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Abstract
We show that background fringe-pattern subtraction is a useful technique for removing static noise from off-axis holographic reconstructions and can enhance image contrast in volumetric reconstructions by an order of magnitude in the case for instruments with relatively stable fringes. We demonstrate the fundamental principle of this technique and introduce some practical considerations that must be made when implementing this scheme, such as quantifying fringe stability. This work also shows an experimental verification of the background fringe subtraction scheme using various biological samples.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
17 August 2020: A typographical correction was made to the body text.
1. Introduction
Digital off-axis holography is an imaging technique that is capable of capturing both the amplitude and phase of light using a single intensity image (hologram) [1]. It suffers from various types of noise, however, which include but are not limited to photon (shot and speckle) noise, and detector noise (dark noise, read noise, and quantization noise) [2]. Removal of the zero-order and twin-image terms is straightforward in off-axis holography [3,4], and much effort has been devoted to reducing residual noise in holographic reconstructions due to phase aberrations [5,6], astigmatism, spherical aberrations, and anamorphism [7–9]. However, one problem with off-axis holography is slight temporal variations, which make the extended background fringe and speckle pattern difficult to remove to deep levels without the use of computationally expensive methods.
Here we present a computationally inexpensive method of removing the dominant static noise terms from off-axis holographic images. This method involves the background subtraction of raw holograms prior to numerical reconstruction. By only reconstructing the residual localized fringe packets remaining after subtraction of the background fringe pattern [Figs. 1(D)–1(F)], noise contributions such as speckle are removed prior to propagation through the reconstruction process. A theoretical explanation of the approach is presented, as well as practical considerations when implementing this technique. Finally, processing of experimental data from cultures of bacterial and protozoa cells near the resolution limit of the instrument is shown, highlighting this technique’s utility.
2. Principle of operation
Figures 1(A) and 1(B) show the optical schematic and laboratory image of a common path DHM instrument described in [10–12], which was used for the experimental data presented in this work. A coherent light source is collimated and passed through two identically sized microfluidic wells. One contains the object of interest while the other contains a reference liquid in order to match optical path lengths with the sample well. The sample and reference beams are then passed through two separate but identical objective lenses. Finally, a relay lens recombines the beams at a detector plane, creating an interferogram (hologram).
Fig. 1. Optical Schematic of the DHM instrument used throughout this work (A), an image of the benchtop instrument (B), and fringe stability of the DHM instrument as a function of time (C). Centerline indicates the mean fringe visibility and dashed lines indicate the 95% confidence interval range. (D) Raw hologram with a single B. subtilis bacterium in the field of view. (E) The same hologram after background subtraction and (F) a map of local fringe visibility of the background subtracted hologram in E, showing the residual localized fringe packets of dynamic objects. Scale bar in D represents 10 µm for D, E, and F.
Let two monochromatic and coherent beams, 0(x,y,t) and R(x,y) corresponding to object and reference beams, respectively, be incident on an optical detector, such that the resulting interference pattern recorded by the digital detector is
3. Practical considerations
Background subtraction of holograms is useful because any stationary artifacts in the image become attenuated, leaving only dynamic objects. If the background fringes of the hologram are stable, these fringes would be effectively removed from the resulting image, except for localized fringe ‘packets’ associated with any particle that has moved through the field of view of the hologram. Figure 1(D) shows a typical raw hologram with a single B. subtilis bacterium in the center of the field of view. Figure 1(E) shows the same hologram after background subtraction and Fig. 1(F) shows the background subtracted hologram as a function of local fringe visibility [see Eq. (5)]. Figures 1(E) and 1(F) show that after background subtraction, only localized fringe packets pertaining to dynamic particles remain in the residual hologram. This background subtraction technique vastly increases the signal to noise ratio of any dynamic particle in an image sequence, making it much easier for automated particle detection algorithm to detect particles of interest in space and time. We show here that rapid motion is not necessary for the method to work; movement as small as bacterial Brownian motion will suffice.
Such a background subtraction technique requires a stable interferometer. Many sources of noise can introduce shifts in DHM fringes, causing them to drift across the field of view of the detector. These noise sources include but are not limited to speckle noise, temporal phase noise caused by uncorrelated variations between the two beams of the instrument, as well as changes in illumination wavelength from instabilities in the illumination source. These sources of noise introduce an upper limit on the timescales where the proposed background subtraction scheme is useful. As the background fringes shift, they will by definition become dynamic objects in the image, thus no longer being removed by the background subtraction scheme.
