Speckle noise suppression in digital holography by angular diversity with phase-only spatial light modulator

Yunxin Wang; Puhui Meng; Dayong Wang; Lu Rong; Spozmai Panezai

doi:10.1364/OE.21.019568

1. Introduction

The speckle pattern, as a spatial random intensity distribution, is introduced when the coherent light either is reflected from a rough surface by random scattering or propagates through a medium with random refractive index fluctuation [1]. The speckle noise and surface information of the tested sample usually exist in the same wave front, and the field amplitude affected by speckle noise obeys Rician distribution. The spatial resolution and signal-to-noise ratio of the coherent imaging system is greatly affected by speckle noise, and the serious image quality degradation can be observed in optical coherence tomography (OCT), laser projection, radar and ultrasound imaging etc [2–5]. Especially in digital holography, the lateral average size of the speckle is similar to the reconstruction resolution [6, 7]. Therefore, it is of particular importance to suppress the speckle noise.

Many methods have been proposed to reduce the speckle noise, and which can be divided into three main categories: digital processing, reducing the spatial or time coherence, and superposing multiple speckle patterns [7–26]. The digital processing methods extract the object information from a single hologram based on an appropriate noise model, e.g. wavelet filtering, Fourier transform domain filtering, and non-local means filtering et al. [7–10]. However, these filtering methods bring inevitable loss of object information and the noise suppression is limited. The spatial coherence was restrained by temporally averaging using a moving diffuser or fast scanning micromirror [11–13]. The partially coherent light with a short coherence length, such as light-emitting diode (LED) or superluminescent diode (SLD), was directly applied to reduce the speckle noise [14, 15]. However, these methods have a drawback of expanding the impulse response and limiting the focus depth and imaging resolution of a system [23, 27]. It is worth mentioning that the optical scanning holography, as an unconventional digital holographic technique, has been proved to generate a complex hologram without the speckle noise [16]. Besides, the speckle noise can be reduced through averaging several reconstructed images with different speckle patterns, and the capability of suppressing the speckle noise is enhanced with the increase of the number of the holograms [1]. Thus many methods have been developed to acquire multiple holograms, e.g., using diverse wavelengths [17–19] and polarizations [20, 21], or changing the illumination angle of object [24–26]. Nomura et al. utilized a wavelength-tunable laser to collect 8 holograms with different wavelengths [17], nevertheless, this method needed the special laser and the number of holograms was limited by the tunable range of the laser. Rong et al. recorded multiple off-axis holograms using a circularly polarized illumination beam and a rotating linearly polarized reference beam [20]. However, in order to ensure the noncorrelation of speckle patterns, the polarization state of the reference beam had to be rotated more than 20°, which also restricted the number of the recorded holograms. Using the diversity of the illumination angle is a remarkable method to reduce the speckle noise. Quan et al. obtained multiple holograms by rotating a mirror to change the incident angle of illumination beam [24]. Pan et al. designed a couple of mirrors to adjust the illumination angle manually [25]. While, a small adjustment to the mirrors is tend to produce a big angle interval between two illumination beams, thus it is difficult to obtain a large number of holograms. Therefore, most of the existing methods have two disadvantages. Firstly, the digital holographic imaging systems mostly need some manual operation applied to the mechanical or optical elements during the recording of holograms, which make the system complex and unable to ensure the repeatability of speckle reduction. Secondly, the number of recorded holograms is restricted greatly, and the speckle noise cannot be suppressed in a high level.

Aiming to the problems above, we propose and demonstrate an approach to automatically reduce the speckle noise with a phase-only spatial light modulator (SLM). The speckle patterns will be uncorrelated if the angle of the illumination beam of the object changes greatly. The illumination angle is produced via a phase-only SLM by adding different modulation phase distributions under the control of a computer. Firstly, the demand of the minimal angle difference is analyzed theoretically and simulated. Then, the off-axis image-plane digital holographic system is built where the object is illuminated by the tilted beam reflected from the phase-only SLM and 117 holograms are recorded separately. The reconstruction leads to a set of object wave fields with uncorrelated speckle patterns. Finally, the speckle noise is reduced greatly by averaging the reconstructed images, and the results show good agreement with the theoretical value.

