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Abstract
This work applies digital holography to image stationary micro-particles in color. The approach involves a Michelson interferometer to mix reference light with the weak intensity light backscattered from a distribution of particles. To enable color images, three wavelengths are used, 430, 532, and 633 nm, as primary light sources. Three separate backscattered holograms are recorded simultaneously, one for each wavelength, which are resolved without spectral cross talk using a three-CMOS prism sensor. Fresnel diffraction theory is used to render monochrome images from each hologram. The images are then combined via additive color mixing with red, green, and blue as the primary colors. The result is a color image similar in appearance to that obtained with a conventional microscope in white-light epi-illumination mode. A variety of colored polyethylene micro-spheres and nonspherical dust particles demonstrate the feasibility of the approach and illustrate the effect of simple speckle-noise suppression and white balance methods. Finally, a chromaticity analysis is applied that is capable of differentiating particles of different colors in a quantitative and objective manner.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Digital holography is a versatile method to obtain images of microscopic objects in a contact-free and flexible manner [1]. In many ways, imaging done this way is analogous to imaging with conventional optical microscopy [2]. For example, the image resolution can reach the single micrometer level [3,4] for trapped or fixed particles, and a wide variety of objects can be examined such as biological species [1,2,5–9], particles moving in fluid or optical traps [3,10], free-flowing aerosol particles [11–17], and surface structures [18,19], to name only a few. The salient difference between digital holography and conventional microscopy is the way in which the image is rendered. From a single measurement of a hologram on an image sensor, it is possible to bring into focus objects at different distances from the sensor without the use of lenses or mechanical adjustments thereof [18–22]. This is possible because the image is rendered computationally via the Fresnel diffraction integral [18,19,23]. Moreover, digital holography often features a significantly larger field of view than conventional microscopy [24,25] and a much larger depth of focus [21], and can render images in monochrome or color [1,4]. These characteristics would make the method ideal for applications where objects are moving and/or distributed throughout a volume.
Fig. 1. Optical arrangement for backscatter multiple wavelength digital holography. Three CW laser beams of wavelengths ${\lambda _{\rm{r}}} = 633$ nm, ${\lambda _{\rm{g}}} = 532$ nm, and ${\lambda _{\rm{b}}} = 430$ nm are cleaned, expanded, and combined to form a single white beam. The beam is focused to a waist at a pellicle beam splitter where approximately half proceeds and expands as it reaches a concave mirror. There it is retro-reflected and collimated, and then reflected again by the pellicle to constitute the reference beam at the sensor. The other half of the beam illuminates a sample of micro-particles on an anti-reflection coated window. Backscattered light from the particles that transmits through the pellicle reaches the sensor, constituting the object beam. Inset (a) shows the spectral response of the sensor’s color channels along with the laser spectra, while inset (b) shows how specular reflection from the window is rejected. Further discussion is provided in the text.
This work is motivated by aerosol science and technology, where a general interest is the determination of individual aerosol-particle size, shape, and material composition [26]. Such information is important in many contexts including atmospheric science, national security, medicine, and industrial manufacturing, among others [26–30]. While a variety of methods are available to provide some of these particle characteristics, such as elastic light scattering, the most effective is an imaging approach. The difficulty, however, is that aerosol particles often cannot be collected or immobilized for analysis with conventional microscopy. Here, digital holography with its contact-free ability becomes an ideal choice because particle size and shape are easily determined provided that a well-resolved image is obtained. A number of examples are available to illustrate the effectiveness of digital holography for this purpose including the imaging of free-flowing particles in the laboratory [11,14,17,31], in the outdoor environment from an unmanned aerial vehicle [13] and aircraft [16], and as a ground-based instrument [15].
As explained below, a hologram of a particle is formed by the interference of coherent light illuminating a particle with the portion of light the particle scatters. In the majority of cases in the literature, a single wavelength is used, wherein the resulting image is monochrome in nature [1]. However, it is possible to employ multiple wavelengths and obtain a colorized image [4,5,19,32]. Again, this can be understood by the similarity of digital holography and conventional microscopy. Colorized images are useful in material analysis, biomedical imaging in the study of tissue structures, and water quality [33–35]. Moreover, a hologram can be formed from light scattered by the particle around either the forward-scattering direction or backward-scattering direction [22,36]. The resulting image then takes the appearance of a light-transmission or reflection image, respectively. Indeed, some authors refer to the forward and backward holograms as transmission and reflection holograms, respectively. There is no physical difference between these terms [37], i.e., between backscattering and reflection, etc.
Backscatter holography presents a unique opportunity for particle characterization. The color of an opaque object that one sees when it is illuminated by white light is due to the spectral components that reflect from the object [32,38]. And this reflection is governed, in part, by the material composition. Thus, it is plausible that backscatter holography carried out with multiple wavelengths could provide information about particle material, which is perhaps the most difficult aerosol characteristic to obtain.
