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Abstract
Due to the sequential nature of data acquisition, it is preferable to limit the number of illuminations to be used in tomographic diffractive microscopy experiments, especially if fast imaging is foreseen. On the other hand, for high-quality, high-resolution imaging, the Fourier space has to be optimally filled. Up to now, the problem of optimal Fourier space filling has not been investigated in itself. In this paper, we perform a comparative study to analyze the effect of sample scanning patterns on Fourier space filling for a transmission setup. Optical transfer functions for several illumination patterns are studied. Simulation as well as experiments are conducted to compare associated image reconstructions. We found that 3D uniform angular sweeping best fills the Fourier space, leading to better quality images.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Tomographic diffractive microscopy (TDM) provides 3D images of unstained samples’ complex refractive index (RI) distribution [1–5]. To do so, the object is scanned with varying illumination directions, with various diffracted fields collected and recombined to numerically reconstruct 3D images. The angular scanning helps to fill the spatial frequency support in Fourier space, leading to a synthetic aperture approach. However, changing the illumination direction while keeping the sample static results in a so-called “missing cone” of unrecorded object frequencies [2], which translates into poor resolution along the optical axis. Approaches such as illumination rotation followed by sample and/or numerical rotation [6–8], combination of transmission and reflection acquisitions in a $4\pi$ microscopy approach [9], mirror-assisted TDM [10,11], as well as learning and iterative algorithms [12–16] have been proposed to recover this missing information, and to improve optical sectioning and longitudinal resolution.
Because of the sequential data acquisition nature of TDM, for speed considerations, a limited number of illuminations is generally preferred (for some configurations, several acquisitions can be performed in parallel [17,18]). However, doing so may result in a lack of frequency components in Fourier space, which degrades the image quality. Furthermore, effective Fourier space filling is also highly dependent on the scanning scheme. Hence the choice of an appropriate scanning pattern is also critical for optimizing the number of angles to be used in TDM experiments.
Controlling the precise angles of illumination is not always straightforward and is highly dependent on the available technology. Several scanning technologies have been implemented in TDM. Rotating Pechan prisms [19] and tip/tilt mirrors [9,20] can be used, but mechanical movements are slow, and they can induce parasitic vibrations, detrimental for rapid acquisitions using interferometric systems, which are very sensitive. Fast steering mirrors [7,21] have been implemented for higher speed acquisitions. Fast galvanometer mirrors used in conjunction with a high-speed camera have even demonstrated an acquisition data rate corresponding to several 3D images per second [22]. Spatial light modulators (SLMs) [12,23] have also been used, especially due to their advantages of providing high mechanical stability, due to the absence of induced vibrations. Digital micromirror devices (DMDs) [24–27] provide the same mechanical advantages, combined with a higher modulation speed. Note that accurate retrieval, not setting, of illumination is in fact key for precise reconstructions: illuminations can present small departures from their design pattern, but as long as one can accurately compute the illumination angle from the acquired data, reconstructed image quality is not affected [28–30].
Fig. 1. (a), (b) Synthetic aperture process in TDM and final OTF shape in the (${k_x} - {k_z}$) and (${k_x} - {k_y}$) planes (gray). Influence of the number of illumination angles in Fourier space filling: (c) single hologram acquired with DHM. (d), (e) Shape of the OTFs in the (${k_x} - {k_z}$) plane using 50 and 100 illumination angles, respectively. (f) 3D OTF obtained considering 100 illuminations.
Authors often (but not systematically [9,30]) describe, albeit without always justifying, the illumination schemes in their experiments. 1D line scanning is fast, but does not permit isotropic gain in resolution [31,32]. Using LED arrays provides an easy way to obtain a 2D, but non-tunable, illumination pattern [33]. Circular patterns, using a scanning device [12,25] or from fixed position LEDs [34] have also been used, and spiral scanning has been implemented using galvanometer mirrors [14].
In this work, we therefore investigate illumination patterns, using a Fourier space filling factor as a metric. Simulations on phantoms and experimental tests are also conducted.
