Underlying principle of resolution-enhanced OCT
RE-OCT is a spatial-frequency bandwidth expansion approach based on the exchange of information between resolution and SNR that is supported by the underlying theorem of invariance of information capacity3-5. We conjecture that the transverse resolution of a tomogram acquired by a beam-scanned OCT system can be enhanced via computational bandwidth expansion (BE), where signal at higher spatial frequencies is raised via multiplication by a magnitude mask in the spatial-frequency domain (i.e., magnitude-based deconvolution). This is feasible due to the under-filling of the objective aperture (i.e., a physical bandwidth limit) that is implemented in the ubiquitous telecentric scanning scheme (Supplementary Fig. 2b) used by most OCT systems worldwide. However, deconvolution inherently amplifies noise and degrades the SNR of the image. To address this problem, we also conjecture that the SNR penalty associated with computational BE can be compensated by first suppressing the system noise—one efficient way in OCT is by coherently averaging multiple successively acquired complex tomograms44-46. The underlying principle of RE-OCT is illustrated in Fig. 1. RE-OCT utilizes coherent averaging for efficient noise suppression to ‘earn’ SNR, which can be used to ‘purchase’ resolution via computational BE.
To understand the underlying principle of RE-OCT in the context of the theorem of invariance of information capacity, we restate Cox and Sheppard’s expression for the information capacity of an optical system4 (Eq. 1).
where L, T, and B denote the spatial field-of-view (FOV), temporal duration, and bandwidth in the associated dimension, respectively. The first three terms represent the SBP along the three spatial dimensions while the fourth term represents the TBP. The last term represents the SNR (in bits), where s and n denote the average signal power and the additive noise power, respectively. Coherent averaging can be considered as the exchange of the temporal information (of an object that is invariant in time) for enhanced SNR. This effectively expands the information capacity of coherent-averaged versus the single-shot tomograms via the increase in the SNR term, which can be sacrificed to expand each of the transverse spatial-frequency bandwidths, Bx and By. (For a full analysis of this process, see Supplementary Section I.) One scenario is to expand the bandwidth equally along each transverse dimension by a factor equal to the square root of the gain in SNR in order to, in principle, enhance resolution in the transverse plane by the same factor, without suffering any SNR penalty relative to the traditional single-shot tomogram (Supplementary Fig. 1). However, RE-OCT is not limited to this scenario; the trade-off between resolution and SNR can be flexibly navigated by tuning the number of tomograms averaged and the BE factor applied (i.e., more SNR than the amount ‘earned’ from coherent averaging can be sacrificed to prioritize further resolution enhancement) (Fig. 1a).
Coherent-average noise suppression in space and spatial-frequency domains
Noise suppression via coherent and incoherent average over N = 1 through 100 acquisitions was investigated in both space and spatial-frequency domain in a silicone phantom containing scattering particles (Fig. 2). Supplementary Section III discusses the effects on the image signal in the space domain, where a coherent average demonstrated a factor of √N superior noise reduction efficiency over an incoherent average (Fig. 2a,b). Our results are consistent with previous work in OCT and theoretical trends45,46 (see Supplementary Section III for caveats of experimentally achieving the theoretical performance).
Here, we additionally investigated coherent-average noise suppression in the spatial-frequency domain (Fig. 2c–e), where the image is a superposition of the detected backscattered signal with a Gaussian magnitude spectrum (when imaged with a Gaussian beam) and the system noise with a uniform magnitude spectrum (circular Gaussian random variable in space). The power spectrum exhibited a dynamic range (DR) in the spatial-frequency domain, as measured from the power at DC to the noise floor (Fig. 2d). A coherent average over N = 100 acquisitions resulted in a suppressed noise floor that led to an increase in DR of 20 dB, corresponding to noise reduction by a factor of 100 (Fig. 2d). In other words, the coherent average has revealed phase-stable but low-magnitude backscattered signal at higher spatial frequencies, which were originally below the noise floor in the single-shot image but are now above the suppressed noise floor. Consequently, signal phase remains correlated (i.e., absence of random phase variation) across spatial frequencies corresponding to a larger bandwidth (Fig. 2c,e). The improved correlation of signal phase after reduction in the noise floor can also be understood by considering the impact of SNR on phase noise in phase-sensitive OCT50,51. Supplementary Section VII investigates (in simulation) the relations between N, DR and phase-correlation limit in the spatial-frequency domain.
