reconstruction of digitally recorded holograms. The quantification of this improvement has been done by means of connecting the SNR value with the lateral resolution of the image. A threshold of normalized SNR value of 0.8 or higher has been used to assert that the system can resolve a given spatial frequency. An improvement of the order of 30% of the lateral resolution can be achieved with the superposition of 60 uncorrelated reconstructions. The method based on the reduction of the contrast of the speckle noise of the reconstructed images, was applied to conventional resolution test targets such as the start and the USAF 1951. In the first, the improvement of the lateral resolution is evident for the decrease of the radius of the circle at which a SNR of 0.8 is achieved. In the latter, the direct observation of element that is fully resolved reveals such an improvement. From both results, it is clear that different number of superimposed holograms are required to achieve a given spatial resolution, the higher spatial resolution the larger the number of required superimposed holograms.
[1] U. Schnars, W. Juptner, Meas. Sci. Technol. 13 (2002) R85, and the references therein. [2] U. Schnars, W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, Springer, 2005. [3] L.P. Yaroslavskii, N.S. Merzlyakov, Methods of Digital Holography Translated from Russian by D. Parsons, Consultants Bureau, New York, 1989. [4] S. Pasko, R. Jo´zwicki, Opto-Electron. Rev. 11 (3) (2003) 203. [5] J. Garcia-Sucerquia, J. Herrera, D. Velasquez, Optik 116 (2005) 44. [6] D. Kim, Opt. Lett. 29 (2004) 611. [7] Y. Pu, H. Meng, J. Opt. Soc. Am. A 21 (2004) 1221. [8] J. Maycock, B.M. Hennelly, J.B. McDonald, Y. Frauel, A. Castro, B. Javidi, T.J. Naughton, J. Opt. Soc. Am. A 24 (2007) 1617. [9] L. Repetto, E. Piano, C. Pontiggia, Opt. Lett. 29 (2004) 1132. [10] F. Dubois, M.-L. Novella Requena, Ch. Minetti, O. Monnom, E. Istasse, Appl. Opt. 43 (2004) 1131. [11] J.H. Massig, Opt. Lett. 27 (2002) 179. [12] T.M. Kreis, M. Adams, W. Jueptner, Proc. SPIE 4777 (2002) 69. [13] F. Le Clerc, M. Gross, L. Collot, Opt. Lett. 26 (2001) 1550. [14] Xiao-ou Cai, Hui Wang, Opt. Commun. 281 (2008) 232. [15] J. Garcia-Sucerquia, J. Herrera, R. Castaneda, Opt. Commun. 260 (2006) 62. [16] T. Baumbach, E. Kolenovic, V. Kebbel, W. Ju¨ptner, Appl. Opt. 45 (2006) 6077. [17] D. Gabor, IBM J. Res. Develop. 14 (1970) 509. [18] E.N. Leith, J. Upatnieks, J. Opt. Soc. Am. 54 (1964) 129. [19] J.C. Dainty, Laser Speckle and Related Phenomena, Springer-Verlag, New York, 1975. [20] J.W. Goodman, Introduction to Fourier Optics, second ed., McGraw-Hill, San Francisco, 1996. [21] L. Xu, X. Peng, Z. Guo, J. Miao, A. Asundi, Opt. Express 13 (2005) 2444. [22] B.M. Welsh, J. Opt. Soc. Am. A 12 (1995) 1364.
Acknowledgements The authors acknowledge the remarks made by an anonymous reviewer that contribute to make this paper more readable, complete and precise.
