A novel technique for axial resolution improvement of Optical Coherence Tomography (OCT) systems is proposed. The technique is based on step-frequency encoding, using frequency shifting, of the OCT signal. A resolution improvement by a factor of ~7 is achieved without the need for a broader bandwidth light source. This method exploits a combination of two basic principles: the appearance of beating, when adding two signals of slightly different carrier frequencies, and the resolution improvement by deconvolution of the interferogram with an encoded autocorrelation function. In time domain OCT, step-frequency encoding can be implemented by performing two scans, with different carrier frequencies, and subsequently adding them to create the encoded signal. When the frequency steps are properly selected, deconvolution of the resulting interferogram, using appropriate kernels, results in a narrower resolution width.
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Optical Coherence Tomography (OCT) is an emerging medical imaging technology with applications in the diagnosis of disease in an ever expanding range of fields such as oncology, ophthalmology, cardiology, etc . The main advantage of OCT is its ability to acquire high resolution images (~1-15 μm), in real-time, non-invasively, and in vivo. OCT technology has significantly improved over the past few years with the introduction of fast, high resolution, OCT systems which are well suited for in vivo applications. However, many disease changes, such as those associated with early stage cancer, are in the micron and sub-micron range so further improvements in resolution are required for their detection. The most straightforward approach to improve the axial resolution is to use a light source with a broader spectral bandwidth. For example, a Kerr-lens mode-locked Ti:sapphire laser, Ti:sapphire pumped super-continuum generation, and thermal light sources were used to obtain 0.5-1 μm axial resolution in biological tissue [2–6]. Additional extracavity spectral broadening in highly nonlinear fibers allows for sub-micrometer (0.85μm) resolution OCT at 800 nm and sub-2-micrometer (1.4 μm) resolution at 1.13 and 1.38 μm wavelength [7,8]. Simultaneous dual-band, ultra-high, resolution OCT imaging with an off-the-shelf all fiber integrated supercontinuum (SC) source enables OCT imaging with 1.7 μm and 3.8 μm axial resolutions, at 840 nm and 1230 nm, respectively . In a recent paper, the generation of ultrabroadband biphotons that span a bandwidth of ~300 nm with center wavelength λ0 = 812 nm has also been reported . Using these ultrabroadband biphotons in conjunction with semiconductor single-photon avalanche photodiodes (APDs), the narrowest axial resolution (0.85μm) was reported in quantum OCT (QOCT). However, there is a limit to the resolution improvement that can be achieved even by these state-of-the-art OCT systems. The inversely proportional, asymptotic, relationship between bandwidth and resolution implies that increasingly broader, often unattainable or unsustainable, bandwidths are required for marginal improvements in resolution. Furthermore, there is an additional penalty of either a significant increase in system complexity and cost or loss of sensitivity and power attenuation as the source bandwidth is extended.
Resolution can also be improved using numerical processing techniques. The use of deconvolution techniques on OCT images was first reported in 1997 . An increase in resolution by a factor of ~2 was obtained, as compared to the original interferogram. Promising results, using the CLEAN algorithm, were also reported in 1998 . Recently, two-dimensional deconvolution methods, for deblurring, were shown to improve the quality of OCT images . The maximum entropy method (MEM) was also shown to enhance the resolution in OCT . Moreover, by digitally reshaping the source spectra to known modes, OCT resolution was improved by a few microns . It was also shown theoretically that a chirped quasi-phase-matching nonlinear crystal structure can significantly enhance the axial resolution in QOCT by increasing the spectral width of the generated entangled photon pairs .
In this paper, a simple technique that effectively improves the axial resolution of an OCT system, without the need of a broader bandwidth light source, is demonstrated. The technique is based on step-frequency encoding of the OCT interferogram. Signal encoding belongs to a group of methods used in radar, sonar, and ultrasound to augment the resolution as well as improve the SNR [15,16]. By linearly increasing, step by step, the frequency of successive pulses and adding them, the resolution can be significantly improved . Since, in OCT, the detection is not based on pulse width and time-of-flight, the application of this technique has been appropriately modified. The proposed encoding scheme is based on the summation of scans with different carrier frequencies and deconvolution.
