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In this paper we propose a robust method to suppress the noise components in digital holography (DH), called SPADEDH (SPArsity DEnoising of Digital Holograms), that does not consider any prior knowledge or estimation about the statistics of the noise. In the full digital holographic process we must mainly deal with two kinds of noise. The first one is an additive uncorrelated noise that corrupts the observed irradiance, the other one is the multiplicative phase noise called speckle noise. We consider both lensless and microscope configurations and we prove that the proposed algorithm works efficiently in all considered cases suppressing the aforementioned noise components. In addition, for digital holograms recorded in lensless configuration, we show the improvement in a display test by using a Spatial Light Modulator (SLM).
© 2012 Optical Society of America
1. Introduction
In DH [1] the amplitude information of the object wavefront can be obtained quantitatively and analyzed digitally without the need for wet chemical processing. Due to this fact, many spectacular applications, such as microscopic imaging and phase-contrast digital holographic microscopy [2], 3D object recognition [3] and 3D dynamic display [4] have been demonstrated. In each case, the quality of the processing outputs is strongly dependent on the noise of the holograms. To obtain full benefits using the imaging techniques listed above for phase reconstruction, object recognition and 3D display, a pre-processing on the digital holograms is necessary in order to optimize the signal to noise ratio. To this purpose, several techniques that suppress/reduce noise components in DH are known in the literature. In off-axis DH the recorded intensity is normally corrupted by a mixture of speckle noise and additive noise. Classic filter-based methods, as Fourier domain denoising [5] and wavelet domain denoising [6], have extensively been studied, but they can be applied only to some transform domains. Moreover, such denoising methods are greatly influenced by the change of signal parameters like frequency, amplitude, etc. Great effort has been spent on removing principally the speckle noise in speckle interferometry [7] and DH [8–10]. In particular, the approaches described in [9,10] can be applied as a denoising methods without the knowledge of noise statistics, but introducing others information about the signal, i.e. different recording of the same object. In order to effectively overcome the above-mentioned shortcomings of these denoising methods, it is relevant to design reconstruction algorithms which are robust even in the presence of moderate or high-power noise level. To this end, we propose a denoising technique, called SPADEDH, that permits to obtain a robust denoising, for both additive noise and multiplicative noise on off-axis digital holograms in the hypothesis of high signal to noise ratio regime. Trying to solve a denoising problem implies that given the context of the noisy data, and defining a distortion measure (e.g., mean square error (MSE), or hamming distance), the objective is to find the reconstruction algorithm that achieves the minimum distortion measure at the receiver, assuming that noise statistics are not known at the receiver. This problem does no admit as optimal solution neither the Lasso estimator, formalized in [11], or other greedy algorithms, like Stagewise Orthogonal Matching Pursuit (StOMP) algorithm [12]. Both algorithms are an ad hoc solution for which a rigorous analysis capable of quantifying how suboptimal they are with respect to the optimal solution it is not available. These approaches are used in denoising scenarios, mainly due to their optimality in noiseless compressed sensing (CS) scenarios [13–15] and the well-known Lipschitz continuity [16], which makes the reconstruction algorithms designed for the noiseless measurements robust with respect to noise, a significant advantage when compared to strategies which incorporate knowledge of noise statistics [17]. Having said that, for speckle noise, statistics can be assumed to be known and various statistical approaches can be applied [18]. These considerations make the two aforementioned algorithms extremely robust with respect to imperfect knowledge of the noise statistics. Additionally, the proposed technique provides the possibility of reducing the number of measurements needed to reconstruct the denoised hologram, which is specially advantageous since the measurement time can be reduced. As a result so that, objects which are difficult to keep in fixed positions can be measured within higher accuracy. In summary, our approach can be defined as a l1 minimization algorithm, solved by StOMP, which is able to suppress the noise components on digital holograms without any prior knowledge or estimation about the statistics of noise. We have tested the SPADEDH method to various types of digital holograms, using different objects, from microscopic to macroscopic size, recorded in different conditions and with different wavelengths (either in visible as well as in IR spectrum [19]), in order to show the robustness of the proposed method, with respect the variation of experimental parameters.