Optics Communications 281 (2008) 3454–3460 www.elsevier.com/locate/optcom
Quantitative assessment of lateral resolution improvement in digital holography Freddy Monroy a, Oscar Rincon a, Yaneth Marcela Torres a, Jorge Garcia-Sucerquia b,c,* b
a Department of Physics, Universidad Nacional de Colombia Sede Bogota´, A.A. 3840, Medellin, Colombia Physics School, Universidad Nacional de Colombia Sede Medellı´n, Ciudad Universitaria Bogota, DC, Colombia c COPL, Department of Physics, Engineering Physics and Optics, Laval University, Quebec, Canada G1K 7P4
Received 2 May 2007; received in revised form 4 March 2008; accepted 5 March 2008
Abstract Controllable experimental features such as wavelength, camera’s specifications and reconstruction distance, determine the theoretical limit for lateral resolution in digital holography; however, due to the speckle noise associated to any coherent imaging system it is not possible to reach this theoretical limit. In this paper, a quantitative study the lateral resolution for digital holography under the effect the speckle noise is carried out. It is shown that by reducing the contrast of the speckle noise the resolution capabilities of digital holography are improved; the signal-to-noise ratio of the reconstructed holograms is the metric use to quantitatively assess the reached resolution in the holographic experiments. Ó 2008 Elsevier B.V. All rights reserved. PACS: 42.40.i; 42.30.Kq; 42.40.My Keywords: Digital holography; Lateral resolution; Speckle noise
1. Introduction Numerical reconstruction of digitally recorded holograms, named as digital holography, has become a field of intense investigation; several cornerstone books have been published [1–3] that accomplish the general theory and applications of this field of research and development on optics. Since its beginnings, the number and kind of applications in many areas of science and engineering have been comparable to those of the optical holography. In digital holography, the wet-chemical process associated with the optical holography has been replaced with the power and versatility of numerical calculation of the recon-
*
Corresponding author. Address: COPL, Department of Physics, Engineering Physics and Optics, Laval University, Quebec, Canada G1K 7P4. E-mail address: [email protected] (J. Garcia-Sucerquia). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.03.011
structed optical field providing it with unique features as for instance the phase-contrast images [2]. As it is common to any coherent imaging system, digital holography suffers of the presence of high contrast speckle noise; some direct consequences of this fact are reconstructed images with smaller lateral resolution than theoretically predicted, low signal-to-noise ratio (SNR), and all the nuisance effects associated with it. Since in digital holography, as in other optics-digital system, the dynamic range for results visualization is smaller than in its optical counterpart, the ruining effects of the speckle noise are even worse. This fact has propelled the development of different ways to somehow reduce the speckle noise and partially alleviate their ruining effects [4–16]; the common denominator to all of these approaches is the use of the versatility and power that the digital component has given to holography. The first and immediate way to reduce the speckle noise is digital image processing [4–8]. By application of conventional and non-conventional filtering techniques some
F. Monroy et al. / Optics Communications 281 (2008) 3454–3460
important reduction of the speckle noise over the images of the reconstructed holograms have been achieve. Convolution filters like media and median [4,5] have shown a qualitative improvement of the reconstructed images, while more elaborated approaches that account for the statistics of the speckle phenomenon [6,7] and filtering methods via the Fourier domain [8] have proven greater efficiency and some degree of quantification of their power has been established. All of these approaches have the disadvantage of diminishing the image intensity and resolution by blurring up edges and details. While the speckle noise is originated by the coherent superposition of randomly scattered light on the rough surface of the object, another different approach to speckle noise reduction is to decrease the spatial coherence of light. This idea was originally