2. Correlation analysis of speckle patterns under angular diversity

As any approach based on angular diversity, the correlation of the multiple speckle patterns should be reduced to suppress the speckle noise efficiently. The feature of the speckle pattern depends on the illumination angle on the rough surface and the observation angle on the optical detector of the hologram. When the angle of illumination beam is changed largely enough, any two speckle patterns obtained will be uncorrelated. A detailed theoretical analysis about the correlation of the speckle field intensity with the changed illumination angle is derived in Ref [1]. A summary of the basic theory is briefly reviewed here.

The scattering vector $\vec{q}$ can be expressed as

\vec{q} = {\vec{q}}_{t} + q_{z} \vec{z} = {\vec{k}}_{o} - {\vec{k}}_{i},

where

{\vec{q}}_{t}

and

q_{z}

are the transverse and perpendicular components of the scattering vector

\vec{q}

respectively, and

\vec{z}

is the normal direction;

{\vec{k}}_{o}

is the wave vector in the observation direction,

{\vec{k}}_{i}

is the wave vector of the illuminating beam on a random surface, and both two wave vectors have length

k = 2 π / λ

.

The intensity correlation reflects the correlation degree between two speckle patterns. In the case of the fixed wavelength and observation angle, the intensity correlation of two speckle patterns ${| μ_{A} ({\vec{q}}_{1}, {\vec{q}}_{2}) |}^{2}$ is denoted by

{| μ_{A} ({\vec{q}}_{1}, {\vec{q}}_{2}) |}^{2} = {| M_{h} (Δ q_{z}) |}^{2} {| Ψ (Δ {\vec{q}}_{t}) |}^{2},

where

M_{h} (Δ q_{z})

is the first-order characteristic function of the surface-height fluctuations h, which is determined by the ratio between the surface-height fluctuations and the wavelength;

Ψ (Δ {\vec{q}}_{t})

describes the translation of the speckle pattern as the illumination angle changes;

Δ {\vec{q}}_{t}

and

Δ q_{z}

are the transverse and perpendicular components of the scattering vector difference

{\vec{q}}_{1} - {\vec{q}}_{2}

respectively. For a reflection-type rough surface,

Δ q_{t}

and

Δ q_{z}

can be written as

{\begin{cases} Δ q_{t} = k [\sin (θ_{i} + Δ θ_{i}) - \sin θ_{i}] \\ Δ q_{z} = k [\cos (θ_{i} + Δ θ_{i}) - \cos θ_{i}] \end{cases},

where

θ_{i}

is the angle between

{\vec{k}}_{i}

and the normal direction

\vec{z}

of the scattering surface, and

Δ θ_{i}

is the difference of illumination angles.

With a view to a surface with Gaussian surface-height fluctuations, ${| M_{h} (Δ q_{z}) |}^{2}$ can be defined as

{| M_{h} (Δ q_{z}) |}^{2} = \exp (- σ_{h}^{2} Δ q_{z}^{2}),

where

σ_{h}

is the root mean square surface height.

Ψ (Δ {\vec{q}}_{t})

is related to the diameter of the scattering spot. If the scattering spot is a uniform light spot with the diameter D,

{| Ψ (Δ {\vec{q}}_{t}) |}^{2}

can be represented by

{| Ψ (Δ {\vec{q}}_{t}) |}^{2} = {[2 \frac{J_{1} (0.5 D Δ q_{t})}{0.5 D Δ q_{t}}]}^{2},

where

J_{1}

is Bessel function with the first kind, order 1. Then the intensity correlation of two speckle patterns can be calculated by

{| μ_{A} ({\vec{q}}_{1}, {\vec{q}}_{2}) |}^{2} = \exp (- σ_{h}^{2} Δ q_{z}^{2}) {[2 \frac{J_{1} (0.5 D Δ q_{t})}{0.5 D Δ q_{t}}]}^{2} .

