1. Introduction

When a small particle, such as an aerosol particle, enters a collimated beam of light, a portion of the beam will be absorbed and scattered. If a sensor is positioned to face the oncoming beam, it will receive a net amount of radiant power that is less than what it would receive in the particle’s absence; this reduction is extinction [1,2]. In other words, extinction is regarded as the shadow cast by the particle, albeit the shadow may not be well-defined and dark, especially if the particle is comparable to the wavelength in size. Extinction is thus quantified by an area, the extinction cross-section $C^{\text {ext}}$ [2]. A closer examination of the physics involved shows that extinction is fundamental; it represents the conservation of energy in classical elastic electromagnetic scattering and is manifest by the interference between incident and scattered light [3–7]. Measurements of extinction are important in many contexts, such as quantifying the influence of aerosols on Earth’s radiation budget, remote sensing including lidar [8,9], and visibility in urban environments [10].

Perhaps surprisingly, there is a fundamental connection between extinction and holography. An in-line, or Gabor-type, hologram is formed by the interference of a particle’s scattered wave with the illuminating wave [11,12]. Recording the intensity of this interference pattern with an image-sensor is the basis of digital in-line holography (DIH), which provides a way to image particles in a contact-free manner and is now widely applied. Notice, however, that it is the same interference, i.e., of the the incident and scattered waves, that gives rise to the phenomenon of extinction. Thus, it should be possible to measure $C^{\text {ext}}$ from an in-line hologram. This connection was first proposed theoretically by Berg et al. [13] for spherical particles and later generalized to arbitrary-shaped particles and demonstrated experimentally in [14]. More recently, Ravasio et al. apply the method to mineral dust particles collected in Arctic ice-core melt water [15]. A related connection, discovered by Pettit et al. [16] and later expanded by Potenza et al. [17] and Moteki [18], permits measurement of the complex-valued forward-scattered amplitude of individual particles. By use of the optical theorem [3], the measurement then provides $C^{\text {ext}}$. However, that method relies on the Rayleigh-scattering approximation, which restricts its application to particles much smaller than the wavelength, although modifications for larger particles have been proposed, see [17].

To obtain $C^{\text {ext}}$ from a recorded hologram, an integral is performed. As explained later, if there is significant noise in the hologram, the integral may either diverge or yield an incorrect value. Unfortunately, it is common for such noise to be present in many practical situations. For example, DIH is performed on individual atmospheric aerosol particles using an airborne instrument by Kemppinen et al. [19] where the recorded holograms exhibit a large degree of noise. The noise is attributed to stray light, speckle, and electronic (sensor) sources, e.g., see Fig. 3 in [19]. While the extent of noise in that example does not significantly degrade the image quality, it would render the $C^{\text {ext}}$ measurement-method of [14] ineffective. The purpose of this work is to demonstrate a computational method, inspired by [20], to reduce the noise in holograms such that $C^{\text {ext}}$ can be better estimated.

2. Background

Consider a single particle in a collimated laser beam that illuminates an image sensor along the $z$-axis as shown in Fig. 1(a). The beam diameter is such that it mostly covers the sensor’s pixel-array, denoted $\mathcal {S}_{\text {h}}$, and the particle is located a distance $d$ from the sensor such that the sensor is in the particle’s far-field zone; conditions that are not difficult to achieve in practice. Two measurements are performed: First, a frame from the sensor is taken when no particle is present. This is called the reference, $I^{\text {ref}}$, and is simply an image of the beam profile across the sensor. In the simplest case, the profile is approximated by a planar wave of amplitude $E_{\text {o}}$, yielding a uniform intensity $I^{\text {ref}}=|E_{\text {o}}|^{2}=I_{\text {o}}$. Then, another frame, the raw hologram $I^{\text {holo}}$, is taken when the particle is present and scattered and unscattered light interfere, producing the hologram interference pattern. These frames are shown in Fig. 1(b) and Fig. 1(c), respectively, for a measurement where the particle is a $50$ µm diameter glass sphere and the beam wavelength is $\lambda =440$ nm. Next, the difference between these frames is formed, yielding the measured contrast hologram, $H^{\text {con}}_{\text {exp}}=I^{\text {ref}}-I^{\text {holo}}$, which is shown in Fig. 1(d). Note that it is important that $I^{\text {ref}}$ and $I^{\text {holo}}$ be measured with the beam intensity and sensor exposure-times being as close to the same as possible. If done, the pixel values of $H^{\text {con}}_{\text {exp}}$ should ideally oscillate symmetrically about zero, i.e., if no particle were present when $I^{\text {holo}}$ is taken, all pixels in the resulting $H^{\text {con}}_{\text {exp}}$ would be zero. Noise in $I^{\text {ref}}$ and $I^{\text {holo}}$ carry-over to $H^{\text {con}}_{\text {exp}}$ and become a source of failure to estimate $C^{\text {ext}}$. Significant noise and beam non-uniformity are present in Fig. 1.

