Speckle denoising in digital holography by nonlocal means filtering

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1 Speckle denoising in digital holography by nonlocal means filtering Amitai Uzan, 1 Yair Rivenson, 2 and Adrian Stern 1, * 1 Department of Electro-Optical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 2 Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel *Corresponding author: stern@bgu.ac.il Received 16 August 2012; revised 30 September 2012; accepted 1 October 2012; posted 2 October 2012 (Doc. ID ); published 13 November 2012 We demonstrate the effectiveness of the nonlocal means (NLM) filter for speckle denoising in digital holography. The speckle noise adapted version of the NLM filter is compared with other common speckle denoising filters and is found to give better visual and quantitative results Optical Society of America OCIS codes: , , , Introduction Speckle noise appears when coherent light passes through a randomly fluctuating medium, or is reflected from a rough surface. The latter case is of particular interest in digital holography (DH) [1]. The speckle noise may impose severe limitations on image quality in any coherent-based imaging technique and, particularly, in DH [1]. It may impair spatial resolution, signal-to-noise ratio, and phase accuracy of the reconstructed images [1]. The speckle noise mechanism and its dependence on the object and on the sensors limitations are modeled in [1 3]. For our purpose, we only point out the fact that speckle noise, unlike thermal and readout noise, is object dependent, with its variance being proportional to the local field intensity. This fact is considered when adapting our denoising method to speckle noise in Section 2.B. Several attempts have been made to overcome the image degradation effects of speckle noise since the first implementations of DH [1 18]. The methods can be categorized into optical and digital methods. In general, optical methods are based on increasing the amount of information gathered about the object, X/13/01A195-06$15.00/ Optical Society of America whereas digital filtering techniques extract the object information contained within a single hologram acquisition by utilizing assumptions about the object (e.g., smoothness, sparsity) and the appropriate noise model. Optical techniques include: averaging of multiple holograms recorded with different wavelengths [11], changing the phase of the illumination [13], projecting a number of different speckle patterns onto the object [14], using different polarization states [15], and slightly shifting the sample [17] to cite just a few. Other optical techniques for reducing the speckle noise in DH have been implemented by using partial spatial coherence sources [4], with the addition of digital image processing [10]. In addition, aperture synthesis techniques [5,9,12,16,18] have also been validated as useful strategies for reducing both the magnitude and the contrast of the speckle noise as a consequence of synthetic aperture enlargement and of overlapping between elementary apertures (which generates the synthetic aperture) respectively. Several digital filtering techniques for the reduction of speckle noise in DH have also been developed [7 10]. For example, wavelet [8] and Fourier transform domain filtering [10]. Numerous digital filtering methods have also been developed for other coherent imaging modalities that encounter speckle noise (e.g., ultrasound, radar, microwave). 1 January 2013 / Vol. 52, No. 1 / APPLIED OPTICS A195

2 Among the best known are the Lee filter [19], Kuan filter [20], Frost filter [21], bilateral filter [22] and wavelet threshold filter [23]. It should be noted that, asymptotically, all coherent imaging modalities (acoustic, radar, microwave, DH in the visible) share the same speckle noise model. However, in practice, there may be slight differences due to different setup geometries and sensor properties [6]. In DH in the visible spectrum, it can be assumed that the only source of distortion is due to the limited resolution power of the sensor [1,6]. The lateral speckle average size can be approximated as the reciprocal of the maximum spatial frequency, λz D, where λ is the wavelength, D is the critical aperture (typically the hologram lateral size), and z is the propagation distance [1,6]. The intensity distribution of the speckle noise obeys negative exponential statistics, having a standard deviation equal to the mean value. The field amplitude contaminated by speckle noise follows a Rician distribution [1]. In this work, we propose the use