Time-frequency analysis – Due to wavelength-dependent scattering and absorption by the different structures in tissue, the spectral content of the OCT signal changes with depth. Therefore direct application of the Fourier transform on either spatial or spectral domain detected signal Eq. 36.2 is not appropriate to obtain localized spectroscopic information because the depth resp. wave number varying information will be lost. Instead, spectral analysis methods, conventionally called time-frequency (TF) [20] analysis must be used (or, for the present context, depth-wavelength analysis). In most studies, the preferred method has been to use short-time Fourier transforms (STFT):
where w is an analysis window confined in space around d with spatial width Δd, for example, a Gaussian function. The multiplication with a relatively short window effectively suppresses the signal outside the analysis point d±Δd. Physically, the STFT can be considered as the result of passing a signal through an array of band-pass filters with linearly increasing center frequency and constant bandwidth which is inversely proportional to Δd. Thus, there is an inherent trade-off between spectral and spatial resolution. A window with short spatial width Δd will localize the signal well in space but will have reduced k-resolution; conversely a signal with long width will be less well localized in space with the benefit of increased spectral resolution. For a Gaussian window, a spatial domain width Δd will yield a spectral resolution of Δk = 1/(2Δd).

The wavelet transform was introduced to partially overcome this trade-off by adjusting the window size to the frequency being considered. The basic difference between the wavelet transform and the STFT is that the duration and the bandwidth of the wavelet are both changed (while shape remains the same). Physically, the wavelet transform can also be seen as an array of band-pass filters with constant relative bandwidth with respect to the center frequency. In contrast with the STFT, which uses a single analysis window, the wavelet transform uses short windows at high frequencies and long windows at low frequencies. Again, there is a trade-off between time and frequency resolutions. However, these resolutions depend on frequency: the frequency (resp. time) resolution becomes poorer (resp. better) with increasing analysis frequency.

The wavelet transform projects a signal on a family of functions deduced from a (complex) window function w, the “mother wavelet,” by translations and dilations:

The variable κ is the scale factor, dilating (|κ| >1) or compressing (|κ| <1) the wavelet w. When mother wavelets are used that are well localized around a wave number k ₀, then a time-frequency interpretation is possible through k = k ₀/κ.

Bilinear TF distributions do not suffer from the resolution trade-off between both domains. The most important member of this class is the Wigner distribution:

The Wigner distribution is the Fourier transform of an autocorrelation measure of the signal i _D(d). Whereas in conventional autocorrelation computations, the result is a function of lag only, here the functional dependence on d is maintained. In concordance with the Wiener-Khinchine theorem, the straightforward interpretation of the WD is as a localized power spectral density of the detector signal.

The drawback of the WD lies in its quadratic nature: whereas the STFT of two signals X and Y yields STFT _x (k, d) + STFT _y (k, d), the WD will contain interference terms, e.g., WD _x+y (k, d) = WD _x (k, d) + WD _y (k, d) + 2Re[WD _x,y (k, d)]. Even though the interference term contains information on the separation of X and Y (e.g., in space), when it overlaps with the signal terms, interpretation of the WD becomes challenging. In practice (like with the STFT), the signal is analyzed using a window w. This pseudo-WD effectively smoothens the time-frequency distribution, suppressing the interference terms. However, a short smoothing window will be narrow in time and wide in frequency, leading to a good time resolution but bad frequency resolution and vice versa. It is possible to add a degree of freedom by considering a separable smoothing function Π(k, d) = w ₁(k, Δk)w ₂(d, Δd) that allows independent control in both time and frequency of the smoothing applied to the WVD; Δk and Δd denote the width of the window functions in the spectral and spatial domain, respectively. The STFT compromise between time and frequency resolution is now replaced by a compromise between the joint time-frequency resolution and the level of suppression of the interference terms. Robles et al. showed that the pseudo-WD can also be obtained by STFT analysis using two k-domain window sizes Δk ₁>> Δk ₂. The result STFT₁(k, d) × STFT₂(k, d) is mathematically equivalent to a pseudo-WD with window widths of W ₁(k, Δk = k ₂/2) in the spectral domain and window W ₂(d, Δd = 1/(2 k ₁)) in the spatial domain. As for the WD, any remaining interference terms may carry information, for example, on average spatial scatterer separation [7–9].

Xu et al. [21] also define a third class of time-frequency distributions, based on the presumed known spectral profiles of the laser source and the sample. This allows for fitting the obtained OCT signal to this model equation, thereby retrieving the sought spectral properties. In practice, there may exist an optimal time-frequency analysis depending on the application. For example, the STFT is completely free of artifacts such as introduced by the quadratic analysis of the energy distributions. The poor depth resolution resulting from the requirement of a high spectral resolution may not be a problem at all when analyzing relatively large tissue structures, where a spatially averaged spectrum over the entire tissue structure is required. For example, a spectral resolution of 6 nm in LCS corresponds to a depth resolution of 22 μm, which is sufficient to measure hemoglobin concentrations from optical property spectra in distinct regions of the epidermis and the dermis [12] (see Sect. 36.3.2).

Confocal point spread function – The confocal point spread function (PSF) for a single-mode fiber-based OCT system was derived in [22] and validated experimentally in [23]. It can be directly imaged using OCT on a highly diluted (e.g., negligible scattering) sample.

