Dispersion, a result of wavelength-dependent index of refraction variations, causes pulse-width broadening with detrimental effects in many pulsed-laser applications. It is also considered to be one of the major causes of resolution degradation in optical coherence tomography (OCT) imaging and, thus much effort has been expended over the years to develop techniques to counteract its effects. As a result, several dispersion compensation methods have been developed. Conventional methods rely on placing the right amount of dispersion balancing material in one interferometer arm of the OCT setup to counteract the effects of the dispersion in the other arm. Initially, dispersion balancing was achieved by a fused-silica prism pair is inserted with faces contacted and index matched to form a variable-thickness window in the reference arm. However, this method is usually only practical for second-order dispersion.1–3 Grating-based phase delay scanners can also be used for second-order dispersion compensation. Using the rapid-scanning optical delay, one can adjust dispersion by displacing the diffraction grating from the focal plane of the lens to achieve transform-limited interferogram profiles.4 Dual optical fiber stretchers can also be used for dispersion compensation while affording some degree of tunability. Using two fiber stretchers made up of different fiber types, an all-fiber tunable dispersion compensator can be implemented with the delay and the dispersion in the two arms of the interferometer can be adjusted independently. However, these approaches require bulky equipment and high voltage so they are not easy to use.5 Recently, a fiber-stretching-based dispersion compensator has been combined with a grating-based, time domain OCT system to compensate for second- and third-order dispersion, but the system becomes increasingly complicated.6 Numerical dispersion compensation, based on the use of the fractional Fourier transform (FT), is also possible providing new perspectives on the nature and role of group-velocity dispersion in Fourier domain OCT.7
Even after an OCT interferometer is optimized, dispersion differences are still present due to the dispersive properties of the material of tissues that are imaged. Since this dispersion is specific to the sample that is causing the effect, the concept of extracting useful material information from the OCT signal has emerged.8,9 The idea of using dispersion, as a source of contrast, is not new. For example, there have already been reports, in the literature, of using the dispersion of biomolecules to quantify their concentration. For example, the dispersion of hemoglobin was used to extract the concentration of hemoglobin in intact red blood cell.10 The relation between dispersion and biochemical composition was further demonstrated using quantitative dispersion microscopy, which has confirmed that the dispersion of live HeLa cells agrees well with the dispersion measured for pure proteins solutions.11 Variations in the dispersion of different types of normal skin have also been identified in vivo with coherent reflection measurements of different skin types.12 Given the dramatic changes in cellular biochemistry caused by cancer,13 which are discernible by other optical techniques such as Raman spectroscopy,14 it is highly likely that dispersion can also be used as a contrast mechanism in OCT imaging of early cancer and result in more accurate disease diagnosis. However, if dispersion is to be used for quantitative measurements, it is very important to know the capabilities and limitations of the different methods of measuring dispersion with OCT. In this paper, we compare different techniques for estimating the group velocity dispersion (GVD) from OCT images, in order to evaluate their accuracy and applicability to highly scattering media such as tissues.
Theory and Methods
There are several methods described in the literature on how to measure GVD from the OCT signals or images. They depend on resolution degradation, image feature walk-off shift, and phase differences (Fig. 1). In addition, two recently developed methods for dispersion estimation in scattering tissues, without the need of a reflector or a strong distinct scatterer, were also included.8,15 Each is described further in the following sections along with the mathematical framework for the GVD calculations for each case. Since the index of refraction of the material was needed for some of the calculations, this was measured using the method described by Tearney et al.16
Dispersion causes degradation of the image point spread function (PSF) and, therefore, of the image resolution. The resolution degradation is evident in the pulse width of the broadened Gaussian resolution envelope . If the width of the original envelope and the sample thickness are known, or can be measured, the GVD can be estimated. The original Gaussian resolution envelope width is related to the pulse FWHM by17,18
When dispersion is present, beams at different wavelengths perceive different path lengths. The result is in an apparent shift in the OCT images taken at different center wavelengths, an effect called “walk-off.” The mathematical relationship between the shift and GVD is derived below starting with18,19
For the experimental verification, a single broad source can be used. After acquisition of the interferometric signal, the spectrum is split into two by multiplication with two shifted Gaussian envelopes. Subsequently, two OCT images are formed, corresponding to the two different spectra, from which the walk-off is measured. This approach results in images with reduced resolution but it assures that the OCT images are acquired at exactly the same location and are, therefore, comparable.