The intensity correlation ${| μ_{A} ({\vec{q}}_{1}, {\vec{q}}_{2}) |}^{2}$ is related to $σ_{h}^{}$ , D, $θ_{i}$ and $Δ θ_{i}$ . The angle difference $Δ θ_{i}$ decides the translation of the speckle pattern with the change of the illumination angle, which should be changed largely to ensure the pairwise noncorrelation of speckle patterns in the reconstructed images of the digital holograms. Therefore, it is essential to analyze the demand of the minimal angle difference. Especially, the angle difference $Δ θ_{i}$ is controlled by the imaging system, and the quantitative calculation of $Δ θ_{i}$ is significant and helpful for the optical design.

The lower intensity correlation ${| μ_{A} ({\vec{q}}_{1}, {\vec{q}}_{2}) |}^{2}$ means stronger noncorrelation of speckle patterns. For a certain intensity correlation ${| μ_{A} ({\vec{q}}_{1}, {\vec{q}}_{2}) |}^{2}$ to be realized, the angle difference $Δ θ_{i}$ is determined by $σ_{h}^{}$ , D, and $θ_{i}$ . $σ_{h}^{}$ is dependent on the property of the object. The diameter D of the scattering spot lies on the size of the illumination beam on the random surface of the object, once the imaging system is built, the diameter D of the scattering spot is constant. Next, let us discuss the parameter $θ_{i}$ . The surface of object is not commonly smooth, so the scatter planes of the diverse illumination locations are different. Correspondingly, the angle $θ_{i}$ between the illumination direction ${\vec{k}}_{i}$ and the normal direction $\vec{z}$ of the scatter plane will change continuously as the illumination location varies. Even when we consider an object with the flat surfaces, the angles $θ_{i}$ between the illumination direction ${\vec{k}}_{i}$ and normal direction $\vec{z}$ are also inconsistent on different planes due to the three-dimensional structure.

Considering that a dice with the size 9 mm × 9 mm × 9 mm will be used as the object, its root mean square surface height σ_h is about 186 nm measured by an optical profilometer. The dice will be illuminated by an even scattering spot with the diameter of 10 mm. Based on Eqs. (3) and (6), we analyze the influence of $θ_{i}$ and $Δ θ_{i}$ on the intensity correlation and the result is shown in Fig. 1. It is worth noting that the grazing incidence is ignored and $θ_{i}$ ranges from 0° to 80°.

Fig. 1 Distribution of ${| μ_{A} ({\vec{q}}_{1}, {\vec{q}}_{2}) |}^{2}$ versus $θ_{i}$ and $Δ θ_{i}$ .

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For a fixed $θ_{i}$ , the intensity correlation ${| μ_{A} ({\vec{q}}_{1}, {\vec{q}}_{2}) |}^{2}$ is reduced quickly as $Δ θ_{i}$ increases. Generally, if ${| μ_{A} ({\vec{q}}_{1}, {\vec{q}}_{2}) |}^{2} < 0.05$ , the two speckle patterns are considered to be uncorrelated [21]. Then we can confirm the requirement of a minimal angle difference for each $θ_{i}$ of illumination beam to ensure the noncorrelation of the speckle patterns. Suppose $α_{i}$ as the minimal angle difference for a specific angle $θ_{i}$ , the relationship between $θ_{i}$ and $α_{i}$ can be illustrated in Fig. 2. We can see that $α_{i}$ becomes larger with the increase of $θ_{i}$ . When the angle change is larger than the maximum value of $α_{i}$ , the two speckle patterns are uncorrelated at any $θ_{i}$ from 0° to 80°. For convenience of description, the maximum value of $α_{i}$ from 0° to 80° is called the minimal angular difference $α_{\min}$ for the object, and $α_{\min}$ for the dice is 0.017° as shown in Fig. 2.

