1. Introduction
Various methods are used to describe and numerically simulate image formation in optical coherence tomography (OCT), among which much attention has been paid to such a powerful approach as Monte-Carlo (MC), the basics of which were recently discussed in overview [
1] and its direct application to OCT can be found, e.g., in [
2,
3]. The strength of MC-methods is related to the very MC principle, in which the image is formed due to a huge number of scattering events of individual photons. This principle implies a rather high computational demand, but at the same time makes the MC approach rather general. In particular, the MC approach allows for the accounting of multiple scattering, which is difficult to introduce in other approaches.
It can be recalled that the principle of OCT-image formation [
4,
5,
6] is based on ballistic single backscattering of the illuminating optical beam (i.e., backscattering described in the first-order Born approximation), so that, among various reasons for OCT-image degradation, multiple scattering is only one such factor, consideration of which for many applications is not of primary importance. In view of this, significant attention was paid to the development of models of OCT-scan formation based on the wave representation and single backscattering of the illuminating beams. In such wave-based approaches, the illuminating field of OCT systems can be described using somewhat different representations. In some cases, important features of OCT-image formation could be understood using simple plane-wave approximations in various forms [
7,
8]. For example, image formation in spectral-domain OCT can be described by representing the illuminating field as a set of co-propagating plane waves with different axial wavenumbers. This approximation, even without special attention to the lateral beam structure, can be sufficient to describe the axial form of the point-spread function (PSF). In some variants, along with the description of the PSF form in the axial direction by considering a set of spectral components with different axial wavenumbers, the lateral form of the OCT beam was also taken into account using the spatial-domain description. Often, the lateral field distribution was described using Gaussian beams [
9,
10,
11,
12]. Such approaches opened the possibility of fairly accurately describing the lateral form of PSF, including the influence of the illuminating beam focusing. However, such Gaussian-beam-based models do not allow one to describe other beam forms (e.g., Bessel beams), which also represent interest for utilization in OCT [
13,
14,
15].
Alternatively, for describing the lateral distribution of the illuminating and scattered fields, other models based on angular spectra (i.e., representation of the field as a set of plane waves with various directions) have also been developed with one or another degree of rigor (see, e.g., [
16,
17,
18,
19,
20]). Such approaches combine the spectral representation of both the axial and lateral structures of illuminating beams and, therefore, can be termed a full K-space description. Recently, a comprehensive overview of various K-space-based approaches developed in various techniques of optical visualization (not only OCT in the conventional sense) was published [
21]. It was pointed out in [
21] that the K-space description is a rather general approach, allowing one to analyze very diverse factors in a unified manner, including various beam shapes, effects of dispersion, and aberrations of the optical-wave front.
In application to conventional point-scanning OCT systems, adequate post-processing of the acquired 3D sets of complex-valued OCT signals opens flexible possibilities for various transformations of the initial OCT data. In particular, due to such processing, it becomes possible to introduce corrections for PSF distortions caused by aberrations and/or dispersion effects during wave propagation [
21]. It also becomes possible to perform digital refocusing by reassembling the spectral components of the illuminating field with corrected phases, such that the depth-independent focusing can be formed over the entire 3D dataset (see examples of such digital transformations in [
19,
21]). Similarly, the problem of digital corrections of aberrations was discussed (e.g., [
21,
22]), so that the quality of images obtained by an OCT system with an inexpensive optical system and appreciable aberrations can be made comparable with the quality of images obtained with a much more expensive aberration-less optics.
Here, using the terminology of paper [
21], we consider a modified version of the K-space description of OCT-image formation in the development of recent paper [
19]. The main limitation of this approach is the use of single scattering, which is a generally accepted approximation [
6,
20,
21] corresponding to the very principle of OCT operation. We analyze the most common point-scanning scheme of OCT. In this scheme, the illuminating and then backscattered optical signal passes forth-and-back through the tissue and the same illuminating/receiving aperture. In particular, in the presence of aberrations at this aperture, the optical field passes twice through the source of aberrations, so that the aberration-related distortions in the illuminating and scattered fields experience nonlinear mixing at the reception. The forth-and-back propagation of the optical wave is considered in the K-space (angular-spectrum) representation rigorously without such often-used simplifying limitations as the paraxial approximation or utilization of Gaussian beams. Even if the derived integral expressions without such approximations cannot be evaluated analytically, they can readily be estimated numerically for arbitrary shapes of illuminating beams, not necessarily possessing circular symmetry, e.g., in the presence of anisotropic aberrations, as demonstrated below.