The proposed method of resolution improvement is based on two basic principles: (i) the appearance of beating when adding two waves of slightly different frequencies, and (ii) the resolution improvement by deconvolution with an appropriately chosen kernel function.
2.1 Frequency summation in Optical Coherence Tomography
The OCT signal, which results from the interference of two partially coherent light beams, can be expressed in terms of the source intensity, I0, asEq. (2), the Gaussian envelope is amplitude modulated by an optical carrier. The peak of this envelope represents the location of a reflector and/or a scatterer in the sample under test. Its amplitude depends on the reflectivity of the structure. The optical carrier is due to the Doppler shift resulting from scanning one arm of the interferometer, and the frequency of this modulation is controlled by the speed of scanning. The Doppler frequency is
In the case of a single scatterer and if k1 = k2 = 0.5, then the interference pattern can be written as:
2.1.1 Frequency summation
Adding two A-Scans with two different but similar carrier frequencies, results in a beating pattern at each peak of the interferogram. For example if:
If the frequencies are chosen correctly, the axial resolution can improve significantly. For example, if the frequencies are chosen such that the beating nodes appear at intervals corresponding to half the coherence length of the system, then the interferogram from a single reflector will have the appearance depicted in Fig. 1 .
Filtering the carrier frequency (f1 + f2)/2, results in a demodulated interferogram with beating frequency (f1-f2)/2:Equation (10) implies that, if the initial phases from A-Scan to A-Scan are different, the beating patterns from A-Scan to A-Scan may be different. This will be further discussed in subsequent sections.
2.1.2 Discrete Reflectors
In the case of a sample consisting of discrete reflectors, the OCT interferograms at two frequencies f1 and f2 are given by:
The beating interferogram resulting from the addition of the two A-Scans is:
Filtering the carrier frequency (f1 + f2)/2 results in an A-Scan with beating frequency (f1-f2)/2:Equation (16) includes a phase term which can affect the beating pattern of the interferogram. The results, as well as remedies, of this term are discussed in sections 2.3.3 and 2.3.4.
2.2 Deconvolution of the OCT signal
The OCT interferometric signal is a cross correlation function between the actual locations and reflection coefficients of scattering sites within the sample and the source autocorrelation. Taking the Fourier transform of both sides of Eq. (16) and solving for the impulse response h(Ls), one can theoretically recover the scatterer locations. This is the simplest deconvolution algorithm in image restoration, called inverse filtering, resulting in:
Basic Fourier transform properties define that:Eq. (15) and Eq. (21) give:
By dividing with the Fourier transform of a kernel from a single peak:
Hence, theoretically, we can successfully extract the reflectivity profile albeit with a distance shift of from the real position of each scatterer. This shift is a result of the different phases of the interferograms, which leads to different beating patterns from peak to peak and from A-Scan to A-Scan. The implications and remedies of this distance shift will be discussed in a section 2.3.3.
Equation (24) is the simplest deconvolution algorithm in image restoration, called inverse filtering. The results of simple deconvolution algorithms are extremely sensitive to noise. An important class of algorithms, which addresses the issue of noise, assumes a model in which the observed image is the sum of a perfect convolution and random noise of some particular distribution. One such method, the Lucy-Richardson algorithm, based on Bayes’ theorem, is an iterative process starting with an estimate of the original image which is updated, after each iteration, leading towards the original image . The algorithm is defined by 
2.3 Step frequency encoding, resolution enhancement, and limitations
The axial resolution of an OCT image can be significantly improved, when step frequency encoding is combined with deconvolution.