The following presents an application of multiple wavelength digital holography (MWDH) in the backscatter configuration with the goal of discriminating between particles of different colors. Due to the difficulty of aerosol handling and control, this initial effort is carried out on particles that are stationary on a transparent stage, a window. However, it is emphasized that immobilizing particles is not required, and most of the methods developed would apply to free-flowing aerosol particles in future work. The particles considered are relevant to the large-size category of atmospheric aerosols, which is called the coarse mode, consisting of particles larger than 1 µm. While MWDH is achieved by others in various contexts [4–6,39,40], this work is the first to apply it to objects of this small scale, i.e., coarse-mode particles. To the authors’ knowledge, the work is also the first to develop a MWDH-image analysis that enables a quantitative discrimination of particles of differing colors. This is achieved by chromaticity analysis applied to the color image, where the magnitude of different wavelength components resolved in the image represent a signature for a given particle class.
2. OPTICAL ARRANGEMENT
Figure 1 shows the optical layout of MWDH. The lasers used are CW and consist of a 21 mW He–Ne ${\lambda _{\rm{r}}} = 633\,{\rm{nm}}$ laser, a 100 mW ${\lambda _{\rm{g}}} = 532\,{\rm{nm}}$ solid state laser, and a 10.3 mW ${\lambda _{\rm{b}}} = 430\,{\rm{nm}}$ solid state laser. For brevity, these lasers are called red, green, and blue (RGB) in the following. Each beam passes a liner polarizer (P) and is then cleaned and expanded by a spatial filter. The lenses of the spatial filter are selected to expand the beams to approximately the same diameter, 5 mm, and are then combined by two dichroic beam splitters, DM1 and DM2. These beam splitters are single-edge filters with a cut-on wavelength of 552 nm for DM1 (IDEX/Semrock, LM01-552-25) and 466 nm for DM2 (IDEX/Semrock, LM01-466-25). The combined beams appear nearly white in color. The white beam passes through another linear polarizer to ensure polarization in the vertical direction, which is the $x$ axis as shown in Fig. 1. Then, the beam passes through an achromatic quarter-wave plate (QWP) with its optical axis oriented at ${45^ \circ}$ to the $x$ axis to produce circular polarization for reasons that will be explained below.
Next, an achromatic lens (ACL) focuses the white beam through an iris to a waist at a 45% reflection, 55% transmission pellicle beam splitter (Thorlabs, BP145B1). The portion of the beam transmitting through the pellicle, along the positive $y$ axis, expands from its waist and passes through another linear polarizer and a neutral density (ND) filter (Thorlabs, NE20A-A). This polarizer resides in a motorized rotation mount (Thorlabs, K10CR1), which allows the direction of the resulting linear polarization of the beam to be rotated by an angle ${\theta _{\rm{p}}}$ in the $x$-$z$ plane without diminishing the beam intensity, aside from the attenuation of the ND filter. The beam, which continues to expand, is then retro-reflected and collimated along the negative $y$ axis by a 100 mm focal length concave mirror (Thorlabs, CM508-100-P01), passes back through the ND filter and polarizer, and then reflects off the pellicle along the positive $z$ axis to the sensor. This collimated beam at the sensor constitutes the reference wave required in digital holography.
The portion of the white beam incident on the pellicle that reflects from it travels, diverging, along the negative $z$ axis to illuminate the particle sample. A small portion of this incident beam is backscattered by the particles [see Fig. 1(b)], and this scattered light travels along the positive $z$ axis, through the pellicle, to the sensor to constitute the object beam in digital holography. The particles reside on a 2 mm thick anti-reflection (AR) coated window. Even with the AR coating, there is an unavoidable small amount of light that specularly reflects, and this is enough to overwhelm the intensity of the weak object-beam at the sensor. For this reason, the window is slightly tilted to divert the specular reflection to a small pick-off mirror (PM) where it is then sent to a beam trap and prevented from affecting the sensor’s measurement. The portion of the beam that does not scatter from the particles and transmits through the window is discarded in another beam trap. The use of a trap here is important to prevent additional portions of the beam from backscattering to the sensor.
The sensor, which receives the object and reference beams, is a three-CMOS prism-based area-scan color camera (JAI, AP-3200T-PGE 3-sensor). This sensor is unique as it has separate CMOS arrays (Sony, IMX265) for each color channel, i.e., RGB, where the colors are separated by coatings on the prisms as shown in Fig. 1. Each array is $2064 \times 1544$ pixels with $3.45\,\,\unicode{x00B5}{\rm m} \times 3.45\,\,\unicode{x00B5}{\rm m}$ pixel size. There are two benefits compared to conventional color sensors using a single array with a Bayer filter. First, the spatial resolution for each channel is enhanced because the absence of a Bayer filter means all pixels are used. Second, the prism coatings prevent so-called cross talk between the color channels, where light in one channel can contribute to another [41,42]. The sensor also has global shutter capability, which means the entire CMOS array for each channel is read out at once. For reference, the spectral response is plotted in Fig. 1(a), where each channel is labeled R-ch., etc., along with the lasers’ spectra, labeled as ${\lambda _{\rm{r}}}$, etc. Note how the response in each channel exhibits a relatively flat plateau in a region overlapping the associated laser spectrum.