The data used here were taken with a common-path off-axis DHM described previously [10], or by a multi-wavelength version permitting simultaneous capture at 3 wavelengths [12], with parameters in Table 1. The valid timescales of this background subtraction technique, with this relatively stable instrument, were quantified by collecting holographic images as a function of time without a sample in the field of view of the instrument. Temporally averaged holograms were calculated at various timescales through the hologram sequence. Fringe stability was inferred by calculating the average fringe visibility of the temporally averaged hologram. This is possible because if the fringes are absolutely stable (static), the temporally averaged hologram would equal the fringe pattern of each hologram in the sequence. As the fringe pattern begins to shift, they will become dynamic artifacts in the image and thus not appear in the temporally averaged hologram. Figure 1(C) shows a plot of fringe stability as a function of time for the DHM instrument used throughout this work. The center line in the plot signifies the mean fringe visibility value while the dotted lines signify the 95% confidence interval value range. By defining the point of fringe decorrelation as the point in time when the fringe visibility decreases by -3 dB, the upper timescale that should be used for this background subtraction scheme using our common mode instrument is roughly 40 seconds. Much shorter decorrelation times are expected for less stable instruments.
Table 1. Properties of the common path DHM instrument.
The average fringe visibility of a temporally averaged hologram was calculated by the distinct block processing of the temporally averaged hologram, where each distinct block is an m by n non-overlapping submatrix within the hologram. The size of these distinct blocks correspond to a single spatial period of the fringes within the hologram. The fringe visibility of the (i,j)th block was calculated such that
In addition to an upper limit on the timescale where this background subtraction scheme is useful, it is important to note a lower limit as well. Depending on the particulate speed, a large enough time scale must be used to allow significant positional shifts. This method was found to be sensitive enough to capture the small Brownian motion of bacterial cells.
It is possible to conduct this proposed background subtraction method using a median calculated hologram. Using the median is an alternative that was also explored, but made no difference in this case, given the low variability between the individual holograms on the timescales used in this work. This is advantageous given the much lower computational overhead associated with a mean calculation as a oppose to a median calculation.
4. Experimental procedure
The DHM was used to image two strains of bacteria (Bacillus subtilis and Vibrio alginolyticus), as well as the protozoan, Euglena gracilis. A 1 mm deep well was filled with a dilution of the biological sample in minimal media. A ‘motility’ medium was used for B. subtilis which would not harm the organisms but hinder their growth (10 mM phosphate buffer pH 7.4, 10 mM NaCl, 0.1 mM EDTA, 0.1 mM glucose). Due to the marine origins of V. alginolyticus, a different minimal media recipe was used (50 mM Tris buffer, 300 mM NaCl, 5 mM MgCl2, 5 mM glucose). E. gracilis was diluted in spring water. Holographic images were recorded at a frame rate of 15 frames per second. Background image subtraction was performed with a timescale of 10 seconds (150 holograms). Both the raw holograms and background subtracted holograms were then spatially filtered and reconstructed via the angular spectrum method using either the commercial software KOALA (Lyncée Tec SA, www.lynceetec.com) or a Fiji-based plug-in developed by our group [3,13,14]. All bacterial data were recorded using a monochromatic 405 nm laser illumination source. DHM images of E. gracilis were acquired using the multi- wavelength implementation of the common path DHM instrument, using a 405, 520, and 685 nm laser illumination source simultaneously.
5. Results
Figure 2 shows a raw hologram and background-subtracted hologram of a single B. subtilis bacterium [Figs. 2(A) and 2(B), respectively] and their respective Fourier spectra [Figs. 2(C) and 2(D)]. Figures 2(E) and 2(F) show the intensity reconstruction of the same bacterium using the raw hologram and background subtracted hologram, respectively. Figure 2(G) shows a single plane intensity reconstruction of a motile B. subtilis bacterium over roughly two seconds. This image is a composite image rendered by the minimum pixel intensity projection of multiple time points and illustrates the organism’s motion through time. Figures 2(H) and 2(I) show vertical line plots of pixel values through the bacterium for both the raw reconstruction and background subtracted reconstruction [Figs. 2(G) and 2(H), respectively]. Employing the background subtraction technique on data sets containing B. subtilis saw an increase in SNR of roughly an order of magnitude (11x improvement in SNR)
Fig. 2. Raw hologram of sample containing B. subtilis (A), the identical hologram after background subtraction (B), and their respective Fourier spectra (C and D). Intensity reconstruction of the same B. subtilis bacterium without (E) and with (F) the use of the residual fringe visibility analysis. (G) Select trajectory of a motile B. subtilis bacterium. This image is a composite image rendered by the minimum pixel intensity projection of 2 seconds of data. Arrow indicates the direction of motion. (H and I) Vertical line plots of pixel values through the bacterium shown in E and F, respectively. Scale bar in A represents 10 µm for A, B, E, and F. Scale bar in G represents 10 µm.