2. BASICS OF TDM DATA ACQUISITION AND RECONSTRUCTION
TDM provides the 3D complex RI distribution by combining 2D holograms acquired from multiple illumination angles [2]. In practice, tens [35] to hundreds [20,31,36] of interferograms have to be acquired, depending on the required image quality, and the reconstruction algorithms to be used. For weakly scattering samples, reconstruction algorithms under Born or Rytov approximations are valid (the interested reader is referred to, e.g., [1,3–5,37–40]). In such cases, for each illumination, object wavevectors ${{\boldsymbol k}_o}$, diffracted wavevectors ${{\boldsymbol k}_d}$, and the incident wavevector ${{\boldsymbol k}_i}$ are linked by the elastic condition
So for N acquisitions, one has $\text{N}$ ${{\boldsymbol k}_i}$ vectors, and N ${{\boldsymbol k}_d}$ and ${{\boldsymbol k}_o}$ vector sets, to be properly relocated in the Fourier space: the object waves are obtained from translation of the diffracted waves by the illuminating wave vector, i.e., ${{\boldsymbol k}_d}$ are translated by ${-}{{\boldsymbol k}_i}$.Due to the objective’s limited numerical aperture (NA), only subsets of the diffracted fields consisting of a cap of spheres (arc of circles in 2D) are collected. The relationship among the radius $R$ of the base of the cap (cord of the arc), the NA, and wavelength $\lambda$ is
The Fourier space synthetic aperture process for transmission mode TDM in 2D (${k_y},{k_z}$) and (${k_x},{k_y}$) planes (for the sake of simplicity) is depicted in Figs. 1(a) and 1(b), respectively. More detailed 3D animations can be found in the supplementary materials of [7].
In transmission digital holographic microscopy (DHM), one uses a single illumination, so a single cap of sphere is recorded, as shown in Fig. 1(d). The main drawback of DHM is its very poor axial resolution, due to the very limited set of recorded object waves [30]. Hence, the main objective of TDM is to enlarge the effective optical transfer function (OTF) via a synthetic aperture process, so as to increase the amount of collected spatial frequencies to improve the reconstructed image quality. This can be achieved by rotating the object or by illuminating the object at various angles, which is the configuration we consider here.
Using a small number of angles may result in a degraded reconstructed image quality. As examples, the synthesized OTFs using 50 and 100 angular illuminations are depicted in Figs. 1(d) and 1(e), respectively. In transmission, the 2D OTF takes the so-called “butterfly” shape, and so-called “doughnut” shape in 3D.
The final image quality is in fact highly dependent on image formation mechanisms [10,11], number of angles used for illumination [41], illumination scanning scheme, reconstruction algorithms, experimental noise reduction [21,42] and/or hypotheses relative to the sample’s shape [8].
This work focuses on the influence of the sample scanning scheme. We consider star-like, grid, circle, spiral, and “flower” scanning patterns, which have already been used in TDM experiments. With the aim of improving Fourier space filling, illumination patterns of Fermat’s spiral, 3D uniform distribution on a cap of sphere (UDCS), and 3D uniform distribution on a hemisphere (UDHS) are also introduced and implemented.
3. ILLUMINATION ANGULAR SCANNING
A. Scanning Patterns
Figure 2 depicts the various illumination patterns that we consider in this work, here with 200 illumination angles for each pattern. Star scanning [Fig. 2(a)] [20,30] and flower scanning [Fig. 2(i)] [7,21,42–44] have been used in our group for transmission or reflection TDM experiments (albeit this was not detailed in the above-cited papers). Grid pattern [Fig. 2(b)] [34], annular illuminations [Figs. 2(c) and 2(d)] [12], and spiral scanning [Figs. 2(e)–2(h)] [14] have also been successfully used by several groups. We also introduce in this context of TDM scanning: Fermat’s spiral [Fig. 2(j)], 3D UDCS [Fig. 2(k)], and 3D UDHS [Fig. 2(l)] illumination patterns.
Fig. 2. TDM scanning patterns under study: (a) three-axis star and (b) grid; annular of four concentric circles: (c) non-regular and (d) regular spaced points. The outermost blue star circle also represents the special case of annular scanning with only one circle at maximum NA; single spiral with four turns: (e) non-even and (f) evenly spaced points; double spiral with: (g) non-even and (h) evenly spaced points; (i) eight-petal rose with ${k} = {4}$ (${c} = {4}$, ${d} = {1}$); (j) Fermat’s spiral; (k) 3D uniform distribution on a cap of sphere pattern and (l) on half of sphere.
The various illumination patterns are generated using either their governing equations or possibly by careful re-sampling of the distributions of points. We have also chosen parameters for each pattern that could impact our results.
- • Star pattern, Fig. 2(a): the number of scanning directions (axes) is the main constraint. Here, three and four scanning directions are considered. The number of illumination is fixed by regularly sampling along the scanning directions. Note that 1D linear scanning [31] corresponds to the special case of a star with only one axis, but as such peculiar scanning cannot provide isotropic $(x - y)$ resolution, we do not consider it in the following.
