## Distortion-free freehand-scanning OCT implemented with real-time scanning speed variance correction |

Optics Express, Vol. 20, Issue 15, pp. 16567-16583 (2012)

http://dx.doi.org/10.1364/OE.20.016567

Acrobat PDF (2755 KB)

### Abstract

Hand-held OCT systems that offer physicians greater freedom to access imaging sites of interest could be useful for many clinical applications. In this study, by incorporating the theoretical speckle model into the decorrelation function, we have explicitly correlated the cross-correlation coefficient to the lateral displacement between adjacent A-scans. We used this model to develop and study a freehand-scanning OCT system capable of real-time scanning speed correction and distortion-free imaging—for the first time to the best our knowledge. To validate our model and the system, we performed a series of calibration experiments. Experimental results show that our method can extract lateral scanning distance. In addition, using the manually scanned hand-held OCT system, we obtained OCT images from various samples by freehand manual scanning, including images obtained from human *in vivo*.

© 2012 OSA

## 1. Introduction

1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science **254**(5035), 1178–1181 (1991). [CrossRef] [PubMed]

2. A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. **12**(5), 051403–051421 (2007). [CrossRef] [PubMed]

3. S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. G. Fujimoto, and M. E. Brezinski, “Forward-imaging instruments for optical coherence tomography,” Opt. Lett. **22**(21), 1618–1620 (1997). [CrossRef] [PubMed]

10. L. Huo, J. Xi, Y. Wu, and X. Li, “Forward-viewing resonant fiber-optic scanning endoscope of appropriate scanning speed for 3D OCT imaging,” Opt. Express **18**(14), 14375–14384 (2010). [CrossRef] [PubMed]

1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science **254**(5035), 1178–1181 (1991). [CrossRef] [PubMed]

10. L. Huo, J. Xi, Y. Wu, and X. Li, “Forward-viewing resonant fiber-optic scanning endoscope of appropriate scanning speed for 3D OCT imaging,” Opt. Express **18**(14), 14375–14384 (2010). [CrossRef] [PubMed]

11. W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. **11**(4), 806–810 (2005). [CrossRef]

12. J.-F. Chen, J. B. Fowlkes, P. L. Carson, and J. M. Rubin, “Determination of scan-plane motion using speckle decorrelation: theoretical considerations and initial test,” Int. J. Imaging Syst. Technol. **8**(1), 38–44 (1997). [CrossRef]

13. P. C. Li, C. J. Cheng, and C. K. Yeh, “On velocity estimation using speckle decorrelation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control **48**(4), 1084–1091 (2001). [CrossRef] [PubMed]

14. A. Ahmad, S. G. Adie, E. J. Chaney, U. Sharma, and S. A. Boppart, “Cross-correlation-based image acquisition technique for manually-scanned optical coherence tomography,” Opt. Express **17**(10), 8125–8136 (2009). [CrossRef] [PubMed]

15. K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. **56**(9), 2318–2321 (2009). [CrossRef] [PubMed]

9. J. Ren, J. Wu, E. J. McDowell, and C. Yang, “Manual-scanning optical coherence tomography probe based on position tracking,” Opt. Lett. **34**(21), 3400–3402 (2009). [CrossRef] [PubMed]

14. A. Ahmad, S. G. Adie, E. J. Chaney, U. Sharma, and S. A. Boppart, “Cross-correlation-based image acquisition technique for manually-scanned optical coherence tomography,” Opt. Express **17**(10), 8125–8136 (2009). [CrossRef] [PubMed]

16. J. K. Barton and S. Stromski, “Flow measurement without phase information in optical coherence tomography images,” Opt. Express **13**(14), 5234–5239 (2005). [CrossRef] [PubMed]

17. B. Lau, R. A. McLaughlin, A. Curatolo, R. W. Kirk, D. K. Gerstmann, and D. D. Sampson, “Imaging true 3D endoscopic anatomy by incorporating magnetic tracking with optical coherence tomography: proof-of-principle for airways,” Opt. Express **18**(26), 27173–27180 (2010). [CrossRef] [PubMed]

