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Abstract
The digital in-line particle holography suffers from speckle noise, for which considerable efforts have been devoted in order to mitigate it by designing post-processing algorithms. This paper proposes a novel approach, which mitigates the speckle noise by increasing the signal-to-noise ratio (SNR). It involves the joint design of optical systems and post-processing algorithms, and enhances the SNR by combining several holograms captured under different illumination angles. The experimental results show that the proposed scheme performs better than the normal scheme in terms of SNR and false-detection-ratio.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The digital in-line particle holography has shown its unique advantages in providing wide-filed, instantaneous measurements of particle fields. It is able to extract the 3D position of particles and provide information on the particle size and shape, and has found a wide range of applications [1–4]. However, it suffers from speckle noise, as summarized in Singh's 2010 paper [5], “the presence of speckle noise makes particle recognition difficult, where the intensity level of speckle noise is comparable with the average intensity of particle images and hence deteriorates the reconstruction effectiveness”.
Here, we focus on the noise from out-of-focus particles. Several approaches have been presented in the literature, which can be used to remove or alleviate the noise from out-of-focus particles. Singh et al proposed a digital holographic reconstruction algorithm based on intensity averaging of pixels within a particle, their method can effectively differentiate between in-focus and out-of-focus particles [5]. Latychevskaia et al applied 3d-deconvolution methods to restore true object distribution from holographic records, and obtained noise free 3D reconstructions [6]. Gao et al overcame the difficulty of recognizing true particle among out-of-focus particles by using the edge sharpness in conjunction with image segmentation methods [7]. All these related works were based on designing post-processing algorithms.
The present work proposes a novel scheme which mitigates the noise from out-of-focus particles by increasing the signal-to-noise ratio (SNR). The proposed scheme involves the joint design of optical systems and post-processing algorithms. Previous studies [8,9] which also tried to improve the SNR in in-line particle holography focused on the analog in-line particle holography. Essentially, the method proposed in [8,9] can be understood as a high-pass filtering technique, and utilizes the high-spatial-frequency components produced by particle side scattering to reconstruct particle images and hence requires a film resolution of ~0.1μm to satisfy the Nyquist limit [8]. Therefore, it cannot be directly extended to digital holography due to the finite size of the CCD pixels (the pixel size of typical image sensors ranges from 1.5μm to 10μm). In addition, the proposed method in [8,9] is effective in reducing speckle noise, but also suffers from the noise from out-of-focus particles. This is due to the fact that out-of-focus particles also produce side scattering. Here, our scheme enhances the SNR in the digital in-line particle holography by combining several holograms captured under on- and off-axis illuminations. Intuitively, the proposed scheme is similar to the pulse integration technique in radar systems [10,11] and the multi-quartz-enhanced photoacoustic spectroscopy technique [12]. The proposed scheme is evaluated and verified using experimentally simulated particle fields. The implementation procedure and performance details of the proposed scheme are discussed in the following sections.
2. Methods
2.1 On- and off-axis illumination
As sketched in Fig. 1(A), a collimated light beam is defined in Cartesian coordinate with relative angles (α, β, γ) corresponding to x, y, z axes, and it can be expressed as,
Here k = 2π/λ denotes the wave number and λ is the light wavelength, A is the amplitude of the plane wave. α, β and γ satisfy,Let z axis align with the optical axis of the optical system used, then on-axis illumination defines γ = 0 and can be written as,Off-axis illumination gives γ≠0 and can be expressed as Eq. (1). According to geometrical optics, off-axis illumination causes the intensity to shift from its original position in the recording plane by a distance of Δr = Δz·tanγ, as shown in Fig. 1(B). The lateral shift of the intensity increases with the increase in the distance between the object plane and the recording plane, as shown in Fig. 1(C). This indicates that different Δz result in different image shifts.
Fig. 1 (A) A light beam defined in Cartesian coordinate with relative angles (α, β, γ) corresponding to x, y, z axes. (B) On-axis illumination (solid line) and off-axis illumination (dashed line) of a particle (noted by p) at the object plane (OP) produce two images in the recording plane (RP), represented by point and circle, respectively. The distance between the two images can be determined by the following equation Δr = Δz•tanγ. (C) Δz>Δz’ results inΔr>Δr’.