- • Grid pattern, Fig. 2(b): we consider only evenly spaced illuminations, such as for example obtained when using a LED matrix located in the back focal plane of the illumination setup [33]. The total number of illuminations is therefore simply tuned by adjusting the grid spacing, and keeping only those angular positions that fit within the illumination NA.
- • Annular pattern, Figs. 2(c) and 2(d): the number of concentric circles and the distance between points are the determining variables. Here, two, three, and four concentric circles have been considered, with the same number of illuminations for each circle [Fig. 2(c)], or with same azimuthal spacing [Fig. 2(d)]. To achieve even azimuthal spacing, the ratio between the circumference of each concentric circle is used. Hence, the number of points in each circle ${p_i}$ can be computed from total number of points $N$ as
Single annular illumination at maximum NA has been introduced in [45] for simplifying data acquisitions. It constitutes a special case we have considered in [11], showing that, if properly performed, it permits a simplified approach for mirror-assisted tomography [10], allowing for a versatile transmission/reflection approach, as it transforms a reflection TDM setup into a transmission one, if special mirrored slides are used. This case is illustrated Fig. 2(d) with blue star points distributed along the outermost circle. This configuration records the highest amount of lateral high frequencies, but also leads to the most asymmetric OTF, capturing only the upper half of the “doughnut” [11].
where $t$ is the number of turns and $\theta$ the polar angle. The main constraints are the number of spiral turns, the single [Figs. 2(e) and 2(f)] or double [Figs. 2(g) and 2(h)] nature of the spiral, and the distribution of points along the spiral: evenly angularly distributed [Figs. 2(e) and 2(g)] or evenly spaced along the curvilinear abscissa [Figs. 2(f) and 2(h)]. To sample along the curvilinear abscissa, first the total length of the curve, $L$, is computed: Then instead of using constant $\Delta \theta$ as in Figs. 2(e) and 2(g), one computes $\Delta \theta$ for each point as To study the effect of these constraints, we perform simulations for two to six turns using both even and non-even sampling along the curvilinear abscissa, and for single and double spiral patterns. Double spirals are simply obtained by applying central symmetry to single spirals. where $k$ is the ratio of two integers $k = c/d$. It is the main parameter to change the nature of the flower, defining the number of petals, the width of these petals, and the overlapping/nonoverlapping of the petals. Roses with 35 various petal arrangements ($c \in [2,7 ]$ and $d \in [1,9]$), from three to 14 petals both overlapping and nonoverlapping, are tested.- • Fermat’s spiral, Fig. 2(j): this generates a uniform 2D pattern using the equations
Patterns depicted in Figs. 2(a)–2(j) are 2D ones, such as required when running an experiment by scanning a focused illumination spot throughout the condenser back focal plane, to be transformed by the illumination setup into a plane wave of appropriate inclination. We therefore also consider direct 3D angular distributions, such as one could obtain when using LEDs distributed onto a cap of sphere, as performed in ptychography experiments [47,48].
- • 3D UDCS pattern, Fig. 2(k): this is obtained by evenly distributing ${N}$ points over a cap of sphere of angular aperture corresponding to the considered objective’s NA. The minimal distance between two points has to be constant. There is no analytical solution for such a problem, so this is performed using the method explained in [49] commonly known as Tammes problem, limited to the cap of sphere. Then, these points are back-projected onto a 2D disk. The radius of the so-obtained disk being smaller than one, it is enlarged to one, for consistency with patterns previously described and depicted in Figs. 2(a)–2(j). This process allows to obtain the positions that would have to be addressed in the back focal plane of the condenser to actually perform such an illumination in a TDM experiment.
B. Evaluation Criteria
To compare the efficiency of these illumination patterns, we quantify the OTF “filling factor” FF% in Fourier space. It is defined as the ratio of the volume (in pixel counts), which is effectively filled by the cap of spheres using the scanning ${V_s}$ (Fig. 1) to that of the theoretical OTF volume ${V_T}$:
${V_T}$ is generated using parameters such as the NA of the objective and condenser lenses, the immersion medium RI, as well as pixel size of the camera sensor [7,9]. This reference OTF is a totally filled masks with doughnut shape in 3D, and whose 2D cross sections are shown Fig. 3(a). Note that this metric holds for Born as well as for Rytov approximations, as they are both based on elastic scattering and Fourier transforms.Fig. 3. 2D (${k_x}-{k_y}$) and (${k_x}-{k_z}$) representations of OTFs. (a) Theoretically, fully filled OTF. (b)–(g) OTFs corresponding to 600 illumination angles: (b) three-axis star; (c) rectangular grid; (d) annular pattern with four concentric circles; (e) spiral with six turns and regular spacing; (f) double spiral with six turns and regular spacing of points; (g) eight-petal flower with ${k} = {4}$; (h) Fermat’s spiral; (i) 3D UDCS; (j) 3D UDHS.