In spectral interferometry, dispersion can be estimated from the interference spectrum produced by two time-delayed beams. The interferometric signal, for a given time or phase delay, can be expressed as20–22
Using the Speckle Degradation to Estimate the GVD
The methods described above are very difficult to apply in vivo and are limited only to samples where strong and distinct reflectors are present. A developed method based on speckle width degradation does not require such reflectors and can thus be applied in vivo.
This method uses the dispersion induced change in the speckle size to estimate the image PSF broadening and then calculate the GVD.8 However, speckle variations are difficult to estimate due to the randomness of the speckle signal. To do so, a portion of an OCT image that contains speckle, at depth from the sample surface, denoted as , is related to a similar portion of the OCT image from the sample surface, i.e., , by a depth-dependent speckle-degrading impulse response such that23 For that purpose, the following least mean square error function is defined
Using the Cross Correlation of Speckle to Estimate the GVD
Another approach to estimate the GVD, without the need for strong distinct reflectors, is the use of the speckle cross correlation to estimate the walk-off shift between OCT images at different wavelengths. The walk-off shift can be estimated from the cross correlation of A-scans from corresponding regions of the two half-spectrum images at different center wavelengths created as described in Sec. 2.2 [Fig. 2(a)]. The cross correlation of corresponding A-scans is calculated and the first peak, after the zero lag, in the cross correlation is detected [Figs. 2(b) and 2(c)]. The walk-off shift is estimated from the distance of the peak from the zero lag location [Fig. 2(c)] and the GVD is calculated using the Eq. (7).
For the experimental verification and comparison of the various GVD estimation methods, a swept source OCT system, with a center wavelength of 1300 nm and a resolution of in air, was used. Samples of various glasses, a collagen gel (2 g of collagen in 10 ml of water let to solidify and then cut into slices) as well as fresh ex vivo sections of porcine muscle and porcine adipose tissue were used. All the samples were placed over a reflector, which served as a reference for the actual thickness and system resolution measurements, and eight images () were acquired from different regions for each type of sample.
The glass samples were chosen so that they spanned a wide range of GVD values. Their well-characterized properties were used to verify the validity of the OCT techniques. The glass dimensions were 12.5-mm diameter and 2-mm thickness in all cases. The biological samples were chosen based on their scattering properties, ranging from minimal scattering (collagen) to very scattering (adipose tissue). They were sliced manually so their thickness varied between 1.5 to 2.5 mm. Fortunately, since the actual thickness was measured for each lateral location (see Sec. 4.1), the thickness variation was included in the GVD calculation and did not affect the comparison. In Fig. 3, a list of all the samples with photographs as well as the values of their index of refraction and GVD, obtained from the literature, is provided.
The OCT data were processed in MATLABTM. Initially, an automated algorithm detected the top surface of the sample as well as the mirror peak location in free-space and below the tissue. The tissue thickness, index of refraction, and mirror reflection Gaussian widths were subsequently calculated assuming a Gaussian envelope shape. From those measurements, the GVD was estimated by combining the values from 250 A-scans in each image. The GVD of each sample was taken as the median of those 250 values. The standard deviation of the GVD values obtained from the eight images of each sample, as well as the mean error between the estimated and measured GVD were used as indicators of the precision and accuracy of each of the different methods described above. Furthermore, the standard deviation of the GVD measurements of single A-scans within an image was used as a measure of each technique’s robustness. Given that value, the minimum number of measurements that must be averaged in order to get a GVD estimate with an error of 10% or less with a confidence level of 95% was calculated by
Figure 4 shows typical OCT images of the samples utilized in this study. Using the reflector location (blue lines) as reference, the sample thickness and the group index of refraction, at each lateral location, were calculated using the technique described in the literature,16 i.e.,Fig. 5).
Figure 6 shows a typical example of how the resolution degradation is used to calculate the GVD. The width of the peak corresponding to a reflector placed below the sample [Fig. 6(a), red line] was measured from each A-scan of the image assuming a Gaussian shape. The sample thickness was measured from the top surface [Fig. 6(a), green line] to the reflector line extension [Fig. 6(a), blue line]. Using Eq. (3), the GVD was calculated and overlaid over the OCT intensity image as a pseudocolor hue scale.