Fig. 2 Relationship between $α_{i}$ and $θ_{i}$ .

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The minimal angular difference $α_{\min}$ is related to the root mean square surface height $σ_{h}^{}$ of the object. For the other objects with different root mean square surface $σ_{h}^{}$ , $α_{\min}$ can also be calculated according to the same procedure. From Eq. (6), we can see that ${| μ_{A} ({\vec{q}}_{1}, {\vec{q}}_{2}) |}^{2}$ reduces as $σ_{h}^{}$ increases, and $α_{\min}$ should also decrease with the increase of $σ_{h}^{}$ accordingly. In order to clarify the relation between $α_{\min}$ and $σ_{h}^{}$ , the numerical simulation is carried out for the objects with different $σ_{h}^{}$ . In the simulation, the wavelength of light source and the diameter of the scattering spot are 532 nm and 10 mm respectively, and $σ_{h}^{}$ ranges from 0.06 μm to 300 μm. The result is shown in Fig. 3. We can see that $α_{\min}$ is a constant 0.017° when the $σ_{h}^{}$ of objects ranges from 0.06 μm to 45.86 μm, which can be applied to a series of objects such as dice used in our experiments. When $σ_{h}^{}$ is larger than 45.86 μm, $α_{\min}$ is beginning to reduce with the increase of $σ_{h}^{}$ .

Fig. 3 Relationship between $α_{\min}$ and $σ_{h}^{}$ of the object.

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3. Experimental results and analysis

The experimental setup based on the image-plane digital holographic system is illustrated in Fig. 4. The light source is a single-longitudinal-mode laser with the center wavelength of 532 nm and the output power of 300 mW. A neutral density filter (NF) is utilized to adjust the total intensity of the imaging system. The illumination beam is collimated as a plane wave by the beam expander (BE, composed by spatial filters and collimating lens), and divided into two arms by the beam splitter (BS1). The transmission arm is used as the reference beam; the other arm is reflected by the phase-only reflective SLM and goes through the BS1 again, and then is reflected by the object (OBJ) as the object beam. Since the light is seriously scattered by the object, a lens (L, with a focus length 150 mm) is added to collect more scattered light. The object is placed at a distance of 335 mm from the lens, and the CCD is adjusted to be set at the image plane of the object. The BS2 combines the object beam and reference beam, and ensures the spatial separation of the reconstructed conjugate images. A polarizer (P) is inserted before the CCD to reduce the incoherent noise; meanwhile, the half wave plate (HWP) and polarizer are combined to optimize the intensity ratio between the reference and object beams. The holograms are recorded by the CCD, which has a resolution of 4016 × 2672 pixels, with the pixel pitch of 9 μm × 9 μm. The transmitted light of polarization beam splitter (PBS) possesses the horizontal polarization state, which can meet the polarization requirement of the phase-only SLM. The phase-only SLM has a resolution of 1920 × 1080 pixels with pixel pitch of 8 μm × 8 μm.

Fig. 4 Schematic diagram of experimental setup. M1, M2: mirrors; NF: neutral density filter; PBS: polarization beam splitter; BE: optical beam expander; BS1, BS2: beam splitters; HWP: half wave plate; OBJ: object; L: lens; P: polarizer.

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For the phase-only SLM, different grey levels will introduce diverse phase delay, and then the phase delay of the incident wave can be controlled freely. To achieve the angular diversity, the SLM works as a wedge-shaped prism in the optical path, and its complex reflection function $F_{S L M} (x, y)$ is expressed as

F_{S L M} (x, y) = \exp [- i k (α x_{s} + β y_{s})],

where x_s and y_s are the pixel coordination of SLM; α and β determine the tilted angle.