*et al.*in OCT systems for the first time [14

14. A. Ahmad, S. G. Adie, E. J. Chaney, U. Sharma, and S. A. Boppart, “Cross-correlation-based image acquisition technique for manually-scanned optical coherence tomography,” Opt. Express **17**(10), 8125–8136 (2009). [CrossRef] [PubMed]

18. J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Speckle in optical coherence tomography,” J. Biomed. Opt. **4**(1), 95–105 (1999). [CrossRef]

*in vivo*.

**17**(10), 8125–8136 (2009). [CrossRef] [PubMed]

## 2. Theory

*x*,

*y*,

*z*) to describe the 3D space.

*z*indicates the axial direction;

*x*is the lateral direction or the direction of the manual scan. For simplicity, we assume the motion of the OCT needle probe is limited to

*x*-

*z*plane and the specimen is static.

### 2.1. Manually scanned OCT imaging

*x*direction, the displacement between adjacent A-scans, ∆

*x*, is a function of the instantaneous scanning velocity

*v*and the A-scan acquisition rate

*f*

_{A}, as shown in Eq. (1).

*v*varies with time for a manual-scan OCT probe;

*f*

_{A}is usually a constant for conventional data acquisition devices such as a frame grabber synchronized with an internal, periodical trigger signal. As a result, ∆

*x*varies with time in the same manner as

*v*. Therefore, the lateral intervals between different A-scans are different for manual scan.

*R,*has to be larger than twice the highest spatial frequency of the specimen (

*F*

_{n}):

*R*= 1/∆

*x*=

*f*

_{A}/

*v*>2

*F*

_{n}. Therefore, the scanning speed has to be smaller than

*v*

_{m}shown in Eq. (2).

*v*

_{m}would lead to oversampling and information redundancy. Under the oversampling condition, there is correlation between adjacent A-scans. The degree of correlation can be measured by Pearson cross-correlation coefficient (XCC) shown as Eq. (3).

*I*(

_{x,y}*z*) is the intensity of an A-scan at (

*x*,

*y*).

*I*(

_{x,y}*z*) is calculated by taking the square of the amplitude of the A-scan. Denote the complex valued OCT signal as

*S*(

_{x,y}*z*); then

*I*(

_{x,y}*z*) =

*S*(

_{x,y}*z*)

*S**(

_{x,y}*z*). Similarly,

*I*

_{x+}_{Δ}

_{x,y+}_{Δ}

*(*

_{y}*z +*Δ

*z*) is the intensity of A-scan that is displaced by (Δ

*x*,Δ

*y*,Δ

*z*).

*σ*

_{Ix,y}_{(}

_{z}_{)}and

*σ*

_{Ix+}_{Δ}

_{x,y+}_{Δ}

_{y}_{(}

_{z+}_{Δ}

_{z}_{)}are the square roots of variance for

*I*(

_{x,y}*z*) and

*I*

_{x+}_{Δ}

_{x,y+}_{Δ}

*(*

_{y}*z +*Δ

*z*).

*x*direction, ∆

*y*= ∆

*z*= 0,

*I*

_{x+}_{Δ}

_{x,y+}_{Δ}

*(*

_{y}*z +*Δ

*z*) becomes

*I*

_{x+}_{Δ}

*(*

_{x,y}*z*) and Eq. (3) becomes:For simplicity, we use

*ρ*to denote

*ρ*

_{Ix,y}_{(}

_{z}_{)}

_{,Ix+}_{Δ}

_{x,y}_{(}

_{z}_{)}in subsequent equations.

