## Abstract

Digital holographic microscopy with compressive sensing (CDHM) has successfully achieved tomography and has been applied in many fields. However, the enhancement of axial resolution in CDHM remains to be elucidated. By deducing accurate formulas for the lateral and axial resolutions without paraxial approximation, we quantized the elongation effect of a digital holography (DH) system in this study. Thus, we revealed that the elongation effect, which is affected only by the system’s numerical aperture (NA), is an inherent property of DH systems. We present a detailed analysis herein on the physical significance of the coherence parameter, which is the ratio of a system’s limit axial resolution to the interlayer spacing more thoroughly than in previous research. Further, we achieved the tomography of a fiber by using a DH system with a 10 × microscope, with CS to eliminate the elongation effect, and experimentally validated our theoretical results. By applying these theoretical guidelines, we distinguished crossed fibers at distances of 36.4 μm and 48.5 μm, respectively, using the same experimental setup. There would be potential applications of this theory in tomography and observation of microscale objects in the areas of biological and fluid.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Direct three-dimensional (3D) observations of the distributions and motions of microscale objects is generally a challenging task owing to the limited depth of focus of commercial optical microscopes, which is disadvantageous for the temporal and spatial analysis of time-lapse microscopy images [1]. Scanning optical methods, such as tomographic phase microscopy [2], optical coherence tomography [3], and confocal microscopy, can image thick 3D samples at high lateral and axial resolutions but are still not suitable for tracking moving objects.

Digital holography (DH), which records 3D samples on a single-exposure
hologram rather than mechanical scanning, involves a high image speed and
provides an extended depth of field [4]. This greater depth of field is achieved by numerically
reconstructing the object images at an adjustable distance. Several
previous studies indicated that [5,6], a 3D object recorded
by a DH system is stretched along the *z*-axis; this is
referred to as the depth of field (DOF) elongation problem. In this study,
the elongation is attributed simply to the limited area and pixelated
structure of the camera sensor that discretely samples the object and
truncates the scattering wave-vector. Generally, the axial resolution is
considered to be two orders of magnitude lower than the lateral dimensions
owing to the elongation caused by the DOF. Researchers have theoretically
analyzed the influence of the camera’s discrete hologram sampling
capabilities on 2D reconstruction. However, the influence of the DH system
parameters has not been considered [6]. Reference [7] derives
the theoretical resolution of DH tomography. Under the assumption of
paraxial approximation, the truncation effect of the detector on the
scattering beams in both the frequency and spatial domains is analyzed.
This allows the expressions for the system’s numerical aperture
(NA), lateral resolution, and axial resolution to be derived.

In addition to improving the axial resolution algorithmically, it could be enhanced by adding a 4f amplification system to in-line compressive holography [8]. Although the axial resolution is increased by the square of the magnification, the magnification defined as the ratio of the focal length of two lenses is typically no more than five, which is much less than the magnification of commercial microscope objectives (MO). The axial resolution could be improved by illuminating the 3D samples with a spherical wave to magnify the diffraction signal for the detector [9]. While the axial resolution is improved owing to the high NA provided by the incident spherical wave, the effective NA is only one-third or one-quarter of the NA of microscope objectives. Moreover, 3D magnification causes a distorted reconstructed object cube because of the variable magnification factors along with the object depth. Hardware with a higher NA and homogeneous magnification could greatly support microscale tomography in compressive holography. The system’s resolution can be improved by inserting an MO into a DH system [10–12]. The DOF of the system can be reduced by an order of magnitude after inserting an MO, the minimum of which can double the object’s scale. The reason for this phenomenon is simply attributed to the magnification of the hologram by the MO, which allows more high-frequency information to be detected. This paper provides references for the application of DHM. However, to improve the accuracy and extend the applications of DHM, systematic investigation on the influence of system parameters on the influence of system parameters on the resolution without any assumption is required. Another method to improve the resolution is multiple angle illumination to obtain holograms from more than one direction [13–15]. Compressive digital holographic microscopy (CDHM) has reconstructed the red blood cell dynamics and microcirculation system of a transparent zebra fish larva [15]. This improvement of resolution relied on the CS algorithm, MO, and multi-angle illumination. Another report is on CDHM that realized tomography of a 90 nm nano-wire [16], the axial resolution of which attained 0.89 μm. Such high resolution relies on the 100x magnification capability of the MO. DHM applied in biology observed prey-induced changes in the swimming behavior of predatory dinoflagellates [17]. In fluids, DHM recorded the simultaneous measurements of 3D near-wall velocity and wall shear stress in a turbulent boundary layer [11]. However, these reports on the extent of DOF enhancement by MO are empirical and lack systematic deduction. An accurate formula for high NA applications is required.