Several grey images loaded onto the SLM are shown in Fig. 5. In order to change the illumination angle in x direction, β should be set to zero, and the grey image with the vertical stripes can be used. Similarly, the grey image with the horizontal stripes can change the illumination angle in y direction, and the tilted stripes loaded onto SLM can alter the illumination angle in a tilted direction. When loading a uniform grey value, the function of SLM is like a mirror. The period of stripes determines the change of angle for the illumination beam. The small period represents a large angle change. Therefore, the illumination angle of the object can be adjusted automatically via the phase-only SLM by adding the different linear phase distributions under the control of a computer. Furthermore, the phase distribution can be modulated in two dimensions, which ensures that many tilted illumination beams can be obtained for the speckle reduction.

Fig. 5 Grey images loaded onto the phase-only SLM.

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According to the analyses in the second subsection, to ensure the noncorrelation of the speckle patterns, the angle change between two arbitrary illumination beams should be larger than 0.017° for the dice, which can be precisely achieved by loading a series of linear phase distributions onto the phase-only SLM. Suppose the normal incident light on SLM is along z-axis, the direction of the reflected light is -z-axis when a uniform grey image is loaded onto the SLM. The reflected light along -z-axis is taken as the reference beam. The other 116 grey images with inhomogeneous grey images are loaded onto SLM to change the illumination beam. Figure 6 describes the absolute angle difference between the reference illumination beam and the other 116 illumination beams. The minimal angle difference is 0.021°, and the maximal angle difference is 0.105°. Besides, the curve is symmetrical as shown in Fig. 6, since the phase modulations loaded in the SLM possess the symmetrical phase distributions in the two dimensions.

Fig. 6 Angle differences between the reference illumination beam and other 116 illumination beams.

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Considering that the hologram is recorded at the image plane of the object, the diffraction propagation of a certain distance can be omitted to improve the imaging efficiency. In the numerical processing, the spectrum filtering is employed to eliminate the zero and conjugate terms [28], and then the inverse Fourier transform is utilized to acquire the complex amplitude distribution of the reconstructed image.

Figure 7(a) displays an enlarged cutout (1080 × 1080 pixels) from the reconstructed intensity image using a single hologram. It can be seen that the image is teeming with the speckle noise, and the blurred details cannot be discerned clearly. The synthesized reconstructed images based on averaging multiple holograms are shown in Figs. 7(b)-7(d), and the number of holograms N for reconstruction is 4, 20, and 117 respectively. Compared with Fig. 7(a), the signal-to-noise ratio of the synthesized images is increased apparently, and the clarity of images is also improved.

Fig. 7 Reconstructed images obtained by averaging of (a) 1 hologram; (b) 4 holograms; (c) 20 holograms; (d) 117 holograms.

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The effect of speckle noise in digital holography is evaluated by the contrast of the speckle pattern [1]. Higher contrast indicates larger speckle noise. The contrast value of a speckle pattern is defined as

C = \frac{σ_{I}}{〈 I 〉} = \frac{\sqrt{〈 I^{2} 〉 - {〈 I 〉}^{2}}}{〈 I 〉},

where I is the light intensity of a speckle pattern;

〈 I 〉

and

σ_{I}

are the mean value and the standard deviation of I respectively. The superposition of N statistically independent random variables returns a random variable whose variance is reduced by a factor of

1 / N

, so the speckle contrast improves a factor of

1 / \sqrt{N}

[29].

The contrast values of the reconstructed images inside a uniform rectangular region with the size 80 × 100 marked in Fig. 7(a) are calculated and compared with the theoretical predictions as indicated in Fig. 8. It is clear that the speckle noise has been reduced with the increase of the number of holograms; meanwhile, the experimental and theoretical results are in good agreement. The contrast value of the reconstructed image by a single hologram is 0.5351, while by synthesizing 4 holograms, 20 holograms and 117 holograms, the corresponding contrast values respectively reduce to 52.48%, 24.57% and 13.14% of the initial contrast value, and their theoretical contrast values are 50%, 22.36% and 9.25% respectively. Therefore, the experimental results are very close to the theoretical one.