proposed and applied by the inventors of holography [17,18] and its statistically formalisation was presented a bit later [19]. The general idea that searches for the reduction of the spatial coherence of light, introduces a rotating ground glass in the recording process so that the effective coherence patch is shrunk. The direct result is a speckle spot with controlled size, which allows one to tune its net effect over the reconstructed image. The same idea has been applied to digital holography [9,10] and important improvement has been reach without diminishing considerably neither resolution nor intensity of the reconstructed images. Studies of the influence of the hologram aperture on the generation of the speckle noise in the reconstructed holograms [11–14] have proven that by engineering the hologram aperture is it possible to reduce the speckle noise. The most effective make use of synthetic apertures generated by either fine positioning [11] of recording systems or by the use of multiple cameras [12]; digitally engineering aperture have as well shown a suitable performance [13,14]. Based on the ideas of the statistics of the speckle phenomenon [19] an additional approach to its reduction has been published on the specialised literature [15,16]. In this attempt, the uncorrelated superposition of reconstructed holograms of the same static scene, leads to diminishing the speckle contrast, increasing the image intensity and recovering the spatial resolution. To our best knowledge, the only quantification of the performance of this methodology has been focussed to the evaluation of the of digital holography interferometry [16], but none report has dealt with a quantitative evaluation of the recovering of the lateral spatial resolution, as stated elsewhere by qualitative inspection [4,7,11,12,15]. In this paper, a quantitative study of the influence of the speckle noise on numerically reconstructed holograms is reported. It is verified that the experimentally achievable lateral resolution in digital holography is always smaller than the predicted by the theory, owing to the presence of the speckle noise. It is demonstrated that the size of the grains of speckle equals the theoretically feasible lateral resolution; then the contrast of speckle noise is reduced by
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means of the superposition of multiple uncorrelated images [15] as a way of improving the experimentally attainable lateral resolution. The SNR of reconstructed holograms is used to quantify the reached lateral resolution; its metrics is used to show that the experimentally achievable lateral resolution of the digital holography can be improved by reducing the contrast of the speckle noise. 2. Speckle noise in digital holography Digital recording of holograms with numerical reconstruction is supported on the same foundations as of optical holography; therefore, it can be modelled into two diffraction steps. Firstly, an object located at the plane z = 0 (Fig. 1) is coherently illuminated. The transmitted or reflected object optical field interferes with a reference field at a distance z = d on the so-called hologram plane. The hologram reconstruction constitutes the second step and it is carried out by numerical evaluation of the diffraction process of a coherent optical field (reconstruction wave) illuminating the recorded hologram. In a similar fashion as of optical holography, the reconstruction wave must be similar to the reference field to recover the object optical field [20]. The recorded intensity at the hologram plane I(xh, yh) is the square module of the amplitude superposition of the object and the reference wave. If the complex amplitude of these waves at the hologram plane are represented by O(xh, yh) and R(xh, yh), respectively, I(xh, yh) is given by Iðxh ; y h Þ ¼ jRðxh ; y h Þ þ Oðxh ; y h Þj2 2
2
¼ jRðxh ; y h Þj þ jOðxh ; y h Þj
þ R ðxh ; y h ÞOðxh ; y h Þ þ Rðxh ; y h ÞO ðxh ; y h Þ
ð1Þ
where the subscript h denotes the hologram plane and * stands for the complex conjugate. The first two terms of Eq. (1) represent the intensity of the reference and the object wave, respectively. These do not provide spatial information about the object optical field and represent the so-called zeroth-order diffraction. The last two terms provide the spatial frequency of the
Fig. 1. Coordinate system in a digital holographic process.
