Random resampling masks: a non-Bayesian one ... - OSA Publishing

March 1, 2013 / Vol. 38, No. 5 / OPTICS LETTERS

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Random resampling masks: a non-Bayesian one-shot strategy for noise reduction in digital holography V. Bianco,1,* M. Paturzo,1 P. Memmolo,1,2 A. Finizio,1 P. Ferraro,1 and B. Javidi3 1

2

CNR—National Institute of Optics, Via Campi Flegrei, 34, Pozzuoli (NA) I-80078, Italy Center for Advanced Biomaterials for Health Care, Istituto Italiano di Tecnologia, L. Barsanti e Matteucci (NA), 53, I-80125, Italy 3

ECE Department, University of Connecticut, U-157, Storrs, Connecticut 06269, USA *Corresponding author: [email protected] Received December 6, 2012; revised January 23, 2013; accepted January 23, 2013; posted January 25, 2013 (Doc. ID 181214); published February 21, 2013

Holographic imaging may become severely degraded by a mixture of speckle and incoherent additive noise. Bayesian approaches reduce the incoherent noise, but prior information is needed on the noise statistics. With no prior knowledge, one-shot reduction of noise is a highly desirable goal, as the recording process is simplified and made faster. Indeed, neither multiple acquisitions nor a complex setup are needed. So far, this result has been achieved at the cost of a deterministic resolution loss. Here we propose a fast non-Bayesian denoising method that avoids this trade-off by means of a numerical synthesis of a moving diffuser. In this way, only one single hologram is required as multiple uncorrelated reconstructions are provided by random complementary resampling masks. Experiments show a significant incoherent noise reduction, close to the theoretical improvement bound, resulting in imagecontrast improvement. At the same time, we preserve the resolution of the unprocessed image. © 2013 Optical Society of America OCIS codes: 090.1995, 100.2980, 110.4280.

Due to the coherent nature of the emitting source, holographic imaging may be severely degraded by the presence of both multiplicative speckle and incoherent additive noise [1,2]. Their mixture is hard to model statistically. In addition, speckle degradation depends on the holographic recording system (i.e., unscattered light or scattered light reaching the detector). In digital holography (DH), Bayesian approaches have been intensively investigated to reduce the incoherent noise, but prior information is needed on the noise statistics [1]. With no prior knowledge, one-shot reduction of noise is very useful, and it can be achieved by spatial domain filtering at the cost of a deterministic resolution loss [3,4]. Alternately, an effective non-Bayesian way to reduce noise while keeping the image resolution unaltered is proposed in [5]. However, this is aimed to estimate and subtract the noise content, and its performance gets worse in the case of low signal-to-noise ratio (SNR). Another strategy consists of combining incoherently multiple holographic reconstructions of the same object conveying uncorrelated noise patterns [6–10]. However, in this case a complex ad hoc setup is required to provide some kind of decorrelation between the acquisitions, which in turn limits the number of available uncorrelated images. In particular, early works appeared in the 1970s where the authors employed a moving aperture to provide a random spatial sampling on the hologram plane [11]. The idea was very interesting, but the method was not simple and its effectiveness was limited by constraints due to the mask implementation. Recently, a simple one-shot method has been proposed [12] that employs multiple different masks in the Fourier plane and can be interpreted as a numerical synthesis of the moving aperture in [11]. However, this is only suited for filtering in-line holograms, whose frequency distributions are not sparse, as happens for off-axis holograms. Afterward, subholograms have been derived from one single off-axis acquisition, and they are reconstructed and averaged [13]. However, in both cited cases, an increase in the number 0146-9592/13/050619-03$15.00/0

of images is accompanied by resolution loss, as the numerical aperture of the imaging system decreases. In this Letter, we propose a fast non-Bayesian denoising technique that breaks the trade-off existing between resolution preservation and setup complexity. Our purpose is to improve the image quality by reducing the incoherent noise contribution instead of the speckle, and by directly extrapolating the useful signal. Multiple uncorrelated reconstructions are obtained by a stack of random resampling masks, which are applied to one single acquired hologram. Hence, their incoherent combination returns an improved image where the incoherent noise is significantly reduced. As a consequence, the image contrast is restored as well. The method is robust and can be applied to various kinds of holograms, as it is independent of prior noise statistics and performs well also in case of low SNR. Since it is a one-shot technique, the acquisition process is fast and simple, as no data capture diversity is required. It is noteworthy that, in contrast to [3,4,11–13], the incoherent noise is reduced while preserving the resolution of the unprocessed image. As we shall demonstrate in our experiments, the proposed method has been tested on a noisy off-axis hologram and the improvement has been measured in terms of both noise reduction and contrast enhancement. Let H
u; v be the hologram signal recorded in the plane
u; v, which is corrupted by a mixture of speckle and incoherent noise. After dropping the zeroth diffraction order, we obtain the single-look (SL) amplitude image as I SL
x; y  jFrH
u; vj;