The holographic imaging measures the 3D objective cube in a low dimension.
The 3D object data are compressed into a 2D hologram. If the object is
sparse in some areas, applying the CS algorithm can reconstruct more
information by compression along the *z-*axis. Compressive
holography (CH) [18–22] that employs a compressive sensing (CS) framework can extract
tomographic images with high fidelity from two-dimensional (2D)
holographic measurements. Theoretical and experimental results show that
the CS algorithm can eliminate the image disturbances from each
reconstructed layer, thereby enhancing both the lateral and axial
resolutions [21–24]. The axial resolution by
backpropagation of a single ideal bubble is equal to its theoretical value
[25], however, if the object is
represented as a line as in [26],
the CS axial resolution equals the theoretical value, which is one-sixth
of that attained by backpropagation (BP). Other researchers are obtaining
higher axial resolutions by using CH and CHM (Compressive Holographic
Microscopy) [16]. This comparison
indicates that complex samples and system configurations can influence the
efficacy of the CS algorithm. It is known that the axial resolution of
compressive holography theoretically depends on the sparsity of the object
[27]. The ability of CS to
eliminate noise is restricted by the coherence parameter of the system
[27,28]. The coherence parameter measures the extent of diffusion on
the imaging plane. Therefore, the efficacy of the CS algorithm is
influenced by the sparsity of the object and the system’s
configuration. The influence of DHM parameters on the coherence parameter
should be re-derived.

In this study, we combined in-line digital holographic microscopy and compressive sensing to accomplish high-resolution tomography from only a single-exposure hologram. This is termed compressive digital holographic microscopy (CDHM). An MO is employed to magnify the in-line hologram formed by the object cube; thus, high and homogeneous magnification of the object cube is achieved. Microscale tomographic images can be reconstructed with either high axial and lateral resolution by minimizing a convex optimization problem by total-variation (TV) regularization. We present an accurate formula for the DHM system’s lateral and axial resolutions and quantitatively analyze the enhancement of the resolution by MO. We analyze the physical significance of the CS coherence parameter for DHM. Finally, we achieve tomography of some continuous objects with a high axial resolution and verify our theory.

This paper consists of four sections. Section 2 presents the optical setup and reconstruction methods and the mechanism for enhancing the DHM system’s axial resolution by MO and CS. In Section 3 the experimental results are presented and discussed. Section 4 provides the conclusion.

## 2. Methods

#### 2.1 Optical setup and reconstruction methods

The experimental setup is shown in Fig. 1(a). The volumetric sample is illuminated by a collimated laser beam (632.8 nm). A bright-field MO (GCO-2102 Daheng, 10×, NA 0.25, working distance 7.316 mm, conjugate distance 195 mm) is used to produce a magnified optical field. The CCD is an MV-EM5 10M/C by Microvision Inc. that has a resolution of 2048 × 2048 pixels and a pixel pitch of 3.45 μm. Figure 1(a) also shows the thin-lens model of the DHM system. Figs. 1(b), 1(c), and 1(d) illustrate the superior properties of DHM compared to the DH system. These will be analyzed in detail in the next sub-section.

As illustrated below, the volume sample is placed near the working plane of the MO and the CCD is placed at the conjugate plane on the further side of the MO. The volume sample is then moved slightly further away from the MO. A magnified hologram is formed on the CCD plane and recorded by the CCD.