of the nonlocal means (NLM) filter [24] for speckle denoising in DH. NLM is a relatively new filtering technique, known for its outstanding ability to preserve edges while efficiently removing white noise. In Section 2, the NLM filter is described, along with its adaptation technique for speckle noise reduction. The adaptation to speckle noise emulates that applied previously to ultrasound image denoising [24]. Here, we test the adapted filter on typical real holograms and on simulated holograms, accounting for speckle evolution in typical hologram setups. The NLM denoising results are shown in Section 3 together with comparison to other commonly used digital speckle filtering techniques. We conclude in Section Nonlocal Means for Speckle Denoising A. The NLM Filter Traditional image denoising techniques are based on local image processing, involving only the local surroundings of the pixel to be denoised. In contrast, the NLM denoising method [24 27] is based on the concept that any noisy pixel located in the center of an image patch may be denoised by building relevant statistics from patches having similar structures located anywhere in the image. Whereas local denoising methods rely on the assumption of local stationary statistics of the noisy signal (e.g., smooth signal and white noise), NLM is based on the assumption that, for any given patch in the image, one may find other patches with similar statistics. This assumption holds true for most of the images [26]. In general, the NLM denoising technique was found to outperform state-of-the-art denoising methods when applied to images contaminated with additive noise [24]. Figure 1 illustrates the concept of NLM denoising. Let us assume that we wish to denoise the central pixel of the square window p. The square patch, p, is referred to as the window of interest. The value p of the pixel of interest is replaced by a weighted average of surrounding pixel values from pixels located in a larger search window. This is achieved by first dividing the search window (which in this illustration is the entire image of Fig. 1) into patches having the same size as the window of interest (e.g., patches q 1, q 2,andq 3 ). These patches are usually referred to as similarity windows. Any similarity window is compared to the window of interest to generate a weight representing the amount of similarity between the two windows. The weight is then used to average the values of the pixels at the center of the similarity window with those in the center of the window of interest. For instance, in Fig. 1, the similarity windows q 1 and q 2 would get higher weights than the similarity window q 3 because they are more similar to p. Mathematically the NLM denoising is formulated as follows: u out x i ;y i X x j ;y j Ω dim w x j ;y j u x j ;y j ; (1) where, u out x i ;y i is the denoised pixel value, w x j ;y j is the weight of the pixels in the similarity window centered around x j ;y j,andu x j ;y j is the gray level of the pixel in the similarity window. The weight is calculated by the equation w x j ;y j 1 Z i q 3 Fig. 1. Illustration of the NLM filter windows. The search window is the entire image, while p is the window of interest, which is the central pixel we wish to denoise. The windows q 1, q 2, and q 3 are the similarity windows. exp u N i u N j 2 2;a h 2 ; (2) where u N i and u N j are the gray level of the pixels in the window of interest and in the similarity windows, respectively. The expression u N i u N j 2 2;a represents the gray level Euclidean distance between the window of interest and the similarity window, convolved with a Gaussian kernel with parameter a. The parameter Z i is a normalizing factor used to ensure that the sum of the weights is equal to 1 and h is a filtering parameter controlling the filter size. In the case of additive random Gaussian noise, h 2 is closely related to the image q 1 q 2 A196 APPLIED OPTICS / Vol. 52, No. 1 / 1 January 2013