Figure 7 illustrates how the GVD was calculated experimentally using the walk-off shift. The interferogram (spectrum) for each A-scan [Fig. 7(a), green] was divided into two parts [Fig. 7(a), red and orange] by multiplying with Gaussian envelopes. Each half spectrum was used to create a separate OCT image [Figs. 7(b) and 7(c)]. The walk-off shift [Fig. 7(e)], measured from the relative displacement of the bottom surface [Fig. 7(d), red and orange] of the two images, was used to calculate the GVD from Eq. (7). The location of the bottom surface was estimated from the local maximum of the image below the level of the reflector.
Figure 8 shows an example of the calculation of the GVD using the phase difference method. The real part of the FT of each A-scan was multiplied by a Gaussian envelope located at the location of the reflector behind that tissue thus isolating that single peak [Fig. 8(b)]. The spectrum of the single peak was obtained from the inverse FT of the single-peak interferogram [Fig. 8(c)] resulting again in a full spectrum but which now contained only the frequencies corresponding to the single peak. Applying Eq. (13) to this clean spectrum, the GVD was calculated as a function of wavelength [Fig. 8(d)]. Unwrapping of the phase was performed by changing absolute phase jumps greater than to their complement. The second derivative was approximated numerically by taking the difference of adjacent values twice.
The measurement of the GVD using the phase derivative did not produce accurate results for highly scattering samples. This was due to the presence of strong discontinuities in the phase [Figs. 9(b) and 9(e)], as a result of the scattering discontinuities, producing erroneous GVD estimations [Fig. 9(f)]. This is consistent with the literature, which predicts minimum and nonminimum phase discontinuities from Mie scatterers.28
Speckle Width Degradation
Figure 10 illustrates the application of the speckle-width-degradation-based approach to estimate the GVD from the OCT images. Portions of the image of from just below the top surface [Fig. 10(a), green lines] and just above the bottom surface [Fig. 10(a), red lines] with a width of twice the system resolution are shown as examples in Figs. 10(a) and 10(b). The impulse response sdf was estimated with a Wiener-type deconvolution [Fig. 10(c)] and the width of the sdf for all A-scans was measured [Fig. 10(d)]. The process was repeated as a function of depth and the mean width, overall A-scans, was calculated as a function of depth [Fig. 10(e)]. Subsequently, the dispersion-degraded image resolution width was approximated from a linear fit of sdf mean width [Fig. 10(f)]. The GVD was calculated using and Eq. (3).
Speckle Cross Correlation
Figure 11 shows an example of the estimation of the GVD from the walk-off shift of half-spectrum images using the cross correlation of corresponding A-scans. Corresponding regions, just above the bottom of the sample [Fig. 11(a), red Lines], were selected from each half-spectrum image [Figs. 11(b) and 11(c)]. The cross correlation of corresponding A-scans was calculated and the first peak in the cross correlation was detected. The walk-off shift was estimated from the distance of the peak from the zero lag location [Fig. 11(d)] and the GVD was calculated using Eq. (7). The walk-off shift estimation was more robust when there was ample speckle in the images to provide a better cross-correlation approximation. Figure 11(d) shows some typical cross-correlation curves with the arrow pointing to a miscalculation of the walk-off shift due to a weak cross correlation between A-scans. This phenomenon is more common in clear samples such as the collagen gel used in the ex vivo experiments.
Summary of Results
Table 1 summarizes the results of the GVD measurements using the techniques described above. The left part of Table 1 lists the results of the GVD calculations over entire samples, estimated from the median of the GVD of 250 A-scans from each image. The accuracy of each technique (i.e., how close the results are to their expected values) for each different type of sample is quantified by the percentage error, the difference from the expected values of Fig. 3. Methods that are inaccurate are highlighted in italics in Table 1. The precision of each technique (i.e., how concentrated the results are around their mean) is described by the standard deviation of the values of the results from complete images (median of 250 A-scans). The right part of Table 1 lists the standard deviations of the GVD values from the individual A-scans within each image. Larger values imply that the technique is not as robust and that more averages are required to get a good estimate of the GVD of the entire sample. This is also evident from the minimum number of averages required to get an estimate of the GVD with an error of 10%, or less, with 95% confidence, which is listed in the last column of Table 1. Methods that are not robust are listed in bold font in Table 1. Further discussion of the results follows in Sec. 5.
|Sample||Complete images (average of 250 A-scans)||Single A-scan|
|Mean GVD (fs2/mm)||Std GVD (fs2/mm)||Std GVD (%)||Error (%)||Std GVD (fs2/mm)||Std GVD (%)||Min num of avgs|
Another important consideration, when calculating the GVD using OCT, is the size of the fast Fourier transform (FFT) to be used for the reconstruction of the images. In order to get a precise measurement of the degradation of the PSF or walk off, each peak in the OCT A-scan must be adequately sampled to avoid sampling errors. The effect of the FT sampling on the GVD was estimated from the images acquired experimentally, by changing the measurements by pixel and is shown in Fig. 12(a). These calculations indicate that for a small sampling error, FFT sizes of or larger are required. This is further demonstrated by the effect of the FFT size on the actual GVD measurements [Fig. 12(b)].