*I*(

_{x,y}*z*)> = <

*I*

_{x+}_{Δ}

*(*

_{x,y}*z*) > =

*I*

_{0}; <

*I*(

_{x,y}*z*)

^{2}> = <

*I*

_{x+}_{Δ}

*(*

_{x,y}*z*)

^{2}> =

*I*

_{RMS}

^{2}. Therefore, we have:

^{2}is the square of the amplitude of a complex value. Signal

*S*

_{x}_{,}

*(*

_{y}*z*) is determined by the physics of OCT image formation mechanism and can be expressed as the convolution of scattering distribution function

*a*(

*x*,

*y*,

*z*) with system's 3D point spread function (PSF)

*P*(

*x*,

*y*,

*z*)

*S*

_{x}_{+}

_{Δx}_{,}

*(*

_{y}*z*) can be expressed as

**indicates integration over (-∞, + ∞) in the expressions of**

*∫**S*

_{x}_{,}

*(*

_{y}*z*),

*S*

_{x}_{+}

_{Δx}_{,}

*(*

_{y}*z*) and in the following derivations.

*S*

_{x}_{,}

*(*

_{y}*z*) and

*S*

_{x}_{+}

_{Δx}_{,}

*(*

_{y}*z*) into Eq. (4) and utilizing the fact that OCT system's PSF is not random, we have:

*a*

_{0}is a constant representing the scattering strength. Using the sifting property of delta function, we have

*P*(

*z*) and the lateral PSF

*P*(

*x*,

*y*) are separable because axial and lateral PSFs are governed by different physical principles: axial PSF is determined by the temporal coherence of the light source while lateral PSF is determined by the imaging optics in the sample arm. Furthermore, in Gaussian optics model,

*P*(

*x*,

*y*) is the product of PSFs in

*x*and

*y*dimensions [18

18. J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Speckle in optical coherence tomography,” J. Biomed. Opt. **4**(1), 95–105 (1999). [CrossRef]

*P*(

*x*,

*y*,

*z*) can be written explicitly as

*P*(

*x*,

*y*,

*z*) =

*P*

**(**

_{x}*x*)

*P*

**(**

_{y}*y*)

*P*

**(**

_{z}*z*) and therefore we have:

*P*

**(**

_{x}*x*) can be expressed as:

*w*

_{0}is the Gaussian beam waist of probing beam [21]. It is worth mentioning that Gaussian beam waist in this definition is the distance from the beam axis where the intensity of OCT signal drops to 1/

*e*. This PSF expressed in Eq. (6) is valid and consistent with literature [22

22. Y. Liu, Y. Liang, G. Mu, and X. Zhu, “Deconvolution methods for image deblurring in optical coherence tomography,” J. Opt. Soc. Am. A **26**(1), 72–77 (2009). [CrossRef] [PubMed]

23. P. D. Woolliams, R. A. Ferguson, C. Hart, A. Grimwood, and P. H. Tomlins, “Spatially deconvolved optical coherence tomography,” Appl. Opt. **49**(11), 2014–2021 (2010). [CrossRef] [PubMed]

*ρ*is merely determined by ∆

*x*for fully developed speckle; therefore, we can calculate the cross-correlation coefficient

*ρ*between adjacent A-scans and use the value of

*ρ*to derive the time-varying ∆

*x*as:

*x*with opposite signs can lead to the same value of

*ρ*, according to Eq. (7). In other words, the scanning direction cannot be determined by calculating XCC.

*ρ*can be calculated with Eq. (9).

*i*is the index of pixel in an A-scan and

*j*is the index of A-scan. Segmentation of signal between

*i*and

_{f}*i*is selected to calculate

_{l}*ρ*.