In this configuration, the object beam consisting of backscattered light from the particles interferes with the reference beam to form three holograms, one for each color channel. These are denoted $I_{\rm{r}}^{{\rm{holo}}}$ for the red channel, $I_{\rm{g}}^{{\rm{holo}}}$ for the green channel, and $I_{\rm{b}}^{{\rm{holo}}}$ for the blue channel. Recall that these interference patterns also depend on ${\theta _{\rm{p}}}$ describing the (linear) polarization direction of the reference beam. Next, the particles are removed, which removes the object beam at the sensor, and measurements of the reference beam are made in each channel. These are called reference measurements, $I_{\rm{r}}^{{\rm{ref}}}$, $I_{\rm{g}}^{{\rm{ref}}}$, and $I_{\rm{b}}^{{\rm{ref}}}$. Finally, the difference between the reference and hologram measurements is formed to constitute contrast holograms [14] as $I_{\rm{r}}^{{\rm{con}}} = I_{\rm{r}}^{{\rm{ref}}} - I_{\rm{r}}^{{\rm{holo}}}$ and so on.
3. IMAGE RENDERING
It is now well established that an image of the objects (particles) producing the object beam can be computationally rendered from the interference pattern of the contrast hologram, e.g., see [18,23]. In the so-called in-line configuration, the hologram is produced by interference of the portion of the illumination beam that scatters from a particle, around the forward direction, with the unscattered portion of the same beam [14]. The image is then rendered from the hologram using Fresnel diffraction theory (FDT) [23]. Here, however, the hologram is formed by interference of backscattered light with the reference beam. The configuration is still in-line as the interfering beams co-propagate along the same axis, and an image can be rendered using the same FDT [36].
Figure 2 depicts the image rendering process. The axes in this figure correspond to those in the experimental arrangement, Fig. 1. A particle is illuminated by an incident beam propagating along the negative $z$ axis. A small portion of the light is backscattered, which along with the reference beam, propagates along the positive $z$ axis to interfere across any of the sensor’s CMOS arrays. An array’s surface is denoted by ${{\cal S}_{\rm{h}}}$. Each array is characterized by an $N \times N$ square array of $\Delta x \times \Delta x$ square pixels. In actuality, the sensor arrays are rectangular, i.e., $2064 \times 1544$ pixels. However, when the image is read from the sensor, it is symmetrically cropped relative to the array center to retain only the $N \times N$ portion, i.e., $1544 \times 1544$. Note that the distance between the particle and sensor is $z = d$, which will be a slightly different value for each array due to the different locations of the arrays in the sensor.
Fig. 2. Image rendering process from a contrast hologram. Light backscattered from a particle interferes with reference light across one of the sensor’s CMOS arrays, denoted by the ${{\cal S}_{\rm{h}}}$ plane, to form a fringe pattern. The separation between the particle and sensor is approximately $d$. The pattern is sampled by the array’s pixels, digitized, and then supplied to the Fresnel diffraction integral, Eq. (1). Evaluation of the integral across ${{\cal S}_{\rm{h}}}$, carried out in primed coordinates, for points across the ${{\cal S}_{{\rm{img}}}}$ plane, using un-primed coordinates, yields the diffracted wave amplitude, $K$ of Eq. (1). The absolute value of this amplitude equates to a silhouette-like image of the particle if $z = {z_{{\rm{img}}}} \approx d$. The size of the sensor’s pixels is $\Delta x^\prime $, whereas the size of the pixels in the image plane is $\Delta x$; these values are equal when Eq. (3) is used.
Once the interference pattern in each color channel has been digitized and the contrast holograms have been formed as explained in Section 2, each is used in the Fresnel diffraction integral to render an image. This integral is the Rayleigh–Sommerfeld solution to the Helmholtz scalar wave equation describing diffraction simplified under the Fresnel approximation [23,43]. Specifically, it reads
A. Speckle Suppression
Before discussing how the images from the various color channels are combined to render a color image, the problem of speckle noise should be briefly reviewed. The highly coherent nature of light sources makes such noise inherent to imaging with digital holography. Indeed, for the red laser, being a He–Ne means that its coherence length is likely of the order of a meter or greater. For green and blue lasers, it is estimated that their coherence length is far smaller, on the scale of millimeters, but still much larger than the particles studied. Speckles are due to the interference of portions of laser light scattering from random, rough features on the surfaces intercepting the beam. In the context of this work, speckle noise manifests as grain-like domains of sharply varying intensity several pixels in size that are randomly distributed across the particle image. Speckle grains exhibit several statistical characteristics that are related to the optical arrangement. As explained by Bianco et al. [44], the mean size of a speckle grain $\Delta \eta$ in terms of the sensor pixel size $\Delta x^\prime $ is
Note from Eq. (4) that the speckle grains are wavelength dependent, and thus, will differ between the color channels. In this work, the grains are $\Delta {\eta _i} \lt 10\Delta x^\prime $ for all channels.Fig. 3. Examples of speckle and its reduction in backscatter MWDH. (a) Backscatter color contrast hologram $I_{{\rm{rgb}}}^{{\rm{con}}}$ formed by the white-beam spot on a reflectance standard in place of the window in Fig. 1. Only a single polarization direction for the reference beam is used here, ${\theta _{\rm{p}}}{= 0^ \circ}$. (b) Image with speckle. Reconstruction of beam-spot image from the hologram in (a). (c) De-speckled image after all polarization directions are included, Eq. (5), and the filtering of Eq. (6) is applied.