2. Noise model
In off-axis DH the noiseless hologram H is formed in the interference plane:
where O is the diffracted field produced by the object and R is the reference. With a simple mathematical manipulation, we can rewrite the Eq. (1) as follow: where The quantity q is called zero-order diffraction term, while the contribution in Eq. (2) contains both +1 and −1 diffraction orders. The hologram H is recorded by a CCD and it is usually corrupted by a mixture of speckle noise and additive noise described by Eq. (2): where ns and na are speckle noise, modeled by a multiplicative uniform noise, and an additive zero mean Gaussian noise respectively. The numerical process produces a digital version of the hologram that is reconstructed using a numerical version of the diffraction integral [20]. In order to suppress both noise components, we consider a zero-order term suppression using a high-pass convolution kernel [20]. Therefore, the hologram that we will consider as input of the SPADEDH algorithm is3. SPADEDH algorithm
CS techniques can guarantee good performance in terms of signal reconstruction error for moderate or low-power noise scenarios, thanks to the Lipschitz continuity of the l0 and l1 norms. Therefore, the l0 and l1-minimization techniques as well as all suitable greedy algorithms typical of the CS literature, can potentially provide a practical solution in all those applications, like medical imaging [21], sub-Nyquist sampling systems [22], compressive sensor networks [23], compressive imaging architectures [24], images analysis [21,25], and DH [26–30] where the number of measurements need to be small. In our setting, we focus on the reconstruction of a digital holograms from its noisy measurements. It is well known that if we could assign to our signal of interest a statistical characterization, the conditional estimator is the optimal denoising algorithm if a MSE distortion measure is considered, while a Maximum A Posteriori probability (MAP) estimator or the Discrete Universal DEnoising (DUDE) [31] algorithm are optimal solutions when different distortion measures are considered. However, each of these approaches explores some statistical priors of the signal which make the signal reconstruction algorithm not robust with respect to a large class of digital holograms, and in most practical scenarios such priors are not available. A more robust approach is to model the digital hologram as a non-random sparse signal and resort to a CS scheme which only explores as prior knowledge the sparsity of the signal. In fact, given measurements y and the knowledge that the original signal x is sparse or compressible on the Fourier basis, it is natural to attempt to recover x by solving an l0-minimization problem. Since the l0-minimization is a not convex objective function, we consider a relaxation of the optimization problem using the l1 norm. When the measurements are not corrupted by noise, the optimization can be written as
while the noisy case can be written as with v = Ψx, where Ψ is the n × n basis matrix that transform the original data x in a sparse signal, Φ is m × n matrix generated by randomly sampling the columns, with m < n, and ε is related to the noise level. In order to obtain a good reconstruction from noisy measurements, parameter ε needs to be optimized based on the noisy statistics (typically second order statistics). Examples of these and similar types of reconstruction algorithms where some knowledge of the noise statistics are required, are presented in [25–27]. However, in most practical scenarios, the statistic characterization of the noise is not available and a typical way to obtain such knowledge is to estimate the statistics through a preliminary processing based on several noisy measurements of the signal. Furthermore, once the noise statistics are obtained, the optimization of the parameter ε in Eq. (7) is by itself quite complex [32]. The SPADEDH algorithm overcomes this limitation solving the denoising problem for digital holograms without any prior knowledge or estimation about the statistics of noise. Specifically, we propose as reconstruction algorithm the following optimization problem: where h̃ = vec(H̃) is the recorded digital hologram, in which vec(·) is an operator that creates a column vector from a matrix and the basis matrix is Ψ = Fδ, where Fδ is the numerical reconstruction of the digital hologram at distance δ through the discrete version of the Fresnel-Kirchhoff integral. The denoised hologram is obviously computed as hden = F−δ vden, i.e. through the back propagation of the solution of the denoising problem in Eq. (8). Note that the choice of Fδ as a basis matrix is related to the fact that the numerical reconstruction of ideal noiseless hologram is a sparse signal. In fact, it is composed by the three diffraction terms without overlapping and they occupy, in general, about half of the pixels available. As shown in Eq. (8), in order to decouple the impact over the distortion by the noise from the one produced by the compression rate, we will just analyze the reconstruction algorithm under the assumption that Φ = In, where In is the identity matrix.4. Numerical results
In our analysis we consider different types of digital holograms in order to show the robustness of the proposed method: speckle holograms of two statuettes acquired with two different laser sources, an infrared laser at 10.6 μm and a visible laser at 532 nm, respectively, and the hologram of a Micro Electro Mechanical System (MEMS). Basically two kinds of experimental set-ups were used to acquire these digital holograms: a lensless set-up to record the speckle holograms and a microscope configuration one to acquire MEMS hologram in reflection mode, that is necessary to acquire holograms of opaque objects as the MEMS. The lensless set-up is described in detail in [34], while the microscope configuration in reflection mode, is shown in [35]. We solve the optimization problem in Eq. (8) for the test cases using StOMP, which is a greedy algorithm similar to Orthogonal Matching Pursuit, but faster in the sense that it requires less iterations in the recovery process. Note that the results from this section were obtained minimizing the ℓ1-norm of the estimated signal via StOMP, as opposed to traditional denoising approaches in which the total variation of the signal estimate is minimized [36]. Since we are considering the case for which Φ = In, a low complex recovery is desirable, due to the fact that each greedy step is computationally more demanding than in the traditional CS setting for which m ≪ n. In addition to that, for n large, StOMP has a performance comparable to that achievable by ℓ1 minimization based in linear programming. This condition holds in our setting since the dimension of h̃ is large in general for images (in our case n = 1024 × 1024 for the speckle hologram recorded at 532 nm and the MEMS hologram, while n = 640 × 480 for the speckle hologram recorded at 10.6 μm), and further increases when we move to higher resolutions.
