[1,2]Gianlorenzo Massaro

[1]Dipartimento Interateneo di Fisica, Università degli Studi di Bari, Via
Giovanni Amendola, 173, Bari, 70125, Italy

2]Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Via Giovanni
Amendola, 173, Bari, 70125, Italy

# Assessing the 3D resolution of refocused correlation plenoptic images using
a general-purpose image quality estimator

<EMAIL_ADDRESS>*  [

###### Abstract

Correlation plenoptic imaging (CPI) is emerging as a promising approach to
light-field imaging (LFI), a technique enabling simultaneous measurement of
light intensity distribution and propagation direction from a scene. LFI
allows single-shot 3D sampling, offering fast 3D reconstruction for a wide
range of applications. However, the array of micro-lenses typically used in
LFI to obtain 3D information limits image resolution, which rapidly declines
with enhanced volumetric reconstruction capabilities. CPI addresses this
limitation by decoupling light-field information measurement using two
photodetectors with spatial resolution, eliminating the need for micro-lenses.
3D information is encoded in a four-dimensional correlation function, which is
decoded in post-processing to reconstruct images without the resolution loss
seen in conventional LFI. This paper evaluates the tomographic performance of
CPI, demonstrating that the refocusing reconstruction method provides axial
sectioning capabilities comparable to conventional imaging systems. A general-
purpose analytical approach based on image fidelity is proposed to
quantitatively study axial and lateral resolution. This analysis fully
characterizes the volumetric resolution of any CPI architecture, offering a
comprehensive evaluation of its imaging performance.

###### keywords:

3D Imaging, Correlation Imaging, Volumetric Imaging, Tomography

##  1 Introduction

Correlations of light have long been a subject of study due to their potential
to enhance the capabilities of traditional measurement techniques [1, 2, 3, 4,
5, 6, 7, 8]. In both classical and quantum contexts, the exploration of
correlations has led to significant advancements, particularly in imaging
technologies [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].
In the quantum domain, the unique properties of entanglement and correlation
have been harnessed to surpass the sensitivity limits of conventional imaging
methods [25]. This has enabled breakthroughs, such as sub-shot-noise
microscopy [26, 27], providing unprecedented precision in imaging amplitude
and phase samples [28]. Interestingly, correlation properties similar to those
obtained with quantum states of light can also be observed in classical
systems. This convergence of quantum and classical approaches has revealed
that many protocols initially designed for quantum applications can be
effectively adapted to classical contexts [9, 29, 30, 31, 32]. Consequently,
the study of correlations continues to bridge the gap between quantum and
classical imaging, offering versatile solutions that transcend the traditional
boundaries of these domains. Correlation plenoptic imaging (CPI) [33, 34, 35,
36, 37, 38, 39] is emerging as a promising correlation-based approach to
light-field imaging (LFI). LFI is a technique that allows for the concurrent
measurement of both light intensity distribution and propagation direction of
light rays from a three-dimensional scene of interest [40]. The extensive
amount of information collected by a light-field device enables single-shot 3D
sampling, a task that would require multiple acquisitions across various
planes with a standard camera [41, 42]. This scanning-free characteristic
makes light-field imaging one of the fastest methods for 3D reconstruction,
with applications spanning diverse fields such as photography [43, 44, 45],
microscopy [46] and real-time imaging of neuronal activity [47]. In its
typical implementation, light-field imaging employs an array of micro-lenses
positioned between the sensor and the imaging device (e.g., the camera lens).
These lenslets generate a series of “sub-images” corresponding to different
propagation directions. However, the presence of the array significantly
limits image resolution, preventing it from reaching the diffraction limit and
causing a rapid decline in resolution as 3D reconstruction capabilities
improve [48, 49]. CPI addresses the main limitation of conventional LFI by
decoupling the measurement of light-field information on two photodetectors
endowed with spatial resolution [33], in a lenslets-free optical design. In
fact, three-dimensional information about the sample is encoded in the four-
dimensional correlation function, obtained by correlating the instantaneous
light intensity impinging on the sensors and performing statistical averaging.
The correlation function can then be decoded, entirely in post processing, to
reconstruct high-resolution images of the object without the loss of
resolution typical of conventional LFI.

In this paper, we shall evaluate the tomographic performance of CPI, showing
that the reconstruction approach called _refocusing_[50] endows CPI with the
same axial sectioning capabilities of a conventional imaging system, in which
fine axial sectioning is obtained by increasing the size of the optical
elements. After a quantitative study of the axial depth of the reconstructed
images, an analytical approach based on the _image fidelity_ [51] shall be
proposed. Through the fidelity analysis, the axial and lateral resolution of
CPI be studied quantitatively, in a more general and formally sound approach,
compared to the past, and can be applied to evaluate the imaging performance
of any CPI architecture. Such mathematical tool will thus be used to fully
characterize, analytically and through numerical analysis, the 3D resolution
of CPI.

##  2 Imagine reconstruction in CPI

In CPI, light-field information about the sample is collected by measuring
intensity (or photon number) correlations on two detectors with spatial
resolution. The measured correlation function reads

$\Gamma(\bm{r}_{a},\bm{r}_{b})=\left\langle
I_{A}(\bm{r}_{a})I_{B}(\bm{r}_{b})\right\rangle-\kappa\left\langle
I_{A}(\bm{r}_{a})\right\rangle\left\langle I_{B}(\bm{r}_{b})\right\rangle,$
(1)

where $I_{A,B}$ is the instantaneous light intensity on the detectors [52],
and $\bm{r}_{a,b}=(x_{a,b},y_{a,b})$ are the two-dimensional coordinate on the
two detectors surface; the symbol $\left\langle I\right\rangle$ denotes the
ensemble average of the statistical quantity $I$. The constant $\kappa=0,1$
can vary according to the illumination source of choice for performing CPI: in
fact, when illuminating with entangled photons, light-field information is
collected by evaluating photon number correlations ($\kappa=0$) [53, 54, 55],
whereas, for thermal and pseudo-thermal light, intensity _fluctuations_ should
be measured ($\kappa=1$) [36, 55, 56, 34]. Without loss of generality, we
shall henceforth assume that CPI is performed with pseudo-thermal illumination
($\kappa=1$), and shall also neglect the four-dimensional dependence of the
correlation function, to limit ourselves only to the $x$-component of the
detectors coordinate.
Despite CPI can be implemented in many possible variations, from a fundamental
point of view, the mathematical description of the correlation function can be
explained by means of a second-order response function $\Phi^{\prime}$: in
fact, regardless of the particular architecture