 

Fig. 1. Arrangement and measurements used to estimate $C^{\text {ext}}$ for a single, fixed microparticle using DIH. In (a), an expanded laser beam illuminates an anti-reflection coated glass window with a single microparticle on its surface. Shown in (b) and (c) are the sensor frames taken without, and with, the particle present. These constitute the reference measurement and raw hologram, $I^{\text {ref}}$ and $I^{\text {holo}}$, respectively. Subtracting $I^{\text {holo}}$ from $I^{\text {ref}}$ yields the contrast hologram, $H^{\text {con}}_{\text {exp}}$, shown in (d). Note the non-uniformity in (b) and (c), and the noise present in (d).

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Figure 2 reviews the method in [14] to estimate $C^{\text {ext}}$ from $H^{\text {con}}_{\text {exp}}$. An integration is performed over the square portion $\mathcal {S}$ of $H^{\text {con}}_{\text {exp}}$ that is centered on the interference pattern. The size of this portion is parameterized by the angle $\theta$, which varies from $\theta =0^{\circ }$ to the angular size of the sensor $\theta =\theta _{\text {s}}$. For a given $\theta$ in these bounds, the integral value divided by the (constant) intensity of the beam, $I_{\text {o}}$, is

 

Fig. 2. Demonstration of the method of [14] to estimate $C^{\text {ext}}$ from a hologram. The plot shows the two $f$-curves resulting from the nested-squares integration of the contrast hologram of a $50$ µm diameter glass microsphere. In (a) is the measured hologram $H^{\text {con}}_{\text {exp}}$ with examples of the square integration areas $\mathcal {S}$ shown in blue. The angular size of these areas is indicated (in degrees) by the red arrows, which are also shown in the plot. The same is shown in (b) except for the cleaned hologram $H^{\text {con}}_{\text {cln}}$. The holograms in (a) and (b), respectively, yield the blue and red $f$-curves in the plot. Notice how the noise present in (a) causes the associated $f$-curve to diverge, while (b) results in a good cross-section estimate.

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Figure 2(a) shows the contrast hologram of Fig. 1(d) with nested squares indicating examples of the integration area $\mathcal {S}$ along with their corresponding angular sizes, $\theta$, which are also indicated by red arrows on the plot axis. Integrating this contrast hologram produces the blue $f$-curve shown in the plot, which quickly diverges (negative) due to the noise present. Obtaining a value for $C^{\text {ext}}$ fails in this case. The divergent behavior can also be positive, in cases, depending on the hologram’s noise content. Shown in Fig. 2(b) is the same hologram following the noise-reduction process described below that yields a cleaned hologram, $H^{\text {con}}_{\text {cln}}$. The same integration squares are shown. The (red) $f$-curve for $H^{\text {con}}_{\text {cln}}$ now shows the behavior described in [14]. Top and bottom envelopes, $f_{\text {top}}$ and $f_{\text {bot}}$, are fit as polynomials to the oscillating curve, from which a trend curve is obtained by averaging the envelopes. The trend curve begins at the first maxima of the $f$-curve, for the $\theta$ closest, but not equal to, $\theta =0^{\circ }$. It is this value of the trend curve that provides the hologram-derived estimate for the cross section, $C^{\text {ext}}_{\text {holo}}$ as shown in the plot in Fig. 2. Because the size and material of the particle is known in this case, Mie theory is used to calculate the true cross section, $C^{\text {ext}}$ and both the red and blue $f$-curves are plotted normalized by $C^{\text {ext}}$. Finally, Fig. 2 also reports the ratio $C^{\text {ext}}_{\text {holo}}/C^{\text {ext}}=0.96$, showing that the method provides the cross section to within $4{\% }$ error for this particle.