Resolution-enhanced OCT in silicone phantom
Resolution enhancement in silicone phantom using coherent average over 100 acquisitions and a computational BE expansion factor of 2.4× is shown in Fig. 3. (See Methods and Supplementary Section IV for a complete description of the RE-OCT reconstruction procedure.) RE-OCT achieved a RE factor of 1.5×, from the traditional aberration-free resolution of 2.1 µm to an enhanced resolution of 1.4 µm (Fig. 3a,b), while the peak signal-to-background ratio (SBR) marginally decreased from 50 dB to 48 dB (Fig. 3a,c). However, when computational BE was performed on the single-shot image, not only was the SBR substantially decreased by 10 dB, but the quality of the point spread function (PSF) also suffered (Fig. 3a,c). This penalty is a result of computational BE indiscriminately amplifying both the backscattered signal and the system noise that dominates at higher spatial frequencies (Fig. 2d,e). Resolution improved at the cost of degraded SBR as a larger BE factor was applied (Fig. 3d,e and Supplementary Movie 1). Noise suppression prior to computational BE was essential in maintaining adequate SBR as well as the quality of the PSF in RE-OCT. However, even with a coherent average over 100 acquisitions, the best achievable resolution from this experiment was limited to 1.4 µm at BE factor of 2.4; applying a larger BE factor only resulted in lower SBR and degraded PSF quality, without further improvement in resolution.
In order to investigate the factors that limit the experimentally achievable resolution enhancement, we performed the RE-OCT procedure on 6 simulated en face planes (3 conditions, each with and without aberrations) with properties representative of the silicone phantom images (Fig. 3d–g). (See Supplementary Section VI for information on the simulated en face planes.) Based on Cox and Sheppard’s information capacity framework4, the achieved RE factor is expected to be equivalent to the applied BE factor (see Supplementary Section VIII). This relationship holds true for the simulated ideal noise-free condition, in which the RE-OCT efficiency (defined as the ratio RE factor/BE factor) remained 1 up to the Nyquist limit (Fig. 3f, noise-free limit). However, the presence of system noise decreased the RE-OCT efficiency with increasing BE factor (Fig. 3f, noise only). The trends as a function of BE factor for resolution, SBR, and RE-OCT efficiency for the noise-only condition are remarkably consistent with the experimental results (Fig. 3d–f). Furthermore, the presence of scattering signal from the silicone background, in addition to system noise, caused a slight decrease in RE-OCT efficiency relative to the noise-only condition (Fig. 3f, noise with background). These results suggest that system noise is the primary limiting factor in RE-OCT. Indeed, both experiment and simulation showed that superior resolution was achieved with coherent average over larger N (i.e., more noise suppression) for a BE factor of 2.4 (Fig. 3g). In addition, optical aberrations degraded resolution, SBR, and RE-OCT efficiency for all simulated conditions (Fig. 3d–g, red). As expected, the simulated condition incorporating all three contributions: system noise, silicone background, and aberrations, most closely matched the experiment.
Resolution-enhanced OCT in biological samples
We implemented RE-OCT in collagen gel and ex vivo mouse brain, and show the best RE-OCT performances that were achieved, corresponding to BE factor of 2.0 (Figs. 4 and 5). In fibrous collagen gel, RE-OCT enhanced the visualization of the collagen fibre architecture by not only narrowing the width of the collagen fibres, but also increasing the peak signal magnitude of each fibre as a result of the resolution enhancement (Fig. 4a,b). Remarkably, low-contrast fine microstructural features, which were not clearly discernible in the traditional single-shot image due to weak signal, are more apparent in the RE-OCT image owing to the improved localization of signal energy in space (Fig. 4a, yellow arrows). In the BE single-shot image, the narrowing of fibre width can still be observed to a certain extent, but the peak signal magnitude did not improve as much (Fig. 4b). Furthermore, the SBR was degraded more severely in the BE single-shot image due to the amplification of noise without prior noise suppression (Fig. 4c). Computational BE with BE factors larger than 2.0 resulted in degraded SBR without further narrowing of the fibre width or improvement to the peak signal magnitude (Supplementary Movie 2), similar the degradation observed with BE factors larger than 2.4 in the silicone phantom (Supplementary Movie 1).
In ex vivo fresh mouse brain, RE-OCT enhanced the visualization of myelinated axonal processes, especially the low-contrast features that were less apparent in the traditional single-shot image (Fig. 5a, yellow arrows). Although the narrowing of fibre width could be observed in the BE single-shot image, the peak signal magnitude was degraded without coherent-average noise suppression (Fig. 5b), similar to the effects in collagen gel. Due to the typical fibre thickness of 1-3 µm (which is comparable to the native OCT transverse resolution of 2.1 µm) of myelinated axons52, some of the fibre narrowing observed here is not as prominent as in the collagen gel. The SNR penalty of the computational BE procedure is most apparent in the neuron (Fig. 5a, green inset), which produces lower OCT intensity than the surrounding brain tissue. Although the neuron remained visible in the RE-OCT image, the noise level in the BE single-shot image was brought up to that of the backscattered signal from the surrounding brain tissue, causing the neuron to ‘disappear’ into the background (Fig. 5a,c). This emphasizes the importance of coherent averaging in RE-OCT, particularly when weak-scattering structures need to be clearly visualized. Computational BE with BE factors larger than 2.0 resulted in degraded SBR and lower contrast between the neurons and surrounding brain tissue, without further narrowing of the fibre width or improvement to the peak signal magnitude (Supplementary Movie 3).