2.2 Illumination angles and recording
As mentioned in the last section, different Δz result in different image shifts under off-axis illumination. This provides the possibility to reinforce the in-focus particles and mitigate the out-of-focus particles by combining multiple holograms captured under different illumination angles. To record holograms under different illumination angles, we physically scan illumination angles, and record one hologram for each illumination angle [see also Fig. 2].
Fig. 2 The measurement process: scanning illumination angles and recording one hologram for each angle. N is the total number of illumination angle.
Under the Born approximation, the scattered field at the recording plane for the mth illumination angle can be expressed as [13,14],
and,Here, t(x',y') represents the scattering density of the particles located at a distance Δz from the recording plane, the interference pattern between Es,m and the reference wave Er,m = Amexp[ik(ξcosαm + ηcosβm)] is recorded in form of intensity Im(ξ, η) as,Where, the reference wave amplitude, Am = 1, and m is a positive integer. Note that propagating Er,m to the recording plane needs to satisfy the bright-field illumination condition,Here, NA is the numerical aperture (NA) of the optical system used.2.3 Reconstruction
Equation (7) implies that, if we neglect the constant term |Es,m|2 and the nonlinearity effect caused by |Es,m|2, the reconstruction of the scattered field can be numerically implemented by multiplication of the digital hologram Im(ξ, η) with reference wave [5,15],
Thus, the complex amplitude of real image in the plane at a distance Δz from hologram plane can be determined by,Here, represent the conjugate of . The corresponding intensity distribution can be expressed as,Note that Eq. (9) can be understood as a light refocusing method [16,17], and thus can undo the intensity shift of the particles located at a distance Δz from the hologram plane, which is caused by the mth illumination in the recording plane. The particles located at a distance Δz from the hologram plane is reinforced by summing the intensity from all illumination angles, as follows,here, N denotes the total number of illumination angles (holograms). As a result, the SNR is also improved.3. Experimental validation
The effectiveness of the method described above was tested on a 3D particle sample which consists of two layers. For each layer, copper particles of average size, ~7.5μm, were randomly deposited on a glass plate (see also Fig. 2). The distance between the two layers was 340μm. A collimated laser beam was used for sample illumination and the incident wavelength was 532nm. A 4-f optical system with a magnification factor of 2 was used to collect the scattered field, the corresponding intensity field was recorded by a CCD camera with 2.5µm pixel size. The experimental procedures can be described as follows: (1) capture the hologram under the on-axis illumination; (2) sequentially tilt illumination and acquire a sequence of holograms.
We considered two cases. In the first case, we put one layer of the 3D particle sample at the in-focus position of the optical system used, and another layer was out-of-focus by the amount of 340μm. In the second case, the two layers were out-of-focus by the amount of 300μm and 640μm, respectively. The first case helps in explaining the shift difference between an in-focus particle and an out-of-focus particle in the recording plane. The second case demonstrates the effectiveness of the proposed method. To demonstrate the advantages of our method, the particle field was reconstructed using both the conventional method and our proposed scheme, and the results were denoted “Conventional” and “proposed”, respectively. In order to compare quantitatively, we calculated the background noise contrast (BNC),
and SNR [18,19],Here, Ip represents the average intensity of the particle images and σbn is the standard deviation of the background noise, Ibn,max and Ibn,min represent the maximum and minimum background noise, respectively.3.1 Lateral shift
To explain the shift difference between an in-focus particle and an out-of-focus particle in the recording plane, five holograms were captured. Two raw images are shown in Fig. 3(A) and Fig. 3(B), and the corresponding illumination angles were (α = π/2, γ = 0.025) and (α = π/2, γ = 0), respectively. In other words, Fig. 3(A) was recorded under off-axis illumination, and Fig. 3(B) was captured under on-axis illumination. According to geometrical optics, off-axis illumination causes the intensity to shift from its original position in the CCD plane by a distance of Δr = Δz·tanγ. Thus, the estimated shift corresponding to the layer at the out-of-focus position is Δr = 340·tanγ = 8.5μm, i.e., ~7 pixels, which agrees with the measured distance, as shown in Fig. 3(C). Both theory and experiment show a linear relation between Δr and tanγ, as shown in Fig. 3(D).
Fig. 3 Raw images. (A) Captured image under off-axis illumination (α = π/2, γ = 0.025); (B) Captured image under on-axis illumination (α = π/2, γ = 0); (C) Intensity profile plotted along the red line in (A) and the green line in (B). (D) Linear relation between Δr and tanγ. The illumination angles corresponding to the blue dots are as follows: (π/2,-0.05), (π/2, −0.025), (π/2, 0), (π/2, 0.025) and (π/2, 0.05).