The filling factor gives an indication of how efficient a given scanning pattern is to fill the Fourier space, but does not quantify the reconstructed image quality. So, in addition to the FF(%), we also make simulations of TDM acquisitions, using the various scanning patterns, and compute the root mean square error (RMSE) of those reconstructed images compared to the test objects:
4. OPTICAL TRANSFER FUNCTIONS
A. OTF Construction
We consider an oil immersion (${n_{{\rm imm}}} = 1.515$) objective and condenser, of the same NA, ${\rm NA} = 1.4$, as in high-resolution TDM experiments [7,21,36,42,43]. First, we simulate the OTFs resulting from the scanning patterns presented in Section 3. Then, simulations using test objects are conducted.
Assuming an ideal objective, its coherent OTF being considered as constant and unity, the OTFs are obtained by filling Fourier space with caps of spheres, whose locations are derived from the scanning positions depicted in Figs. 2(a)–2(l) under the mechanism described in Figs. 1(a) and 1(b). We consider 600 illumination angles [20,31,36] in each case, while Fig. 3(a) recalls the theoretical transmission OTF, obtained if 100% filling could be achieved.
Scanning with the three-axis star pattern in Fig. 3(b) and eight nonoverlapping petals of a flower as in Fig. 3(g) allows for good (${k_x}-{k_y}$) frequency coverage, but privileged directions appear, which reflect those of the corresponding scanning patterns, Figs. 2(a) and 2(i). Note that the star pattern can be considered as a special case of a flower pattern, with zero-width petals. Along these privileged directions, an accumulation of low-frequency components is observed, seen on the (${k_x}-{k_z}$) cut of the 3D OTF. Using a grid illumination [Fig. 3(c)] allows for better acquisition of high lateral frequencies, but empty regions appear in the OTF, which reflect the periodic structure of the illumination pattern [Fig. 2(b)]. Annular scanning with four concentric circles [Fig. 3(d)] and six-turn spiral [Fig. 3(e)] and six-turn double-spiral scanning [Fig. 3(f)] also induce accumulation of low-frequency components, but without exhibiting privileged directions, due to their circular nature. Spiral illuminations, however, produce non-rotationally symmetric OTFs, which may be detrimental if imaging samples indeed exhibit such symmetries, such as beads or natural or artificial fibers, e.g., textile or optical fibers [8], or scanning near-field optical microscope (SNOM) tips obtained from tapered optical fibers [7].
Scanning with Fermat’s spiral, 3D UDCS, and 3D UDHS patterns allows for more regular filling [Figs. 3(h)–3(j)] without presenting privileged directions or peculiar structures, as can be seen on both (${k_x}-{k_y}$) and (${k_x}-{k_z}$) cuts.
B. Filling Factor
Figure 4 shows the filling factor, computed as a function of the number of illuminations for the scanning patterns described in Figs. 2(a)–2(l). Except for the rectangular grid, Fermat’s spiral, 3D UDCS, and 3D UDHS, each scanning scheme in fact depends on constraints such as number of axes [Fig. 2(a)], number of petals and their arrangements [Fig. 2(i)], number of circles [Figs. 2(c) and 2(d)] or turns [Figs. 2(e)–2(h)], as well as the distribution of points, being evenly or non-evenly distributed along the curvilinear abscissa. For simplicity, Fig. 4(a) therefore depicts only the curves obtained for those patterns giving the best results for Fourier space filling in their respective families: star pattern with three axes, flower pattern with eight petals (${k} = {4}$), regularly sampled double spiral along the curvilinear abscissa with six turns, and regular azimuthal spacing annular pattern with four concentric circles.
Fig. 4. OTF filling factor plots: (a) for the best in each considered scanning pattern family; (b) mean FF(%) of star, spiral, annular, and flower patterns with their maximal variations due to the generation parameters. (c) Same as (a), taking into account Hermitian symmetry.