By using the first Born approximation the optical field on the working
plane of the MO (*z* = 0) can be
defined as

*Δ*is the lateral sampling pitch in the object space. The first term is the transmitted reference beam, and the second term is the scattered field of the object

*E*(

_{l}*nΔ*) located at a distance

*z*from the working plane. The “${\otimes} $” is the convolution operator and

_{l}*h*(

*nΔ,z*) is the angular spectrum-diffraction point-spread function with the form $exp \left( {jk\sqrt {{{({n\Delta } )}^2} + z_l^2} } \right)$.

_{l}A virtual hologram $I({m\Delta }
)$ formed on the working plane of the MO
would have an intensity of
*U*(*mΔ*),

The compound lens system of the MO can be treated as a perfect thin
lens, as shown in Fig. 1(a). The optical field at a distance
*d _{i}* on the image plane behind the MO is
related to the optical field

*U*(

*mΔ*) on the object plane by

*M*=

*d*/

_{i}*d*is the magnification,

_{o}*Δ’*=

*MΔ*is the sampling pitch of CCD, and $\alpha \textrm{ = }\frac{\pi }{{\lambda {d_i}}}\left( {1 + \frac{{{d_o}}}{{{d_i}}}} \right)$ [29] is the parameter of the quadratic phase factor exp[

*jα*(

*nΔ’*)

^{2}] introduced by the MO. The intensity distribution obtained by the CCD simply becomes the magnified hologram,

*I’*(

*mΔ’*) =

*I*(-

*mΔ’*/

*M*)/

*M*

^{2}. Assuming

*A*= 1, the magnified hologram on the CCD has a linear relationship with the volume object ${E_l}({n\Delta } )$,

*e*includes the zero-order, autocorrelation, and conjugate terms as well as other sources of noise associated with the measurement. The quality of the reconstructed image can be improved by mostly removing the zero-order term and background noise by subtracting a background hologram that does not contain objects. The CS framework is introduced to further remove the twin-images, suppress defocused noises, and achieve high resolution. A linear model relating to the vector form of the object ${\textbf E}$ and the hologram ${\textbf I^{\prime}}$ can be expressed in the following form:

*H*is the diagonal matrix of the

*h*Fourier transform. On the assumption that the edges of objects are sparse, Eq. (5) can be inverted by enforcing the sparsity constraint that adopts total-variation (TV) [30] regularization. Therefore, the object ${\textbf E}$ can be estimated by solving

The algorithm flow of TwIST is present as follows:

- (1) The initial value ${{\textbf E}_0}$ is obtained by BP:${\textbf E} = {\Gamma ^T}{\textbf I^{\prime}}$, and substitute it into Eq. (6) to get optimized function $F({{\textbf E}_0}).$
- (2) Substitute ${{\textbf E}_0}$ into Eq. (7) to get ${{\textbf E}_\textrm{1}}$ to eliminate the noise, then optimized function $F({{\textbf E}_\textrm{1}})$ is calculated; Compare the value of $F({{\textbf E}_0})$ and $F({{\textbf E}_\textrm{1}})$. If $F({{\textbf E}_\textrm{1}}) > F({{\textbf E}_0})$, the threshold shrinkage parameter is adjusted to 2d. Step (2) is repeated until $F({{\textbf E}_\textrm{1}}) < F({{\textbf E}_0})$, then update the number of iterations $t = t + 1$ and move to step (3).
- (3) ${{\textbf E}_t}$ is obtained by two-steps iteration with Eq. (8), then we get $F({{\textbf E}_t})$. Compare the value of $F({{\textbf E}_t})$ and $F({{\textbf E}_{t - 1}})$, if $F({{\textbf E}_t}) > F({{\textbf E}_{t - 1}})$, let ${{\textbf E}_0} = {{\textbf E}_{t - 1}}$ and step (2) is repeated again; move to step (4) if $F({{\textbf E}_t}) < F({{\textbf E}_{t - 1}})$.
- (4) Calculate the relative variations amounts of the optimized function value:$k = |{F({{\textbf E}_t}) - F({{\textbf E}_{t - 1}})} |/F({{\textbf E}_t})$. If $k > \sigma $, let $t = t + 1$ and jump to step (3); if $k < \sigma $, end the iteration, where $\sigma $ is the tolerance error.

