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Abstract
Improving the axial resolution for multiparticle three-dimensional (3D) holographic tracking is crucial but challenging. Here we study the impacts of incident light power, uniformity of the illumination as well as image pixel size on the axial tracking resolution for digital holographic microscopy (DHM). We demonstrate that the resolution highly depends on the image pixel size and the uniformity of the illumination. A 3D localization algorithm based on local-intensity-maxima searching and a Gaussian fit to the integrated intensity of the reconstructed lateral images along the axial direction proves a robust strategy to enhance the axial resolution for colloids and bacteria within a wide depth of field over several tens of micrometers.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is still a challenge to measure the three-dimensional (3D) near-surface dynamics of colloids or microorganisms by using optical microscopy today. Confocal microscopy can acquire 3D locations of multiple colloidal particles through axial scanning [1], but it is limited by the scanning speed in recording spatial and temporal information simultaneously. Digital holographic microscopy (DHM) is capable of tracking 3D location of small particles in real time [2–4]. It records holograms formed by the interference between the optical waves scattered by the objects and the unscattered incident light. The volumetric optical field of the scattered light is reconstructed by Kirchhoff-Fresnel equation from each hologram [5,6], where the instant 3D position of the objects can be determined by object localization criteria.
For digital holographic tracking, object localization is critical, especially along the axial direction. Since the localization procedure finds the optimal focus plane from a series of reconstructed images, it determines the spatial resolution for 3D tracking. Recently, some focusing criteria have been developed for the localization [7–10]. Image cross-correlation approach locates a specific object by cross-correlation to a reference hologram library with known axial positions [7]. The accuracy along the axial direction depends on the spatial precision of the library [8]. Lorenz-Mie fitting takes a least-squares fit of the interference fringes in a holographic image based on Lorenz-Mie scattering theory. It can achieve an axial resolution of 10 nanometers [11]. However, it needs the structural information about each object, which is difficult for diverse colloidal and microorganism systems. The extremum of the amplitude local variance searching can determine 3D localizations without any prior knowledge of the object [9,10]. When searching local amplitude extremum for 3D localization, the holograms are firstly reconstructed slice by slice using Kirchhoff-Fresnel equation or a simplified procedure, i.e., Rayleigh-Sommerfeld back-propagation [5,7]. The volumetric optical wave front containing the features associated with the positions and orientations of objects is obtained. The local variance of integrated wave amplitude [9,12] or intensity [10] along the axial direction can be used as a criterion to determine the focus plane. However, most of these works failed to locate objects dispersed at different axial depths simultaneously. To address this problem, Antkowiak et al. [13] and Sheng et al. [14,15] attempted to develop multiparticle holographic tracking by scanning the amplitude or intensity of each pixel locally.
Recently, inspired by the approach for 2D tracking, we located the local intensity maximum of the reconstructed optical field and determine the centroid for each object by a later trajectory linking procedure and an axial resolution of 0.4 μm was achieved for 1 μm latex particle [16, 17]. Our approach is suitable for multiparticle cases and arbitrary shaped samples. Although the local-intensity-maximum criteria were used in holographic tracking, the tracking resolution and its limiting factors, especially in the axial direction, has not been quantitatively investigated. Herein, we assess several factors associated with the holographic tracking resolution, i.e., the incident light power, the uniformity of the illumination and the image pixel size. We reveal that the axial accuracy for tracking multiple micronsized particles and bacteria can be improved by optimizing these factors.
Meanwhile, curve fitting of object intensity distribution has been applied to refine 2D or 3D locations [2,7,18]. Gaussian fit has been widely adopted to acquire high-accuracy 2D locations, as the intensity distribution of a point source in the focus plane is well approximated by the Gaussian function [19,20]. On the other hand, for 3D location refinement, Xu et al. applied a second-order polynomial fitting to the maximum intensity of reconstructed images for a colloidal particle to enhance the axial resolution of their in-line DHM [2]. Huang et al. located 3D coordinates of a 250 nm virus with an axial resolution less than 5 nm over a range of ~200 nm by utilizing a sinusoidal function [21]. Soulez et al. proposed an inverse-problem approach based algorithm to iteratively refine the 3D coordinates of spherical particles in a wide axial range (~5 mm) with localization accuracy of sub μm [22,23]. Nevertheless, previous works mainly focused on 3D localization of spherical samples. Here, for the first time, we use a localization procedure based on a Gaussian fit for effectively improving the axial resolution for holographic tracking of rod-like bacteria over a large depth of field.