$\Gamma(x_{a},x_{b})=\left|\int
A(x_{s})\int\left[A^{*}(x_{s}^{\prime})\right]^{m}\Phi^{\prime}(x_{s},x_{s}^{\prime},x_{a},x_{b})\,dx_{s}^{\prime}\,dx_{s}\right|^{2}.$
(2)

Eq. (2) establishes the relationship between the electric field at the
detectors coordinates $x_{a,b}$, and the field at the sample coordinates
$x_{s}$111Eq. (4) in Ref. [52]; the function $A(x_{s})$ represents the complex
electric field transmittance of a flat sample, and the coefficient $m$ can
either be $0$, if only one of the two detectors collects light from the
object, or $1$ if light from the sample illuminates both sensors [52]. In the
most general case, the optical response of CPI is thus strongly non-linear
with respect to the input function $A$, both because $A$ is involved twice due
to a second-order response function, and because of the square module taken
after the integrals. In many cases of interest, however, such degree of
complexity is not required to fully predict the optical performance of CPI.
For instance, when only one detector collects light from the sample ($m=0$),
the system is described by a _complex_ first-order response function (or
complex point spread function, PSF)
$\Phi(x_{s},x_{a},x_{b})=\int\Phi(x_{s},x_{s}^{\prime},x_{a},x_{b})$. On the
other hand, for CPI schemes characterized by $m=1$, a second-order response
function results from effects of partial coherence on the sample surface,
namely, from non-negligible coherence area on the object. Such effects,
however, can either be neglected, as is the case __ when the sample itself is
the source of thermal light, or should be avoided through careful optical
design, _e.g._ when working with transmissive samples. By neglecting partial
coherence on the source, the response function of the system becomes
$\Phi^{\prime}(x_{s},x_{s}^{\prime},x_{a},x_{b})\sim\delta(x_{s}-x_{s}^{\prime})\Phi(x_{s},x_{a},x_{b})$,
so that Eq. (2) can be written as

$\Gamma(x_{a},x_{b})=\left|\int
A(x_{s})\left[A^{*}(x_{s})\right]^{m}\Phi(x_{s},x_{a},x_{b})\,dx_{s}\right|^{2}.$
(3)

From Eq. (3), one immediately recognizes that the only difference between
architectures characterized by $m=0$ and $m=1$ is that the former are
sensitive to the phase content of the complex function $A$, whereas the latter
are only sensitive to the intensity profile of the object
$\left|A\right|^{2}$. This distinction does not have any influence on the
optical performance of the technique, so we shall limit our discussion to
phase-insensitive architectures, characterized by negligible coherence area on
the sample surface.

To some extent, some information about the sample can be inferred already from
the correlation function itself [52]; however, in order to gain axial
localization of the sample features and dramatically improve the signal to
noise ratio (SNR)[57, 58], the $\Gamma$ function must be _refocused_ to obtain
a sharp image of the sample. As outlined in Refs. [50, 52, 54], the refocusing
procedure is based on a reasoning that is based entirely on geometrical optics
arguments, _i.e._ on ray-tracing. Through refocusing, a specific object
coordinate $(x_{s},z)$ is reconstructed by summing together all the correlated
optical paths leading to the two detectors, crossing the object plane at axial
coordinate $z$ and transverse coordinate $x_{s}$. In CPI, the geometrical
locus of detector points corresponding to a single coordinate $(x_{s},z)$ is a
straight line in the $(x_{a},x_{b})$ plane, so that the reconstruction is
obtained through the line integral

$\Sigma_{z}(x_{s})=\int_{\gamma(x_{s},z)}\Gamma\,d\ell,$ (4)

where the integration path $\gamma(x_{s},z)$ has equation

$\gamma(x_{s},z):\alpha(z)\,x_{a}+\beta(z)\,x_{b}=x_{s}$ (5)

in the $(x_{a},x_{b})$ plane. As specified in Eq. (5), the axial plane that is
reconstructed is determined by the coefficients $\alpha(z)$ and $\beta(z)$,
which, according to the optical design of the experiment, can be associated to
the optical distances and and experimental parameters.

##  3 Axial sectioning

Because of the geometry of the integration path reconstructing a sample
coordinate $x_{s}$, Eq. (4) can be recognized as a Radon transform of the
correlation function at an angle
$\theta(z)=-\arctan\left[\alpha(z)/\beta(z)\right]$ so that, in the formalism
conventionally used for techniques based on tomographic reconstruction through
Radon transformation [59],

$\Sigma_{z}(x_{s})=R\Gamma(\theta,x_{s})=\int_{-\infty}^{+\infty}\Gamma\left(x^{\prime}\,\sin\theta+x_{s}\,\cos\theta,\
-x^{\prime}\,\cos\theta+x_{s}\,\sin\theta\right)dx^{\prime}.$ (6)

Such an analogy allows us to promptly infer that, as with any tomographic
technique based on back-projection reconstruction, the “depth” of the
reconstructed images can be expected to depend on the number of directions in
the Fourier domain available for the reconstruction [60, 44], or,
equivalently, on the length of the integration path $\gamma(x_{s},z)$ in Eq.
(4). In other words, the axial sectioning of CPI can be associated to the
extension of the support of the correlation function which, as demonstrated
both theoretically [52, 50] and experimentally [35], depends on the size of
the optics, namely, on the numerical aperture (NA).

Figure 1: Schematics of a single-lens CPI scheme

For definiteness, all the results presented in the paper are obtained on a CPI
architecture based on a single-lens design [55], schematically represented in
Fig. 1. In this architecture, the two detectors $A$ and $B$ are optically
conjugated, by means of an imaging lens, to two planes in the region of space
occupied by the sample. We shall indicate with $z_{A}$ ($z_{B}$) the optical
distance between the object plane imaged on detector $A$ ($B$) and the lens,
so that the plane is imaged with magnification $M_{A}$ ($M_{B}$), as defined
by the optical distances involved and the focal length $f$ of the lens. After
the lens, light coming from the sample is separated into to two optical paths
leading towards the detectors by means of a beam-splitter.

The tomographic capabilities of CPI are better understood through direct
comparison with a conventional imaging system, based on a single-lens design
and on intensity measurement (for instance, if only measuring the intensity
$I_{A}$ impinging on detector A). In such systems, the axial scanning of the
3D space surrounding the sample is performed mechanically, by varying the
relative distance between the detector and lens, so that a stack of different
planes at focus is available after the measurement. The quality of the axial
scanning depends on how effectively the imaging system suppresses, at any
given focusing position, spurious contributions from unfocused planes, namely,
on the thickness of the depth of field (DOF). Suppression of background planes
in conventional imaging is knowingly due to the circle of confusion (COC)
[61], a mechanism that is entirely explained in terms of ray optics, which
makes so that point-like objects in the background produce a circle-shape
projection onto the plane on focus. Such effect is increasingly more relevant
as the NA of the system increases, so that high-NA designs not only beneficial
because of the high resolution they provide at focus, but also because of
their shallow DOF [62]. We shall now demonstrate that CPI benefits from the
same high-NA design, as far as axial sectioning is concerned.