For lensless holograms, in the Fresnel-Kirchhoff integral Fδ, we use the in-focus distance, while for the hologram recorded in microscope configuration we use the Back Focal Plane (BFP) distance because, in this case, we have the biggest degree of sparsity in that plane. In particular, for the hologram acquired at 10.6 μm, as it is in-focus in the Fourier plane, we use δ = ∞. The results of the SPADEDH algorithm are shown in Fig. 1. In all cases, the final holograms are obtained by the back propagation, of the denoised complex field vden, in the hologram plane. In order to evaluate the results, for lensless holograms, we perform also a display test, by using a SLM as a (potentially 3D) projection system (see Fig. 2), while, for the hologram recorded in microscopic configuration, we compute the amplitude in-focus reconstruction. We compare SPADEDH algorithm with the Discrete Fourier Filter (DFF) [5]. This method consists into numerically compute the propagation from the discrete hologram plane to the discrete Fourier planed. Then, the Fourier plane data are filtered and inverse discrete Fourier transformed to the image plane. This is repeated N times and the resulting N intensities are summed. In our implementation of this method, we set N = 100. Note that, this method is based on the perfect knowledge of the hologram bandwidth. In order to quantify the efficiency of the SPADEDH method, in terms of how much noise we are able to remove from the original reconstruction, we define two different parameters: the signal to distortion ratio (SDR) and a measure of contrast C, given in [37]
where, in the SDR equation, || · ||2 is the l2 norm, Ĩ = |Fδh̃| is the amplitude of the original infocus digital reconstruction and Î = |Fdhden| is the processed one and, in Eq. (10), σ(I) and 〈I〉 represent the image gray-level standard deviation and mean, respectively, and I is the amplitude of the in-focus digital reconstruction of original hologram in one case and in the other case is the amplitude of the processed one. In Table 1 and Table 2 we report the computation of the two parameters of efficiency for the cases under analysis.
Fig. 1 In-focus digital reconstructions of lensless test cases (a) and (c), and digital reconstruction in the BFP and in the in-focus plane (red boxes) of the MEMS hologram (e). (b), (d) and (f) are the denoised reconstructions using SPADEDH method.
Fig. 2 Projections of lensless holograms using SLM. (a) and (c) are the projections of the original holograms, while (b) and (d) are the projections of the denoised holograms obtained by the SPADEDH algorithm.
Table 1. Calculation of SDR
Table 2. Calculation of contrast C
These results show that the SPADEDH algorithm provides gains both in terms of SDR and image contrast C and this shows its effective efficiency and robustness with respect the variation of experimental parameters. Finally, in order to evaluate the improvement of projections in the display test, we compute the of intensity into signal regions (SR) percentage increase, shown in the Fig. 2
For the statuette of astronaut we have G ≈ 31%, while for the statuette of Venus results G ≈ 16%.5. Conclusion
We propose an efficient and robust algorithm for noise removal in off-axis DH based on l1 minimization problem, that does not consider any prior information of the noise statistics. Numerical results show the goodness and robustness of the SPADEDH algorithm for both lensless holograms of centimeters size objects at Vis as well as at long IR wavelengths and for holograms recorded in microscopic configuration, with respect the DFF technique. In the first case, the display of denoised holograms shows the improvement in terms of the efficiency of the optical reconstruction. In fact, an increasing of intensity of the reconstructed in object image was observed. Also for the holograms recorded in microscope configuration we obtained good results in terms of denoising in the BFP spectrum as well as an improvement in the reconstruction efficiency in terms of both SDR and image contrast.
Acknowledgments
We are grateful to Dr. A. Gertrude and Dr. M. Locatelli of CNR-Istituto Nazionale di Ottica of Firenze to have acquired the hologram of Venus.
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