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recorded hologram and are responsible for the twin-images [20]. A two-dimensional discrete detector (CCD or CMOS camera) located at the hologram plane records a sampled version of Eq. (1). If we consider a detector with Nx Ny pixels with their centers separated Dxh(Dyh) and pixel size aDxh(aDyh) (0 < a 6 1) along the x(y) coordinate, the sampled version of Eq. (1) is given by xh yh I S ðxh ; y h Þ ¼ Iðxh ; y h Þrect ; N x Dxh N y Dy h xh y h xh yh comb ; ; rect Dxh Dy h aDxh aDy h ð2Þ with the convolution symbol, rect and comb the rect and comb function, respectively [20]. The first rect function represents the window of the whole sensor, i.e. accounts for the finite size of the detector chip. The curl-bracketed functions are the responsible for the hologram sampling; the comb function represents the location of each pixel and the rect function regards to the finite size of each individual sensor element (pixel). The hologram reconstruction is carried out by numerical calculation of the diffraction of Eq. (2) when it is illuminated by the reference wave. In many practical applications the reference wave is a homogeneous plane wave impinging perpendicular to the hologram plane, then it can be represented as a constant field of amplitude E0. The used experimental arrangement was set such that can be correctly described by the Fresnel diffraction theory; the hologram reconstruction can be performed by calculating the Rayleigh–Sommerfeld diffraction formula in the Fresnel–Fraunhofer approximation [20]: iE0 ip 2 2 Eðxi ; y i ; zÞ ¼ exp ðxi þ y i Þ kz kz Z 1 Z 1 ip 2 2 I S ðxh ; y h Þ exp ðxh þ y h Þ kz 1 1 i2p exp ðxh xi þ y h y i Þ dxh dy h kz iE0 ip exp ðx2i þ y 2i Þ ¼ kz kz ip F I S ðxh ; y h Þ exp ðx2h þ y 2h Þ ð3Þ kz with z the propagation distance, k is the wavelength of the reconstruction field, the subscript i stands for the image plane, F{} represents a two-dimensional Fourier transform and E(xi, yi, z) is the complex amplitude of the reconstructed field at the distance z. Eq. (3) states that the reconstructed complex amplitude is the result of multiplying 2 the Fourier transform ip F I S ðxh ; y h Þ exp kz xh þ y 2h times Fresnel’s phase at
ip 2 and times a the reconstruction plane exp kz xi þ y 2i phase factor due to the propagation distance iEkz0 . Both
phase factors outside the Fourier transform are essential in the calculation of phase-contrast images, but can be neglected in the case of amplitude or intensity contrast images [2]. By introducing the sampled intensity IS(xh, yh) (Eq. (2)) into the expression for the reconstructed optical field (Eq. (3)) one obtains iE0 ip Eðxi ; y i ; zÞ ¼ exp ðx2i þ y 2i Þ kz kz ip F Iðxh ; y h Þ exp ðx2h þ y 2h Þ kz xi y N x Dxh N y Dy h sinc N x Dxh ; N y Dy h i kz kz n xi yi 2 Dxh Dy h comb Dxh ; Dy h a Dxh Dy h kz kz o xi y sinc aDxh ; aDy h i ð4Þ kz kz The terms in the first line are similar to
the explained in the ip x2h þ y 2h repreformer paragraph, provided that exp kz sents the Fresnel phase at the hologram plane. The sinc function on the second line blurs-up the reconstructed optical field. Its extension, first minimum of it, is given by kz N x Dxh kz dy ¼ N y Dy h dx ¼
ð5Þ
for the x and y coordinates, respectively, and have been recognized as the speckle size [3,21]. The curl-bracketed functions in Eq. (4) are the result of the sampling functions in Eq. (2). Their effect over the reconstructed amplitude is determined by the value of a as it follows: (i) if a = 1, the pixel size equals the distance between their centers and the zeroes of the sinc function coincides with the pitch of the comb function. In this situation, the curl-bracketed functions reduces to a delta function at the origin and the hologram reconstruction suffers minimum distortion; (ii) when there is a distance between the pixels of the sensor (0 < a < 1) the sinc produces a modulation of the reconstructed amplitude. It has an extension along x coordinate given by kz/Dxh (first minimum of the sinc function), and therefore to guarantee the smallest distortion