(a)

(b)

Figure 2: _Panel_ a.: measured correlation function in the case of a Gaussian-
slit object, placed on the optical axis at $z_{s}=(z_{a}+z_{b})/2$;_Panel_ b.:
comparison between a proper refocusing $\sqrt{\Sigma_{z}}$ with $z=z_{s}$ and
an incorrect refocusing at $z\neq z_{s}$

Fig. 2a reports an example of a measured correlation function
$\Gamma(x_{a},x_{b})$, obtained by simulating the optical setup in Fig. 1. To
obtain the correlation function, we set the focal length of the imaging lens
to $f=75\,$ mm; detectors A and B image their respective conjugate planes with
magnifications $M_{a}=1.1$ and $M_{b}=0.9$, thus resulting in $z_{a}\simeq
143$ mm and $z_{b}\simeq 158$ mm. For ease of calculation, we have assumed the
lens pupil to have a Gaussian apodization, with a width of $\sigma_{l}=25$ mm.
The plot is obtained by assuming that the imaged object is a Gaussian
transmissive slit having intensity profile

$\left|A(x_{s})\right|^{2}=\exp\left[-\frac{x_{s}^{2}}{2\sigma_{s}^{2}}\right],$
(7)

with $\sigma_{s}=200\,\mu$m. For the case shown in Fig. 2a, the object is
placed on the optical axis at a distance $z_{s}=(z_{a}+z_{b})/2$ from the
lens, so that neither A nor B yield a focused image of the sample. For this
CPI architecture, the object coordinate $x_{s}$ of a sample placed at a
generic axial distance $z$ is reconstructed by integrating along the line

$\gamma(x_{s},z):x_{s}=-\frac{z-z_{b}}{z_{a}-z_{b}}\,\frac{x_{a}}{M_{a}}+\frac{z-z_{a}}{z_{a}-z_{b}}\,\frac{x_{b}}{M_{b}}.$
(8)

Therefore, an object placed mid-way between $z_{a}$ and $z_{b}$ is correctly
reconstructed via a Radon transformation at an angle $\theta(z=z_{s})=-\pi/4$,
as is evident from the picture (the dashed green line is parallel to the
object features). As is the case for standard imaging, however, one does not
typically know _a priori_ the correct axial position of the sample. Hence, a
complete $z$-scan of the 3D space surrounding the sample is usually needed.
This entails calculating Radon transformations at many different angles
$\theta(z)\neq\theta(z_{s})$ which can be expected to result in a less sharp
image than perfect refocus $\theta(z_{s})$, and also to a suppression of the
intensity peak, as is the case for out-of-focus imaging in a conventional
scenario. Rather intuitively, this effect must be the analogous of the COC,
since the blurring of misfocused plane can easily be envisioned to depend on
how extended the integration paths are (as defined by the NA), and on how
misfocused the integration angle is with respect to the correct one. Such
picture can be graphically verified from Fig. 2a, where we reported the lens
boundaries (yellow lines, corresponding to the projection of the lens
coordinates at $\pm 3/2\,\sigma_{l}$ onto the $(x_{a},x_{b})$ plane), and two
integration angles corresponding to $z=152\,$ mm $\neq z_{s}$ (dashed blue
line), and $z=z_{b}\neq z_{s}$ (dashed red line). In fact, the extent of
blurring can clearly be expected to be directly proportional to both the
difference of the lines slope, with respect to the green line, and to how
large the NA of the lens is (distance between the yellow lines). This is
verified in Fig. 2b, demonstrating that the best reconstruction (green line)
happens when integrating at $\theta(z_{z})$ and results in an image
indistinguishable from the reference object (solid salmon area). For the other
integration directions (solid blue and red lines), slight and severe blurring
occur, respectively, so that the final reconstruction results in a blurred and
fainter image.

###  3.1 Quantitative analysis of the axial sectioning

From a qualitative point of view, the results shown in Figs. 2a and 2b hint
that the tomographic capabilities of the refocusing algorithm are related to
the COC. This can also be proved quantitatively, by studying how faithfully
can CPI reconstruct the image of an object placed at a given axial coordinate.
In order to do so, we shall use a newly introduced tool for evaluating image
quality, named the _image fidelity_ [51]. If an imaging system produces an
image $I(x_{s})$ of a sample with intensity transmittance
$\left|A(x_{s})\right|^{2}$, the fidelity $F_{A}[I]$ is defined as

$F_{A}[I]=\int\sqrt{I(x)\left|A(x)\right|^{2}}\,dx;$ (9)

where the quantities involved must be properly normalized ($\int
I(x_{s})\,dx_{s}=1$ and $\int\left|A(x_{s})\right|^{2}\,dx_{s}$). $F_{A}[I]$
saturates to unity for perfectly accurate imaging systems, namely
$I(x)=\left|A(x)\right|^{2}$, and approaches $0$ for very unfaithful imaging.
The analysis in terms of fidelity enables us to obtain _fidelity curves_ that
fully characterize the optical performance of the technique in terms of the
experimental parameters, and replace the concept of _resolution_ in imaging
modalities where such concept cannot be as clearly defined as in standard
imaging [51].

To fully characterize the analogies with the COC of a misfocused imaging
system, we shall compare the fidelity of a CPI refocusing $z$-scan directly
with the fidelity of the mechanical $z$-scan of a standard imaging system,
having the same optical design as a single arm of our CPI scheme. If an object
of size $\sigma_{s}$ is fixed at a distance $z_{s}$ from the lens, an axial
scan is obtained by moving the detector-to-lens distance so as to change the
$z$ in focus; in this context, the resulting image depends on both parameters
$z_{s}$ and $z$. Overall, the image resulting from an intensity measurement
depends on three parameters, and we shall indicated as it as
$I_{\text{std}}(x;\,z,z_{s},\sigma_{s})$, where $x$ is the image coordinate.
If an object is placed at $z_{s}$, also the image reconstructed through CPI
refocus $\Sigma_{z}(x;\,x_{s},\sigma_{s})$ depends on the same three
parameters, with the difference that $z$-scanning is obtained through Radon
transformation and not by mechanical movement of the detector. For our
analysis, both $z_{a}$ and $z_{b}$ will be considered to be fixed. The
fidelity of a standard and refocused image are evaluated, respectively, as