2. Materials and methods
2.1 Materials
Deionized (DI) water with a resistivity of 18.2 MΩ·cm was produced by Millipore. Negatively charged polystyrene latex particles (PLPs, diameter 0.2 and 0.8 μm, Thermo Fisher) and gold nanoparticles (GNPs, diameter: 50 and 80 nm, Sigma-Aldrich) were dispersed in DI water. Besides, wild-type Escherichia coli (E. coli, strain HCB1) was used to assess the performance of our in-line DHM for holographic tracking of microorganisms. The culture of bacterial strain was detailed elsewhere [17]. Briefly, E. coli cells were cultured with fresh tryptone medium (1% tryptone, 0.5% NaCl in DI water) to the mid-log phase (OD600 = 0.4) and diluted into a mobility buffer (MB; 10 mM potassium phosphate buffer containing 0.1 mM EDTA-2Na and 10 mM glucose; pH 7.2) to ~106 cells/mL. The sample chamber in the DHM experiments is composed by two clean coverslips (thickness: 0.17 mm, Fisher Scientific) with a polydimethylsiloxane (PDMS) spacer (22 mm × 22 mm × 0.5 mm) in between. The coverslips were soaked in a fresh piranha solution (H2SO4:H2O2 = 3:1, v/v) bath at 90 °C for 2 h, sonicated in DI water, ethanol and acetone respectively, and dried with nitrogen gas. Afterwards, the coverslips and the PDMS spacer were treated in a plasma cleaner (Harrick Plasma, PDC-002) for 5 minutes before use. Surface modification of the glass with polyethylenimine (PEI, MW = 70,000 g/mol) was made by dipping it into a 1 mg/mL PEI solution for 3 h. The coverslip was rinsed with DI water and dried with nitrogen gas. One drop of sample was added in the sample chamber to make particles/bacteria stuck to the PEI-coated coverslip. 150 μL of DI water was injected into the sample chamber. Finally, the chamber was carefully sealed and mounted on the DHM for further observation.
2.2 Reconstruction and localization
3D holographic tracking was realized by an in-line DHM equipped on an inverted microscope (IX83, Olympus) with two objectives (NA = 0.6, 40X; NA = 1.4, 100X, Olympus). Collimated LED lamps (S1: λ = 455 nm, M455L3-C1, 900 mW; S2: λ = 450 nm, M450LP1, 1850 mW, Thorlabs) provided parallel illumination through the sample. Holograms formed by the interference between the unscattered and scattered waves from the colloidal particles were recorded by a sCMOS camera (Zyla-5.5-CL3, Andor Technology). To eliminate the stationary noise, the removal of background without particles was performed. The volumetric reconstruction of the distribution of scattered light is based on Rayleigh-Sommerfeld back-propagation [5,7]. Briefly, the complex scattered field Es(r,z) can be reconstructed at any height z above the focal plane by solving a convolution,
where Es(r, 0) is the scattered amplitude in the focal plane, h(r,-z) is the Rayleigh-Sommerfeld propagator,The intensity of the scattered light Is(r) can be estimated by
2.3 Localization and refinement of z coordinates
We employed a procedure similar to 2D particle tracking to locate the 3D coordinates for multi-particle system. Briefly, the candidate particle locations are identified as the local brightness maxima within the volumetric reconstructed images. In practice, an intensity threshold permits around the upper 10-30th percentile of brightness to be compared. A volumetric pixel is considered as a candidate if no other pixel within a cubic of length w is brighter. Furthermore, a trajectory linking algorithm is performed, which recognizes the neighboring positions in the same trajectory as long as their distance is smaller than a characteristic length scale (L). L varies for different systems, and is determined before the experiments. Finally, the candidate 3D coordinates (xi, yi, zi) for each object are obtained. Curve fitting of object intensity distribution based on the Gaussian function was then applied to refine axial locations. Figure 1(a) demonstrates the reconstructed images of 0.8 μm PLPs at different defocus distances (z) and the corresponding intensity profile along x axis.