3. Method

The process to produce a noise-reduced contrast hologram begins by finding the center of the interference pattern. This step is required because the center defines the $\theta =0^{\circ }$ direction [Fig. 1(a)] around which the successive integration squares in Fig. 2 are defined to ultimately estimate $C^{\text {ext}}$, recall [14]. The center of the pattern also identifies the region in the reference measurement $I^{\text {ref}}$ where the intensity of the incident beam $I_{\text {o}}$ is found (more detail below). Consider Fig. 3(a), which shows the same $H^{\text {con}}_{\text {exp}}$ of Figs. 1,2, except in its full uncropped form, i.e., portraying the sensor’s full format, here rectangular. One can see a small region with concentric fringes due to the particle. Because it is easier to work with square arrays in code, the hologram is cropped symmetrically about the hologram’s center, resulting in Fig. 3(b). Next, an image of the particle is generated from the cropped hologram by first evaluating the Fresnel transform, $\text {Fr}_{d}\left \{\ldots \right \}$, to generate the diffracted wave amplitude $E^{\text {img}}$ that would be produced across a plane a distance $z=d$ from the hologram plane $\mathcal {S}_{\text {h}}$ by illuminating the hologram with a plane wave [12]. Referring to the geometry in Fig. 1(a), this amounts to evaluating:

 

Fig. 3. Steps used to crop and center the interference pattern in a contrast hologram. A contrast hologram $H^{\text {con}}_{\text {exp}}$ obtained from the sensor is shown in (a), which is then cropped square to yield (b). Next, the particle image $I^{\text {img}}$ is reconstructed by applying the Fresnel transform $|\text {Fr}_{d}\left \{\ldots \right \}|^{2}$ to the cropped hologram. The resulting image, shown in (c), is then cropped by a square window (red) to yield the zoomed-in view in (d). From this view, the center $(x_{\text {c}},y_{\text {c}})$ of a circumscribing circle is determined, which then allows (d) to be centered, resulting in (e). The cropped hologram, (b), is then centered via the same translation to yield (f).

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The process of generating the particle image is generally called image reconstruction and is explained well in many references, e.g., see [12] or [21]. In practice, one does not evaluate the integrals in Eq. (2) directly, but rather, evaluates them by means of Fourier transforms, Eq. (38) in [12], due to the speed at which such transforms may be computed. Open-source code is available in [12] to compute the Fresnel transform. To determine the value for $d$ required to produce a well-focused image, the reconstruction process can be iterated following a variety of so-called auto-focus routines, e.g., see [22].

The particle appears as a speck in Fig. 3(c) and the next step is to crop the image in an approximate manner to the point where the particle is clearly visible. That is done in Fig. 3(d), which shows the portion of Fig. 3(c) outlined by the red square. The center of the particle $(x_{\text {c}},y_{\text {c}})$ is then determined by circumscribing the particle image by a circle. With knowledge of the particle’s center, both $I^{\text {img}}$ and $H^{\text {con}}_{\text {exp}}$ can then be translated in the $x$-$y$ plane so that the image and the hologram fringe-pattern appear at the center of their respective cropped arrays as shown in Fig. 3(e) and Fig. 3(f). The extinction estimation method, Eq. (1), requires knowledge of the incident beam intensity $I_{\text {o}}$, which is now readily found by averaging the pixel values in the portion of reference measurement $I^{\text {ref}}$ contained within the circle determined from $I^{\text {img}}$. Due to the small size of the particles compared to the typical nonuniformity of the beam, e.g., Fig. 1(b), the intensity across the particle is uniform to good approximation.