$\displaystyle F_{A}[I_{\text{\text{std}}}](z,z_{s},\sigma_{s})$
$\displaystyle=\int\sqrt{I_{\text{std}}(-M(z)\,x_{s},\,z,z_{s},\sigma_{s})\,\left|A(x_{s},\,\sigma_{s})\right|^{2}}dx_{s}$
(10) $\displaystyle F_{A}[\Sigma_{\text{z}}](z,z_{s},\sigma_{s})$
$\displaystyle=\int\sqrt{\sqrt{\Sigma_{\text{z}}}(x_{s};\,z_{s},\sigma_{s})\,\left|A(x_{s},\,\sigma_{s})\right|^{2}}dx_{s}.$
(11)

The two definitions are adapted so as to compare the results as fairly as
possible. In fact, the optical magnification of the imaging system $M(z)$ has
been introduced in Eq. (10) to scale the image to its original size on the
object side of the lens; such operation is not necessary in CPI, since it is
included in the refocusing procedure [50]. Since the $z$-scanning is obtained
by changing the detector position, each plane in focused is image with a
different magnification, originating the dependence of $M$ on $z$. The
additional square root on $\Sigma_{z}$ has been introduced in Eq. (11) to
compensate for the known fact that, even in perfect imaging conditions, CPI
returns the _squared_ intensity profile of the sample when both detectors see
light from the object (see Eq. (2) when $m=1$).

Figure 3: Curves characterized by image fidelity larger than $90\%$ in the
$(z,\sigma_{s})$ space

In Fig. 3, we report the results of the fidelity analysis for standard imaging
and CPI refocusing. The analysis has been carried out by considering a
Gaussian object of width $\sigma_{s}$, placed at a fixed distance
$z_{s}=(z_{a}+z_{b})/2=150$ mm from the lens, for both CPI and standard
imaging. The experimental parameters $z_{a}$, $z_{b}$, $f$ and $\sigma_{l}$
have been fixed to the same values as in Fig. 2a. Since the object is at a
fixed coordinate, the fidelity of Eqs. (10) and (11) reduces to a two-variable
function of the object size $\sigma_{s}$ and scanning coordinate $z$. The blue
area, corresponding to refocusing, and orange area, corresponding to standard
imaging, identify the region of the $(z,\sigma_{s})$ space in which a sample
can be imaged, by the corresponding technique, with fidelity larger than
$90\%$. Hence, at any given object size $\sigma_{s}$, the difference between
the largest and smallest $z$ at which an object is imaged with $90\%$ fidelity
can be regarded as the axial resolution, or DOF, of the technique. As we can
see, for large object sizes and large displacements (leftmost panel),
refocusing shows exactly the same performance as conventional imaging, which
is knowingly limited by the COC. Considering the COC is a ray optics concept,
it is worth investigating the behavior of $F_{A}[\Sigma_{z}]$ in the
geometrical optics approximation for refocusing, namely

$F_{\text{geom}}(z,z_{s},\sigma_{l})=\lim_{\lambda\rightarrow
0}F_{A}[\Sigma_{z}](z,z_{s},\sigma_{l}).$ (12)

The implicit equation $F_{\text{geom}}(z,z_{s},\sigma_{l})=c$, identifying the
curve at fidelity $c$ in the $(z,z_{s},\sigma_{l})$ space, can be easily
inverted to obtain the minimum size of the object that can faithfully be
reconstructed, as a function $z$ and $z_{s}$

$\sigma_{\text{geom}}(z,z_{s})\simeq\left|z-z_{s}\right|\frac{\sigma_{l}}{z_{s}}\,f(c),$
(13)

where $f(c)$ is a function of the threshold value $c$, arbitrarily chosen to
discriminate between faithful and unfaithful images. As predicted, the
expression of $\sigma_{geom}$ is exactly the same as the COC in conventional
imaging, being directly proportional to both the _effective_ numerical
aperture $\sigma_{l}/z_{s}$, and to the relative displacement from the plane
of sharpest reconstruction. As can be seen from Fig. 3, where Eq. (13) is
reported as a dashed black line, when the object size is far from the lateral
resolution limit of the techniques at the given $z_{s}$, the optical
performance of both standard imaging and refocusing is determined by ray
optics. In this conditions, one obtains that the thickness of a reconstructed
image at any given object size is

$\text{DOF}_{\Sigma}(\sigma_{S})\propto\frac{\sigma_{s}}{\text{NA}},$ (14)

as for a standard imaging system.

##  4 Lateral resolution of CPI

In the middle panel of Fig. 3, we see that when the object size approaches the
resolution limit of CPI at the object position $z_{s}$, the optical
performance of refocusing detaches from the geometrical trend, which would
incorrectly prescribe an point-like resolution
($\sigma_{\text{geom}}(z=z_{s},z_{s})=0$). A conventional $z$-scanning imaging
system, however, still behaves according to geometrical optics at that scale,
detaching from it only when the Rayleigh limit ($\propto\lambda/$NA) is
reached (right panel). The fact that, at any given transverse plane $z_{s}$,
the resolution of CPI refocusing is worse than a conventional imaging system
is a known fact. For a CPI architecture such as the one we are considering, in
fact, only two planes can be knowingly reconstructed at Rayleigh-limited
resolution, namely, the planes at distance $z_{a}$ and $z_{b}$, which are
optically conjugated to the detectors [55].

As opposed to the previous section, we shall now analyze the image quality of
a refocused image as the object is moved along the optical axis, by
disregarding the axial depth of the reconstructions. This corresponds to
studying the case in which, for any given object placement $z_{s}$ one does
not consider the whole $z$-scanning $\Sigma_{z}$, but only the sharpest
reconstruction $\Sigma_{z=z_{s}}.$ Through Eq.(11), we can thus evaluate the
performance of CPI in the plane of best refocusing, namely
$F_{A}[\Sigma_{z}](z=z_{s},z_{s},\sigma_{s})$. For reference, we shall also
report the image quality obtained by detectors $A$ and $B$ without the use of
correlations, to show the performance improvement granted by the correlation
measurement. The fidelity of the images collected on the detectors,
separately, can be evaluated through Eq. (10), as
$F_{A}\left[I_{\text{std}}\right](z=z_{a,b},z_{s},\sigma_{s})$, at the two
fixed experimental values $z_{a}$ and $z_{b}$.