2.3.1 Axial resolution improvement with step frequency encoding and deconvolution
By adding two A-Scans at two different frequencies, beating will appear and the interferogram from a single reflector will have the appearance depicted in Fig. 1. Already there is an improvement in the full width half maximum (FWHM) of the central lobe and, therefore, the resolution. The resolution improvement results from the fact that the encoded central lobe is narrower than the equivalent envelope of a standard OCT scan. The degree of the improvement depends on the frequency steps. In the encoded case the FWHM of a peak is the width τ of the central lobe which can be calculated by:
After deconvolution an additional resolution improvement of a factor of ≥ 2 occurs which results in:
From the relation x = vτ, where v is the velocity of the scanner, the improved resolution is at least
The resolution improvement depends on the beating frequency and higher beating frequencies results in narrower peak widths. However, increasing the frequency difference between steps is not without a penalty since the interferogram’s sidelobes increase.
2.3.2 Sidelobe amplitude
One of the important limitations of this technique is the appearance of sidelobes. The relative size of the sidelobes can be calculated assuming a Gaussian envelope for the interferogram, i.e.:
From Eq. (33) we can see that, if the frequency steps are larger, the amplitude of the sidelobe As is larger. In addition there are more sidelobes in the range of a coherence length. Figure 2 illustrates how the amplitude and number of sidelobes can increase with increasing frequency steps (f1-f2). In Fig. 2(a), the frequency difference is smaller (3kHz) than in Fig. 2(b) (6kHz). The increase in the number and amplitude of the sidelobes is evident in the latter case.
The sidelobe limitation is to a large extend remedied by deconvolution with appropriate kernels and further multiplying by he deconvoluted envelope of the standard OCT signal. In the case of an encoding scheme similar to that of Fig. 2(a), where the sidelobes are at ~05-0.6 of the central lobe, a reduction of as much as 45 dB (20-45 dB typically) was achieved after deconvolution and multiplication. In an encoding scheme similar to that of Fig. 2(b) such a reduction would be more challenging, hence the tradeoff between resolution improvement and sidelobe intensity.
2.3.3 Location shift
Since OCT is not detecting the reflected replicas of a single pulse, as in ultrasound or radar, there is a fundamental problem. When externally modulating the light, as proposed in the following experimental section, each A-Scan can have a different phase from the next A-Scan, and each peak can have a different phase from the next peak in the same interferogram. This leads to different beating patterns from A-Scan to A-Scan and from peak to peak where the nodes of the beating can be at random locations relative to the center of the interferometric peak. The location shift from the real location of the scatterers, as is described by Eq. (25), is a result of the different phases of each interferogram which lead, after addition, to different beating patterns (Fig. 3 ). The phase of each interferogram is:Fig. 3 (a)), whereas the maximum shift at π/2 (Fig. 3 (b)). From the relation Δφ = ω0Δt, where ω0 = 2π(f1-f2)/2, the maximum location shift is:
This can be converted to distance given the relation Δx = vΔt
The problem of the location shift can be partly alleviated by performing deconvolution with several kernels corresponding to different beating pattern, and subsequently multiply the results. It was noted experimentally that using 3 such kernels was adequate to reduce the positional variation. Even after multiple deconvolutions the uncertainty related to the initial phase of each A-Scan remains. This uncertainty is small, in the range of vT/2, as described theoretically in Section 2.3.3, and evaluated experimentally. An algorithm for motion correction was used to alleviate this problem. It could even be completely removed if there was a practical method of synchronizing the modulation phase to the OCT scanning.