Factors that limit achievable resolution enhancement
We revisit the image signal power and phase in the spatial-frequency domain (Fig. 2c–e) to further understand the role of system noise, background, and aberrations on RE-OCT resolution. System noise limits not only the available DR of the spatial-frequency-domain image, but also the spatial-frequency bandwidth over which signal phase (associated with any given point scatterer in space) remains correlated (Fig. 2d,e and Supplementary Fig. 7). Phase correlation in the spatial-frequency domain has a direct implication on the spatial resolution of a coherent image—in order to achieve the best localization of signal energy in space, signal at different spatial frequencies must be able to constructively interfere (i.e., be in-phase with each other). Thus, the phase-correlation limit (which is limited by SNR in our experiment) determines how much of the expanded spatial-frequency bandwidth (determined by the BE factor) can support constructive interference and contribute to enhancing the resolution in RE-OCT. Computational BE far beyond the phase-correlation limit only serves to amplify the contribution of phase-decorrelated higher spatial frequencies, which degrades the SBR and the quality of the PSF without further improving the resolution (Fig. 3d–f and Supplementary Movie 1).
In contrast to system noise, background is composed of backscattered (single- (SS) and multiple-scattering (MS)) signal from the sample medium (silicone in this case). The spatial-frequency spectrum of the SS background is bandlimited and obeys the imaging bandwidth support of the system (determined by the illumination beam width in our system). Meanwhile, evidence has shown that frequency content of MS background may extend beyond the imaging bandwidth of the system53. In either case, background may exhibit uncorrelated phase as opposed to a flat profile of an ideal PSF (Supplementary Fig. 6b) and contribute to the disruption of phase correlation within (SS case) as well as outside (MS case) of the imaging bandwidth. As a result, resolution may be degraded by the presence of background compared to if the medium were completely transparent. In this respect, the role of background on OCT resolution is similar to that of optical aberrations—while aberrations contribute slowly varying phase inside the pupil, background contributes uncorrelated phase that results in the OCT speckle. Importantly, both effects imply that the sample itself may limit the achievable resolution; there can be contribution from sample-induced aberrations in addition to system aberrations, and the degradation of resolution by uncorrelated background phase becomes more severe when the structure of interest has lower SBR (e.g., due to weak scattering from the structure or strong scattering from the medium, or both). Furthermore, both background and aberrations are factors that cannot be mitigated by coherent-average noise suppression.
Expanded framework of information capacity and resolution in coherent imaging
A fundamental limit to resolution enhancement by RE-OCT is governed by the disruption of phase correlation in the spatial-frequency domain—due to system noise, background, aberrations, and other factors (e.g., sample instability, mechanical vibration, etc.). Among other factors, system noise played the most significant role in our experiments by determining the available DR of the image and the phase-correlation limit in the spatial-frequency domain. Notably, system noise is also the only factor that can be suppressed via coherent averaging in our experiments. Thus, the basis of RE-OCT lies in navigating the trade-off between resolution (in the space domain) and DR (in the spatial-frequency domain) of the image via coherent-average noise suppression and computational BE (Fig. 1a), where DR represents the impact on SNR that manifests in the spatial-frequency domain (Fig. 2d). In Fig. 3a–c, we prioritized resolution enhancement and applied a BE factor of 2.4, which sacrificed more DR than the 20 dB earned with coherent averaging (Supplementary Fig.8b). Alternatively, we could apply a BE factor of only 1.4 and simultaneously improve both resolution and SBR by a smaller margin (Fig. 3d,e), where the SNR penalty was offset by the increased peak PSF intensity as a by-product of improved localization of the PSF in space (note the maxima in Fig. 3e).
In order to reconcile the predictions of information capacity and our experimental RE-OCT results, we propose an expanded framework of information capacity and resolution in coherent imaging (Fig. 6). The expanded framework emphasizes phase correlation in the spatial-frequency domain (in addition to SNR, FOV and spatial-frequency bandwidth in Cox and Sheppard’s framework4) as an important facet of the information capacity of a coherent imaging system. In theory, resolution is governed by the imaging bandwidth of the optical system. In practice, however, phase-correlation limit in the spatial-frequency domain (Fig. 2e and Supplementary Fig. 7) must also be considered when determining the best achievable resolution. Supplementary Section VIII computes the resolution enhancement that is theoretically supported by Cox and Sheppard’s information capacity framework4, and exemplifies the additional practical limit imposed by the SNR-limited phase-correlation limit (Supplementary Fig. 8a). Importantly, any factors—whether associated with the optical system or the sample itself—that can disrupt phase correlation in the spatial-frequency domain may prevent the optimal resolution (determined by the bandwidth support of the optical system) from being experimentally realized. By extension, in a time-dependent system whose information capacity includes the time-bandwidth product4, the temporal resolution of such a system would also be subjected to the correlation of phase in the temporal-frequency domain.