Figure 4: Fidelity analysis in the plane of sharpest reconstruction
$\Sigma_{z=z_{s}}$

The fidelity analysis in the plane of sharpest reconstruction is reported in
Fig. 4. As already well known, the DOF of CPI is much larger than the DOF of
the two conventional single-lens systems measuring intensity on A and on B
(blue and green areas, respectively). As in the previous case, we define the
DOF as the difference between the largest and smallest $z_{s}$ that allow for
a faithful image reconstruction at a given object size. We should point out,
however, that the meaning of DOF in this context is different from the
previous section. In fact, the previous DOF was an estimation of the “depth”
of the reconstruction, whereas now it is the axial range in which CPI can
successfully reconstruct an object of given size $\sigma_{s}$. The difference
can be understood by considering that the previous analysis was carried out by
considering a _fixed object_ at coordinate $z_{s}$, while the analysis
reported in Fig. 4 evaluates the best possible refocusing at any object
coordinate. In other terms, if the analysis of Fig. 3 is useful for evaluating
the DOF of a refocused image, Fig. 4 allows one to evaluate the DOF of the
technique itself. The DOF of the reconstruction should then be regarded as the
_axial resolution_ of the technique, so that the ratio between the two
determines the number of independent planes in the overall DOF of the
technique that can be _isolated_ through refocusing.

The _fidelity curves_ of Fig. 4 must be intended as a replacement for the
_visibility_ or _resolution curves_ of CPI, presented in previous literature
[33, 55, 56, 34, 35, 54]. For reason that will be clarified shortly, however,
the fidelity curves represent a much more general way of estimating the
optical performance of CPI. The lower boundary to the orange areas of the
figure represents the parametric curve
$F_{A}[\Sigma_{z}](z=z_{s},z_{s},\sigma_{s})=0.9$, namely, the $90\%$-fidelity
curve in the $(z_{s},\sigma_{s})$ plane. Such curve thus establishes a
correspondence between the axial coordinate $z_{s}$ and the minimum object
size $\sigma_{\text{min}}(z_{s})$ that can be reconstructed faithfully enough
at that position. It is already known that the functional relationship
$\sigma_{\text{min}}(z_{s})$ is for the most part independent of the numerical
aperture of the setup, with the exception of the planes in focus
($z_{s}=z_{a}$ and $z_{s}=z_{b}$), which are available with Rayleigh-limited
resolution. Such independence of the optical performance of CPI from NA sets
the technique apart from conventional imaging for which, as we detailed in the
previous section, the defocused optical performance is defined by the NA-
proportional COC. One can thus check whether the performance of CPI can be
predicted independently on NA even from an analytical point of view, by
studying the fidelity of an ideal system

$F_{\infty}(z_{s},\sigma_{s})=\lim_{\text{NA}\rightarrow\infty}F_{A}[\Sigma_{z}](z=z_{s},z_{s},\sigma_{s}).$
(15)

Also in this case, the implicit equation of the $c$-fidelity curve
$F_{\infty}(x_{s},\sigma_{l})=c$ can be inverted, so that the fidelity curve
in the infinite-NA case read

$\sigma_{\infty}(z_{s})\simeq\sqrt{\lambda\left|\frac{1}{z-z_{a}}-\frac{1}{z-z_{b}}\right|^{-1}}\,f^{\prime}(c).$
(16)

The infinite-NA fidelity curve is reported in the two insets in Fig. 4,
showing the details of the CPI fidelity in very close proximity of the two
axial planes in focus. The plot demonstrates that, apart from a very small
region of space in which the resolution is defined by the NA-dependent
Rayleigh limit ($\propto\lambda/$NA), the optical performance of CPI
reproduces with extreme accuracy trend associated to the infinite-NA regime.
The mechanism responsible for the loss of resolution of CPI outside of the
natural DOF of the lens is thus completely independent of the optics size.

##  5 Conclusions

The surprising independence of the lateral resolution of CPI on the NA of the
imaging system, with the exception of the plane (or planes) in focus, was
already known in previous literature and even proven experimentally [35]. The
origin of such property was, however, not clearly understood, mostly because
of two aspects: firstly, the exact mathematical relationship between the form
of the refocused image and the object is not known, and, secondly, because the
image quality of CPI has been so far assessed by using conventional image
quality estimators (_e.g._ “two-point” resolution, maxima-minima visibility,
modulation transfer function (MTF) analysis). Such estimators are unfit to
correctly evaluate the performance of CPI: from Eq. (3), it is rather evident
that the technique, even without considering the added degree of complexity
introduced by refocusing, is a _non-linear_ imaging method, in the sense that
the input signal $\left|A(x)\right|^{2}$ is not related to the output image
through a simple _transfer function_ , as is the case for conventional imaging
systems. In standard imaging, in fact, the only mechanism222If aberrations are
neglected. responsible for image quality degradation is blurring or,
equivalently, the fact that the final image is obtained by convolution of the
input with a _positive_ PSF. Because of this property, all the merit figures
conventionally used for image quality assessment can be reduced to the
underlying linearity of the imaging formation process, and can thus be applied
to a limited extent to predict the performance of nonlinear imaging [51]. In
this work, we have decided to base our analysis of the optical performance of
CPI on the image fidelity, which, being independent of the details of the
image formation process, allows for a direct and _unbiased_ comparison to
other techniques. Through the fidelity analysis, we have demonstrated that the
same results and NA-independence of the lateral resolution in the plane of
best refocusing can be obtained in a mathematically consistent formalism,
which has the advantage of not relying on assumptions on the imaging formation
process. Compared to an assessment through image visibility, typically used in
literature to evaluate the performance of CPI [55, 54], the fidelity has the
advantage is based on the _global_ features of the image, and not on the very
local nature of maxima and minima. Furthermore, compared to visibility, the
fidelity is always an increasing function of the object size, when all the
other parameters are fixed,

$\frac{\partial F_{A}[\Sigma_{z}]}{\partial\sigma_{s}}(z,z_{s},\sigma_{s})>0,$

and does not show any unphysical fluctuating behavior [63, 64, 55, 54]. Most
importantly, the results that we presented in this work only for the case of
the optical design of Fig. 1, have been obtained also by extending the study
to other CPI architectures; in all cases, the fidelity analysis has confirmed
the NA independence and square-root trend of the lateral resolution of CPI
that was already known in literature. Through the fidelity analysis we have
confirmed that the fidelity curves in the plane of best refocusing are always
given by a square-root law, independently of the architecture

$\sigma_{\infty}(z_{s})\propto\sqrt{\lambda\,\left|d(z_{s})\right|};$ (17)

the “equivalent distance” $d(z_{s})$ is, however, a function of the CPI
architecture at hand. For instance, for CPI architectures based on position-
momentum correlations [56], $d(z_{s})=z_{s}$, where $z_{s}$ is the distance
from focus, so that the resolution is given by a pure square-root law. For the
architecture discussed here, instead, the infinite-NA trend which correctly
reproduces the fidelity curves

$d(z_{s})=\left(\frac{1}{z-z_{a}}-\frac{1}{z-z_{b}}\right)^{-1},$ (18)

as can be deduced from Eq. (16). In all cases, the performance of CPI outside
of the planes in focus can be mathematically obtained by evaluating the
infinite-NA fidelity $F_{\infty}(z_{s},\sigma_{s})$, and then inverting the
implicit equation $F_{\infty}=c$.