(1)

where Fr  denotes the Fresnel transform and
x; y is the image reconstruction plane. Taking inspiration from the setup described in [11], we implemented a numerical synthesis of a random diffuser to perform N different resamplings of H
u; v, whose combination (see Fig. 1 and Media 1) results in the multi-look (ML) image © 2013 Optical Society of America

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OPTICS LETTERS / Vol. 38, No. 5 / March 1, 2013

Fig. 1. (Media 1) Block scheme of the proposed denoising method.

I ML
x; y 

N 1X jFrH
u; v⊙M n
u; vj; N n1

(2) Fig. 2. (Color online) Reconstructed amplitude images of a toy car. (a) SL image. (b) ML image processed with FCM. (c) (Media 2) ML image processed with ROMs. (d) Details of the edges corresponding to the red boxes in (a) for various reconstructions.

where
M n
u; v 

0 if
u; v ∈ S n 1 otherwise

(3)

is a binary resampling mask ∀n  1; …; N, ⊙ denotes the Hadamard product, and S n are 2D domains jointly covering the whole image area. In contrast to [12,13], the masks are built in order to avoid reduction of the numerical aperture when N increases. For each obscured pixel, the missing information is recovered from the homologous samples in the stack. Once the ML image in Eq. (2) is obtained, we measured the noise reduction by means of the dispersion index D and the relative deviation R
x; y, respectively, defined as D

R
x; y 

σ ML μSL ; μML σ SL

(4)

I 2
x; y − μI 2 ; μI 2

(5)

where σ and μ, respectively denote the standard deviation and the mean value of the gray levels calculated over a homogeneous segment of the image. In our first experiment, we applied the proposed processing to an off-axis hologram of a toy car, choosing the 2D domains in order to respect the following constraint: N X n1

M n 
N − 1O;

of Fig. 2(b), the incoherent noise and, in turn, the gloss effect is removed, thus restoring the speckle contrast typical in digital holograms. An analysis has been carried out in order to find the best stack of random binary masks satisfying the fairness constraint (6). Figure 3(a) shows the improvement in terms of dispersion index obtainable when N increases from one, achieved with masks at different ratios, A, between the dark and bright areas. Due to the constraint, once A is set, the number of available masks N max is set as well. We found that the best fair configuration corresponds to A  10%, reaching an improvement of 63% with only 10 reconstructions. It is noteworthy to point out that this choice allows us to improve the image quality with a very fast processing. However, the slope of the curves increases with A, suggesting that a further improvement can be reached by removing the constraint (6). A different stack of random overlapping masks (ROMs) (Media 1) has been tested, and their performance is shown in Fig. 3(b). As expected, when A increases, a higher noise reduction can be reached, approaching the 90% saturation value when A  50%. Figure 4 shows the dispersion index D versus N for both tested stacks. As expected, when N increases, a reduction of D occurs as a result of the incoherent noise mitigation. In Fig. 4 we also plotted the boundary curve

(6)

where O is the matrix whose elements are all equal to 1 (Media 1). That is, we imposed that during the complementary sampling process each hologram pixel is obscured just once. This is a fairness criterion, herein named fair constraint masking (FCM), which assures the independence of the technique on the hologram being processed. Figure 2(a) shows the SL image, where the degradation introduced by the mixture of speckle and incoherent noise is apparent. In particular, in the SL case the incoherent noise acts as a sort of thin gloss superimposing on the entire image. In contrast, in the ML image

Fig. 3. (Color online) ML improvement in noise reduction, measured by the dispersion index of Eq. (4). (a) FCM: improvement [%] versus N. Different curves are obtained by varying the ratio A. (b) ROM: improvement [%] versus A [%].