Fig. 1 (a) Reconstructed images of 0.8 μm PLPs at various defocus distances (z) and their intensity profiles in x direction. A Gaussian function was adopted to fit the intensity profile in a selected region. (b) The integrated intensities (Iint) of the selected lateral regions (ROI) at z ranging from 10.25 to 16.75 μm (zi = 13.5 μm) are well fit by the Gaussian function, and the refined value zr is 12.9 μm.
It should be noted that the in-focus image always appears as bright points since the background is removed. The intensity profile along x axis around the particle (the center peak) at each z can be fit by a Gaussian function to determine the refined xi and yi coordinates. Besides, the integrated intensity (Iint) (after removal of the background) in a selected lateral region of interest (ROI) of reconstructed images near zi can be obtained. We found that Iint along z axis in a proper ROI is well described by Gaussian function to yield a refined zi coordinate as the center position from the fitting. Figure 1(b) shows Iint in a region with a dimension of 10 × 10 pixel2 (corresponds to 1.62 × 1.62 μm2) which is centered by (xi, yi) of a specific particle for 41 images reconstructed near zi (13.5 μm).
2.4 Determination of the axial resolution for tracking
The spatial resolution for particle tracking was defined as the standard deviation of the locations of a fixed particle at different sampling times [18], or the root mean square error of the coordinates of fixed particles at various z [7]. To access the spatial resolution in the axial direction, we recorded holograms with 50 - 100 particles randomly dispersed and immobilized on the PEI coated coverslip in DI water. By translating the motorized objective in the axial direction with a certain step length (with a precision of 10 nm), we recorded defocused holograms at various z. On the other hand, the holograms were reconstructed to obtain zi. For each of these holograms, zi was plotted against the objective displacement (d) to deduce the localization errors (zi - d). The root mean square (rms) localization errors were acquired for all the particles and the mean value is defined as the axial resolution [7].
Figure 2 shows the axial resolution for holographic tracking of 0.8 μm PLPs. Holograms of 100 immobilized particles from 0 < z ≤ 40 μm with a binned width of 1 μm were recorded by DHM equipped with 40 × objective. zi agrees well with d from Fig. 2(a). Figure 2(b) represents localization errors together with the standard deviation obtained at all z. Figure 2(c) shows the rms error for each z, where a localization accuracy (416 nm) is acquired in 8 ≤ z ≤ 28 μm which is the range of interest.
Fig. 2 The axial resolution of DHM for multiparticle tracking. Holograms of 0.8 μm PLPs were recorded by DHM equipped with a 40 × objective and the light source S1. (a) Mean reconstructed distance of a PLP from the focal plane (zi) as a function of the objective displacement (d). The error bars show the standard deviation of 100 particles. The dash line denotes zi = d. (b) The localization errors (zi - d, blue) ± standard deviation (cyan) plotted against d with a binned width of 1 μm. (c) The root mean square (rms) localization errors plotted against d. In the range of 0 < z ≤ 40 μm, the axial resolution is 476 nm for 0.8 μm PLPs. In practice, 8 < z < 28 μm is the observation range for experiments, where the axial resolution is 416 nm.
3. Results and discussion
In our experiments, we used high intensities (photon N > 104) to obtain a better signal to noise ratio (SNR). However, the localization accuracy in our experiments can be still influenced by the detection number of photons which is proportional to the incident light, the size of the pixel as well as the uniformity of the light source respectively.
3.1 Effect of intensity and uniformity of light source on the axial resolution
A stronger incident intensity helps to improve the spatial resolution. For instance, Kusumi et al. performed 3D particle tracking for 40 nm GNPs by video-enhanced contrast microscopy utilizing a 100 W mercury arc lamp and a 100 × Plan-Neofluar objective [24]. To explore the effect of light intensity on the axial resolution, we used two light sources (S1 and S2) with different intensities for holographic tracking of 108 PLPs with 0.8 μm diameter. Holograms and background images with a dimension of 1024 × 1024 pixel2 were acquired from 0 < z ≤ 40 μm with a step length of 1 μm using the 40 × objective. The incident intensity profiles were measured from the background images (see Fig. 3(a) and 3(b)) and the axial resolution was determined (Fig. 3c). As shown in Fig. 3, when the intensity of S2 is 2.5 times of S1, the axial resolution is enhanced 1.27 times for tracking 0.8 μm PLPs, smaller than the square root value (1.58) predicted by the shot-noise limit [20,25]. The nonuniform illumination [Fig. 3(a)] in holographic tracking might account for this deviation.