3 noise variance [25]. The above procedure is repeated for all pixels u x j ;y j in the image. There are three parameters controlling the NLM filter performance. The first one is the size of the search window; the larger it is, the better is the denoising that can be achieved. However, choosing a large search window may significantly increase the run time. It can be shown that, beyond a certain size of the search window, the improvement becomes asymptotically constant [24,26,27]. Typically, the search window is set to be between pixels and pixels. The second parameter is the size of the similarity window, which is typically chosen to be between 3 3 and 9 9 pixels [24,26]. The last parameter is h, which sets the degree of filtering; this parameter has to be adapted to the noise level of the pixel of interest. B. Adaptation of the Nonlocal Means Filter for Speckle Denoising The NLM filter was originally developed for suppressing additive signal-independent noise, such as Gaussian white noise. As mentioned in Section 1, this is not the case for speckle noise. The speckle intensity is dependent on the local signal. This implies that the filtering parameter value, h, in Eq. (2) should depend on the pixel to be filtered. In [24], the NLM filter was adapted for speckle denoising in an ultrasound imaging application. Here, we follow a similar strategy for speckle denoising in DH. By formulating the NLM as a Bayesian inverse problem, it was shown in [24] that the weight term in Eq. (2) for the speckle noise case is given by distributions and with speckle patterns appropriate to typical DH setups. 3. Results A. Experimental Results In our study, we tested the NLM filter with the appropriate weight function, as denoted in Eq. (3). We then compared the results with those obtained with several filters that are commonly used for speckle suppression: median filter, Lee filter [19], Frost filter [21], bilateral filter [22], and wavelet thresholding (WT) [23]. The median filter is well known for its efficient spike removal. Lee and Frost filters are considered to be standard filters for speckle denoising in SAR and ultrasound imaging. Bilateral filters and WT are considered among the most efficient edgepreserving denoising filters. First, we applied the NLM filter to object images reconstructed from digitally recorded holograms. Figure 2 shows a representative image. The Gaussian kernel parameter in (1) is set to a 0.5, and γ in (3) is set to 0.5. The dice object in Fig. 2 was captured with phase shifting DH in the Fresnel setup. The four phases were generated with w x j ;y j 1 Z i exp u N i u N j 2 2;a u N j 2γ ; (3) where γ is a noise factor depending on the imaging system parameters [24]. Its nominal value is γ 0.5. Comparing Eq. (2) to Eq. (3), it is evident that, while in the additive noise case, the degree of filtering was set by a constant value of h, in the speckle denoising case the degree of filtering depends also on the values of the pixels in the similarity window centered around x j ;y j. This is appropriate for the speckle model, where higher pixel values indicate stronger noise levels. According to [24], the division in the exponent term of Eq. (3) should be a Hadamard (pixelwise) division. Alternatively, the denominator in the exponent term of Eq. (3) may be chosen to be the mean value of the similarity window. This choice is more computationally efficient and it is appropriate for images that are relatively smooth within the window of interest. In our simulations we found that the pixelwise division method gave slightly better results. The adapted NLM filter (3) was found to be effective for ultrasound image denoising. In the next section, we demonstrate its usefulness for digital holograms that involve objects with different structural Fig. 2. Comparison of speckle reduction filters applied on an object s image, which was reconstructed from its digital holographic recording: (a) back propagation from the original hologram, (b) denoised with median filter, (c) denoised with Frost filter, (d) denoised with Lee filter, (e) denoised with bilateral filter, (f) denoised with WT filter, (g) denoised with NLM filter. 1 January 2013 / Vol. 52, No. 1 / APPLIED OPTICS A197

4 Table 1. Performance Comparison Between NLM, Median, Lee, Frost, Bilateral, and Wavelet Thresholding Filters for the Digitally Holographic Recorded Cube Object of Fig. 2 a NLM Median Lee Frost Bilateral Wavelet ENL SSI SMPI a Best results are designated in bold. quarter-wave retarder plates, and the field was reconstructed according to [28]. The illuminating wavelength was λ nm, imaging distance z mm, and the CCD camera used pixels of size 9 μm. The following parameters were used with the NLM filter procedure: search window size of pixels, similarity window size of 3 3 pixels, degree of filtering parameter h 8, and noise factor γ 0.5. The object s image was then denoised using all six of the above-mentioned filters (Frost, median, Lee, bilateral, WT, NLM), with the results shown in Fig. 2. From Fig. 2, the superiority of the NLM filter is evident; the object s image in Fig. 2(f) is much clearer, when compared with the other filters. For a quantitative comparison of the various filters, we