Now consider Fig. 4(a) and Fig. 4(b) where the cropped, centered hologram and particle image are shown with scale bars to clarify the range of pixel gray-level variations. Although the particle image appears with good quality, apparently despite any noise in the hologram $H^{\text {con}}_{\text {exp}}$, the histogram of the $I^{\text {img}}$ pixel gray-levels in Fig. 4(c) clearly displays a distribution between zero (white) and 0.4 (black). The smaller gray-levels in particular are due to the hologram’s noise even though it is not apparent in the image, Fig. 4(b). A mask is then generated by selecting all pixels with gray-levels in $I^{\text {img}}$ below a threshold $\eta$ and re-assigning them to zero, while setting the remaining pixel values to one. In most cases, the particle-image pixel values in the histograms will be obviously separated from the background-noise pixels, which appear at the smallest values. Such is the case in Fig. 4(c) where the noise shows gray-levels smaller than 0.1. Note the log scale in the histogram, revealing that most pixels are noise. As such, the specific value chosen for $\eta$ is largely arbitrary as long as it is larger than the noise values but not so large as to exclude particle-image pixels, which appear at the largest gray-levels. Often a value for $\eta$ that is midway between the smallest and largest gray levels in $I^{\text {img}}$ will perform well. To illustrate the insensitivity of the analysis to the specific value of $\eta$, we note that increasing (decreasing) $\eta$ from its value of $\eta =0.2$ here by 50% results in a maximum change of 3% (-3%) in the eventual value for $C^{\text {ext}}$ obtained from the cleaned hologram.

 

Fig. 4. Demonstration of the noise-reduction method. In (a) and (b) are the hologram and reconstructed image of Fig. 3, except with scale bars revealing the pixel gray-level variations. A histogram of the pixel gray-levels is shown in (c). Setting pixels with levels $>\eta$ to one while re-assigning those $\le \eta$ to zero gives the mask $M$ in (d). Multiplication of $M$ with the $I^{\text {img}}$ gives the cleaned image $I^{\text {img}}_{\text {cln}}$ of (e) where the background noise is now removed. Here, the complex-valued image is plotted with colors representing phase and color-saturation representing magnitude. Finally, a cleaned contrast hologram $H^{\text {con}}_{\text {cln}}$ is generated via the inverse Fresnel transform in Eq. (8).

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Figure 4(d) shows the resulting mask $M$, which is essentially a binarization of $I^{\text {img}}$. Next, the mask array is multiplied element-by-element with the complex-valued diffracted wave amplitude corresponding to the image, i.e., $M\odot E^{\text {img}}$. The operation results in a new, complex-valued image amplitude $E^{\text {img}}_{\text {cln}}$ where the (complex) pixel-values within the particle-image portion of $E^{\text {img}}$ are unaltered, but all surrounding pixels, which may possess noise, are zeroed. Figure 4(e) shows a plot of $E^{\text {img}}_{\text {cln}}$ where each pixel is a complex number and the colors denote the phase while the color saturation denotes magnitude. The particle-image portion of the complex image has values with phase of approximately $-\pi /2$. The final step is to evaluate the inverse Fresnel transform of the complex image $E^{\text {img}}_{\text {cln}}$ to produce a new noise-reduced hologram in the original hologram plane, $\mathcal {S}_{\text {h}}$. The result, $H^{\text {con}}_{\text {cln}}$, is shown in Fig. 4(f) and earlier in Fig. 2(b). In cases where the particle in $I^{\text {img}}$ does not exhibit a sharp boundary with the background, it is advisable to slightly dilate, by several pixels, the particle-image feature in the mask. This is a conservative step to prevent the mask from clipping the particle image and adding unwanted aberrations to the resulting hologram. Such dilation is done in Fig. 4(d), but would likely be superfluous for larger particles where a sharper particle image-boundary would be expected.