Figure 4(a) shows the computed filling factor FF(%), considering from one to 600 illuminations. Note that for low numbers of illuminations, FF(%) is very similar for each scanning, but curves plot farther apart with an increasing number of illumination angles. This reflects the ability of some scanning patterns to fill the Fourier space without inducing much redundancy in the captured object frequencies when registering cap of spheres as explained in the synthetic aperture process (Fig. 1). On the contrary, some scanning patterns exhibit accumulations of the same captured frequencies, as can already be observed in Figs. 3(b)–3(j). 3D UDCS and 3D UDHS scanning always give the best FF(%). Annular, flower, Fermats’s, and grid scanning deliver very similar results in terms of Fourier space filling, while spiral and star scanning appear less efficient. Scanning along a single circle at maximum NA in fact delivers the lowest filling factor, because it fills only the upper half of the OTF ($0 \le {k_z} \le {k_{{\max}}}$) {see also Fig. 4(b) and corresponding discussion in [11]}, leading to high redundancy, resulting into a FF(%) of only 44% with 600 angles, for a theoretical maximum FF(%) of 50%. Note that in an actual experiment, redundancy can help to increase the signal-to-noise ratio by averaging those frequencies, which has been recorded several times [21,30,36,42].
Star, spiral, annular, and flower scanning represent families of patterns depending on various parameters. Hence, we plot in Fig. 4(b) error bars around the mean value of each family, giving the maximum and minimum values we obtained when computing FF(%) for all the studied members of this family.
The spiral pattern is characterized by more variations as a function of the constraints (number of turns, simple or double spiral, type of point separation). The minimum is obtained using a two-turn non-regularly sampled single spiral, the maximum being for a six-turn regularly sampled double spiral. For star patterns, a narrow difference is found between the maximum filling obtained for a three-axis star, and the minimum filling obtained with a four-axis star. Considering the so-called flower pattern family, an increase in the number of petals translates into a better distribution of illumination angles, which results in better filling of Fourier space with reduced redundant information. However, the arrangement of petals as nonoverlapping as well as overlapping, with more than eight petals, showed almost no difference in the filling factor. Considering the same number of petals, an odd number of petals results in higher redundancy compared to an even number of petals. The maximum and minimum filling are achieved using eight and three petals, respectively, in which the maximum difference between them is at 600 angles. Annular scanning for a various number of concentric circles induces a change in the filling factor, the maximum and minimum values being at four evenly sampled and two non-evenly sampled concentric circles, respectively.
Summarizing, in terms of Fourier space filling, 3D UDHS scanning patterns deliver the best results.
C. Special Cases
There are special configurations for which the OTF filling can be improved, taking benefits of peculiar properties of either the observed sample, the microscopic setup, or both.
In the first case, the observed sample presents morphological symmetries, for example, samples exhibiting top-down symmetry with respect to the focal plane, or rotational symmetries, for which tomographic data can be replicated in Fourier space to reflect the object’s symmetry. In the extreme case of axis-symmetric objects such as rods or fibers, a single holographic data set recorded with inclined illumination can even be sufficient to efficiently reconstruct the sample [8].
In mirror-assisted tomography, introduced by Mudry et al. [10], the special mirrored slide to be used induces a symmetry in the illumination system as well as in the detection setup. It has been shown that such a device acts in fact as a $4\pi$ system, originally introduced in the framework of diffractive tomography by Lauer [9]. Two transmission OTFs, as well as two reflection OTFs, can indeed be recorded using a single objective [11]. For samples that are mostly transparent and very weakly back-diffracting light, it was shown that a supplemental simplification is possible, and one annular illumination is then sufficient to acquire the entire transmission OTF [11], when properly taking into account both the acquisition system symmetries and the specific sample properties.
A third case also takes into account a sample’s optical properties. For example, for a purely transparent, non-absorptive sample, its spectrum must obey Hermitian symmetry [11,50]. Hence, if the scanning induces a final asymmetry in the OTF, as can be seen in Figs. 2(b)–2(j), this asymmetry can be compensated for by forcing Hermitian symmetry on the object’s spectrum [11,50]. Doing so results in a more filled spectrum, which is equivalent to considering an instrument having a larger or better filled OTF. This is of particular interest for annular scanning at maximum NA, with a doubled filling factor by forcing Hermitian symmetry. But applying symmetry is then also valid to improve the filling factor of the other scanning patterns, with a lower gain, but starting from higher values. Figure 4(c) shows filling factors in that case. The most remarkable feature is that annular scanning at maximum NA is now as efficient as 3D UDHS, which justifies using this simple scanning method when Hermitian symmetry holds.