3.2 Effectiveness
To investigate the effectiveness of the proposed method, three holograms (the number of holograms will be discussed below) were captured, the corresponding illumination angles were as follows: (π/2, 0), (π/2, 0.025) and (π/2, 0.05). The first two holograms are shown in Fig. 4(A) and Fig. 4(B) as examples. The captured images were processed using the typical method and the proposed scheme. The recovered results corresponding to the first layer of the sample were compared, as shown in Fig. 5. The smooth and bright background in Fig. 5(B) demonstrates fewer artifact in the image produced by the proposed scheme than that by the conventional scheme. Figure 5(C) is a composite image which shows Fig. 5(A) and Fig. 5(B) overlaid in different color bands. Gray regions in Fig. 5(C) show where Fig. 5(A) and Fig. (B) have the same intensities. Magenta and green regions show where the intensities are different. The gray regions in Fig. 5(C) state that the proposed scheme can be used to extract the particle field as the conventional scheme. The BNC value is decreased from 0.24 in Fig. 5(A) to 0.16 in Fig. 5(B), and the SNR value is increased from 4.1 to 8.3. For the second layer, one can obtain similar results.
Fig. 5 3D particle sample experimental results. (A) The holographic reconstructed result corresponding to the first layer of the 3D particle sample. (B) As (A), but showing the results from the proposed scheme. (C) Composite image showing (A) and (B) overlaid in different color bands. (D) Intensity profile plotted along the blue line in (A) and the red line in (B).
Theoretically, increased SNR translates into decreased false detection ratio, Rf, which is defined as the ratio of the number of false particles Nf and number of reconstructed particles Nd. Here, a false particle refers to a detected particle that does not correspond to one particle in the actual particle field. Nf and Nd were manually determined by examining the results. Particle detection is usually carried out by applying a threshold to segment particle images from background. We chose the threshold as [20,21],
Applying the threshold determined by Eq. (14) to Fig. 5(A) and Fig. 5(B) yielded Fig. 6(A) and Fig. 6(B). Figure 6(C) is a magnified composite image which shows Fig. 6(A) and Fig. 6(B) overlaid in different color bands. Gray regions in Fig. 6(C) show where Fig. 6(A) and Fig. 6(B) have the same intensities. Magenta and green regions show where the intensities are different and are respectively from Fig. 6(A) and Fig. 6(B). Evidently, the number of false particles in Fig. 6(A) is more than that in Fig. 6(B), and Rf is decreased from 51:129 in Fig. 6(A) to 8:86 in Fig. 6(B). Thus, we can conclude that the proposed scheme performs better than the normal scheme in terms of SNR and Rf.Fig. 6 (A&B)The particle field images after applying a threshold to Fig. 5(A) and Fig. 5(B), respectively. (C) Magnified composite image showing (A) and (B) overlaid in different color bands.
The slight difference in particle size, as shown in Fig. 5(D) and Fig. 6(C), is attributable to illumination misalignments, thus the authors believe that one can further increase the effectiveness of the proposed scheme by optimizing illumination angles, though the work reported here has not been focused on this point. Additionally, since three holograms can be captured in a single recording by use of a color CCD [22,23], the proposed scheme can be easily extended into single-shot imaging. Further investigation is warranted to determine the optimal illumination angles and extend into single-shot imaging.
Theoretically, the greater the number of illumination angles Nill is, the higher the recovered image quality is, as shown in Fig. 7. However, Nill >3 prevents the study of samples with fast events and dynamics. This is a major reason for capturing only three holograms in the experiments.
4. Conclusions
We proposed a novel approach that mitigates the noise from out-of-focus particles by improving the SNR. It was achieved by combining several holograms captured under different illumination angles. The proposed scheme was evaluated and verified using experimentally simulated particle fields. The experiment results show that the proposed scheme performs better than the normal scheme in terms of SNR and Rf, and the proposed scheme can be easily extended into single-shot imaging. Our future work in this area is to determine the optimal illumination angles and take the proposed scheme into single-shot imaging.
Funding
Natural Science Foundation of China (11702275, 11802289, 11672275); Science Challenge Project of China (TZ2016001); CAEP Foundation (CX2019006).
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