Fig. 3 Incident intensity profiles of two light sources (S1 and S2) in (a) the lateral and (b) the axial direction. It is shown that the intensity is uniform in the axial direction, but nonuniform in the lateral plane. For S1: the mean intensity (IA) is 20707, the standard deviation of the intensity (σI) is 1459; For S2: IA = 52611, σI = 2444. (c) Root mean square (rms) localization error plotted against δd = d-d0, d0 denotes the smallest defocused distance to the focal plane in the range of interest. Holograms of 0.8 μm PLPs were recorded using S2 and a 40 × objective. The axial resolution is 328 nm.
From Fig. 3(a), the local illumination intensity increases as the lateral coordinate increases in our DHM. As a result, holograms with a reduced dimension recorded in the region has a better uniformity. To explore the effect of nonuniform illumination on the axial resolution, two series of holograms of 48 PLPs (0.8 μm diameter) with a dimension of 512 × 512 pixel2 with the same intensity but different uniformity were recorded. As the uniformity increased for 1.19 times, the axial resolution of DHM for holographic tracking was enhanced 1.70 times. We define a parameter to better describe the degree of illumination uniformity as u, where u −1 = ApσI/A, Ap: the area occupied by particles in each hologram, A: the area of the hologram, σI: standard deviation of the intensity) in the case that the amount of particles and the dimension of the hologram are different. Consequently, since u of holograms with 56 different 0.8 μm PLPs (IA = 55199, σI = 3.1% IA, 512 × 512 pixel2, Δz = 348 nm) is 0.7 times of that with 108 particles (IA = 52611, σI = 4.6%, 1024 × 1024 pixel2, Δz = 328 nm), the axial resolution of the latter is higher than the former.
3.2 Effect of the image pixel size
Thompson et al. proposed that the optimal image pixel size is equal to the standard deviation of the point spread function of a light spot [20]. If the pixel size is too small, the localization accuracy is not reliable. In contrast, if the image pixel size is too large, the pixilation noise becomes significant. We explored the impact of image pixel size on the axial resolution by comparing the resolutions with different objective magnifications (M) because the image pixel size corresponds to the real pixel size of the camera (6.5 μm) divided by M. Replacing 40 × objective with the one of 100 × magnification, the image pixel size was reduced 2.5 times and the axial resolution was expected to be refined. However, IA and u are reduced at the same time when increasing the objective magnification (IA: from 52611 to 15871; u−1: from 4.5 × 10−5 to 7.2 × 10−5). As discussed above, these changes correspond to a reduced axial resolution. As a result, the axial resolution is increased from 240 to 148 nm (~1.7 times). This fact implies that the image pixel size has a stronger impact on the resolution than incident intensity and illumination uniformity.
3.3 Refinement of z coordinates by Gaussian fitting
Although remarkable enhancement can be achieved by optimizing pixel size, generation of uniform illumination and increase of the incident intensity, these approaches are difficult to be implemented in practice for commercial microscope and diverse samples. Here we applied the localization algorithm based on a Gaussian function to improve the axial resolution for holographic tracking, particularly for 3D tracking of bacteria.
For holographic tracking of bacteria, holograms and background images of 88 E. coli cells at various z were recorded using light source S2 and 40 × objective. As shown in Fig. 4(a) and 4(b), intensity profiles along the lateral plane near the center of a specific bacterium can be well described by the Gaussian function. Moreover, Gaussian fitting was further applied to Iint of ROI in the lateral plane (1.62 × 1.62 μm2) at different z to get the axial location shown in Fig. 4(c). After the refinement, the axial resolution changes from 737 to 571 nm. The dimension of ROI of the reconstructed images selected for Gaussian fitting was also explored in Fig. 5. Under the 40 × objective, the intensity profiles in ROI with 5 × 5 and 40 × 40 pixel2 fail to be modeled by the Gaussian function, while the intensity profile in the selected region with both 20 × 20 and 10 × 10 pixel2 fits quite well. Since the length of a bacterium is around 12 pixels for 40 × magnification, the proper dimension of this region is roughly equal to or slightly smaller than the bacterium size. The size of ROI was selected in such a way to avoid the effect of the outer bright rings and loss of information. Furthermore, this approach has been successfully applied to refine locations of swimming E. coli with various orientations (Fig. 6).