### 2.2. Scanning speed correction based on speckle decorrelation

*N*spectra acquired at lateral locations:

*x*

_{1},

*x*

_{2}, ...,

*x*

_{N}. Although ∆

*x*, the interval between

_{i}*x*

_{i}and

*x*

_{i + 1}, is not a constant for different A-scans due to non-constant scanning speed, we could extract ∆

*x*using

_{i}*ρ*, the XCC between

_{i}*I*(z) and

_{i}*I*

_{i}_{+1}(

*z*) (A-scans obtained at spatial coordinate

*x*

_{i}and

*x*

_{i + 1}). As a result, we were able to estimate ∆

*x*, the displacement (in x direction) between the first and the last A-scan in a frame by summing up ∆

_{total}*x*: ∆

_{i}*x*= ∑∆

_{total}*x*. With a pre-set sampling interval ∆

_{i}*x*, we could calculate the number of A-scans required for this particular frame of data by dividing ∆

_{s}*x*with ∆

_{total}*x*:

_{s}**= ∆**

*M**x*/∆

_{total}*x*. Afterwards, we performed interpolation to obtain A-scan data at spatial points 0, ∆

_{s}*x*, 2∆

_{s}*x*,...,

_{s}**∆**

*M**x*to obtain A-scans that were evenly distributed in x dimension.

_{s}## 3. OCT system and software implementation

24. K. Zhang and J. U. Kang, “Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT,” Opt. Express **18**(22), 23472–23487 (2010). [CrossRef] [PubMed]

*w*

_{0}. From our calculation,

*w*

_{0}equaled 6μm and this indicated a 12μm 1/

*e*lateral resolution of our OCT. This was further verified by using our OCT system to image 1951 USAF resolution target. The obtained

*en face*OCT image clearly showed that our OCT system can resolve the 5th element in group 6 which corresponds to a 10μm FWHM lateral resolution and therefore 12μm 1/

*e*lateral resolution. Therefore, we assumed

*w*

_{0}to be 6μm for this system when using Eq. (7) to correlate

*ρ*and ∆

*x*. With the 12μm lateral resolution,

*F*= (1/12)(μm

_{n}^{−1}); therefore the largest scanning speed that satisfies the requirement of Nyquist sampling is about 420mm/s, as implied by Eq. (2). For our calibration experiments, a high-speed galvanometer was used to perform lateral scanning.

*N*spectra was acquired and transferred to GPU memory. Afterwards, we re-sampled the spectral data from wavelength (

*λ*) space to wavenumber (

*k*) space using cubic spline interpolation and then performed fast Fourier transformation (FFT) to obtain A-scans. With the obtained A-scans, we calculated the XCC between adjacent A-scans using the OCT signal intensity and re-distributed the A-scans using algorithm shown in Fig. 2 to achieve uniform spatial sampling. Before calculating

*ρ*, we processed each A-scan with a moving average filter that averages three adjacent pixels in axial direction and subtracted the output of the filter from each A-scan to reduce low spatial frequency components in the A-scan and therefore increase the sensitivity of cross correlation calculation for lateral motion estimate [14

**17**(10), 8125–8136 (2009). [CrossRef] [PubMed]

*w*

_{0}is about 20μm for the single mode fiber probe and this lateral scanning speed is significantly larger than moderate manual scanning speed (several millimeters per second). With the software optimization, we will be able to further increase the processing speed in future studies.

*w*

_{0}in our software for our single mode fiber probe, we have first used an estimation of

*w*

_{0}based on the experimentally measured lateral resolution of our CP OCT system from our previous work [26

26. X. Liu and J. U. Kang, “Progress toward inexpensive endoscopic high-resolution common-path OCT,” Proc. SPIE **7559**, 755902, 755902-11 (2010). [CrossRef]

27. J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic functional Fourier domain common path optical coherence tomography for microsurgery,” IEEE J. Sel. Top. Quantum Electron. **16**(4), 781–792 (2010). [CrossRef]

*w*. Afterwards, we manually scanned a highly scattering phantom for 1cm (

**= 1cm) and acquired a certain number of A-scans that were uniformly distributed. We performed such scanning for 10 times and calculated the average A-scan number in all the measurements which is indicated by**

*L***. According to Eq. (8), we were able to obtain a better estimation of**

*M**w*

_{0}that equals (

*w*

**)**

*L**/*(

**∆**

*M**x*). We varied the value of ∆

_{s}*x*and obtained a consistent value of

_{s}*w*

_{0}. All images in section 5.2 were acquired based on

*w*

_{0}= 18.5μm which was constant for different values of ∆

*x*.