#### 2.2 Enhancement of axial resolution by the microscope objective

In a DH system, the axial resolution is restricted by the limited area
and digitalized pixels of the recording element, usually a CCD or a
CMOS. In the DHM system shown in Fig. 1(a), the virtual hologram formed on the working
plane of the MO is magnified by the MO that also can be considered as
the recoding element, shrinks because of the MO. The reduced recording
element has the same number of pixels but a pixel pitch
*Δ* = 1/*M ·
Δ′*.

With the small pixel pitch, the reduced CCD can record high spatial
frequency components of the hologram, although by sacrificing the
recorded hologram size. However, it can be compensated for by reducing
the distance *z* from the object plane to the hologram
plane.

Hereafter, we discuss quantitatively the MO’s enhancement of the
system resolution, Generally, a plane wave parallel to the optical
axis is used to illuminate the object in a DHM system. The wave
vectors scattered by the object are distributed over a spherical
surface having a radius of 1/*λ*, as does the
red sphere shown in Fig. 2. Because the backscattering light cannot be recorded, we
only consider that *f _{z}* > 0. Most
recording elements have identical pixel distribution in the

*x*and

*y*directions, so if the passband of the recording element is 2Δ

*f*, the recorded wave vectors form a spherical crown shown with the height of Δ

_{x}*f*in Fig. 2.

_{z}The pixel pitch of the CCD shrinks, for it to record high spatial
frequency. The system’s NA can be large. The paraxial
approximation of $\textrm{sin}{\theta
_u}$ and $\textrm{tan}{\theta
_u}$ is ${\theta _u}$, which is acceptable when it is less
than 30°. In DHM systems if the aperture angle is larger than
30°, the expressions deduced by paraxial approximation [7] are not applicable, otherwise,
super-resolution results that are not correct maybe achieved, As shown
in Fig. 3, if the
paraxial approximation is adopted, the system NA’s
sin*θ _{u}* is larger than 1 when the

*θ*is larger than 55°, which is unrealistic.

_{u}Therefore, the NA of a DH system must be defined by:

where*W*=

_{x}*mΔ*is the lateral dimension of the reduced recording element,

*z*is the distance from the object plane to the hologram plane.

The system’s NA can also be expressed in the frequency domain:

where $|f |= {1 / \lambda }.$According to the equations above, the lateral resolution in the object
plane Δ*x*_{0} can be expressed by

*z*is determined by The axial resolution Δ

*z*

_{0}can be expressed as

If the sample has a feature size larger than
Δ*x*_{0}, it simply can be considered as
a linear combination of Δ*x*_{0}-sized
particles. Therefore, the DM system’s effect of elongating the
sample is similar to elongation by lateral resolution, i.e., the
length along the *z* axis of the sample is the
multiplication of the lateral size of the sample and the elongation
factor expressed by Eq. (12).

For a DH system with a microscopy, i.e., a DHM system, the imaging of the hologram is not only restricted by the DH system’s aperture, but also by the MO’s aperture. The DHM system is formed by a DH imaging system and an MO imaging system. The light distribution on the Fourier plane of the MO in Fig. 1(a) is:

where $\tilde{U}$ is the Fourier transform of*U*in Eq. (1). ${H_1}({{f_x}} )= circ({{{{f_x}} / {\Delta {f_x}}}} )$ and ${H_2}({{f_x}} )= circ({{{{f_x}} / {\Delta {f_{x0}}}}} )$ is the coherent transfer function of the DH imaging system and the MO imaging system, respectively. circ() is defined by:

*NA*

_{0}is the numerical aperture of the MO.

From the analysis above, we can conclude that the NA of the DHM system equals to the smaller one of the DH’s NA and the MO’s NA. The axial and lateral resolution of a DHM system can be calculated by the system NA.