We pause to explain several details regarding the generation of $H^{\text {con}}_{\text {cln}}$ from $E^{\text {img}}_{\text {cln}}$. Using the scalar-wave approximation [12], the distribution of light intensity $I^{\text {holo}}$ across the sensor $S_{\text {h}}$ in Fig. 1(a) when a particle is present is governed by the total wave amplitude $E(\mathbf {r})$:

Upon reconstruction, the first term in Eq. (6) generates the original object wave while the second term generates the conjugate of that wave. The image associated with the first term has the appearance of originating from a virtual object to the left of the hologram in Fig. 1(a). Similarly, the image associated with the second term is the real image appearing to originate from an object on the right of the hologram [24,25]. Thus, reconstruction will always contain both images overlapped due to the in-line configuration. When one is in focus, its twin is blurred. The masking process largely suppresses the blurred twin image in addition to the background noise.

Just as application of the Fresnel transform to $H^{\text {con}}_{\text {exp}}$ generates the wave amplitude corresponding to either of the twin images, application of the inverse transform $\text {Fr}^{-1}\left \{\ldots \right \}$ to the complex particle image-amplitude generates the (total) wave amplitude across the sensor. Taking the simple case where $|R|^{2}=1$, representing a planar reference wave, the measured hologram can be re-generated from the complex particle-image as

Notice, however, that no masking is done in Eq. (7). If masking is done, then half of the information in the re-generated hologram is lost, that due to the blurred twin, and a factor of two is required to bring the re-generated hologram into agreement with that which is measured, i.e.,

If the factor of two is not included in Eq. (8) and the cross section is estimated from the resulting cleaned hologram, the result will be half of the correct value.

To provide a clearer sense for the performance of the noise-reduction method, Fig. 5, top row, shows the degree to which noise is removed from the measurement depicted in Fig. 1(a) at wavelength $\lambda =440$ nm. Shown in (a) is $H^{\text {con}}_{\text {exp}}$ again except with a red line passing through the center of the pattern. In Fig. 5(b) is the reconstructed complex-valued particle image $E^{\text {img}}$, which better displays the noise present, i.e., the diffuse background surrounding the particle. In (c) is the noise-free hologram, $H^{\text {con}}_{\text {cln}}$, again with a red line through the center. Finally, in Fig. 5(d) is a plot of the hologram pixel values along the red lines, where $H^{\text {con}}_{\text {exp}}$ is shown in blue and $H^{\text {con}}_{\text {cln}}$ in black. Noise is evident in $H^{\text {con}}_{\text {exp}}$ and is mostly removed in $H^{\text {con}}_{\text {cln}}$.

 

Fig. 5. Performance of the noise-reduction method of Fig. 4 for two wavelengths, $\lambda =440$ nm (top row) and $\lambda =1040$ nm (bottom row). The plots in (d) and (h) show the noisy oscillations of the measured contrast holograms $H^{\text {con}}_{\text {exp}}$ in blue and the resulting noise-reduced holograms $H^{\text {con}}_{\text {cln}}$ in black. The color scale for plots (b) and (f) is the same as in Fig. 4(e).

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While leaving the particle undisturbed, the laser wavelength is then changed to $\lambda =1040$ nm. Repeating the noise-reduction method on the contrast hologram yields the data presented in Fig. 5(e)–5(h). Again, the method is effective at removing the noise in $H^{\text {con}}_{\text {exp}}$. Notice the increased spacing between the interference fringes, e.g., compare Fig. 5(c) and 5(g). This increase is due to the change in wavelength. One can also see a slight degradation in the resolution of the particle image from $\lambda =440$ nm to $\lambda =1040$ nm, i.e., compare Fig. 5(b) to 5(f). Such loss in resolution is characteristic of imaging with DIH [12].

Figure 6(a) shows the cross section values resulting from the noise-reduction method applied to the spherical particle above and two non-spherical particles. The latter are an ash particle collected from the eruption of the Cumbre Vieja volcano on La Palma, Spain in the fall of 2021, and an iron(III) oxide, $\text {Fe}_{2}\text {O}_{3}$, particle taken from stock powder of the compound manufactured by Sigma Aldrich. Images of the particles reconstructed from the measured holograms are shown in Fig. 6(b) along with a $50$ µm scale bar applying to all images. The cross sections determined from the contrast holograms, denoted $C^{\text {ext}}_{\text {holo}}$, are plotted against theoretical estimates of the true values, $C^{\text {ext}}$. Both $\lambda =440$ nm and $\lambda =1040$ nm are considered for each particle.