of the reconstructed image, the latter amount must be greater than the largest lateral dimension of the image [21]; the same analysis applies to the y coordinate with kz/Dyh the constraining amount. The comb function has a pitch given by the extension of the complete reconstructed field along each image plane coordinate. The extension of this function must fit the size of the desired reconstruction dimension, in order the avoid nuisance effects such as aliasing. The practical implementation of hologram reconstruction via the calculation of the diffraction integral (Eq. (3)) relies upon the numerical implementation of it. Eq. (3) can be numerically calculated via its discrete version called discrete Fresnel transform (DFT) [1,3] given by
F. Monroy et al. / Optics Communications 281 (2008) 3454–3460
" !# iE0 m2 n2 exp ipkz Eðm; n; zÞ ¼ þ kz N 2x Dx2h N 2y Dy 2h Ny Nx X X ip 2 2 2 2 Iðk; lÞ exp ðk Dxh þ l Dy h Þ kz k¼1 l¼1 km ln exp i2p þ ð6Þ Nx Ny In the discrete version, the coordinates of sampled hologram (Eq. (2)) have been replaced for integer indices (k, l) and the hologram intensity I(k, l) spreads over an area of Nx Ny pixels. The indices m = 1, . . . , Nx and n = 1, . . . , Ny account the extension of the reconstructed field and the pixel size Dxi Dyi in the image (reconstructed hologram) is related to the camera pixel size Dxh Dyh by kz N x Dxh kz Dy i ¼ N y Dy h Dxi ¼
ð7Þ
Eq. (7) determines the size of the smallest square detail that can be reconstructed, hence it has been regarded as the lateral resolution of the digital holographic process [1–3]. The calculated object field (Eq. (6)) is a complex function of the discrete reconstruction coordinates (m, n) for a particular distance z. Then, it is possible to numeri2 cally evaluate intensity, Iðm; n; zÞ ¼ jEðm; n; zÞj ¼ 2 2 Re½Eðm; n; zÞ þ Im½Eðm; n; zÞ as well as phase
/ðm; n; zÞ ¼ arctan 2
Im½Eðm;n;zÞ Re½Eðm;n;zÞ
of the reconstructed opti-
cal field; here, Re and Im stand for the real and imaginary parts of complex field in that order; and the function arctan2 accounts for the signs of the Im½Eðm; n; zÞ and
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Re½Eðm; n; zÞ in the computation of the inverse tangent function. Eqs. (5) and (7) equal, this means that the speckle size as well as the achievable lateral resolution are determined by the same experimental parameters. The smaller the speckle size the finer the details are possible to reconstruct. In digital holography, the presence of speckle noise ruins the reconstruction quality. It leads to images with low signal-to-noise ratios (SNR) and the theoretical lateral resolution limit dictated by Eq. (7) is not experimentally reachable. In order to decrease the effects of speckle noise in digital holography several procedures have been proposed and demonstrated [4–16]. In this work, the reducing of the contrast of the speckle noise has been adopted as a method to improve the SNR in digital reconstructed holograms. This procedure is based upon the superposition of N decorrelated reconstructed holograms as it is described in detail in ref [15]. By imaging a resolution test targets and by evaluating the SNR of the reconstructed holograms it is verified that the experimentally achievable resolution is always below the limit stated by Eq. (7). Then, it is demonstrated that the actual experimental resolution in digital holography can be improved by reducing the contrast of the speckle noise. 3. Experimental method and results To study the influence of the speckle noise on the resolution of digitally recorded and numerically reconstructed holograms, it has been implemented the experimental setup illustrated on Fig. 2. A 22.5 mW He–Ne laser (k = 632.8 nm) is split into the reference (R) and the object (O) beams. The variable beam (VBS) splitter allows one to set the proper reference-to-object intensity ratio,
Fig. 2. Experimental setup. (VBS) variable beam-splitter (CSP) Collimator and spatial filter, (RGG) rotating ground glass, (M) mirrors, (BSP) beamsplitter cube, (PC) personal computer (O) object beam; (R) reference beam.