The resolution is also restricted by the pixel pitch of the recording element being reduced. The maximum lateral resolution is Δ′, which equals the reduced CCD’s pixel pitch. The corresponding aperture angle is calculated by solving Eq. (9), and the maximum axial resolution is calculated by Eq. (11). The upper limit of the aperture angle of a DHM system is:

*θ*

_{u}and the wavelength

*λ*, as is the detectable lateral extent

*L*

_{0}. If the lighting wave is a plane wave, the lateral size of the detector element should be equal to the detectable lateral extent

*L*

_{0}. i.e., $\mathrm{\Delta }{x_0} = \Delta ^{\prime}$. If we assume that the number of pixels in a row is 800, the relationship of the detectable lateral extent

*L*

_{0}and the aperture angle

*θ*is shown in Fig. 4. When the aperture angle

_{u}*θ*is greater than 40°, the

_{u}*L*

_{0}is less than 300 μm. The lateral size of the typical state-of-art detector element is several millimeters. Therefore, to construct a high NA DH system that can use almost the full size of the detector element, an MO with a suitable magnification factor should be used. As the

*L*

_{0}approaches 189.9 μm when the

*θ*approaches 90°, a 10∼40 × MO is recommended. To achieve a high system NA, which can be much larger than the NA of MO, the distance

_{u}*z*should be shortened. If the

*z*is constrained by a cover glass or container, an MO with a smaller magnification factor is recommended.

We further explore the influence of the system’s NA on the
lateral and axial resolutions. The curves of the lateral and axial
resolutions relative to the aperture angle
*θ _{u}* are shown in Fig. 5(a). To show clearly the variance of
the system’s depth of focus from the NA, we calculate the ratio
of axial resolution to lateral resolution by changing with aperture
angle

*θ*. This is shown in Fig. 5(b) and also that, when using a small NA DH system, the axial resolution is much larger than the lateral resolution, which proves the elongation effect of a DH system. The depth of focus is much greater than the lateral resolution when the aperture angle is less than 20°. However, when the aperture angle is greater than 40°, the ratio of axial resolution to lateral resolution is less than 5. From the last paragraph, we know that for a system to have an aperture angle greater than 40°, an MO must be employed. When an MO is employed, the system’s depth of focus is shortened. The ratio reduces further when the NA is increased. The lower limit of the ratio is 2, because only the forward scattering light can be detected. Further, for each DHM system, the upper limit of the aperture angle is restricted by the magnification and pixel pitch of the CCD, as expressed in Eq. (12).

_{u}To conclude, the improved NA of the DHM leads to the enhancement of axial resolution. The reason why NA improves is that the microscale hologram is magnified by the MO. The way how the MO increases the microscale hologram equals to the way how a reduced scale of CCD working on the front working plane of the MO records a microscale hologram. The improvement of the NA enhances both lateral and axial resolutions, so that tomography of microscale objects can be achieved. Besides, along with an increase of the system’s NA, the axial resolution is improved, which demonstrates that the MO enhances the system’s resolution and depth of focus.

#### 2.3 Enhancement of axial resolution by compressive sensing

In compressive sensing, the coherence parameter is defined to quantize
the correlation between measurement and sensing matrices. In
holographic tomography of a 3D object, the coherence parameter
reflects the correlation between two object points having the same
lateral position in two adjacent layers [23]. The axial resolution of the system includes the
influence of the distance between the object and the detector and the
size of the detector. Therefore, in holographic tomography, the
coherence parameter is the ratio of the interlayer spacing
Δ*z* to the limit axial resolution
Δ*z*_{min} of the system

*z*can be obtained by substituting Eq. (13) into Eq. (11):

_{min}The ratio of the number of object features that can be accurately reconstructed to the number of detector pixels is bounded by [26]:

When Δ*z*equals axial resolution Δ

*z*

_{min},

*S*= 1. The sample in DH tomography is in a 3D cube, so, when the first layer of the sample is located at the position where the axial resolution on the first layer reaches the limit z resolution, the z resolutions of the other layers increase with the distance to the sensor. Therefore, the coherence parameter increases along with the increase of the distance, which means it is more difficult to eliminate the noises. This conclusion is in accordance with the analysis in Sec. 2.2, i.e., the elongation effect is more obvious when the aperture angle is smaller, which means more noise. In DH tomography, the purpose of the CS algorithm is to eliminate the elongation effect, make the reconstructed length of the sample on each layer equal the width.