_{s}## 4. Calibration of the relationship between cross-correlation coefficient and lateral displacement

*x*and

*ρ*(XCC), shown as Eq. (7) and (8). We used a galvanometer to perform lateral scans with known scanning speeds. We applied a periodical sawtooth voltage

*V*from a function generator to the galvanometer and synchronized

*V*with the acquisition of a frame of data which contained

*N*A-scans (

*N*= 1000). For a 100% duty cycle sawtooth driving voltage, ∆

*x*, lateral interval between adjacent A-scans stays constant because the driving voltage increases linearly during signal acquisition. Therefore, we could calculate the displacement between adjacent A-scans directly from the amplitude of the sawtooth function: ∆

*x*=

*γV*/

*N*. Here

*γ*is a coefficient that relates the driving voltage (V) applied to galvanometer and the probing beam displacement (D) at the focal plane of the imaging lens:

*γ*= D/V.

*γ*was measured to be 1.925mm/V in the OCT setup for our calibration experiments. As a result, by applying different

*V*, we could achieve different scanning speeds and thus different ∆

*x*. We acquired B-scans at various scanning speeds. One example of the image obtained is shown 5(a) which contain 1000 A-scans, with a sampling interval ∆

*x*equal to 0.96μm.

*ρ*, XCC between the

_{i}*i*

^{th}and (

*i*+ 1)

^{th}A-scan, using pixels within the range as indicated by the double-head red arrow in Fig. 5(a) . Afterwards, with all the

*ρ*obtained, we took the ensemble average of XCC:

_{i}*ρ*from each B-scan corresponding to a certain ∆

*x*. The result is shown as red circles in Fig. 5(b). As a comparison, we also plotted the theoretical relationship between

*ρ*and ∆

*x*shown in Eq. (7) as a black, dashed curve in Fig. 5(b). By calculating ensemble average of XCC using A-scans with different offsets from the same B-scan, we obtained decorrelation curve as shown in Fig. 5(c). Similarly, theoretical relationship between

*ρ*and ∆

*x*is shown as black, solid curve in Fig. 5(c). The consistency between experimental results and the analytical model described by Eq. (7) and (8) implies that we can use

*ρ*to quantitatively extract the lateral sampling interval and thus correct non-constant scanning speed.

*ρ*from B-scans acquired experimentally at different spatial sampling intervals (

_{i}*δx*). The results are shown in Fig. 5(d). With each obtained

_{i}*ρ*, we used Eq. (8) to calculate a corresponding displacement value Δ

_{i}*x*and then assess the variation of Δ

_{i}*x*,

_{i}*σ*

_{xi}^{2}. Assume that the displacement between each pair of the A-scans follows the same statistics and the probe travels for a given distance Δ

*x*. In this case,

_{total}**, the number of A-scans acquired is Δ**

*M**x*/

_{total}*δx*. Based on these assumptions,

_{i}*σ*

_{xtotal}^{2}, the variance of the estimated displacement approximately equals

*M**σ*

_{xi}^{2}. In Fig. 5(e), we show the ratio between

*σ*

_{Δ}

*and Δ*

_{xtotal}*x*, for different values of Δ

_{total}*x*and

_{total}*δx*.

_{i}*δx*) and larger lateral distance travelled would result in a higher accuracy of displacement estimation. In other words, a large number of sampling points for a given displacement can provide a better displacement estimation, due to the inherent statistics of XCC. Other than the random nature of XCC, there are several reasons for displacement calculated using XCC and actual displacement to be different. First, OCT signal suffers from optical and electrical noise, such as shot noise, excess noise, thermal noise, etc; in addition, OCT signal decorrelates overtime due to random environmental disturbance. Second, the waist of probing beam varies at different lateral displacement due to lens abbreviation. Third, parts of A-scans with low signal intensity produced high correlation due to signal absence, so that the measured XCC was higher than the theoretical values for large displacements in Fig. 5(b) and 5(c). However, those factors might be negligible as compared to the inherent random noise of XCC, because OCT is a high speed and high sensitivity imaging modality.