Fig. 8 Contrast value of the reconstructed intensity images versus the number of synthesized holograms.

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The small deviations between the experimental results and theoretical calculations are mainly caused by the residual partial correlation of reconstructed images. The correlation of reconstructed images can be evaluated by the correlation coefficient [30]:

c . c_{p, q} = \frac{| \sum_{i}^{m} \sum_{j}^{n} [I_{p} (i, j) - 〈 I_{p} 〉] [I_{q} (i, j) - 〈 I_{q} 〉] |}{{\sum_{i}^{m} \sum_{j}^{n} {[I_{p} (i, j) - 〈 I_{p} 〉]}^{2} \cdot \sum_{i}^{m} \sum_{j}^{n} {[I_{q} (i, j) - 〈 I_{q} 〉]}^{2}}^{1 / 2}},

where

c . c_{p, q}^{} \in [0, ​ 1]

, p and q are the two images to correlate, and p, q = 1,2,3...117; n and m represent the size of the reconstructed image in pixels;

〈 I_{p} 〉

and

〈 I_{q} 〉

are the average values of the intensity images

I_{p}

and

I_{q}

respectively.

If the speckle patterns are uncorrelated with each other, the correlation coefficient should be zero. Actually, the reconstructed images are difficult to be entirely uncorrelated due to the presence of incoherent noise and parasitic noise caused by imperfect optical elements. Figure 9 reveals a three-dimensional diagram of the correlation coefficient between pairwise reconstructed images p and q. There are 117 peak points with a constant value 1, which denote the self-correlation coefficient of each reconstructed images. Except these points, the average correlation coefficient is 0.0087, and 98% of the correlation coefficients are lower than 0.05. All of these data demonstrate that the angular diversity based on the phase-only SLM can almost ensure the noncorrelation of the speckle patterns to a great extent; meanwhile, a large number of holograms can be recorded automatically with the help of the programmable SLM, so this method can reduce the speckle noise effectively and easily.

Fig. 9 A three-dimension diagram of the correlation coefficient between every two pairs of reconstructed images.

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4. Conclusion

In this paper, a speckle noise suppression approach is proposed based on angular diversity. In order to ensure the noncorrelation of the speckle patterns, the minimal angular difference of two illumination beams is quantitatively analyzed and obtained based on the root mean square surface of the object, and the simulation results show that the minimal angular difference is object-dependent. The phase-only SLM is modulated by the different phase distributions in two dimensions to precisely produce a large number of tilted illumination beams. More than one hundred of holograms with different speckle patterns are recorded based on an image-plane digital holographic system. The experimental results demonstrate that the speckle noise can be greatly decreased by averaging a series of reconstructed images. The proposed method has two obvious advantages. Firstly, the entire holographic imaging system needs no manual operation, which reduces system complexity and improves its practicability. Secondly, the accurate control of the tilted illumination beams is achieved by the phase-only SLM, which makes the recording of a large number of holograms possible and ensures good effectiveness and repeatability for the speckle reduction. Therefore, the presented method of speckle noise suppression enhances the progress of digital holography from the laboratory to the practical application.

Acknowledgment

This work is financially supported by National Natural Science Foundation of China (No. 61077004, and 61205010), Beijing Municipal Natural Science Foundation (No. 1122004), Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20121103120003), and Science and Technology Project of Beijing Municipal Commission of Education (No. KM201310005031).

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Speckle noise suppression in digital holography by angular diversity with phase-only spatial light modulator

Abstract

1. Introduction

2. Correlation analysis of speckle patterns under angular diversity

3. Experimental results and analysis

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (9)

Equations (9)

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