March 1, 2013 / Vol. 38, No. 5 / OPTICS LETTERS

Fig. 4. (Color online) Dispersion index versus N. A Comparison between FCM and ROM is shown. Dotted curve: theoretical improvement bound.

Fig. 5. (Color online) Relative deviation [Eq. (5)] in a homogeneous area of reconstructed image. (a) SL image. (b) ML image. (c) Normalized Laplacian versus A. Blue (triangles), ML-FCM. Red (circles), ML-ROM.

p 1∕ N , which should be expected when superimposing N uncorrelated images. With both configurations, the trends approach the theoretical improvement bound, with a better behavior obtained with the ROMs, at the cost of an increase in computational time. In Fig. 5, we show R
x; y for the SL and the ML-ROM case. As expected, since the measure is performed over a homogeneous segment of the reconstructed image, noise reduction results in a smoother trend of this function. The ML gain achievable with the ROM stack is also apparent in the reconstructed image of the full object in Fig. 2(c) (Media 2) and in Fig. 2(d), where the details of the toy car corresponding to the red boxes of Fig. 2(a) are displayed. As a consequence of the incoherent noise reduction, the image contrast on the edges significantly improves. Moreover, we extract from the mixture the speckle distribution, which is object dependent. As a contrast metric, we measured the image Laplacian, defined as in [14], whose behavior is plotted in Fig. 5(c), normalized to the Laplacian of the SL image. We found that A ∈ f10; 50g% assures a contrast enhancement, or at least maintains this parameter while reducing the noise.

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Fig. 6. (Color online) Normalized image amplitudes plotted at a fixed row along the columns of the test area indicated by the red dashed box in Fig. 2(b). The SL and ML images show the same trends.

The improvement in image quality is achieved preserving the resolution of the unprocessed SL image. This observation is also supported by the results presented in Fig. 6, where the normalized image amplitudes are plotted at a fixed row along the columns of a test area (red dashed box in Fig. 2(b)). We selected the toy car radiator as a test area, so that a wider dynamic corresponds to a better resolution capability. It is remarkable that no significant differences are recognizable between SL and the two ML trends. In conclusion, we proposed a robust method for rapidly reducing the incoherent noise in DH. It uses resampling binary masks to obtain multiple uncorrelated reconstructions from one single acquisition. Since it is a non-Bayesian method, it is independent of prior knowledge of the noise statistic. Experiments show an improvement close to the theoretical bound. Importantly, this is achieved while preserving the image resolution. References 1. S. Sotthivirat and J. A. Fessler, J. Opt. Soc. Am. A 21, 737 (2004). 2. A. Uzan, Y. Rivenson, and A. Stern, Appl. Opt. 52, A195 (2013). 3. J. G. Sucerquia, J. A. H. Ramirez, and D. V. Prieto, Optik 116, 44 (2005). 4. N. Bertaux, Y. Frauel, P. Réfrégier, and B. Javidi, J. Opt. Soc. Am. A 21, 2283 (2004). 5. P. Memmolo, I. Esnaola, A. Finizio, M. Paturzo, P. Ferraro, and A. M. Tulino, Opt. Express 20, 17250 (2012). 6. B. Javidi, P. Ferraro, S. Hong, and D. Alfieri, Opt. Lett. 30, 144 (2005). 7. L. Rong, W. Xiao, F. Pan, S. Liu, and R. Li, Chin. Opt. Lett. 8, 653 (2010). 8. S. Kubota and J. W. Goodman, Appl. Opt. 49, 4385 (2010). 9. J. I. Trisnadi, Proc. SPIE 4657, 131 (2002). 10. Y. Kuratomi, K. Sekiya, H. Satoh, T. Tomiyama, T. Kawakami, B. Katagiri, Y. Suzuki, and T. Uchida, J. Opt. Soc. Am. A 27, 1812 (2010). 11. F. T. S. Yu and E. Y. Wang, Appl. Opt. 12, 1656 (1973). 12. J. Maycock, B. M. Hennelly, J. B. Mc Donald, Y. Frauel, A. Castro, B. Javidi, and T. J. Naughton, J. Opt. Soc. Am. A 24, 1617 (2007). 13. M. Abollasshani and Y. Rostami, Optik 123, 937 (2012). 14. Y. S. Choi and S. J. Lee, Appl. Opt. 48, 2983 (2009).