The main goal of the present study is the use of the K-space approach, basically similar to [
19], to derive a general form of a filtering function that is intended to transform the initial 3D set of complex-valued OCT data into a “desired”(or “target”) form. The latter can be understood in a rather general sense. An essential feature of the proposed filtering procedures is the utilization of both phase and amplitude transformations, unlike often discussed procedures based on only phase transformations. After presenting the basic equations describing the initial OCT-signal formation and deriving the general form of the filtering function, we illustrate the generality and efficiency of the proposed approach by considering several instructive examples. In particular, we demonstrate a non-Gaussian example of the transformation of a 3D set of OCT data (initially corresponding to a beam with a Lorentzian profile to the 3D image for a Bessel beam). Examples will be given to demonstrate that the influence of arbitrary (including anisotropic) aberrations at the illuminating/receiving aperture can be digitally compensated (despite the nonlinear mixing of aberrations acquired during forward and backward propagation). Next, in addition to conventionally discussed digital refocusing intended to obtain the same lateral resolution as at the depth of physical focus, we consider an example of digital filtering enabling “super refocusing”. This super-refocusing does not yet overcome the diffraction limit imposed by the illuminating-light wavelength, but allows one to overcome the lateral resolution limited by the initial radius of the focal waist of the illuminating beam, although the increased resolution in such super-refocused images is obtained at the expense of some reduction in the signal-to-noise ratio (SNR).
2. Rigorous K-Space Formulation of OCT-Image Formation in the 1st-Order Born Approximation of the Scattered Signal
The spectral approach in various forms has been widely used for describing OCT-image formation (e.g., [
16,
17,
18,
21,
23]). In what follows, we will use the K-space formulation presented in [
19] as a basis for the further analysis. In the framework of ballistic first-order scattering (which corresponds to the 1st-order Born approximation), this description is rather rigorous and does not need other popular approximations, such as the widely used paraxial approximation or mandatory utilization of Gaussian beams.
Each spectral component of the illuminating beam with wavenumber
is characterized by a complex-amplitude distribution
at the output lens aperture with the axis passing through lateral coordinates
. The source is characterized by the discrete spectrum
over the wavenumber. The lens is orthogonal to
z-axis, corresponding to the axial direction, the lens plane has the axial coordinate
as shown in
Figure 1. Coordinate
is the distance between the aperture and the tissue boundary (and often this gap is filled with an immersion layer). Scanning of the beam position with lateral coordinates
is used for obtaining the 3D volume of OCT data. We also use a widely utilized approximation of discrete scatterers [
24] that are characterized by coordinates
. Such localized scatterers are convenient for representing the initial and transformed images, but the discussed spectral transformations of optical fields are rigorous and independent of the scatterer density. Like in real-spectral-domain OCT, we represent the illuminating beam as a set of spectral components with wavenumbers
.
For a given wavenumber
, the distribution
of the complex-valued field over the input/output lens with the axis passing through lateral coordinates
is characterized by the angular spectrum
defined by the following Fourier-transform over lateral coordinates
:
where
are the lateral components of the total wavenumber
;
is the angular spectrum of the illuminating-beam distribution
over lateral coordinates
for the axis position
. The representation of the illuminating beam via its angular spectrum
has the advantage that it allows one to describe an arbitrary beam profile as a set of plane waves with the wave vector components
.
During propagation between the input-lens plane
and plane
corresponding to the depth of
s-th scatterer each of the plane-wave components acquires the additional phase
, which means that the complex-valued amplitude of each propagating components is multiplied by the following propagator function
:
Here, subscript “
s” means that
depends on the depth
of
s-th scatterer and it is assumed that
corresponds to propagation along
z-axis (i.e., in-depth propagation). Therefore, according to Equations (1) and (2), the angular spectrum
of the field propagated from the illuminating aperture with the lateral coordinates
to the location of the
s-th scatterer with coordinates
is given the following product
Notice that spectrum
in Equation (3) depends only on the axial coordinate of the scatterer via function
. In contrast, lateral coordinates
of
s-th scatterer do not enter the incident-wave spectrum (3). Its inverse Fourier transform over spectral coordinates
gives the complex amplitude
of the incident beam with lateral coordinates
in the location
of the scatterer:
We emphasize that subscript “s” in means that the incident-wave amplitude depends not only on the position of the beam axis , but also on coordinates of the s-th scatterer.