Figure 3 shows a raw and background subtracted hologram of a select V. alginolyticus bacterium [Figs. 3(A) and 3(B), respectively], and their respective Fourier spectra [Figs. 3(C) and 3(D)]. Figures 3(E) and 3(F) show a single plane intensity reconstruction of the same hologram from Figs. 3(A) and 3(B), respectively. These intensity reconstructions also demonstrate an order of magnitude increase in SNR. Figures 3(G) and 3(H) show a volumetric rendering of intensity reconstructions using the raw and background subtracted holograms, respectively. These volumetric renderings have been thresholded to the same value for comparison. The implementation of the background subtraction technique can be seen to increase the SNR of volumetric intensity reconstructions which can allow for the 3D localization of near diffraction limited objects such as V. alginolyticus. Due to the standard thresholding used in the volumetric renderings of Figs. 3(G) and 3(H), vertical lines can be seen in Fig. 3(G), which are the residual artifacts of the point spread function of the V. alginolyticus. Downstream processing techniques, such as deconvolution, would be effective in removing such artifacts.
Fig. 3. Raw hologram of V. alginolyticus (A), the identical hologram after background subtraction (B), and their respective Fourier Spectra (C and D). Intensity reconstruction of the same V. alginolyticus bacterium without (E), and with (F) the use of the residual fringe visibility analysis, and a 3D rendering of the reconstructed intensity z-stack of the raw (G) and background subtracted hologram (H). Scale bar represents 10 µm for (A, B, E, and F).
Figure 4 shows a raw and background subtracted hologram of multi-wavelength DHM data containing E. gracilis [Figs. 4(A) and 4(B)], and their respective Fourier spectra [Figs. 4(B) and 4(C)]. Figures 4(D) and 4(E) show a pseudo-colored composite intensity reconstructed image of a select E. gracilis using the raw and background subtracted hologram, respectively. The use of background subtracted holograms for multi-wavelength DHM data dramatically reduces the need for other post-processing unique to multi-wavelength data such as white balancing. Furthermore, the reduction in noise enables sub-cellular features of the E. gracilis to be much more visible, namely, the stigma (eyespot) and nucleus.
Fig. 4. Raw multiwavelength hologram of sample containing E. gracilis (A), the identical hologram after background subtraction (B), as well as their respective spectra (C and D, respectively). A pseudo-colored composite intensity reconstruction of a select E. gracilis using the raw hologram (E), and background fringe subtraction technique (F). Scale bar in (A) represents 50 µm for both (A and B). Scale bar in (E) represents 20 µm for both (E and F).
6. Conclusion
A novel method of enhancing image contrast of intensity reconstructions of digital off-axis holographic images is presented, including a theoretical justification, practical considerations in its implementation, and an experimental verification using biological samples. With roughly an order of magnitude of increased contrast provided by this method, standard image thresholding and clustering techniques become possible and enable high throughput and low computational overheard volumetric tracking. Furthermore, by conducting background subtraction on the raw holograms as opposed to the volumetric reconstruction, computational overhead is reduced proportionally to the number of axial planes reconstructed, potentially significantly decreasing computation times.
It is critical to note that fringe stability is necessary for this approach to work. We implemented it using a common-path off-axis DHM designed to be stable against vibrations. With a traditional Mach-Zehnder interferometer, the fringes may not be stable on the timescales reported in this work. Thus, the fringe stability and resultant number of frames used for background fringe subtraction will need to be determined for each instrument and light source combination.
Funding
Jet Propulsion Laboratory.
Disclosures
The authors declare no conflicts of interest.
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