Fig. 4 (a) Reconstructed images of an E. coli cell and its intensity profile in the lateral plane at different z, where zi = 32.7 μm is the candidate axial location determined by searching the local intensity maximum of the reconstructed volume. (b) Intensity profiles in the lateral plane at zi = 32.7 μm (red line) was well fitted by a Gaussian function (blue line). (c) Iint of lateral ROI (10 × 10 pixel2) at various z agrees with the Gaussian function.
Fig. 5 The reconstructed images of E. coli under 40 × objective in ROI with a dimension of 40 × 40, 20 × 20, 10 × 10 and 5 × 5 pixel2 and their integrated intensity profiles (Iint).
Fig. 6 (a) Background-free holograms and reconstructed images of swimming E. coli at various orientations and (b-d) their integrated intensity profiles (Iint). Iint were well modeled by a Gaussian function.
As shown in Fig. 7(a) to 7(c), Gaussian fit is effective for acquiring refined lateral and axial locations without any hardware developments. The finest axial resolution for holographic tracking that our in-line DHM can achieve is presented in Fig. 7(d), where the axial resolution for high-throughput tracking of 0.8 and 0.2 μm PLPs, 50 and 80 nm GNPs, as well as E. coli is 85, 58, 159, 116 and 318 nm respectively within an wide observation depth of tens of micrometers. The difference between these values might be resulted from the signal to noise ratio as well as particle size dispersity. Among these samples, the signal to noise ratio for bacterial tracking is poor as the difference between the refractive index of the medium and bacteria is minimal while their size dispersity is much broader than PLPs and GNPs. Therefore, their localization accuracy is the lowest. For PLPs, the localization accuracy was improved from 103 to 58 nm with a decreased diameter from 0.8 to 0.2 μm. In contrast, it drops from 116 to 159 nm as the diameter of GNPS decreases from 80 to 50 nm, where the reduction in signal to noise ratio with the decreasing particle size might be responsible for it.
Fig. 7 The axial resolution of DHM for holographic tracking of 0.8 μm PLPs (Δz) and that refined by Gaussian fitting (Δz’) in dependence of (a) incident light power, (b) illumination uniformity and (c) objective magnification. The error bars show the standard deviations of ~20 positions. (d) The optimized axial resolution of DHM for different samples. The error bars show the standard deviation of 80-100 locations.
Note that although much higher axial resolution was reported for holographic tracking techniques [2,18,24], most results were acquired within quite a short range along the z direction (~hundreds of nm) with a few particles concerned. Bae et al. applied Gaussian fitting to measure the average standard deviations of the 3D locations of a PLP with a size of 3 μm recorded by off-axial DHM. They reported a localization accuracy of approximately 2 nm in the lateral direction and 5 nm in the axial direction for a specific particle [18]. Since only one particle is concerned, the effects of nonuniform illumination were neglected. However, this effect may become severe for high-throughput tracking. We recorded holograms of 0.2 μm PLPs and calculated the localization accuracy in axial direction for an individual particle and 64 particles randomly dispersed in different positions. We are able to achieve a localization accuracy of 13 nm within an axial range for tens of micrometers in the case that only one particle is concerned, while it sharply increases to 78 nm for 64 particles. Consequently, the nonuniform illumination in the sight of view as well as the location depths of the objects is the main reason that limits the axial resolution for high-throughput tracking.
4. Conclusions
Factors responsible for determination of the axial resolution for holographic tracking, including intensity and uniformity of light source, image pixel size were examined utilizing DHM. The image pixel size and the nonuniform illumination greatly limit the localization accuracy for high-throughput holographic tracking. Meanwhile, the axial resolution is susceptible to the differences in refractive index between medium and sample as well as the size and morporlogy of the sample. We illuminated that Gaussian fit to the reconstructed integrated scattered intensity in a proper ROI along the axial direction is highly effective to enhance the axial resolution of holographic tracking over a wide depth of field for spherical particles with different sizes and nature as well as aspherical bacteria.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Funding
National Natural Science Foundation of China (21574046, 21637001); Fundamental Research Funds for the Central Universities.
Acknowledgment
The authors acknowledge Dr. Yingzi Yang and Prof. Feng Qiu from Fudan University for intellectual contributions and suggestions.
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