Propagation of the backscattered signal can be conveniently represented, as shown in
Figure 2, in which the back-propagation path is mirrored, as if the back-propagated signal effectively propagates in the same direction along the
Z-axis. In this representation, the symmetry between forward and back propagation is especially clear. The distance of the two paths is exactly the same, and the axial back-propagation from the
s-th scatterer towards the lens again corresponds to multiplication by the propagator
given by Equation (2). Next, for each position
of the lens axis, when passing through the lens, the amplitudes of received spectral components are multiplied by the factor
given by Equation (1). Thus, the spectrum transformation during the back propagation is described by the same total spectral factor
as in Equation (3) for the forward propagation. Consequently, the transition from the spectral representation to the received signal amplitude is again given by the Fourier transform, similar to Equation (4).
It should also be taken into account that in the plane
passing through
s-th scatterer the spectral components of the scattered signal with the wavenumber
can be represented as
, where the scattering coefficient
characterizes the proportionality between the scattered-wave amplitude and the amplitude
of the wave incident onto the scatterer located at
. Notice that coefficient
may be different for different scatterers. Taking this remark into account the amplitude of the received signal
from
s-th scatterer can be written similarly to Equation (4):
so that recalling Equation (4) for
, one can rewrite Equation (5) in a simple and compact form
Here, we assume that for widely used OCT systems with a fairly narrow bandwidth of the illuminating source of the order of several percent and for nearly point-like scatterers with broad scattering diagrams, the dependence of the coefficient on the wavenumber and wavenumber projections can the neglected. Consequently, can be considered approximately constant. However, if necessary, the above-mentioned effects can be incorporated into the dependence and this coefficient should enter the Fourier integral in Equation (5). It should also be emphasized that the symmetry of the received and incident at s-th scatterer signals in Equations (5) and (6) holds for beams of arbitrary form, not only Gaussian ones.
To find the total form of an A-scan corresponding to the position of the illuminating-beam axis
the contribution of all scatterers should be summed by performing summation over index
of scatterers, and the (inverse) Fourier transform should be performed to combine the contributions of all received spectral components with the wavenumbers
:
Here, the abbreviation (Digital Fourier Transform) emphasizes that this Fourier transform is made numerically, for example, using efficient FFT algorithms realized in many numerical packages; the additionally introduced function describes the amplitudes of spectral components with wavenumbers of the illuminating-source spectrum.
Equations (1)–(7) constitute a convenient, computationally highly efficient framework for simulating various types of OCT scans for exactly controllable and flexibly variable conditions. Depending on the particular problem, the simulated visualized regions with realistic millimeter-scale sizes may contain a few scatterers for demonstration. Also, configurations with
scatterers separated by several micrometers to imitate the spatial density of scatters typical of real biological tissues can be readily simulated, as was recently demonstrated in [
19]. In particular, simulations of regions with either regularly or stochastically moving scatters can efficiently be made, for which alternative methods (such as Monte-Carlo-based ones) are not well suitable.
Concerning the further discussed procedures for digital transformation of 3D sets of OCT data composed of A-scans given by Equation (7), it should be noted that for a given position of the illuminating-beam axis, we obtain only one 1D A-scan given by Equation (7). The latter contains complex amplitudes of the received wave components with wavenumbers . Amplitudes and received for a given position of the beam axis do not yet give information about the distribution of amplitude and phase of the OCT signal in the lateral directions, which is indispensable if we want to digitally transform the OCT image (e.g., for performing digital refocusing). In the K-space description, such transformations correspond to the reassembling of angular spectral components of the received signals. Therefore, to find the set of spectral components one needs to perform scanning of the illuminating beam in x- and y-directions to obtain a sufficiently large and densely spaced set of acquired complex-valued A-scans for various . These coordinates should be equidistantly spaced to efficiently perform the Fourier transform from the spatial domain to the K-space. Next, to enable the desired lateral resolution, the initial set of spatial data should be sufficiently dense to satisfy the Nyquist-Kotelnikov criterion. In particular, to enable OCT-image refocusing for a focused beam, the distance between adjacent positions of A-scans in both lateral directions should be smaller than the focal radius of the beam. The corresponding examples will be given in the following sections.