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such that maximum contrasted holograms are recorded. Two collimating and spatial filtering systems (CSP) produce clean plane waves for R and O beams. These beams are directed by the beam-splitter cube (BSP) towards a CCD camera HitachiÒ KPM-32. The CCD camera records the intensity generated by the interference of R and O. This intensity is the hologram and its reconstruction is carried by implementing Eq. (6) in a PC. The object beam passes through a rotating ground glass (RGG), which allows for the implementation of the speckle reduction technique [15]. The core of this technique is the superposition of reconstructed holograms of the same static scene with uncorrelated speckle patterns, into one unique resulting image. This superposition produces a reduction of the contrast pffiffiffiof ffi the speckle noise in the final image that follows the 1= m law, with m the number of superimposed images. The reconstructed images with the uncorrelated speckle are obtained from the reconstruction of uncorrelated holograms; these holograms are generated by keeping still the RGG during the recording of each of the m individual holograms while by rotating it between the consecutive registers. In this experiment, 61 decorrelated holograms have been acquired and labelled h1,h2,h3, . . . , h61. Consecutive pairs (h1 h2, h2 h3, . . . , h50 h61) are pixel-wise subtracted to fully eliminate the zeroth-order diffraction and the resulting 60 holograms are individually reconstructed. The superposition of up to 60 reconstructed holograms leads to one unique image with reduced speckle noise, and therefore improved SNR. As object, a matte star test target with smallest (highest) spatial frequency 1.27 (10) lp/mm has been used. The target has been split into regions by means of circles of different radii, the smaller the radius the higher the spatial frequency. The largest radius is 15 mm and the target was set at a distance of 47 cm from the CCD. Theoretically, such a setup, allows one to reconstruct objects with a maximum spatial frequency of 8.7 lp/mm and for the used configuration, the maximum recordable lateral dimension is 35 mm [21].
The improvement the lateral resolution by means of reducing the contrast of the speckle noise is quantify by means of the SNR; it measures the randomness of the quantity of interest about its mean value. In the reconstructed hologram the intensity distribution is the random quantity x and it can be assumed wide sense stationary and consequently its mean equals x. The variance is defined as customary to end up with a definition of SNR given by [22] x ffi SNR ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8Þ P N 2 1 ðx x Þ i i¼1 N PN 1 with x ¼ N i¼1 xi . To determine the value of the SNR of the image of the start test target at different spatial frequencies, donut-like areas are delimited by concentric circles of different radii. The calculation of Eq. (8) is carried out within areas like the one shown in panel A of Fig. 3 between the two white circles; the spatial frequency is given by the radius r of circle halfway between the smallest and the largest circle through 1 the relationship lp=mm ¼ 120 . To test the algorithm, the 2p r SNR of start test target shown in Fig. 3 panel A has been calculated with the results shown in panel B. By contrasting the measurement of the SNR at any given spatial frequency with the corresponding profile of the image for the same spatial frequency, we conclude that a particular spatial frequency is fully resolved if the SNR is equal or greater than 0.8. As it could be expected, the maximum SNR for the start test target is one. The SNR value drops to 0.8 when the spatial frequency of the image is of the order of 11 lp/mm, i.e. this is the maximum number of lp/mm that one can expect to image from this star test target. The maximum (minimum) spatial frequency one can obtain from the image in panel A of Fig. 3 corresponds to circle of radius 1.7 mm (15 mm). Fig. 4 shows the results of superimposing different number of decorrelated reconstructions. For panel A, B and C we have superimposed 2, 30 and 60 images, respectively. To test the achievable lateral resolution, the SNR has been calculated according to the former description. The circle for which the value of the SNR drops to 0.8 has been drawn in
Fig. 3. Calculation of the SNR value over an image of a star test target. Panel A is the image of the star test target with a sample of the area over which the calculation is done. In panel B, it is shown how the SNR changes for different values of spatial frequency for the image shown in panel A.