Speckle noise in digital holography is a well-studied topic, and numerous methods are available to suppress the effect to varying degrees [44–51]. The approach taken here relates to the varying linear polarization directions produced by the rotating polarizer following the QWP in Fig. 1. For each polarization direction, the speckle distribution in a given channel’s particle image changes slightly because reflection (or backscattering) is polarization dependent. Due to the random nature of the distribution, it is then plausible that averaging the images as the polarization is changed will reduce the overall impact on image quality. This is the reason that the polarizer is mounted in a motorized rotation stage. A set of holograms and reference measurements is taken as the polarizer rotates ${\theta _{\rm{p}}}$ from 0° (positive $x$ axis) through 180° in 20° steps. Recall that because the polarization state of the white beam is circular following the QWP, the rotating polarizer does not alter the beam intensity. Then, contrast holograms are formed for each color channel, and polarization direction and images are rendered via Eq. (3) for each. The $i$th channel’s average image is then given by
Even with polarization-based speckle suppression, significant speckle noise remains in the images. Further suppression of the speckle is possible by convolving a channel’s image $| K |$ from Eq. (5) with a two-dimensional Gaussian kernel $G$ to generate a de-speckled image $| {K^\prime} |$ as
To illustrate the effect of the de-speckling method, Eq. Eq. (5), the gray-level images corresponding to the RGB color channels are combined to form a color image. This is done by basic color addition where the red channel’s (gray-level) image is colored red and then added to the green image colored green and then to the blue image colored blue. The luminance of each color is defined by the gray level of the corresponding channel’s image. For example, if a pixel in the red channel’s image has a gray level of zero, which is black, the luminance of red color assigned to that pixel is zero, again black. If the gray level is one, which is white, then the red color has maximum luminance, a value of one. This same procedure can be applied to the contrast holograms $I_{\rm{r}}^{{\rm{con}}}$, $I_{\rm{g}}^{{\rm{con}}}$, and $I_{\rm{b}}^{{\rm{con}}}$ to assemble a color contrast hologram, $I_{{\rm{rgb}}}^{{\rm{con}}}$. Figure 3(a) shows an example of $I_{{\rm{rgb}}}^{{\rm{con}}}$ measured when the particle sample is replaced by a white diffuse-reflectance standard. This color hologram is only one of the ${N_{\rm{p}}}$ holograms collected as the polarization direction is rotated; here, the direction is ${\theta _{\rm{p}}}{= 0^ \circ}$. The white illumination beam backscatters from the standard, and the resulting MWDH image is that of the beam spot on the standard. Figure 3(b) shows this image, again for only the single polarization direction, and when the blurring method of Eq. (6) is not applied. That is, this image displays full speckle, as does the hologram in Fig. 3(a). Finally, Fig. 3(c) shows the image after the full de-speckle method is applied. One can clearly see that the speckle effect is diminished to some degree and the beam-spot feature is enhanced.
B. White Balance
One challenge associated with combining multiple single-color (holographic) images to mimic a full-color image is white balance (WB). Simply, WB is the recognition that if equal-luminance single-color images are added, meaning RGB images here, the result should be a gray or white image depending on the specific luminance value. This is the motivation for using the reflectance standard in Fig. 3. The standard (Ocean Optics, WS-1) is a white surface with Lambertian characteristics, i.e., matte white, with a nearly uniform reflectance ${\gt}98\%$ over a broad spectral range of 400–1500 nm. Thus, the addition of the color channel images in Fig. 3 should yield a beam spot with a white appearance. While the spot does appear white, the balance between colors is not perfect. For this reason, a WB procedure is justified, which will be especially important for later considerations when particles are examined.