5. IMAGE SIMULATIONS
Figure 5 illustrates the influence of the scanning pattern onto simulated image reconstructions of a synthetic sample. We consider four scanning patterns: 3D UDHS, with the best filling factor, Fermat, with average one, and star scanning and annular scanning at maximum NA, which give the lowest ones. The considered object is suspended in a background medium of RI ${n_0} = {n^{\prime} _0} + i{n^{\prime \prime} _0} = 1.49 + 0i$, and consists into a microsphere of diameter $2\,\,\unicode{x00B5}{\rm m}$ and refraction index ${n_1} = 1.46 + 0i$, and presents two absorptive inclusions of $0.5\,\,\unicode{x00B5}{\rm m}$ diameter, with complex RI ${n_2} = 1.46 + 0.02i$, and ${n_3} = 1.46 + 0.01i$, and located along the optical axis, at $z = \pm 0.5\,\,\unicode{x00B5}{\rm m}$, respectively.
Fig. 5. Image simulations of a synthetic object. $(x - y)$ plane sections (top rows) and $(x - z)$ plane sections (bottom rows). (a)–(e) Real part $n^\prime $ of the index of refraction. (f)–(j) Imaginary part $n^{\prime \prime} $. (a), (f) Original bead. (b)–(e), (g)–(j) Reconstructions using (b), (g) three-axis star, (c), (h) Fermat, (d), (i) 3D UDHS, and (e), (j) annular scanning at maximum NA.
Figures 5(a) and 5(j) depict cross sections in the $(x - y)$ and $(y - z)$ planes, for the synthetic object index of refraction real and imaginary part images. Those are simulated under Born approximation by simply filtering the object’s Fourier spectrum by the OTFs generated by the four considered scanning patterns, and using 60 illuminations. Figures 5(b)–5(d) and 5(f)–5(h) depict, for these scanning patterns, the cross sections in the $(x - y)$ and $(y - z)$ planes of the real and imaginary parts computed images.
Simulated images using star, Fermat’s spiral, and 3D UDHS illuminations are of similar, but increasing quality, as expected from their filling factors (Fig. 4). The absorbing inclusions are detected, and the shape of the bead is correct (star scanning images exhibiting low-level hexagonal-shaped artifacts). Annular scanning at maximum NA delivers the lowest quality images. This is attributed to its strongly asymmetric OTF (half the so-called theoretical doughnut), which translates into a mixing of real and imaginary parts when performing Fourier transforms, as can be seen in Fig. 5(e), where residuals of the absorption inclusions superimpose into the index of refraction image.
Fig. 6. Refractive index profiles along (a) $y$ axis and (b) $z$ axis obtained for: star, Fermat, 3D UDHS, and annular scanning at maximum NA.
Figures 6(a)–6(c) show profiles through the object, laterally and longitudinally, for the index of refraction $n^\prime $ as well as longitudinally for the imaginary part. For $n^\prime $, Figs. 6(a) and 6(b), the best estimation of the profile, as well as of the bead’s index, is obtained for 3D UDHS scanning, Fermat’s spiral giving the smoothest profile, and star scanning delivering a lower index estimation, while also exhibiting stronger residual oscillations. As for profiles, results are opposite for both inclusions, 3D UDHS delivering the best estimation for the weakest-absorption bead, but worst for the strongest one, the results being inverted for star scanning, Fermat’s spiral giving again the smoothest profile. In $(x - y)$ plane slices, errors also appear, due to the first-order Born approximation, as demonstrated in [37], but here are very weak, noticeable only on the star scanning image. Note also that a residual of the $2\,\,\unicode{x00B5}{\rm m}$ non-absorbing bead is visible around the absorbing inclusions on the images of the star and 3D UDHS scanning, but not on the Fermat scanning. This effect is, however, very weak. With annular scanning, index of refraction estimation is the poorest, the absorption also exhibiting large negative (with no physical meaning) values.
We conclude that annular scanning at maximum NA should be limited to samples for which Hermitian symmetry, or top-down symmetry, can be applied, combined with advanced reconstruction methods, contrary to mirror-assisted tomography, for which data can be recorded without a priori hypotheses [11].
To evaluate the efficiency of star, Fermat, and 3D UDHS scanning patterns, we also compute the RMSE Eq. (4) in a $25 \times 25 \times 25\,\unicode{x00B5}{\text{m}^3}$ cubic volume around the object, given in Table 1 for the RI and absorption. 3D UDHS scanning gives the overall lowest RMSE for these simulations.