Unlike the NA-independent image quality in the plane of sharpest refocusing,
the axial sectioning enabled by refocusing is defined entirely by the NA,
specifically by the same COC defining the imaging depth of conventional
imaging systems. This results in a very interesting fact, namely, that the
lateral and axial resolution are decoupled from each other. This property
enables the designs of optical setups characterized by high resolution, as
defined by the CPI scheme of choice, with the sectioning capabilities that can
be tuned independently by selecting the appropriate lens size. Such operation
can even be performed in post-processing, by limiting the size of the
refocusing integration path (Eq. (4)) to simulate a reduced _effective_ NA,
with no effect on resolution, for a deeper image reconstruction and improved
computational efficiency. Such versatility sets apart CPI from conventional,
lenslet-based, plenoptic imaging, which suffer from a strong trade-off between
lateral and axial resolution [65, 66, 49].

Acknowledgements

This article has benefited from the contributions of prof. M. D’Angelo, who
obtained project funding and reviewed the final draft.

Funding The author acknowledges funding from Università degli Studi di Bari
under project ADEQUADE, and from Istituto Nazionale di Fisica Nucleare under
projects Qu3D and QUISS.

Project ADEQUADE has received funding from the European Defence Fund (EDF)
under grant agreement EDF-2021-DIS-RDIS-ADEQUADE (n° 101103417). Project Qu3D
is supported by the Italian Istituto Nazionale di Fisica Nucleare, the Swiss
National Science Foundation (grant 20QT21187716 “Quantum 3D Imaging at high
speed and high resolution”), the Greek General Secretariat for Research and
Technology, the Czech Ministry of Education, Youth and Sports, under the
QuantERA programme, which has received funding from the European Union’s
Horizon 2020 research and innovation programme.

Funded by the European Union. Views and opinions expressed are however those
of the author(s) only and do not necessarily reflect those of the European
Union or the European Commission. Neither the European Union nor the granting
authority can be held responsible for them.

Conflict of interest The author has no competing interests to declare that are
relevant to the content of this article.