An example of this procedure is shown in Fig. 36.4 for LCS, in a weakly scattering medium (0.038 vol.% of ø198 polystyrene spheres, μ _s <<1 mm⁻¹). The PSF is then given by
where α is a scaling factor, d _focus is the geometrical position of the focus in the sample, and Z _R is the Rayleigh length of the system (Fig. 36.4b). From Z _R , the beam waist can be computed as ω = (Z _R λ/2π)^½, and from that the NA can be derived, using NA = sin(θ) ≈ sin(λ/(π ω)). In the preceding definitions, Z _R ω and NA are defined in the medium, e.g., Z _R = nZ ₀ where Z ₀ is the Rayleigh length of the system measured in air and n is the refractive index. Clearly, the PSF is therefore wavelength dependent. When possible the calibration should be performed wavelength resolved (Fig. 36.4).

The confocal PSF can also be exploited to optimize the trade-off between spatial and spectral resolution by restricting the spatial extent of the collected data region. Xu et al. used high-numerical-aperture optics, geometrically restricting the spatial extent of the signal while using long analysis windows to extract high-resolution spectral information [24].

Sensitivity roll-off – SOCT suffers, when using spectral domain detection (Chaps.​ 5, “Spectral/​Fourier Domain Optical Coherence Tomography”, 6, “Complex and Coherence-Noise Free Fourier Domain Optical Coherence Tomography”, and 7, “Optical Frequency Domain Imaging”), from the inherent loss of sensitivity with depth due to the finite resolution of the detecting spectrometer (or the finite instantaneous bandwidth/sampling time in swept-source implementations). This causes unwanted signal attenuation that can, similarly to the confocal PSF correction, be accounted for in post-processing. Here too, the effect can be turned to advantage by limiting the spatial extent of the collected data. Figure 36.5 shows the theoretical and measured roll-off of an LCS system based on a commercially available Ocean Optics USB4000 spectrometer. The “low” spectral resolution of 8 nm of this device results in roll-off function with full width at half maximum of FWHM ∼10 μm, so that meaningful interference signals are only collected in a window of approximately 20 μm around the equivalent position of the reference arm in the sample. Details of spectral domain LCS can be found in [13].

Retrievable properties using spectroscopic low-coherence interferometry – In general, all methods described above result in a wavelength-resolved power spectrum S acquired within a depth window Δd around a depth d in the sample. We will indicate all wavelength-dependent parameters in the remainder of this chapter by a bold-faced character. The following theory assumes validity of the 1st Born approximation, e.g., the illuminating field is much stronger than the scattered field. Under this assumption, the LCI signal is formed by single backscattering, so that the amplitude of S (d) decreases exponentially with measurement depth and the attenuation coefficient μ _LCI of the sample. Hence, we can describe S (d) using Beer’s law:

The factor 2 in Eq. 36.7 accounts for round-trip attenuation to and from depth d. We note that if the amplitude E(d) of the LCI signal is considered, rather than backscattered power, this factor drops from Beer’s law since E(d) is proportional to the square root of S (d). We also assume that the influence of the PSF and sensitivity roll-off have been accounted for in preprocessing. The LCI-attenuation coefficient μ _LCI is given by
with μ _t the attenuation coefficient, defined as the sum of the scattering coefficient μ _s and the absorption coefficient μ _a . The latter 3 parameters are formally defined in textbooks, e.g. [25], and are discussed in more detail below. We introduce the LCI-attenuation coefficient μ _LCI (the experimental outcome) because in practice, even when correction for PSF and roll-off is not (optimally) performed, and/or when the first Born approximation does not hold, a single exponential decay model Eq. 36.7 is often suitable for fitting the OCT system [23]. In these cases, the simple relation of Eq. 36.8 breaks down, i.e., μ _LCI ≠ μ _t . However, the μ _a can still be retrieved because absorption takes place along the photon’s path (regardless of its trajectory), but values for “μ _s ” should be interpreted with caution in this case.

The parameters ζ and μ _{b, NA} determine the amplitude of S (d) at d = 0. The system-dependent parameters are defined by ζ = S ₀ ∙ T ∙2Δd, with S ₀ the source power spectrum incident on the sample and T the axial point spread function Eq. 36.6. The backscattering coefficient μ _{b, NA} is a sample-dependent parameter, which is defined as the product of the scattering coefficient μ _s and the scattering phase function p (θ), integrated over the numerical aperture (NA) of the detection optics:

To quantitatively determine the μ _{b, NA} of the sample using Eq. 36.8, knowledge of ζ is required. A method to determine ζ is by a separate calibration measurement on a sample with a known μ _{b, NA} , e.g., using Mie theory on well-defined scattering particles [11].

To determine μ _LCI the same calibration measurement may be used, although it is not always necessary since μ _LCI can be obtained directly from the slope of the exponential decay between two or more chosen depths d (according to Eq. 36.7, see also Fig. 36.3c). As a consequence, μ _LCI can be determined in any depth region of interest. The sample’s attenuation is composed of the losses due to both scattering and absorption. When correcting S (d) for the attenuation, also the μ _{b, NA} can be determined at any depth of interest.

Interpretation of measured properties – The absorption coefficient μ _a is directly related to the individual absorption spectra and concentrations of chromophores (e.g., water, hemoglobin, bilirubin) present in the probed volume. The diagnostic value of μ _s measurements depends on its relation to tissue morphology and organization. Ideally, tissue classification based on μ _s would be highly correlated with classification by the pathologist based on microscopic evaluation. The classic approach is to model tissue as an ensemble of spherical scatterers with an effective size to match the experiment. The scattering cross section σ _s and phase function p(θ) are then obtained, e.g., by Mie theory [25]. The scattering coefficient follows from