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Fig. 4. Superposition of hologram reconstructions of a star test target. The number of superimposed holograms is 2, 30 and 60 for panels A, B and C, respectively. The white circle has the radius r at which the SNR drops to 0.8, i.e. it gives the spatial frequency that is fully resolved.
white denoting the maximum spatial frequency of the corresponding image. For panel A this value corresponds to 1.6 lp/mm. As the number of superimposed reconstructed images raises the maximum achievable spatial resolution increases as well, for Panel B it corresponds to 2.6 lp/mm and for panel C to 3.6 lp/mm. To quantitatively study the behaviour of the improvement of the lateral resolution by this method, one can make a plot of the SNR vs. the number of superimposed reconstructed holograms for different spatial frequencies. Fig. 5 shows how this improvement grows up rapidly up to an almost stationary value; the plot is shown for 1.3, 1.6, 1.9, 2.6 and 3.7 lp/mm. Since the improving method of lateral resolution is based upon the reduction p offfiffifficontrast of speckle ffi noise, the ideal curve must follow a m shape [15,19,22], with m the number of superimposed reconstructions. The ideal behaviour is not completely followed due to the
Fig. 5. SNR vs. number of superimposed reconstructed holograms for different spatial frequencies.
slightly correlation existent among the different holograms, a situation that has been already recognized before [22]. According to Eq. (7) and the experimental setup that was used one should be able to reconstruct images with spatial frequencies up to 8.1 lp/mm. From Fig. 5, one can see that images with just 1.3 lp/mm can be resolved if at least two reconstructed images are superimposed. This value corresponds to the 16% of the theoretically expected value. This percentage grows up to a maximum value of approximately 46% (3.7 lp/mm) when 60 reconstructions are superimposed. Besides there is an important improvement of the lateral resolution with the superposition of non-correlated reconstructions, its maximum achievable value still diverges a 54% of the expected value. According to this result, one then must be sure that the experimental setup has resolution capabilities (Eq. (7)), that exceed almost in a 60% the maximum spatial frequency of the object to be imaged with digital holography. As another example of the technique of superimposing decorrelated reconstructions to improve lateral resolution in digital holography, in Fig. 6 are shown the results over reconstructed images of an USAF test target 1951. In Panel A only the element 3 of group 1 (2.52 lp/mm) can be resolved, whereas with the superposition of 30 images (panel B) the element 1 of group 2 (4 lp/mm) is fully resolved. The ultimate resolution is achieved when 60 holograms are superimposed as it shown in panel C; there a frequency of 4.49 lp/mm is fully resolved (element 2 of group 2). According to Eq. (7) the lateral resolution for the experiment from which those images were taken was 14.7 lp/ mm; these results confirm the previous criterion that in order to experimentally achieve a given lateral resolution, the experiment must be setup up to surpass at least in an 60% the expected resolution. Here it is observed the same behaviour of a rapid improvement up to an almost stationary value of the resolution as it is exhibited in Fig. 4. This example emphasizes the fact that with just the superposition of uncorrelated reconstructions of the same holograms one can improve the lateral resolution of the numerically reconstructed images from digitally recorded holograms.
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Fig. 6. The same technique of superimposing uncorrelated reconstructions to improve lateral resolution applied to a USAF test target 1951. Panel A, B and C 2, 30 and 60 holograms superimposed, respectively.
4. Conclusions
References
The recently proposed method of superimposing decorrelated reconstructions of a unique hologram has been successfully applied to study the improvement of the lateral resolution of the numerical reconstruction of digitally recorded holograms. The quantification of this improvement has been done by means of connecting the SNR value with the lateral resolution of the image. A threshold of normalized SNR value of 0.8 or higher has been used to assert that the system can resolve a given spatial frequency. An improvement of the order of 30% of the lateral resolution can be achieved with the superposition of 60 uncorrelated reconstructions. The method based on the reduction of the contrast of the speckle noise of the reconstructed images, was applied to conventional resolution test targets such as the start and the USAF 1951. In the first, the improvement of the lateral resolution is evident for the decrease of the radius of the circle at which a SNR of 0.8 is achieved. In the latter, the direct observation of element that is fully resolved reveals such an improvement. From both results, it is clear that different number of superimposed holograms are required to achieve a given spatial resolution, the higher spatial resolution the larger the number of required superimposed holograms.
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Acknowledgements The authors acknowledge the remarks made by an anonymous reviewer that contribute to make this paper more readable, complete and precise.