used three blind speckle denoising metrics. (By blind metrics, we mean metrics based solely on the images themselves without access to the ground truth images.) The metrics are (a) Equivalent Number of Looks (ENL) [29] given by ENL mean 2; (4) std where mean and std denote the mean gray level of the image and its standard deviation, respectively. The ENL is a good metric for the quality of noise reduction in homogenous areas. The ENL increases with noise reduction. (b) Speckle Suppression Index (SSI) [29] given by SSI std I f mean I f mean I o std I o ; (5) where I o denotes the noisy image and I f the filtered image. The SSI metric tends to be less than 1 for effective filter performance. (c) Speckle Suppression and Mean Preservation Index (SMPI) [29] given by SMPI 1 jmean I f mean I o j std I f std I o : (6) The SMPI metric tends to be smaller for better filtering with regard to noise reduction, while preserving the average gray level of the image. A comparison between the six filters in terms of the above metrics is shown in Table 1. From Table 1 it can be seen that the NLM filter provided better results in terms of ENL and SSI, while the Wt filter has the best result in terms of SMPI. The latter result can be associated with the fact that the DC level (average gray value) of the NLM-denoised image is slightly biased and to the fact that the SMPI measure is sensitive to DC changes. B. Denoising Comparison with Synthetically Noised Images The ENL, SSI, and SMPI are blind metrics in the sense that they estimate the speckle reduction only Fig. 3. Simulation results: (a) and (d) original images, (b) and (e) noisy images, (c) and (f) respective denoised images using NLM. A198 APPLIED OPTICS / Vol. 52, No. 1 / 1 January 2013

5 Table 2. Performance Comparison (in terms of PSNR improvement) Between NLM, Median, Lee, Frost, Bilateral, and Wavelet Thresholding Filters for Electronic Chip and Coins Images a Image name NLM Median Lee Frost Bilateral Wavelet Chip Coins a Best results are designated in bold. from the image with no access to the ground truth image. As seen in the previous subsection, these metrics do not necessary correlate with the visual image quality. For a more precise quantitative comparison, we have applied the filters on synthetically noised images. Since in this case we have access to the original image, we can evaluate the filters performances using nonblind metrics, such as peak signal-to-noise ratio (PSNR). We simulated a coherent imaging process and added speckle noise. For our simulation, we used the imaging method described in [30], addressing all propagation effects, i.e., blurring due to the limited lens aperture and speckle noise. With the algorithm in [30], the noise is developed in accordance with propagation through the imaging system. Therefore, it is more relevant to DH than speckle noise generators based solely on first order statistics (e.g., MATLAB s speckle noise generator). For the simulation, we used the following parameters: wavelength λ nm, object distance from lens z mm, focal length f 100 mm, z mm, and pixels in the detector with pixel size of 5 μm. We tested the performance of the six above-mentioned filters on two reference images (electronic chip [31] and MATLAB s coins image) shown in Figs. 3(a) and 3(d). The results are summarized in Table 2, showing the PSNR improvement obtained by applying the filters on each image. The PSNR is calculated with comparison to the original, non-noisy image. The PSNR improvement is calculated by subtracting the PSNR of the filtered image from the PSNR of image with speckle noise. From Table 2, the advantage of the NLM filter relative to the other filters is apparent; the NLM method yields significantly higher PSNR improvement compared to the other filters. However, it should be noted that the improved denoising performance is achieved on account of the computation time; typical run times for NLM, median, Lee, Frost, bilateral, and wavelet filters were 646, 3, 3, 14, 7, and 5 s, respectively. 4. Conclusion In this work we presented the NLM filter for speckle denoising in DH. This filter is an adapted version of the conventional NLM filter, which was originally developed for additive white noise suppression. We compared the adaptive NLM filter performance to the results obtained with other commonly used filters by applying them to an object s image that was acquired experimentally with a phase-shifting holography technique. The comparison was performed using blind metrics and visual inspection. 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