3. A Remark about Independence of Manipulations with the Angular Spectrum of OCT Signals on the Lateral Positions of Scatterers
In what follows, we perform various transformations (such as refocusing, etc.) with the acquired 3D sets of OCT data. For practical usage, a key point is that, for such transformation of OCT images as refocusing, the procedures certainly should account for the specific depth but should not require a priori knowledge of the lateral positions of scatterers. Mathematically, this means that when performing the corresponding manipulations in K-space with OCT data, the lateral positions of scatterers can be factorized independently of the axial positions of these scatterers.
To demonstrate this, we consider Equation (7) describing the complex-valued A-scans in the acquired 3D-pack of OCT data. To manipulate the complex-valued quantities
in K-space, they should be transformed to the spectral form by applying the Fourier transform
to Equation (7). Since the Fourier transform
, as well as the summations and the inverse Fourier transform
in Equation (7) are linear operations, their order can be arbitrary. Therefore, in Equation (7), we can apply
to amplitudes
of the received signal from
s-th scatterer:
. The structure of
is given by Equation (6). Bearing in mind that the Fourier transform of a product corresponds to the convolution of the Fourier-transform of the multipliers, one can write:
Recalling that
is given by Equation (4), one can notice that in Equation (8), the convolution can be explicitly represented in the following form
where we denote
It is clear from the structure of Equations (9) and (10) that and the lateral coordinates of the scatterer enter the Fourier transform of the received amplitude only as a factor independent of the convolution integral . The latter is given by Equation (10), in which the axial position of s-th scatterer enter via the dependence of function on .
4. Spectral Form of the Filtering Function to Transform the Initial Set of A-Scans to Arbitrary Target form
Now, we consider how the above-derived relationships can be used to solve the following problem. Let us assume that the illuminating field has the complex-valued amplitude
after the illuminating-lens aperture and the acquired A-scans are
. Our goal is to transform the initial 3D set of A-scans
to another (“target”) form
. This means that the target form of scans corresponds to another field distribution
at the illuminating aperture:
In particular, the distribution may contain some aberrations at the illuminating/receiving aperture, whereas is a desired aberration-free distribution. In other cases, the desired distribution may correspond, for example, to refocusing to another depth.
We recall that the field distribution in the K-space is described by the angular spectrum given by integral (1). Similarly, the target distribution in the spectral representation corresponds to , to obtain which one should use instead of in Equation (1). Then, in integral relationship (10) one should substitute by to obtain the target convolution function .
Next, we recall that Equations (9) and (10) define the angular spectrum
of the received signal from
s-th scatterer for the initial field distribution
. Equations with the same structure as (9) and (10) also define angular spectrum
for the target distribution
. Consequently, the angular spectrum
can be transformed to the desired form
via the following filtering function:
An important feature of the filtering function (12) is that it does not depend on lateral coordinates of scatterers and their scattering strengths , which is due to the factorized form of Equation (9). It also worth noting that for avoiding the occasional division by nearly zero values, in the denominator of Equation (12) some regularization term should be added .
Rigorously speaking, the filtering function in Equation (12) depends on the wavenumber of every component of the illuminating optical field. Equation (12) also indicates that the filtering depends on the scatterer depth . However, we recall that unlike coordinate of s-th scatterer, which is a continuous quantity, the depth coordinate in the reconstructed OCT scans (see Equation (7)) is discreet. Thus, for transformation of OCT scans it makes sense to define the depth-dependent filtering function (12) for discreet depth points .
Now, one can obtain the corrected (target) A-scan
by transforming the initial A-scan
given by Equation (7) to the K-space, then multiplying the so-found spectrum by the filter function (12) and then again applying the Fourier transform to return to the corrected spectrum to the spatial domain. This procedure looks as follows:
Now, one may recall that the spectral composition
of the illuminating beam for OCT devices is well localized around the central wavenumber
. Taking this into account, Equation (13) can be significantly simplified by eliminating the inverse and forward Fourier transforms over the wavenumbers
. The latter can be substituted by the central wavenumber
. Then, the simplified version of Equation (13) takes the form:
In comparison with Equation (13), the simplified Equation (14) strongly reduces the computational expenses with only an insignificant influence on the accuracy of the entire correction procedure.
Thus, the derived Equations (13) and (14) (together with Equation (12) for the filtering function in the k-space) describe the OCT-scan transformation to the desired form. The transformation steps can be summarized in the diagram shown in
Figure 3. In the next section, we will present some examples of application of the described transformation procedure.