2.3.4. Design trade-offs
The tradeoff between sidelobe intensity and number, location shift error, and resolution improvement is obvious from Eqs. (31), (33), and (37). The greater the frequency difference of the A-Scans, the smaller the distance shift error,and the greater the resolution improvement. Unfortunately, however, the number and intensity of sidelobes is also increased. A careful selection of frequency steps is necessary to obtain optimal results for resolution improvement while maintaining tolerable levels of sidelobe amplitude and distance shift error. From simulation and experimental data, it appears that an appropriate choice for the frequency steps is such that the beating pattern has a period of half the coherence length of the system. Such a value, results in a resolution improvement of a factor of 3 with a total improvement of a factor of ≥ 6 with subsequent deconvolution. In addition, the main sidelobe level is ~0.5 that of the main peak which can be sufficiently reduced by multiple deconvolutions to avoid significant detail degradation. If T is the time it takes for one coherence length to be scanned by the TD OCT system (T = coherence length / velocity of scanning), then the beating frequency must be fbeat = (f1-f2)/2 = 2/T. As with any design trade-offs, there values will highly depend on system design and imaging study specifics.
4. Experimental Method
Figure 4 depicts the experimental setup of the Time Domain (TD) OCT system used in this study. It consisted of a superluminscent source (SLD) and a fiber optic-based Michelson interferometer. The SLD, operating at a centre wavelength λ0 ≈1300 nm and spectral FWHM of ~50 nm, resulted in an axial resolution of 13.5μm. The transverse resolution of the system was 16.5 μm, power of light incident to the sample was 8mW, and sensitivity of the system was measured to be 106 dB. The reference arm was scanned by translating a retro-reflector with a galvanometer at a velocity of 17.5mm/sec. The system employed two acousto-optic frequency shifters, (FS1 and FS2, 25 MHz, variable 5MHz bandwidth, 1st-order diffraction, Brimrose Inc.), one in the reference and one in the sample arm. The two frequency shifters were driven with a dual channel RF driver to produce a net frequency shift and added to the system a total insertion loss of 5 dB. The signal from the detector was then digitized using a 16-bit 1.2 MHz data acquisition board.
Encoded images were acquired by taking two A-Scans at each point, with f1 = 100kHz and f2 = 103kHz, and subsequently adding them, to produce a beating frequency of 1.5 kHz which corresponded to about half the coherence length for the giving reference arm scanning scheme. This beating frequency satisfied the requirements, discussed before, to achieve optimal results for resolution improvement while maintaining tolerable levels of location shift and sidelobe amplitude.
A reference sample (three microscope cover slips of ~170 μm thicknesses) was used to extract single peak interferograms subsequently used as kernels for the deconvolution (Fig. 5 ). Standard and encoded OCT images were collected, processed and compared.
A limitation of the experimental approach described above is that the acquisition time is twice that of standard OCT since two A-Scans are acquired at each point in the sample. This limitation can probably be addressed in the future by using more than one acousto-optic modulators in the reference arm.
The resolution of a system, as defined by Rayleigh, is its ability to discriminate adjacent structures in an image. In order to illustrate the improvement in OCT resolution, using the proposed method of encoding, images were acquired from two glass microscope slides tightened together. The two adjacent, inner, glass surfaces, with an air gap of ~8 μm between them (measured from the encoded OCT A-Scans after processing) provided a suitable target for the evaluation of the system’s axial resolution. The kernels used for deconvolution are shown in Fig. 6(a) and 6(b.) They were collected experimentally from the target described in Fig. 5. Figure 6(a) shows the kernels used for the deconvolution of a standard OCT image whereas Fig. 6(b) shows the kernels for the deconvolution of the encoded image.
Figure 6(c-e) illustrates the results of the proposed method and provides a quantitative measure of the resolution improvement achieved. Figure 6(c) is a plot of a single OCT A-Scan showing the reflections from the faces of the two glass microscope slides. The middle peak is the reflection from the two adjacent middle surfaces, a close-up of which is shown in Fig. 6(d). The blue line is a standard OCT scan (with no deconvolution), the red line is the standard OCT scan after deconvolution with the kernels of Fig. 6(a), and the black line is the encoded OCT scan after deconvolution with the kernels of Fig. 6(b). The two adjacent surfaces were not discernible using standard OCT and appeared as a single peak. Even when deconvolution was performed on the standard OCT signal the two peaks where still not clearly resolved. With step-frequency encoding and deconvolution the two peaks are unmistakably separated.