 

Fig. 6. Estimated extinction cross sections $C^{\text {ext}}_{\text {holo}}$, plot (a), obtained from the noise-reduced contrast holograms $H^{\text {con}}_{\text {cln}}$ of three particles at two wavelengths, $\lambda =440$ nm and $\lambda =1040$ nm. In (b) are the images $I^{\text {img}}$ of the particles reconstructed from their holograms. The scale bar applies to all of the images. The values of $C^{\text {ext}}_{\text {holo}}$ are plotted against theoretical estimates based on either Mie theory for the sphere, $C^{\text {ext}}_{\text {Mie}}$, or the geometric area of the image $C^{\text {ext}}\simeq 2C^{\text {geo}}$ for the volcanic ash and $\text {Fe}_{2}\text {O}_{3}$ particle.

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In the case of the $50$ µm diameter glass sphere, Mie theory is used to determine $C^{\text {ext}}_{\text {Mie}}$ using a refractive index of $m=1.526+0i$ for $\lambda =440$ nm and $m=1.507+0i$ for $\lambda =1040$ nm based on [26] and a particle diameter of $D=49.3$ µm, which is based on the manufacturer’s reported mean particle size. In the case of the ash and $\text {Fe}_{2}\text {O}_{3}$ particles, the theoretical cross section $C^{\text {ext}}$ is based on the so-called extinction paradox, i.e., $C^{\text {ext}}\sim 2C^{\text {geo}}$ where $C^{\text {geo}}$ is the particle’s area projected along the beam direction, its geometric cross section. This area is established by summing the pixels constituting the reconstructed particle image. The paradox value may be used when the beam is collimated, i.e., is essentially a plane wave, and the particle is much larger than $\lambda$ [27]. Such conditions apply here. The error bars represent $\pm 10{\% }$ error in both the theoretical and hologram-derived values for $C^{\text {ext}}$, which is based on the general error found in [14] for a wide variety of similar particles. In all, Fig. 6 demonstrates that the hologram-derived cross sections, $C^{\text {ext}}_{\text {holo}}$, are close to the theoretical values, within approximately $15{\% }-20{\% }$ error for most cases. Whether such error is acceptable or not would depend on the application. However, as is clear from the blue curve in Fig. 2, cross section values could not be obtained at all without use of the method here.

4. Discussion

One may wonder what effect the particle size has on the noise-reduction method, namely, the mask generation in Fig. 4(d). Several factors are important to consider. First is the image-resolution limit, which is often approximately $2p$ for a single-shot lens-free holography arrangement like Fig. 1 [12]. Particles smaller than $2p$ are not well-resolved and any resulting mask would not accurately represent the true particle size and shape. Second is the relative strength of the noise. Referring to the histogram in Fig. 4(c), if the noise gray-levels are not clearly separated from the particle-image levels [unlike Fig. 4(c)], then it may be difficult to choose a suitable $\eta$ value that prevents noise from appearing in the mask. This could happen if excessive noise is present or if the particle does not scatter with sufficient intensity to bring the interference pattern above the noise. Notice, however, that a surprisingly strong level of noise can be present, and an accurate mask is still produced, e.g., compare the curves in Fig. 5(h).

The aim of this work is to reduce the noise present in measured holograms for the purpose of rendering the extinction cross section method of [14] more effective. That is, our aim is not to necessarily improve the quality of the reconstructed image, although that does appear to occur to some degree. Indeed, there are many examples in the literature of effective methods to enhance the image quality, such as so-called pixel super-resolution [28,29] and twin image suppression [30]. Our need is to have a method that does not require more than one hologram measurement, which we achieve here, so that aerosol particles may eventually be considered. The important point is that with an aerosol, the particles are inherently in motion, and thus, the multiple hologram-measurements that are required in some image-enhancement methods (namely super-resolution) are not feasible for aerosols. In future work, we envision applying the noise-reduction method to aerosols and test its ability to provide acceptable values for single-particle $C^{\text {ext}}$. If successful, an new way to measure this important parameter would be available and may aid studies in aerosol science and remote sensing.