An important factor that contributes to the need for WB is that the lasers are not equal in power [52]. This results in $I_{\rm{r}}^{{\rm{con}}}$, $I_{\rm{g}}^{{\rm{con}}}$, and $I_{\rm{b}}^{{\rm{con}}}$ with differing magnitude features producing a color image with unequal color addition for an object that is otherwise white. In other words, the three beams do not constitute an ideal true-white-light source. A simple WB method is developed to correct for the unequal laser power and is shown in Fig. 4. Here, the de-speckled image in Fig. 3(b) is analyzed in terms of the probability distribution function (PDF) for each color channel in the image. The PDF is a way to visualize the distribution of RGB pixel values in a given color channel’s image. In Fig. 4(a), the distribution of pixel values clearly shows that the mean pixel value for the red channel is greater than the other channels. The WB shifts the pixel values to the average of the means of each channel’s PDF. Figure 4(b) shows the PDFs after this shift of pixel values along with the resulting image. Note that this WB is not as sophisticated as others used in the literature for related applications [32,53].
Fig. 4. Demonstration of the simple white balance (WB) method. (a) Before WB. Probability distribution function (PDF) for the pixels in a backscatter color hologram image for the white beam on the reflectance standard; the image is included inset. Here, one can see that the mean of each color channel’s PDF is different. (b) After WB. Same distribution after the WB procedure along with its effect on the beam-spot image. Both images are de-speckled.
To further illustrate the cumulative effect of the de-speckle and WB methods, consider Fig. 5, which shows how the hologram-derived images are layered to generate a colorized image. Here, the reflectance standard is removed, and the AR-coated window in Fig. 1 is reinstalled. A small amount of white polyethylene micro-spheres is spread across the window. These spheres (Cospheric LLC, WPMS–1.00 45–53 µm–5 g) are non-fluorescent and range from $45 - 53\,\,\unicode{x00B5}{\rm m}$ in diameter. The window is placed as close as possible to the former location of the reflectance standard, with the help of an $xyz$ translation stage, so that the focus distance $d$ in Section 3 is nearly the same, and thus, the same WB used for the reflectance standard can be applied. Figures 5(a)–5(c) show the backscattered contrast holograms in each color channel, $I_{\rm{r}}^{{\rm{con}}}$, $I_{\rm{g}}^{{\rm{con}}}$, and $I_{\rm{b}}^{{\rm{con}}}$, colored accordingly for the ${\theta _{\rm{p}}}{= 0^ \circ}$ polarization direction. While it is difficult to see, each hologram features a rich pattern of interference fringes. Figures 5(d)–5(f) show the particle images resulting from the application of Eq. (3) to these holograms, again for each color channel and a single polarization direction. In Fig. 5(g) is the result of layering the individual RGB images of Figs. 5(d)–5(f) via additive color mixing to form a single color image. This image shows strong speckle because only a single polarization direction is considered. Figure 5(h), however, shows the color image after all polarization directions are considered; the images are averaged via Eq. (6), and the simple WB is applied. Here, it is evident that the speckle is reduced, resulting in a notably improved image. Finally, Fig. 5(i) shows a microscope image of the same particles.
Fig. 5. Synthesis of color particle images via additive color mixing. (a)–(c) Individual color channel backscatter holograms, $I_{\rm{r}}^{{\rm{con}}}$, $I_{\rm{g}}^{{\rm{con}}}$, and $I_{\rm{b}}^{{\rm{con}}}$, respectively; white $50\,\,\unicode{x00B5}{\rm m}$ diameter spheres on the window in Fig. 1. The particle images resulting from reconstruction of these holograms are presented in (d)–(f), respectively. (g) Result of adding (d)–(f). Only one of the ${N_{\rm{p}}}$ polarization directions is considered for (a)–(g), and strong speckle is evident. (h) Result of applying the speckle suppression and WB methods, and (i) microscope image of the same particles under epi-illumination.
Figure 5 demonstrates several important aspects of MWDH. At the most basic level, it is clear that digital holograms can be formed from the comparatively weak light scattered from micro-particles around the backward direction, and this can be done for multiple particles at several wavelengths (R, G, and B) in the visible spectrum. While this has been demonstrated previously for a single wavelength in [36], in most work considering color digital holography of objects at this (small) scale, the holograms are formed with the significantly more intense forward-scattered light, i.e., the conventional in-line configuration, e.g., see [5]. Indeed, the forward-scattered light intensity for a $50\,\,\unicode{x00B5}{\rm m}$ diameter glass sphere illuminated by ${\lambda _{\rm{r}}} = 633$ nm light is four orders of magnitude greater than the backward-scattered intensity [54]. The resulting color image following partial speckle suppression and WB, i.e., Fig. 5(h), does represent the particle size and shape well in addition to a fair degree of correct color rendition. This is seen via comparison to the microscope image. However, it is clear that the color rendition is not optimal, meaning that the particles in Fig. 5(h) still display uneven pixels of R, G, or B, preventing a truly uniform white appearance. The reason for this is the fact that the speckle is wavelength dependent, and the de-speckle method used is not able to completely remove the effect. We remind the reader that more sophisticated speckle suppression can be considered, e.g., see [32,53,55,56].