The improvement in the axial resolution of the system was quantified by examining the reflections from one of the single surfaces, in this case the peak at 2250 μm, a close up of which is shown in Fig. 6(e). The standard OCT resolution was measured to be 13.5 μm. After deconvolution of the standard OCT interferogram, the resolution improved to 6 μm and, after applying step-frequency encoding and deconvolution, was further improved to 1.9 μm, approximately a 7 times improvement. This is consistent with the theoretical analysis of the technique which predicted:
The technique was also tested on a biological sample. Figure 7(a) shows a small region (1.5x0.5 mm) of an onion image acquired with standard OCT. Figure 7(b) shows the same region after deconvolution of the standard OCT image. Axial resolution improvement is evident. Figure 7(c) is the onion image acquired with encoded OCT after deconvolution and Fig. 7(d) is the same image with the distance shift corrected using a simple motion correction algorithm. In this image, there is an improvement in resolution which results in the appearance of the characteristic onion double wall structures as evident from the zoomed regions of the same images in Fig. 8 . The presence of the wall structure is also recognized in Fig. 8(e) which is a light microscopy image of onion cells. The distance between the cell membranes, measured from the encoded OCT images, was found to be between 8.0 and 8.4 μm, which is consistent with the literature (6-10 μm corresponding to the thickness of two cell walls.). This is a case of new information, revealed using the proposed technique, which would have been previously unavailable. The fact that this is indeed a feature of the sample and not an artifact, as might be suggested below when discussing Fig. 10 and 11 , is further substantiated by the fact that the double line artifact produces lines at a distance of 3-4 μm (Fig. 10 & 11) and it is not present in other biological samples, such as those of lung (Fig. 9 .)
Rabbit lung parenchyma was also imaged, ex vivo, to further demonstrate the applicability of the proposed technique to biological samples. The tissue was harvested immediately post termination and preserved in phosphate buffered saline at 4 °C. Imaging was performed within about half an hour after harvesting. The zoomed images in Fig. 9 (d, e, and f) illustrate how the additional resolution improvement revealed small alveoli which in standard OCT appeared as highly backreflecting dark areas. In vivo imaging of tissue samples was not performed due to the speed limitations of this system. However, given the effect of this method on resolution and speckle, there is no reason to believe that it would not be applicable to in vivo tissue imaging.
The proposed method is heavily depended on deconvolution. As a result, the effect of the method on speckle might be a concern. This effect was also investigated experimentally based on images consisting entirely of speckle. These images were acquired from samples consisting of polysterine microspheres (2 μm diameter) embed in acrylamide gel in a concentration such that each imaging voxel included ~10 spheres. These samples were prepared based on the technique described in the literature . The results, shown in Fig. 10 and 11, indicate that neither the Lucy-Richardson deconvolution nor the step-frequency encoding and deconvolution significantly affect the amplitude of the speckle. However, they result in a progressively finer texture.
Figure 10 and 11 also illustrate a possible artifact of the proposed technique which must be seriously considered when interpreting the improved OCT images. Highly intense, specular, reflections, such as those present at the top surface of flat, highly reflective samples, can be mistakenly identified as a double layer. This is a result of the increased intensity of the sidelobes of the actual source, which in the case of the source used here, is not perfectly Gaussian (see blue line in Fig. 11.) Fortunately, this artifact is not present in biological tissues, such as the example of the lung images shown in Fig. 9.
By encoding the OCT interferogram and subsequently demodulating and deconvoluting, significant axial resolution improvement was achieved without the need of a broader bandwidth light source. The resolution improved significantly in this case by a factor of ~7. Despite the limitations of this very preliminary implementation of step-frequency encoding, it is evident that this technique has the potential to dramatically enhance the resolution of OCT systems. In addition, different forms of encoding can be implemented, both in the time and Fourier domain, which will not suffer from the limitations uncovered in these experiments.
This research was supported in part by the Research Promotion Foundation of Cyprus.
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