1.

Introduction

Spatial frequency domain techniques use spatially modulated illumination patterns to quantify absorption and scattering properties of tissue and other diffusive media across visible and/or near-infrared ranges.^1–4 These techniques can be implemented in a variety of ways such as spatial frequency domain imaging (SFDI)² or spatial frequency domain spectroscopy (SFDS).³ Through the quantification of these optical properties, SFDI/S has demonstrated the potential to non-invasively assess skin cancers,^5,6 wound severity,^7,8 and tissue structure and viability^9,10 in the skin. In many of these clinical applications, such as burn wound assessment and melanin quantification, depth sensitivity and subsurface discrimination approaches enabled by SFDI/S are critical features toward accurate characterization of specific tissue volumes.^11–15

Spatial frequency domain techniques utilize a light transport model-based approach to relate the measured remitted reflectance from turbid media, such as tissue, to quantitative values of absorption and reduced scattering. These models presume a uniform distribution of light upon which the spatial frequencies are encoded, and the propagation of this planar illumination pattern will only be impacted by the absorption and scattering properties within the tissue volume. To that end, the optical characteristics of the instrument must be accounted for to isolate measured reflectance signals resulting from the interaction of these illumination patterns with the tissue itself. This task is typically performed in a calibration procedure where a fully characterized reference phantom is measured either prior to or after the target medium.^2,3 These reference phantoms play a critical role, as their absorption and scattering properties are known in advance and hence can characterize these instrumentation contributions to the detected reflectance signals.

Though not often acknowledged explicitly in the literature, this calibration approach using turbid reflectance standards not only will correct for inhomogeneities in illumination intensity over the field of view at each individual spatial frequency, but also any out-of-plane degradation of the pattern integrity (independent from the tissue’s intrinsic optical properties) as the planar illumination pattern propagates into the tissue volume. These out-of-plane degradations are a result of the optical design of the SFDI/S system, primarily its depth of focus and chromatic aberrations, which change as a function of distance relative to the ideal image plane. Since many SFDI/S systems are built around pre-existing projection units, the optical design is often embedded with the digital micromirror device and optimized for wide-field projection applications in either the visible or near-infrared wavelength regimes.^16–20 This has led to systems that reduce the projection field of view with additional lenses. This type of modification also reduces the working distance between the instrument and the target medium, which has advantages toward the development of compact handheld systems. This reduced working distance, however, can also reduce the depth of focus. This could rapidly blur the projected pattern as it propagates into the tissue volume and compete with the effects of light scattering. Likewise, several SFDI/S approaches span both visible- and near-infrared regimes,^3,6,8,13,17 which also can be complicated by chromatic aberrations as most achromatic designs only compensate for this type of aberration over a few hundred nanometers.

It has been shown that when the target tissue has a distinct surface shape and depth (i.e., topology), errors in the determination of optical properties may occur due to the tissue surface’s distance relative to the ideal imaging plane of the SFDI/S instrument.²¹ Previous studies have employed sample surface profile information to adjust for venous imaging²² or imitate diffuse reflectance of biological tissues.²³ Similar approaches, combined with multi-step calibrations procedures, have also been developed to correct for this displacement error in the context of SFDI/S.^13,24

In this work, we will present a simple method using a single tissue-simulating phantom to evaluate the influence of out-of-plane optical aberrations in the context of the resulting absorption and reduced scattering coefficients. As different spatial frequency instrument designs will exhibit their own distinct combinations and magnitudes of optical aberrations, this characterization method can provide vital feedback on the accuracy and spectral integrity of depth-specific optical properties. Additionally, once this optical characterization is obtained, we will also demonstrate how this data can be used to compensate for these aberration errors in post-processing for a spatial frequency domain setup that is particularly susceptible to these types of aberrations (e.g., a compact SFDS system). This compensation approach is evaluated both in measurements of a phantom with differing optical properties of that used in characterization and in multi-layer phantom constructs. The former one demonstrates that this empirical method can be applied to arbitrary tissue optical properties over a range of positional displacements from the ideal image plane and in the latter case demonstrates that sub-surface optical properties can be extracted when the top layer properties are known.

2.

Material and Method

3.

Results

The scaling errors from measured data relative to when the phantom was placed in the focused position are shown in Fig. 1 with solid lines. Here, there is a progressive underestimation of scattering properties as a function of increasing distance from the focal plane. The dashed lines show the fit curves [Eq. (1)] across the full spectral range at all depths. These fittings were done to remove the measurement noise artifacts and establish a smooth mesh that can describe the non-linear influence that the aberrations convey on the resulting reduced scattering coefficient. An LUT was generated from these fitted curves that could be interpolated to any arbitrary depth up to 15 mm. To compensate for aberrations, measured scattering values would be divided by the interpolated depth-specific curve from this table, absorption values would be multiplied by it. Additionally, the offset term for the absorption values was determined to be $0.00025{\mathrm{mm}}^{-1}$ per millimeter depth for this specific instrument setup. This offset value was highly linear over this 15-mm range, with $R^2=0.999$.

Fig. 1

The calculated ratios of each reduced scattering coefficient spectra at all depths relative to when the phantom was in the focused position ($z=0$), dashed lines are the data fitted using Eq. (1).