4. CHROMATICITY ANALYSIS
A persistent difficulty with analyzing color images is to have a method that can characterize the color characteristics of one image to others in a quantitative way, in particular, a method that avoids the subjective nature of human vision as much as possible. Much work exists on the general topic of color science, color rendering, color perception, and the like, and the reader may consult [57–59] for context and details. The approach taken here is that of chromaticity, which has the advantage of characterizing the colors in an image in a way amenable to visual perception while simultaneously doing so with quantitative measures.
Because human color vision involves three types of color receptors, the cones, the perception of color exhibits a property known as trichromacy. This is the concept that given three sources of light, each being a primary color such as red, green, or blue, any other perceptible color can be rendered via simple mixing of these primaries, i.e., via color addition. The choice of primaries involves a degree of freedom; they may be nearly monochromatic like the lasers used here, or they may be sources of light with finite bandwidths. An important property of the primaries is that a mixture of any two cannot produce the same perceptual response (color) as the third. The color that one perceives can be described by three tristimulus values, denoted $X$, $Y$, and $Z$ by convention [60], which depend on the primaries, and describe the eye’s cone sensitivities. The tristimulus values amount to coordinates of a three-dimensional vector in a color space, where the direction specifies the color, and the length of the vector specifies the amount of said color, i.e., the color’s luminance. By normalizing the tristimulus values by their sum, a new set of values can be defined that does not depend on the luminance and specifies only the color. The result is a new color space with coordinates, denoted $x$, $y$, and $z$ by convention, and these are known as chromaticity coordinates. To avoid confusion with the spatial coordinates of Figs. 1 and 2, these will be specified as ${x_{\rm{c}}}$, ${y_{\rm{c}}}$, and ${z_{\rm{c}}}$ here.
Because they are normalized, i.e., ${x_{\rm{c}}} + {y_{\rm{c}}} + {z_{\rm{c}}} = 1$, only two of the chromaticity coordinates are needed to specify any perceptible color as a single point $({x_{\rm{c}}},{y_{\rm{c}}})$ in a two-dimensional plot, which is called a chromaticity plot. The color is represented in this way independent of its luminance, which is an advantage as one can analyze the colors present in an image regardless of the intensity of any given color. An example is shown in Fig. 6(a) where the chromaticity plot of the beam spot on the reflectance standard before WB, i.e., Fig. 4(a), is shown. Here, each pixel corresponds to a point on the plot where the corresponding color is indicated by the background pallet. There is a special point, called the white point, where the primaries balance to produce white. The large spread of points around the white point indicates that the image (inset) contains a diversity of colors other than white. Following the WB procedure above and a thresholding of the pixel luminance to remove the speckle noise background, the result is Fig. 6(b). One can see that the points are more tightly clustered around the white point, indicating that the image has been transformed to a more balanced color appearance. The luminance of each color point can be recovered from $Y$ and used as a vertical coordinate to produce the three-dimensional chromaticity plot in Fig. 6(c), where the points for both Figs. 6(a) and 6(b) are shown in black and red, respectively. Here, one sees that the majority of the points in Fig. 6(a) most spread from the white point belong to the dim speckle noise background seen surrounding the beam spot in the inset in Fig. 6(a). Thresholding of this noise and the slight shift of color of the surviving pixels produces the distribution of red points in Fig. 6(c).
Fig. 6. Chromaticity analysis of backscatter MWDH images. The distribution of pixel colors in the MWDH image of the beam spot on the reflectance standard is shown in (a) as points in the chromaticity plot. (b) Same plot following the WB and noise-thresholding operations. (c) Data of (a) and (b) together in a three-dimensional chromaticity plot where the vertical coordinate is the luminance of a given pixel’s color. Here, black dots correspond to (a) and red dots to (b).
5. PARTICLE DISCRIMINATION
The purpose of the chromaticity analysis in this work is not to render the most accurate color representation of a particle as possible with respect to human vision, but rather to enable a quantitative, objective method to discriminate particles of different colors. To explain how this is done, first consider Fig. 7. Here, non-fluorescent colored polyethylene microspheres $45 {-} 53\,\,\unicode{x00B5}{\rm m}$ in diameter are deposited on the AR-coated window in Fig. 1. The colors of the spheres include the white spheres of Fig. 5, red spheres (Cospheric LLC, REDPMS–0.98 45–53 µm–5 g), green spheres (Cospheric LLC, GPMS–0.98 45–53 µm–5 g), and blue/cyan spheres (Cospheric LLC, BLPMS–1.00 45–53 µm–5 g). Figure 7(a) shows the MWDH image reconstructed from a backscattered color hologram $I_{{\rm{rgb}}}^{{\rm{con}}}$ of a collection of these spheres following the de-speckle and WB procedures. For comparison, the same group of spheres as imaged with a microscope in epi-illumination mode is shown in Fig. 7(b). This is achieved without disturbing the particles or the window by imaging them via reflection from the PM in Fig. 1. When imaged this way, the lasers are shuttered, and the microscope’s own white-light source is used. Because the microscope’s light source is broadband with low spatial coherence, there is no speckle. One can see that while the quality of the MWDH image is not perfect and speckle noise remains, the image does successfully render a color image of the group where the different colors of the spheres are obvious.