6. Discussion and Conclusions
In this paper, in the development of a series of earlier works related to the use of angular-spectrum approach to the description of OCT-image formation, we illustrate some possibilities of transformation of OCT data opened by this powerful approach. In the general sense, this approach develops the ideas of Fourier optics formulated several decades ago (see, e.g., textbooks [
40,
41]). For the realization of these principles, the progress in the development of powerful numerical packages (such as Matlab, Mathematica, etc.) opened rather convenient possibilities.
The main result of this paper is the derivation of a general form of the filtering function that enables various transformations of 3D sets of OCT data. The developed rather general and rigorous formulation of spectral-filtering procedures is summarized in
Figure 3 and mathematically presented by Equations (10)–(14). To the best of our knowledge, they have not been derived before. The derived expressions in some special cases can be evaluated analytically. In more general cases, they are intended for numerical evaluation, for the realization of which such conventional assumptions as the paraxial approximation or the necessity of Gaussian beams are not required. The formulated approach, in particular, made it possible to clearly reveal conditions implicitly assumed in the conventionally discussed refocusing (compare the filtering functions in Equations (15) and (27)). We demonstrated that although this refocusing looks like the application of a phase-correcting factor in the spectral domain, actually this procedure corresponds to the digital correction of both phase and amplitude to shape the illuminating beam. Without such a simultaneous correction, the lateral size of the digitally refocused point-spread functions (i.e., the lateral resolution after refocusing) becomes dependent on the depth and may be either somewhat greater or smaller than in the initial focus (see
Figure 4). To demonstrate the generality of the derived spectral filtering function (12), we presented an example of the transformation of a highly focused beam with an initial Lorentzian profile into a narrow Bessel beam.
Next, besides conventional refocusing, we demonstrated that the filtering function (12) allows one to simultaneously manipulate the phase and amplitude shape of the illuminating beam for “super-refocusing”. Certainly, this does not yet mean overcoming the diffraction limit, but the lateral resolution after such super-refocusing may become several times higher than in the initial physical focus. The ultimate resolution attainable using such super-refocusing, similarly to the physical-focus radius, is limited by the well-known diffraction limit. In real-life situations, the attainable resolution is additionally affected by the level of noise, which may be the main limiting factor in super-refocusing rather than the diffraction limit. However, for quite a realistic OCT initial signal-to-noise ratio ~25–30 dB, it is still possible to obtain an increase in the lateral resolution ~several times, as illustrated in
Figure 6 and
Figure 7.
Generally, the same procedures of digital filtering in the K-space may be used for other transformations of OCT signals, in particular, for compensation of aberrations as discussed in
Section 5.3. Concerning other possibilities for analysis of OCT-signals and studies of various distorting factors, it was pointed out in [
19,
21] that the K-space description of OCT-scan formation makes it possible to simulate dispersion effects.
The procedures for forming OCT scans described here and in [
19] enable a flexible and computationally efficient framework for realistic simulations of OCT signals in various situations, for which direct experimental tests are too difficult/expensive. All examples demonstrated in the manuscript required a few seconds of computation for the initial generation and transformations. Even for rather heavy simulations of forward problems, such as 3D data volumes containing large amounts of scatterers, computations can be made reasonably rapidly, although the required time increases proportionally with the number of scatterers. For example, with a volume of 128 × 128 × 128 pixels, corresponding to 192 × 192 microns laterally and 512 microns in depth with 250,000 scatterers (which imitate the density of biological cells ~5–6 microns in size), the simulations required 59 min using 12 cores of the AMD Ryzen 9 3900X CPU. However, for subsequent transformations of such an image, the computation time
does not depend on the number of scatterers and requires only a few seconds, like in the above-presented examples with several scatterers. Numerous examples of problems in which such simulations can be used are mentioned in [
19,
21]. In the latter work, specific examples related to studying non-Gaussian beams or such modalities as OCT-elastography and OCT-angiography are given. Overall, the K-space-based approach opens rather diverse possibilities for numerical modeling of OCT data and their digital transformations, as well as for deep analysis of OCT-signal features for usage in biomedical imaging and other applications. We also emphasize that the utilization of such simulated (but fairly realistic) OCT data opens unique possibilities for the development and testing of various signal-processing methods in fully controllable and flexibly variable conditions, which may be very difficult/expensive, or even impossible in real physical experiments.