_{i}*x*dimension is necessary for accurate scanning speed correction. Comprehensive evaluation of the error in distance measurement using XCC will be our future work and is beyond of the scope of this manuscript.

## 5. Results

### 5.1 Quantitative lateral sampling interval extraction

*γ*from our previous measurement, we were able to calculate the displacement of the probe beam with regards to its neutral position when the voltage applied to the galvanometer was 0. The calculated displacement of the probe beam at different time is shown in the upper inset of Fig. 6(a) . Instantaneous ∆

*x*was calculated by taking the absolute value of the difference between the displacements of adjacent sampling points, as shown in the lower inset of Fig. 6(a). In this experiment, 5000 A-scans were acquired sequentially. By stacking the A-scans, we obtained the pseudo B-scan shown in Fig. 6(b). When the driving voltage/displacement reaches their extreme points, the interval between adjacent A-scans became smallest, which can be clearly seen in Fig. 6(a). With a small sampling interval, data is redundant, as in areas indicated by the arrows in Fig. 6(b). We calculated the XCC between the adjacent A-scans in Fig. 6(b). XCC as a function of time is shown in the upper inset of Fig. 6(c) which was processed with a low pass filter for noise reduction and normalized to the maximum value. We further calculated ∆

*x*using Eq. (8) with XCC obtained in the upper inset of Fig. 6(c) and show the result as the red curve in the lower inset of Fig. 6(c). In the lower inset of Fig. 6(c), we also plot ∆

*x*calculated from the known driving voltage applied to the galvanometer, as the black curve. The consistency between the red and black curve verified our assumption that ∆

*x*could be extracted from XCC quantitatively. There are several reasons for the red and black curve in the lower inset of Fig. 6(c) to be slightly different, as discussed in the previous section. However, the inherence statistics of XCC plays the most significant role in resulting errors. As shown previously in Fig. 5(d), if the sample is laterally homogenous and the displacement between the adjacent A-scans is small, for example less than 1μm, the errors in displacement due to the inherent randomness of XCC are small. Therefore, the errors might be mostly due to other random noise in OCT signal. With a larger sampling interval, errors come from the inherent statistics of XCC rather than other noise in OCT signal. As shown in the lower inset of Fig. 6(c), difference between estimated and actual interval is smaller when interval between A-scans is smaller.

*x*to be 5μm and performed nearest neighbor interpolation as described in section 2.2. The resultant image is shown in Fig. 6(e) in which the oversampling artifact is removed.

_{s}### 5.2. Images obtained from manually scanned OCT probe with real-time correction

*x*to be 1μm, 2μm, and 4μm and show the corresponding images obtained in Fig. 8(a) , 8(b), and 8(c). The plastic covering film and the underneath fluorescence materials of the IR card can be clearly seen from Fig. 8. With different spatial sampling interval ∆

_{s}*x*, the same physical length is represented by different numbers of A-scans. As the lateral axis of Fig. 8 is A-scan index, the scale of porous structure of the fluorescence materials decreases from Fig. 8(a) to 8(c) due to the increasing sampling interval. Results in Fig. 8 verify that we were able to achieve uniform spatial sampling interval during manual scan and the sampling interval is explicitly determined through ∆

_{s}*x*a which is parameter in our software.