## 3. Experimental results

To verify our theoretical results, we use the optical setup illustrated in Fig. 1, and focus on tomography of slim continuous fibers, rather than discrete particles or objects segregated in layers. First, we show the result of tomography on one fiber and discuss the results to verify the results theoretically calculated in the equations in Section 2. Second, we show that following the theory, we achieve tomography of several crossed fibers, with all the fibers separately and continuously imaged. The reconstructions were performed on a personal computer with Intel Core i5-4590 at 3.3 GHz and 16 GB of RAM. MATLAB R2016a with 64-bit application was used for algorithm implementation.

#### 3.1 Tomography of one single fiber and discussion

We record the magnified holograms of the background and melt-blown cloth fibers from a facemask, by using the optical setup illustrated in Fig. 1. The holograms are cropped to 800 × 800 pixels as shown in Fig. 6(a), and are then zero-padded with 20 pixels on all sides. The fibers’ cuboid space of 290μm × 290μm × 150 μm is split into 100 layers ranging from ${z_1} = 100\textrm{\; }\mathrm{\mu }\textrm{m}$ to ${z_{100}} = 250\textrm{\; }\mathrm{\mu }\textrm{m}$. It is indicated that, because the least aperture value of the system is larger than 50°, the approximation formulas presented in [7] are not suitable in this case.

In the TwIST algorithm, the regularization parameter is 4, and the
number of iterations is 80. Subsequently, the estimation of the 3D
object with the desired axial resolution by CS reconstruction, is
displayed sequentially in Visualization 1. The CS
reconstruction time is 523 s. Figs. 6(d) and 6(e) shows the results of BP and CS reconstruction at the
*z*_{5}, ${z_{50,}}$ and ${z_{95}}$ planes. Because the interlayer
spacing of 1.5 μm is much larger than the theoretical
elongated sample, the estimations of a 3D object by BP reconstruction
are disturbed by the defocused fields and the twin-images at one
hundred different *z* planes where it is difficult to
distinguish which portion belongs to the in-focus images of the single
fiber. Contrastingly, the results of reconstruction by CS display the
in-focus portion of the single fiber in each *z* plane
due to the effective reduction of the defocused contributions and
twin-images. Figure 6(b)
exhibits the noise intensities of the BP and CS reconstructions along
the intersecting yellow lines at the ${z_{50}} = 170
$μm reconstruction plane. The
noise level in the CS reconstruction is seen to be much lower than for
the BP reconstruction. More importantly, we quantify the improvement
of the depth of focus of the digital holographic microscopy (DHM)
system by calculating the intensity values of objects along the
*z-*axis from the BP and CS reconstructions. Without
loss of generality, the object portion of the imaging plane is
selected from the blue squares at the ${z_5}$, ${z_{50}}$, ${z_{95}}$ plane in Fig. 6 for which the background area is not
considered. The intensity values of the object portion in all one
hundred layers are then calculated. Finally, the curves of the
normalized average intensity along *z*-direction are
shown in Fig. 6(c) for
comparing the BP and CS reconstructions.

According to Eq. (13) the maximum aperture value of this DHM system is
calculated to be 66.55°. By counting the number of pixels in
the in-focus images of a single fiber, the diameter of the fiber
(approximately 13.8 μm; 4 pixels) is estimated. Subsequently,
we calculate the aperture angle values and the elongation effect of
the system on the layers. Based on that, we calculate the
*z* resolution of the elongated samples. In
Fig. 6(c), we show the
measurements of the full-width-half-maximum (FWHM) of the two curves
and compare the experimental *z* lengths of the
elongated sample with those obtained by calculation. The results are
shown in Table 1.