Table 1. RMSE of Reconstructed Images for Refractive Index and Absorption in a $25 \times 25 \times 25\,\,\unicode{x00B5}{\rm m^3}$ Volume around the $2\,\,\unicode{x00B5}{\rm m}$ Bead
6. EXPERIMENTAL RESULTS
For experimental investigations, we use our previously developed TDM system, built from off-the-shelf opto-mechanical elements, and based on a Mach–Zehnder interferometer. A detailed presentation of this experimental setup can be found in [7] (supplementary materials) and [8,41]. We here use illumination at $\lambda = 633\;{\rm nm} $, angularly scanned with a Newport FSM 300 fast steering mirror. Two $100 \times$ NA = 1.3 oil immersion objectives are installed as condenser and imaging objectives, in a symmetric configuration. These are slightly lower NA objectives than in our previous experiments [7,20,30,41,43,44], but they present the advantage of a larger working distance, which simplifies handling of samples sandwiched between two high-magnification objectives, while still keeping a high NA, for which investigations presented in Sections 4 and 5 are still valid.
Beads with well-defined diameters are used as test samples (Sigma Aldrich 904465-2G monodisperse non-porous $2\,\,\unicode{x00B5}{\rm m}$ diameter silica beads). Note that while monodisperse beads are (by definition) well defined in diameter, depending on their fabrication method (from a Stöber–Fink–Bohn process), the index of refraction of the silica constituting them can greatly vary, from ${n_{{\rm bead}}} = 1.38$ to 1.45, while ${n_{{\rm bead}}} = 1.46$ for pure fused silica beads (see [51] and references therein). Beads are prepared between two $170\,\,\unicode{x00B5}{\rm m}$ thick coverglasses, embedded in Eukitt (Sigma Aldrich 03989-100ML) with index of refraction close to that of the coverglasses: ${n_{{\rm Eukitt}}} = 1.49$ to 1.50 [52]. The absorption of silica and Eukitt at $\lambda = 633\,\,\rm nm$ is negligible.
We first mimic fast acquisition conditions for star, Fermat, and 3D UDHS scanning by considering a low number of 60 illumination angles. By using such a low number of acquisitions, coupled to fast GPU reconstructions, one could indeed envisage a real-time combined acquisition, reconstruction, and 3D display of tomographic images, as presented in [41]. Such a system would not deliver high-quality images, but could be very useful for fast investigations of preparations in view of selecting samples of best interest, for which slower, but high-quality, high-resolution images could then be taken. For examples, our current system [7,8,41], as well as commercial ones [53,54], take up to a few seconds for data acquisition, and around 10 s for image reconstructions, which limits the speed at which a slide can be explored. But one has to ensure that such fast images would be of adequate quality.
Fig. 7. Refractive index images of a silica bead in Eukitt. (a) ($x - y$) and (b) ($y - z$) slices, along with index profile curves. Green: 3D UDHS. Black: Fermat’s spiral. Blue: star scanning patterns.
Figure 7 shows images of a silica bead in ($x - y$) and ($y - z$) planes, as well as index of refraction profiles along lateral and longitudinal axes. The 3D UDHS and Fermat’s spiral scanning give slightly better contrast images, which we attribute to the acquisition of more frequency components, as they present less redundancy than star scanning (Section 4). Star pattern scanning leads to slightly smoother images (less noise), as this redundancy translates into averaging those redundant frequency components. In this transverse plane, these three scanning methods, however, achieve similar lateral resolutions, as can be concluded from the index of refraction profiles. Since the beads are spherical, these profiles have been obtained by radial averaging in the ($x - y$) plane. Along $z$ axis, the situation is slightly different: images are noticeably affected by the “missing cone”; hence, the beads are significantly elongated along the optical axis, so that radial averaging cannot be used anymore. As an alternative, we averaged four different beads for each scanning pattern. One can note that star scanning delivers slightly more elongated, less contrasted images than Fermat or 3D UDHS scanning. One should therefore favor such more elaborated scanning patterns for TDM experiments. Beads are very simple test samples, so it would be interesting to also investigate more elaborate, index-calibrated, 3D phantoms such as described in [55].
As more complex test samples, we have therefore imaged Helianthus tuberosus pollen grains. Pollen grains from this family can constitute interesting 3D test objects, as they present structures at various scales, such as pores and external ornamentations, which range from large echinus (spikes of several micrometers) to small perforations ranging down to the 100 nm range. They also present inner structures such as double walls, the outer exine, and the inner intine, with the exine itself often being a layered structure, exhibiting columellae (pillars) between its footlayer and tectum [56]. Pollen grains are also often colored, and so present absorption. In that case, a simplified hypothesis or reconstruction algorithms neglecting absorption cannot be used, but absorption in itself represents an interesting [7,9,20,30,43,57–59], but often neglected, contrast mechanism in TDM. For such thicker samples, we used Rytov reconstructions (Figs. 8 and 9), which provide better results than using the Born approximation.
Fig. 8. Slices through a Helianthus tuberosus pollen grain TDM image. (a) Index of refraction $n^\prime $. (b) Imaginary part $n^{\prime \prime}$.