Data availability Not applicable

## References

  *   * Giovannetti et al. [2011] Giovannetti, V., Lloyd, S., Maccone, L.: Advances in quantum metrology. Nature Photonics 5 (2011)
  * Zavatta et al. [2006] Zavatta, A., D’Angelo, M., Parigi, V., Bellini, M.: Remote preparation of arbitrary time-encoded single-photon ebits. Physical Review Letters 96 (2006)
  * Iskhakov et al. [2011] Iskhakov, T., Allevi, A., Kalashnikov, D., Sala, V., Takeuchi, M., Bondani, M., Chekhova, M.: Intensity correlations of thermal light: noise reduction measurements and new ghost imaging protocols. The European Physical Journal Special Topics 199, 127–138 (2011)
  * Avella et al. [2016] Avella, A., Ruo-Berchera, I., Degiovanni, I.P., Brida, G., Genovese, M.: Absolute calibration of an emccd camera by quantum correlation, linking photon counting to the analog regime. Optics Letters 41, 1841–1844 (2016)
  * Agliati et al. [2005] Agliati, A., Bondani, M., Andreoni, A., De Cillis, G., Paris, M.G.A.: Quantum and classical correlations of intense beams of light investigated via joint photodetection. Journal of Optics B: Quantum and Semiclassical Optics 7(12) (2005)
  * Allevi et al. [2012] Allevi, A., Olivares, S., Bondani, M.: Measuring high-order photon-number correlations in experiments with multimode pulsed quantum states. Phys. Rev. A 85, 063835 (2012)
  * Pittman et al. [1995] Pittman, T.B., Shih, Y.-H., Strekalov, D.V., Sergienko, A.V.: Optical imaging by means of two-photon quantum entanglement. Phys. Rev. A 52, 3429 (1995)
  * Pittman et al. [1996] Pittman, T., Strekalov, D., Klyshko, D., Rubin, M., Sergienko, A., Shih, Y.: Two-photon geometric optics. Physical Review A 53(4), 2804 (1996)
  * Agafonov et al. [2009] Agafonov, I.N., Chekhova, M.V., Iskhakov, T.S., Wu, L.-A.: High-visibility intensity interference and ghost imaging with pseudo-thermal light. Journal of Modern Optics 56, 422–431 (2009)
  * Aspden et al. [2013] Aspden, R.S., Tasca, D.S., Boyd, R.W., Padgett, M.J.: EPR-based ghost imaging using a single-photon-sensitive camera. New J. Phys. 15, 073032 (2013)
  * Cassano et al. [2005] Cassano, M., D’Angelo, M., Garuccio, A., Peng, T., Shih, Y.H., V., T.: Spatial interference between pairs of disjoint optical paths with a single chaotic source. Optics Express 25 (2005)
  * Bai and Han [2007] Bai, Y., Han, S.: Ghost imaging with thermal light by third-order correlation. Physical Review A 76(4), 043828 (2007)
  * Bina et al. [2013] Bina, M., Magatti, D., Molteni, M., Gatti, A., Lugiato, L.A., Ferri, F.: Backscattering differential ghost imaging in turbid media. Phys. Rev. Lett. 110, 083901 (2013)
  * Brida et al. [2011] Brida, G., Chekhova, M., Fornaro, G., Genovese, M., Lopaeva, E., Berchera, I.R.: Systematic analysis of signal-to-noise ratio in bipartite ghost imaging with classical and quantum light. Phys. Rev. A 83, 063807 (2011)
  * Bromberg et al. [2009] Bromberg, Y., Katz, O., Silberberg, Y.: Ghost imaging with a single detector. Physical Review A 79(5), 053840 (2009)
  * Chan et al. [2009] Chan, K.W.C., O’Sullivan, M.N., Boyd, R.W.: Two-color ghost imaging. Phys. Rev. A 79, 033808 (2009)
  * D’Angelo et al. [2005] D’Angelo, M., Valencia, A., Rubin, M.H., Shih, Y.: Resolution of quantum and classical ghost imaging. Physical Review A 72(1), 013810 (2005)
  * Erkmen and Shapiro [2010] Erkmen, B.I., Shapiro, J.H.: Ghost imaging: from quantum to classical to computational. Advances in Optics and Photonics 2(4), 405–450 (2010)
  * Ferri et al. [2005] Ferri, F., Magatti, D., Gatti, A., Bache, M., Brambilla, E., Lugiato, L.A.: High-resolution ghost image and ghost diffraction experiments with thermal light. Phys. Rev. Lett. 94, 183602 (2005)
  * D’Angelo et al. [2017] D’Angelo, M., Mazzilli, A., Pepe, F.V., Garuccio, A., V., T.: Characterization of two distant double-slits by chaotic light secondorder interference. Scientific Reports 7 (2017)
  * Ferri et al. [2010] Ferri, F., Magatti, D., Lugiato, L., Gatti, A.: Differential ghost imaging. Physical Review Letters 104(25), 253603 (2010)
  * Shih [2016] Shih, Y.: The physics of turbulence-free ghost imaging. Technologies 4(4), 39 (2016)
  * Xu et al. [2018] Xu, Z.-H., Chen, W., Penuelas, J., Padgett, M., Sun, M.-J.: 1000 fps computational ghost imaging using led-based structured illumination. Optics express 26(3), 2427–2434 (2018)
  * Paniate et al. [2024] Paniate, A., Massaro, G., Avella, A., Meda, A., Pepe, F.V., Genovese, M., D’Angelo, M., Ruo-Berchera, I.: Light-field ghost imaging. Phys. Rev. Appl. 21, 024032 (2024) https://doi.org/10.1103/PhysRevApplied.21.024032
  * Moreau et al. [2019] Moreau, P.-A., Toninelli, E., Gregory, T., Padgett, M.J.: Imaging with quantum states of light. Nat. Rev. Phys. 1, 367–380 (2019)
  * Brida et al. [2010] Brida, G., Genovese, M., Ruo-Berchera, I.: Experimental realization of sub-shot-noise quantum imaging. Nat. Photonics 4, 227–230 (2010)
  * Samantaray et al. [2017] Samantaray, N., Ruo-Berchera, I., Meda, A., Genovese, M.: Realization of the first sub-shot-noise wide field microscope. Light: Science & Appl. 6(7), 17005 (2017)
  * Ortolano et al. [2023] Ortolano, G., Paniate, A., Boucher, P., Napoli, C., Soman, S., Pereira, S.F., Ruo-Berchera, I., Genovese, M.: Quantum enhanced non-interferometric quantitative phase imaging. Light: Science & Applications 12(1), 171 (2023) https://doi.org/10.1038/s41377-023-01215-1
  * Allevi et al. [2017] Allevi, A., Cassina, S., Bondani, M.: Super-thermal light for imaging applications. Quantum Measurements and Quantum Metrology 4(1), 26–34 (2017)
  * Gatti et al. [2004] Gatti, A., Brambilla, E., Bache, M., Lugiato, L.A.: Ghost imaging with thermal light: comparing entanglement and classical correlation. Phys. Rev. Lett. 93, 093602 (2004)
  * O’Sullivan et al. [2010] O’Sullivan, M.N., Chan, K.W.C., Boyd, R.W.: Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging. Phys. Rev. A 82, 053803 (2010)
  * Valencia et al. [2005] Valencia, A., Scarcelli, G., D’Angelo, M., Shih, Y.: Two-photon imaging with thermal light. Phys. Rev. Lett. 94, 063601 (2005)
  * D’Angelo et al. [2016] D’Angelo, M., Pepe, F.V., Garuccio, A., Scarcelli, G.: Correlation plenoptic imaging. Phys. Rev. Lett. 116, 223602 (2016)
  * Massaro et al. [2022] Massaro, G., Giannella, D., Scagliola, A., Di Lena, F., Scarcelli, G., Garuccio, A., Pepe, F.V., D’Angelo, M.: Light-field microscopy with correlated beams for high-resolution volumetric imaging. Scientific Reports 12, 16823 (2022)
  * Massaro et al. [2023] Massaro, G., Mos, P., Vasiukov, S., Lena, F.D., Scattarella, F., Pepe, F.V., Ulku, A., Giannella, D., Charbon, E., Bruschini, C., D’Angelo, M.: Correlated-photon imaging at 10 volumetric images per second. Scientific Reports (2023)