The elongated *z* lengths on the three layers of the
sample reconstructed by BP are consistent with those calculated by
using the theory presented in Sec. 2.3, thus verifying the elongation-effect theory. The
elongated *z* lengths reconstructed by CS are very
close to the diameter of the fiber, i.e., 13.8 μm. And the CS
algorithm enhances the *z* length on each layer by
factors of 3.8, 5.4, and 7.7 respectively, which correspond to the
elongation factors on the three layers. Both the two results show that
the CS algorithm eliminates the elongation effect caused by the DHM
system. Therefore, the axial resolution equals the lateral resolution
in this CHM system.

#### 3.2 Tomography of three crossed fibers

A tomogram of three crossed fibers from a piece of melt-blown cloth is demonstrated. By using the same setup, magnified holograms of the background and crossed microscale fibers are captured. The hologram with background subtraction of 640${\times} $640 pixels (including 20 zero-padding pixels on all sides) is shown in Fig. 7(a). We set the interlayer spacing to $\mathrm{\Delta }z = 2\; \mathrm{\mu }\textrm{m}$ in the object space. The crossed fibers’ 221 μm × 221 μm × 200 μm sized cuboid space is then divided into 100 layers ranging from ${z_1} = 330\; \mathrm{\mu }\textrm{m}$ to ${z_{100}} = 530\; \mathrm{\mu }\textrm{m}$. The regularization parameter and the number of iterations are kept the same as in Sec. 3.1. The CS reconstruction time is 532 s. Figs. 7(e) and 7(f) depict the results of BP and CS reconstruction at the ${z_{50}} = 430\; \mathrm{\mu }\textrm{m}$ plane. Figs. 7(c)–7(d) shows the intensity distribution along the intersecting yellow lines in (e) and (f), respectively. In Figs. 7(c) and 7(d), it can be seen that the distances between the crossed fibers in the green and blue squares are 36.4 μm and 48.5 μm, respectively. The result of the CS algorithm’s reconstruction is that the crossed fibers are clearly distinguishable. The depth-of-field extended images (DEI) reconstructed by BP and CS are shown in Fig. 7(g) and 7(h), respectively. The DEI reconstructed by BP in Fig. 7(g) has noise interference while the DEI reconstructed by CS in Fig. 7(h) is quite clear. We also produced a pseudo color 3D image of the DEI sample in Fig. 7(h), as shown in Fig. 7(i). In the overlapped areas, the lines have two colors. The segmented positioning along the skeleton method [32] is used in the DEI to solve the positioning problem of spatially continuous objects. The complete tomographic images by CS reconstruction are sequentially displayed in Visualization 2. From the 360° rotating view in the video, it can be observed that the three fibers are continuous and can be distinguished.

## 4. Conclusion

Without paraxial approximation, we deduced accurate formulas for the lateral and axial resolutions in the DH system. We found that the elongation effect is an inherent property of the DH system. In this study, the elongation effect was quantized. The upper limit of the system’s NA is determined by the recording element’s pixel pitch; while it is determined by the shrunken pixel pitch of the image sensor in a DHM system. By theoretical analysis, we concluded that, in a DHM system, the MO shrinks the CCD so that the NA of the DHM system is increased. By improving the NA both the lateral and axial resolutions are enhanced and the elongation effect is reduced while the reduced pixel pitch limits the system’s NA. We analyzed the physical meaning of the coherence parameter being the ratio of a system’s limit axial resolution to the interlayer spacing more thoroughly than in previous research. The experimental results verify our theoretical findings. We achieved continuous fiber and crossed fibers tomography by using CHM, with a 10 × MO. By applying CS, the elongation effect by the DH system was eliminated, and the crossed fibers at distances of 36.4 μm and 48.5 μm were separately visible. The theory and experiments proposed in this paper show the potential applications of CHM in many areas, e.g., water quality and air quality detections in environmental science, 3D imaging of neuronal cells and threadlike micro-creatures in biology, 3D imaging of fibers, etc.

## Funding

Shenzhen general research fund (JCYJ20190813172405231); Shenzhen International Cooperation Research Project (GJHZ20180929162202223); Cross-disciplines Project by Shenzhen International Graduate School, Tsinghua University (JC2017002).

## Disclosures

The authors declare no conflicts of interest.

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