The most computationally intensive steps in the processing of the data were the required FFTs. To create an image from the raw interferogram, 512 FFTs of length at least were required. In the case of the phase difference method, FFTs were required in order to accurately reconstruct the spectra from a single peak. The complexity of the speckle degradation method was burdened by an additional 190 two-dimensional () Weiner-type deconvolutions, whereas the final method, speckle cross correlation, required 250 A-scan cross correlations. The average execution time on a PC (i7, quad core, 2 GHz) was , 1.15, 8.79, 1.97, and 1.23 s per image for the psf-degradation, walk-off shift, phase difference, speckle degradation, and speckle cross-correlation methods, respectively.
All three mirror-based techniques performed well in low-scattering and fairly uniform samples, such as the glasses or the collagen gel, with the exception of the walk-off shift and the phase difference when interrogating , which is characterized by very low GVD. Based on these observations, only the PSF degradation method can be used to estimate GVD values below . The measurements from all collagen gel samples were comparable and exhibited little variation ( to 6%). In muscle tissue, which is significantly more scattering, only the PSF degradation and walk-off shift methods produced sufficiently consistent results with to 14%, while the phase derivative resulted in . The measurements of the width and walk-off shift were degraded by the presence of speckle noise. Finally, in adipose tissue, which is even more scattering, the PSF degradation and walk-off shift method results were accurate resulting in a of to 19% while the phase derivative was completely wrong for the reasons described earlier. In general, the methods that did work were sufficiently accurate with an error always . The speckle-based techniques were not as precise as the reflector-based methods ( to 20%) but they are the only methods applicable to in vivo imaging. With the exception of the speckle walk-off method when applied to low-scattering samples (collagen gel), the remaining estimates were accurate with an error to 5%. In addition to the methods that result in erroneous estimates (described above and italized in Table 1), there are other methods that are impractical due to the unfeasibly high number of individual GVD values that must be averaged to get a good estimate of the GVD of the sample (bold font in Table 1).
GVD is presented in all tissues and could potentially provide diagnostically useful information. OCT can be used to estimate the GVD and, therefore, enhance the effort for early diagnosis of serious diseases such as cancer. Given the results presented above, for ex vivo GVD estimation, the resolution degradation method is the best choice since it is less sensitive to tissue scattering properties in contrast to the other techniques evaluated. This method performs particularly well for larger GVDs with an error of . If the GVD is to be used to provide sensitive diagnostic information from highly scattering human tissues in vivo, it would be preferable to use the speckle degradation as an estimator of GVD. Given that tissue GVD is usually , this method is expected to perform well with an error of . Whichever the case, the use of the GVD as a disease marker is an exciting prospect which should be further investigated.
The authors have no relevant financial interests in this article and no potential conflicts of interest to disclose.
This study was funded by the European Union’s Horizon 2020 Research and Innovation Program under grant Agreement No. 739551 (KIOS CoE) and from the Republic of Cyprus through the Directorate General for European Programs, Coordination, and Development.
Christos Photiou received his BSc degree in physics from the University of Patras and his MSc degree in environmental health from the Cyprus University of Technology (in association with the Harvard School of Public Health). He is currently a PhD student in the Department of Electrical and Computer Engineering at the University of Cyprus and a researcher at the KIOS Research and Innovation Center of Excellence. His research interests include biomedical image processing, machine learning, and optical diagnostics.
Costas Pitris received his BS and MS degrees from the University of Texas at Austin, his PhD from Massachusetts Institute of Technology, and his MD from Harvard Medical School. He is an associate professor in the Department of Electrical and Computer Engineering and the KIOS Research and Innovation Center of Excellence at the University of Cyprus. His main research interests are in the area of biomedical optics applied to medical diagnosis and monitoring.