Enhancing Final Image Contrast in Off-Axis Digital Holography using Residual Fringes

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


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][8][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.

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][11][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). 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 where x d , y d are discrete spatial coordinates within the hologram, n is an integer value describing a discrete time sample of the intensity incident on a digital detector at a sampling interval of T seconds. Because relative phase shifts occur between beams, unstable fringe phases can in fact be taken into account by attributing them entirely to the object beam, with the reference beam assumed ideal. The object beam can be expressed as the superposition of a temporally static beam O s (x,y) and spatiotemporal deviations from that static beam caused by dynamic objects ε(x, y, t), such thatO(x, y, t) = O s (x, y) + ε(x, y, t). Thus, the hologram can be expressed as The temporally averaged hologram is defined as For a sufficiently large timescale, it can be assumed that.ε = 0 The residual hologram is defined as the difference between Eqs. (2) and (3): The O s and ε beams are on-axis with each other and thus their interference terms only contribute to the zero-order term of Eqs. (3) and (4). The term |ε| 2 − |ε|2 is also a zero-order term. The residual hologram in Eq. (4) is then binned and quantized into an 8-bit format in order to be compatible with other downstream image processing software. Using typical techniques of Fourier-space spatial filtering, which removes zero-order and twin image artifacts from the reconstructed images [3], the spatially filtered hologram that becomes reconstructed can be expressed as I F R = R * ε .

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  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. 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 where I max and I min are the maximum and minimum pixel value within the (i,j) th block. The mean fringe visibility value across all distinct blocks can then be calculated and becomes the average fringe visibility for that particular temporally averaged hologram. 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.

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 MgCl 2 , 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 multiwavelength implementation of the common path DHM instrument, using a 405, 520, and 685 nm laser illumination source simultaneously.    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.

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.