Fig. 9. External TDM refractive index images of the pollen grain presented in Fig. 8. (a) 600 illumination angles with 3D UDHS pattern, and 60 angles. (b) Star, (c) Fermat’s, and (d) 3D UDHS scanning. (e) Index contrast profiles along the $z$ direction through an echinus at locations indicated by colored arrows.
Figures 8(a) and 8(b) show slices at the same level through real and imaginary part images of a Helianthus tuberosus pollen grain. Note the very different aspects of the pollen in both images. The double-wall structure appears, with the intine (arrow) visible as a filled layer, while columellae (circle) are visible in the exine (arrow). A cavea in the exine is also visible (arrow), more easily distinguished in the index image than in the imaginary image. The wall structure, which is almost homogeneous in the index, appears on the contrary with very marked borders on the imaginary component image (rectangle). This suggests that the absorptive components (at $\lambda = 633\,\,\rm nm$) are concentrated in the outer region of the exine and in the intine layers: Fig. 8(b). This is different from what was observed in other types of pollen grains [7,30], while in [60,61], pollen membrane and inner-structure characterization in terms of index of refraction was reported, but not about absorption. The “bubble” attached to the pollen seems to correspond to ejection of inner material through the pore. Note the small, non-absorbing particles visible only in the index of refraction image (see also Visualization 1).
Figures 9(a)–9(d) show four outer views of this pollen grain (see also Visualization 2). Figure 9(a) is a high-resolution image obtained with 600 angles of illumination, while Figs. 9(b)–9(d) have been obtained using 60 angles of star, Fermat, and 3D UDHS scanning, respectively. Acquisitions have been performed sequentially, changing the FSM300 programming to ensure that no drift of the object occurs. Images are then post-acquisition reconstructed. This guarantees that the same voxel in each image corresponds to the same point of the object under examination. Note that the high-resolution image [Fig. 9(a)] clearly depicts large echinus (E), smaller perforations (P) at the basis of these echinus, as well as the smallest (here seen as 200–250 nm structures) pollen ornamentation (O) on the exine surface [62]. The 60-angle star, Fermat and 3D UDHS images are less resolved. Taking Fig. 9(a) as a ground truth, note that the 60-angle 3D UDHS image is marginally more detailed/contrasted (double arrows) than the 60-angle star and Fermat scanning images. This is confirmed by plotting a $z$-direction profile through one of the echinate (colored arrows): Fig. 9(e) shows that for the fast scanning images, the 3D UDHS image indeed depicts better contrast/resolution than the Fermat one, the star images being of lower quality. These results confirm those obtained with beads, that for fast imaging using a low number of illuminations, 3D UDHS helps to deliver more defined images.
For very fast acquisitions, one could even further decrease the number of illuminations. However, image quality then degrades very fast, rendering sample recognition difficult. Visualization 3 shows the same views as for Fig. 9, with 10 illuminations. Images are computed by selecting proper illuminations out of the 600 illuminations in Fig. 9, to mimic star, Fermat, and 3D UDHS with only 10 illuminations. The overall pollen shape is recognizable, but all smaller details, still visible with 60 angles, are now lost.
7. CONCLUSION
We have studied eight classes of scanning patterns in transmission TDM experiments, with the aim of optimizing Fourier space filling in the low-diffraction regime. From simulations, we have found that the 3D UDHS illumination pattern best fills Fourier space, leading to better quality reconstructions. Understanding and selecting a favorable sample scanning pattern in TDM experiments can greatly contribute in speeding up image acquisition by reducing the number of illumination angles needed to achieve the required image quality. These investigations can complement hardware optimizations, in view of improving performances of TDM systems and minimizing image reconstruction errors [63].
These predictions have been confirmed in experiments using glass beads and highly decorated pollen grains, even with a low number of 60 acquisitions, and most basic, direct inversion reconstruction algorithms based on Rytov or Born approximations, which are of the same computational complexity. Such a configuration would allow for a real-time 3D acquisition/reconstruction/display system [41].
Optimizing the illumination scanning pattern would also be of interest for improving acquisitions in other TDM configurations, such as combined illumination/sample rotation experiments [7,64], TDM in reflection [21,35,42], combined 4Pi transmission/reflection TDM [9] or mirror-assisted TDM [10,11].
Funding
Agence Nationale de la Recherche (ANR-18-CE45-0010, ANR-19-CE42-0004); Région Grand Est (FRCR 18P-07855, FRCR 19P-10656).
Disclosures
The authors declare no conflicts of interest.
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