Fig. 7. Color resolution in backscatter MWDH images. (a) MWDH image for a collection of colored $50\,\,\unicode{x00B5}{\rm m}$ diameter micro-spheres with the de-speckle and WB methods applied. (b) Microscope image of the same spheres under white-light epi-illumination. The colors of the spheres include white, red, green, and blue/cyan.
The success of the color-resolved image from a single $I_{{\rm{rgb}}}^{{\rm{con}}}$ measurement in Fig. 7 is more notable than it may appear. Recall that the hologram is formed from backscattered light, which is far less intense than forward scattered (or transmitted) light. Being approximately $50\,\,\unicode{x00B5}{\rm m}$ in diameter, these particles are significantly smaller than most objects in similar attempts at color imaging with digital holography, e.g., [53]. Perhaps the salient feature of Fig. 7(a) is that neighboring particles of different colors, which are separated by less than their respective sizes, can be distinguished without the color content of one interfering with the other. This can be seen most dramatically in Fig. 7(a) for the red and cyan particles.
Chromaticity analysis is applied to Fig. 7(a) to objectively discriminate particles of different colors in Fig. 8. Four sub-images are formed from Fig. 7(a) where each is cropped to display only a single particle from the right half of the image in Fig. 7(a). Although not required, the black background of each of these sub-images is removed to leave only the particle image without background. Each of these sub-images is shown inset in Fig. 8. The same process is applied to the microscope image of Fig. 7(b), and these are also shown inset. Then, points are plotted on a chromaticity plot for each sub-image, where the MWDH image corresponds to black points, and the microscope image corresponds to the red or green points. In each case, the points are clustered in a portion of the chromaticity plot that is representative of the overall color of the particle. The spread of points indicates a variety of colors are present, which can be verified by close inspection of the sub-images themselves. Overall, however, it is clear that particles of different colors are differentiable from the chromaticity analysis in terms of the region of the plot in which the majority of the points appear. Two striking examples are the red and green particles of Figs. 8(b) and 8(d), respectively. The points are relatively tightly clustered in the red and green portions of the chromaticity plot, and one could confidently assign each particle to one of the two colors without visual interpretation of the image. Notice that the darkness of a pixel color in the sub-images has no effect on the position of its corresponding point’s location in the chromaticity plot. In other words, the distribution of pixel points in the chromaticity plot is independent of the variation in brightness across the corresponding sub-image.
Fig. 8. Particle discrimination based on color with chromaticity analysis. Each plot presents the chromaticity points for the pixels of an individual particle in Fig. 7. (a) Blue/cyan particle, (b) red particle, (c) one of the white particles, and (d) green particle. Black points correspond to the MWDH images, shown inset in a black frame, and the red or green points correspond to the microscope images, shown in a dashed frame of the same color. Green points are used in (b) to provide contrast with the chromaticity diagram in the background.
Fig. 9. Volumetric discrimination of particles by color with backscatter MWDH. White micro-spheres $50\,\,\unicode{x00B5}{\rm m}$ in diameter are deposited on side 1 of the window in Fig. 5, and yellow chalk-dust particles are deposited on side 2. The particles are thus separated by 2 mm along the axial direction, i.e., the $z$ axis. (a) Image resulting from the backscattered color hologram where the focus distance in Eq. (3) is selected to bring this side into focus. White particles are evident, one of which is outlined in red. (b) Microscope image of the same region of the window, again focused (manually) to bring side 1 into focus. Chromaticity analysis is applied in (c) to the particle in the red outline in (a) and (b). (d)–(f) Same analysis is applied to side 2 where the chalk particles reside.
If a chromaticity plot of the pixels of the entire image in Fig. 7 were made, the result would be equivalent to combining the point distributions of all the plots in Fig. 8. In that case, the spread of points in Fig. 8, which is due to the remaining speckle noise in the sub-images, would make it difficult to know that there are multiple objects of distinctly different colors, i.e., without engaging an observer to make the determination. Yet, as long as the MWDH image has sufficient spatial resolution, it is usually possible to partition it into single particle sub-images and perform the chromaticity analysis without the need for an observer to subjectively assign particles to a given variety of color categories. It is in this sense that chromaticity analysis is regarded as a quantitative means for differentiating particles of different colors independent of how an observer may perceive the image.
6. VOLUME IMAGING WITH DISCRIMINATION
The reader may wonder what is the advantage of MWDH imaging over conventional epi-illumination microscopy, especially given the inferior image quality of MWDH in terms of both color rendition and spatial resolution, e.g., recall Fig. 7. The answer relates to a number of unique abilities of digital holography and the specific context in which the imaging is done.