The measured reduced scattering spectra for the reference phantom, test phantom, and the multi-layered constructs at different distances to the projection lens is shown in Figs. 2(a)–2(c). As shown, the reduced scattering coefficient at different distances from the projection lens progressively deviates from the in-focus measurement. The compensated reduced scattering spectra are shown in Figs. 2(d)–2(f) for the reference phantom, the test phantom, and the multi-layered construct, respectively. These spectra were corrected using the characterization data obtained from reference phantom only. After applying the aberration compensation method, the scattering spectra of the reference phantom from all distances are within an average of 0.1% root-mean-square (RMS) error relative to the in-focus values (0.2% at 10 mm from the focal plane). Left uncorrected, these RMS errors would be 18.2% on average and 40.2% at 10 mm. The average for the test phantom scattering was 1.4% over all distances (uncorrected average: 17.1%, 36% at 10 mm) and $\sim 0.6\%$ for the multi-layered construct.

Fig. 2

Reduced scattering coefficient calculated at each distance and the respective aberration compensated data for the (a) and (d) reference phantom; (b) and (e) test phantom; and (c) and (f) multi-layered phantoms. $z=0$ refers to the ideal, in-focus reduced scattering spectra.

The measured absorption spectra from the reference, test and multi-layer phantoms are shown in Figs. 3(a)–3(c), respectively. Figures 3(d)–3(f) shows the recovered optical properties for absorption coefficient after correcting for the errors, using the characterization data obtained from reference phantom only. There is on average a $\sim 32.2\%$ RMS error in uncorrected absorption spectra relative to the in-focus values for the reference phantom (45.5% at 10 mm). After applying the compensation method, the absorption RMS error reduced to within 0.03% on average. The test phantom showed an average $\sim 49.1\%$ RMS error (69.5% at 10 mm) that was reduced to an average of 3.8% over all distances (3.4% at 10 mm) and for the multi-layered construct’s average of $\sim 43.7\%$ was reduced to 4.7%.

Fig. 3

Absorption coefficient calculated at each distance and the respective recovered aberration compensated data for the (a) and (d) reference phantom; (b) and (e) test phantom; and (c) and (f) multi-layered phantoms. $z=0$ refers to the ideal, in-focus absorption spectra.

Overall, the data shows that the recovery of both out-of-focus and subsurface optical properties can be achieved even though these optical aberrations can have a considerable impact on the directly measured spatial frequency dependent reflectance values.

4.

Discussion

We have presented a method to characterize and subsequently compensate for the depth-specific impact that optical aberrations can have in spatial frequency domain measurements. The goal of this investigation was to demonstrate that while these aberrations can significantly impact the determination and spectral integrity of subsurface optical properties, they can be treated independent from the intrinsic values of the turbid medium. Using a test phantom different from that used to characterize the systems aberrations, we have shown that the compensation method can be applied as a scaling factor (and an additional offset, in the case of weak absorption) irrespective of the target's optical properties, reducing errors on average by a factor of $\sim 12$. While not typical in most practical settings, this compensation approach was shown to remain effective even at such extremes as 15-mm displacement errors. Additionally, to further illustrate this independence of aberrations from the target’s properties, multi-layer phantoms were used where the bottom layer optical properties were recovered to within $<5\%$ its target.

This approach does not directly address the effect of aberrations on the spatial frequency dependent reflectance, but rather on its resulting estimation of absorption and reduced scattering coefficients. One consequence of this approach is that it assumes that as the patterns of light propagate into the turbid medium, the spatial frequency remains constant. This is not true as defocus errors will broaden the effective area of the projected image as it propagates farther way from the ideal image plane, effectively reducing the spatial frequency. In this case, the measured broadening of the project pattern at a depth of 10 mm was an additional 2 mm, effectively decreasing the spatial frequency by $<5\%$ at its most extreme depth, even if all spatial frequencies were to reach that depth, that would result in a calculation error $<5\%$.²¹ We acknowledge that this assumption is a limitation to this approach. Given its magnitude relative to the other aberration induced errors, however, we consider this residual error to be minimal.

Considerations to the optical design of the SFDI/S can also play a significant factor in these types of aberration induced errors. Rather than utilizing this empirical approach to compensate for these errors, careful lens design choices can minimize these aberrations and hence negate the potential need to utilize this method. The fitting equation [Eq. (1)] was derived empirically to describe the specific combinations of aberrations present in the system used in this investigation. It is expected that different systems would contain different combinations of aberrations and therefore may require another functional form to describe their depth-specific errors. Note that the terms in this equation do appear to correlate with chromatic aberration and depth of focus errors from a simple lens design. The first term in this equation is the most dominant and correlates with the expected behavior of chromatic aberration when only a single glass material is used in the optical system. If chromatically compensating lenses are used (e.g., achromats, apochromats, etc.), this simple power law term would no longer be sufficient as higher-order terms would be required to describe the increasingly complex chromatic dependence of focal length. The second term represents the slight wavelength dependent change in depth of focus due to the chromatic aberration shift in focal length and the last term represents the general depth of focus related to the numerical aperture of the imaging system.

From this investigation, the impact of the depth of field of the projected illumination has been shown to impact the recovery of subsurface and out-of-plane optical properties. There are several optical designs that can mitigate this type of aberration, e.g., high F/# systems, telecentric designs, or aspheric extended field approaches. These optical solutions, however, often come at a price in terms of practical clinical integration and use. Typically, these optical designs require long working distances and/or low light throughput that compete with clinical space and data acquisition times. It is for these reasons that this characterization and compensations methods are presented here as a pragmatic compromise on the instrumentation design while maintaining spectral integrity of the data collected.

5.

Conclusion

We have provided a method to quantify primary optical aberrations' potential impact on spatial frequency domain measurements with respect to depth and proposed a simple empirical approach to compensate for these errors in post processing. We have shown that, even though these aberrations can considerably affect the spectral integrity of subsurface optical characteristics, they can be addressed independently from the turbid medium's intrinsic optical properties. This compensation method can be applied as an LUT to turbid media whose surface is displaced from the ideal focus as well as the subsurface tissue structures in layered media.