  * Abbattista et al. [2021] Abbattista, C., Amoruso, L., Burri, S., Charbon, E., Di Lena, F., Garuccio, A., Giannella, D., Hradil, Z., Iacobellis, M., Massaro, G., et al.: Towards quantum 3d imaging devices. Applied Sciences 11(14), 6414 (2021) https://doi.org/10.3390/app11146414
  * Scattarella et al. [2023] Scattarella, F., Diacono, D., Monaco, A., Amoroso, N., Bellantuono, L., Massaro, G., Pepe, F.V., Tangaro, S., Bellotti, R., D’Angelo, M.: Deep learning approach for denoising low-snr correlation plenoptic images. Scientific Reports 13(1), 19645 (2023) https://doi.org/10.1038/s41598-023-46765-x
  * Petrelli et al. [2023] Petrelli, I., Santoro, F., Massaro, G., Scattarella, F., Pepe, F.V., Mazzia, F., Ieronymaki, M., Filios, G., Mylonas, D., Pappas, N., Abbattista, C., D’Angelo, M.: Compressive sensing-based correlation plenoptic imaging. Frontiers in Physics 11 (2023) https://doi.org/10.3389/fphy.2023.1287740
  * Pepe et al. [2017] Pepe, F.V., Di Lena, F., Mazzilli, A., Edrei, E., Garuccio, A., Scarcelli, G., D’Angelo, M.: Diffraction-limited plenoptic imaging with correlated light. Phys. Rev. Lett. 119, 243602 (2017)
  * Adelson et al. [1992] Adelson, E.H., Wang, J.Y.A., Technology. Media Laboratory. Vision, M.I., Group, M.: Single Lens Stereo with a Plenoptic Camera. M.I.T. Media Lab Vision and Modeling Group technical report. Vision and Modeling Group, Media Laboratory, Massachusetts Institute of Technology, ??? (1992). https://books.google.it/books?id=umWhGwAACAAJ
  * Pawley [2006] Pawley, J.: Handbook of Biological Confocal Microscopy vol. 236. Springer, ??? (2006)
  * Ihrke et al. [2016] Ihrke, I., Restrepo, J., Mignard-Debise, L.: Principles of light field imaging: Briefly revisiting 25 years of research. IEEE Signal Processing Magazine 33(5), 59–69 (2016)
  * Ng et al. [2005] Ng, R., Levoy, M., Brédif, M., Duval, G., Horowitz, M., Hanrahan, P.: Light field photography with a hand-held plenoptic camera. Computer Science Technical Report CSTR 2, 1–11 (2005)
  * Ng [2005] Ng, R.: Fourier slice photography. ACM Transactions on Graphics 24(3), 735–744 (2005). ACM
  * Birklbauer and Bimber [2014] Birklbauer, C., Bimber, O.: Panorama light-field imaging. In: Computer Graphics Forum, vol. 33, pp. 43–52 (2014). Wiley Online Library
  * Broxton et al. [2013] Broxton, M., Grosenick, L., Yang, S., Cohen, N., Andalman, A., Deisseroth, K., Levoy, M.: Wave optics theory and 3-D deconvolution for the light field microscope. Opt. Express 21, 25418–25439 (2013)
  * Prevedel et al. [2014] Prevedel, R., Yoon, Y.-G., Hoffmann, M., Pak, N., Wetzstein, G., Kato, S., Schrödel, T., Raskar, R., Zimmer, M., Boyden, E.S., et al.: Simultaneous whole-animal 3d imaging of neuronal activity using light-field microscopy. Nature Methods 11(7), 727–730 (2014)
  * Georgeiv et al. [2006] Georgeiv, T., Zheng, K.C., Curless, B., Salesin, D., Nayar, S., Intwala, C.: Spatio-angular resolution tradeoffs in integral photography. In: Proceedings of the 17th Eurographics Conference on Rendering Techniques. EGSR ’06, pp. 263–272. Eurographics Association, Goslar, DEU (2006)
  * Goldlücke et al. [2015] Goldlücke, B., Klehm, O., Wanner, S., Eisemann, E.: Plenoptic cameras. Digital Representations of the Real World: How to Capture, Model, and Render Visual Reality, eds. M. Magnor, O. Grau, O. Sorkine-Hornung, and C. Theobalt (CRC Press, 2015) (2015)
  * Massaro et al. [2022] Massaro, G., Pepe, F.V., D’Angelo, M.: Refocusing Algorithm for Correlation Plenoptic Imaging. Sensors 22, 6665 (2022)
  * [51] Massaro, G., Barile, B., Scarcelli, G., Pepe, F.V., Nicchia, G.P., D’Angelo, M.: Direct 3d imaging through spatial coherence of light. Laser & Photonics Reviews, 2301155 https://doi.org/10.1002/lpor.202301155 https://onlinelibrary.wiley.com/doi/pdf/10.1002/lpor.202301155
  * Massaro et al. [2022] Massaro, G., Di Lena, F., D’Angelo, M., Pepe, F.V.: Effect of finite-sized optical components and pixels on light-field imaging through correlated light. Sensors 22(7) (2022) https://doi.org/%****␣sn-article.bbl␣Line␣900␣****10.3390/s22072778
  * Pepe et al. [2016] Pepe, F.V., Di Lena, F., Garuccio, A., Scarcelli, G., D’Angelo, M.: Correlation plenoptic imaging with entangled photons. Technologies 4, 17 (2016)
  * Di Lena et al. [2018] Di Lena, F., Pepe, F., Garuccio, A., D’Angelo, M.: Correlation plenoptic imaging: An overview. Appl. Sci. 8(10), 1958 (2018) https://doi.org/10.3390/app8101958
  * Di Lena et al. [2020] Di Lena, F., Massaro, G., Lupo, A., Garuccio, A., Pepe, F.V., D’Angelo, M.: Correlation plenoptic imaging between arbitrary planes. Opt. Express 28, 35857–35868 (2020)
  * Giannella et al. [2024] Giannella, D., Massaro, G., Stoklasa, B., D’Angelo, M., Pepe, F.V.: Light-field imaging from position-momentum correlations. Physics Letters A 494, 129298 (2024) https://doi.org/10.1016/j.physleta.2023.129298
  * Massaro et al. [2022] Massaro, G., Scala, G., D’Angelo, M., Pepe, F.V.: Comparative analysis of signal-to-noise ratio in correlation plenoptic imaging architectures. The European Physical Journal Plus 137(1123) (2022)
  * Scala et al. [2019] Scala, G., D’Angelo, M., Garuccio, A., Pascazio, S., Pepe, F.V.: Signal-to-noise properties of correlation plenoptic imaging with chaotic light. Phys. Rev. A 99, 053808 (2019)
  * Clackdoyle and Defrise [2010] Clackdoyle, R., Defrise, M.: Tomographic reconstruction in the 21st century. IEEE Signal Processing Magazine 27(4), 60–80 (2010) https://doi.org/10.1109/MSP.2010.936743
  * Levoy [1992] Levoy, M.: Volume Rendering Using the Fourier Projection-slice Theorem. Computer Systems Laboratory, Stanford University, ??? (1992)
  * Stokseth [1969] Stokseth, P.A.: Properties of a defocused optical system$\ast$. J. Opt. Soc. Am. 59(10), 1314–1321 (1969) https://doi.org/10.1364/JOSA.59.001314
  * Murphy and Davidson [2012] Murphy, D.B., Davidson, M.W.: 6\. Diffraction and Spatial Resolution, pp. 103–113. John Wiley & Sons, Ltd, ??? (2012). https://doi.org/10.1002/9781118382905.ch6
  * Scattarella et al. [2022] Scattarella, F., D’Angelo, M., Pepe, F.V.: Resolution limit of correlation plenoptic imaging between arbitrary planes. Optics 3, 138–149 (2022)
  * Scattarella et al. [2023] Scattarella, F., Massaro, G., Stoklasa, B., D’Angelo, M., Pepe, F.V.: Periodic patterns for resolution limit characterization of correlation plenoptic imaging. The European Physical Journal Plus 138(8), 710 (2023) https://doi.org/10.1140/epjp/s13360-023-04322-5
  * Dansereau et al. [2013] Dansereau, D.G., Pizarro, O., Williams, S.B.: Decoding, calibration and rectification for lenselet-based plenoptic cameras. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1027–1034 (2013)
  * Georgiev et al. [2009] Georgiev, T.G., Lumsdaine, A., Goma, S.: High Dynamic Range Image Capture with Plenoptic 2.0 Camera. In: Frontiers in Optics 2009/Laser Science XXV/Fall 2009 OSA Optics & Photonics Technical Digest, p. 7. Optical Society of America, Washington, DC (2009)