Recall from Section 3 that an image can be rendered from a single color hologram in any number of image planes ${{\cal S}_{{\rm{img}}}}$ by simply re-evaluating Eq. (3) for different values of the focus distance $z = d$. Thus, if particles reside in different axial locations (different $z$), one can bring each into focus from $I_{{\rm{rgb}}}^{{\rm{con}}}$ post-measurement. This has led to a variety of volume imaging applications [61–64], and in the case of a single wavelength, Subedi et al. [22] show this can be done over an axial distance greater than 4 cm. For the same to be done with microscopy requires that multiple measurements (images) be taken with mechanical adjustments of the position of the microscope, or optical element within, to have a focused image for each axial location. For the stationary particles here, there is little practical advantage for MWDH in this regard, as it is straightforward to simply capture multiple images with the microscope. However, if the particles were moving, as would be the case in an aerosol, for example, imaging with a microscope would become highly challenging. This is because it is often difficult to control the trajectory of aerosol particles to flow within a volume confined to the microscope’s depth of focus. Moreover, the particles generally move too fast for a microscope to be mechanically re-focused.
Figure 9 demonstrates the advantage of MWDH in this regard. Here, different particle samples are placed on both sides of the window in Fig. 1, called side 1 and side 2. On side 1, which is closest to the sensor, the particles are the ${\sim}50\,\unicode{x00B5}{\rm m}$ diameter white micro-spheres of Fig. 5, and on side 2, an axial distance of 2 mm farther from the white particles are yellow-colored chalk-dust particles, with an average size of ${\sim}50 {-} 100\,\,\unicode{x00B5}{\rm m}$. A single backscattered color hologram $I_{{\rm{rgb}}}^{{\rm{con}}}$ is captured, and an image is reconstructed at the two different focus depths corresponding to the axial locations of side 1 and 2. The WB and de-speckling methods are also applied. Then, the particles on each side are imaged with the microscope. In Fig. 9, the first row considers the white particle on side 1 with Fig. 9(a) showing the MWDH image, Fig. 9(b) the microscope image, and Fig. 9(c) a chromaticity plot of the image pixels for the particle inside the red outline in the two images. The second row in Fig. 9 shows the same analysis for the chalk particles on side 2. Comparison of the MWDH image quality to that in Fig. 7 shows a degree of degraded quality, which is attributed to the increased speckle noise due to the greater concentration of particles used and the greater degree of particle-surface roughness of the chalk. Nevertheless, comparison of the MWDH and microscope images in Fig. 9 shows that holography is able to render both the morphology and color of the particles well enough that they can be associated unambiguously with the corresponding microscope images.
Examination of the chromaticity plots in Fig. 9 reveals that white particles are clearly differentiable from the yellow chalk, and the MWDH colors are close to those rendered by the microscope. Perhaps as important is the spatial color resolution. While Fig. 7(a) shows that particles of different colors are resolved for neighboring particles less than $50\,\,\unicode{x00B5}{\rm m}$ apart in the lateral directions, i.e., $x$ and $y$ axes, here, one sees that different colors are also differentiated well along the axial direction. That is, the yellow particles in Fig. 9(d) on side 2 do not prevent the rendering of a mostly white particle on side 1. Of course, if yellow and white particles were to be at the same lateral location on sides 1 and 2, then it would be unlikely that this degree of color separation would be seen, as the images would overlap. As a final note, Figs. 9(d) and 9(e) demonstrate that nonspherical particles are rendered well in terms of size and shape with MWDH, at least for particles of this size (approximately $50\,\,\unicode{x00B5}{\rm m}$).
7. DISCUSSION
This work has demonstrated the feasibility of imaging individual particles approximately $50\,\,\unicode{x00B5}{\rm m}$ in size and in color over a three-dimensional volume from a single backscattered color hologram. The significance of this demonstration lies in the fact that most applications of color digital holography consider far larger objects and that given the comparatively small size of the particles here, the backscattered light is orders of magnitude weaker than related demonstrations. Because of the backscatter nature of the measurements, speckle noise is most pronounced, and it significantly impacts efforts to WB the images. However, we emphasize again that the objective is not to develop the most perfect color image of particles, but rather to have enough color resolution in an image to discriminate particles of substantially different colors. This is achieved via chromaticity analysis (see Section 5). The unique ability of digital holography to bring particles at different axial positions into focus after the fact has useful applications. These would include imaging moving particles, such as aerosols, and the development of instruments capable of imaging particles in a contact free manner without the need to move optical elements. Our work adds color to these abilities and a method to distinguish particles of different colors in a quantitative, objective manner.
Funding
National Science Foundation (AGS-210775); Air Force Office of Scientific Research (FA9550-19-1-0078); Army Research Office (W911NF2120053).
Acknowledgment
We thank Robert Eastlund of Graftek Imaging Inc. for providing the JAI sensor spectral response data.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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