_{s}*M*

_{i}) shown in Fig. 9(c). Afterwards, we extracted zero-crossing points of

*M*

_{i}to detect the edge of each line, as indicated by red circles in Fig. 9(c). Then we calculated widths of the lines, their mean and standard deviation (STD). The ratio between STD and mean of the width was 0.025, which indicates that our method effectively removed artifacts induced by non-constant manual scanning speed. In comparison, in Fig. 9(d), we show the image obtained from manual scan, but without correction using cross-correlation. Artifacts due to non-uniform scanning speed are clearly visible in Fig. 9(d).

## 6. Discussion

*x*from XCC, it requires that we know

*w*

_{0}, the Gaussian beam waist of probing beam which can be calculated or experimentally measured. As a result, the calibrating decorrelation curve shown in Fig. 5(b) is only valid for an OCT system with a certain

*w*

_{0}. If

*w*

_{0}used in the calculation for lateral interval between adjacent A-scans is different from the actual beam size, the image reconstructed from our algorithm will be different from the “true” image by a scaling factor in the lateral dimension. However, under such circumstance, uniform sampling can still be achieved to obtain an image that is easy for human to comprehend. In fact, the size of the imaging beam from the probe changes as the beam propagates, and the lateral PSF of OCT system depends on the image depth. Therefore, the speckle decorrelation curve has depth dependency as well. Considering the overall effect, the lateral resolution defined by the Gaussian beam waist is always slightly different from the decorrelation length of OCT signal. Moreover, to reduce the effect of the diverging beam, we took only part of an A-scan to calculate XCC when implementing our software. As a result, the statistics of different pixels within the segment of an A-scan does not vary significantly.

*x*axis and that there is no axial motion from the scanning probe, which is not exactly true in a realistic scenario. For example, human hands suffer from physiological tremor and this makes the probe to move randomly in both lateral and axial directions. However, experimental results have shown that the tremor during retinal microsurgery has low frequency motion (<1Hz) with amplitude in the order of 100μm [28]. As a result, with high data acquisition rate, adjacent A-scans usually do not have offset in axial direction for more than a few pixels. Moreover, a cross-correlation maximization-based shift correction algorithm was recently proposed to suppress artifact due to axial motion [29

29. J. Lee, V. Srinivasan, H. Radhakrishnan, and D. A. Boas, “Motion correction for phase-resolved dynamic optical coherence tomography imaging of rodent cerebral cortex,” Opt. Express **19**(22), 21258–21270 (2011). [CrossRef] [PubMed]

## Acknowledgment

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6. | S. Han, M. V. Sarunic, J. Wu, M. Humayun, and C. Yang, “Handheld forward-imaging needle endoscope for ophthalmic optical coherence tomography inspection,” J. Biomed. Opt. |

7. | J. Han, M. Balicki, K. Zhang, X. Liu, J. Handa, R. Taylor, and J. U. Kang, “Common-path Fourier-domain optical coherence tomography with a fiber optic probe integrated Into a surgical needle,” Proceedings of CLEO Conference (2009). |

8. | M. Balicki, J. Han, I. Iordachita, P. Gehlbach, J. Handa, J. U. Kang, and R. Taylor, “Single fiber optical coherence tomography microsurgical instruments for computer and robot-assisted retinal surgery,” Proceedings of the MICCAI Conference, London, 108–115 (2009). |

9. | J. Ren, J. Wu, E. J. McDowell, and C. Yang, “Manual-scanning optical coherence tomography probe based on position tracking,” Opt. Lett. |

10. | L. Huo, J. Xi, Y. Wu, and X. Li, “Forward-viewing resonant fiber-optic scanning endoscope of appropriate scanning speed for 3D OCT imaging,” Opt. Express |

11. | W. G. Jung, J. Zhang, L. Wang, P. Wilder-Smith, Z. P. Chen, D. T. McCormick, and N. C. Tien, “Three-dimensional optical coherence tomography employing a 2-axis microelectromechanical scanning mirror,” IEEE J. Sel. Top. Quantum Electron. |

12. | J.-F. Chen, J. B. Fowlkes, P. L. Carson, and J. M. Rubin, “Determination of scan-plane motion using speckle decorrelation: theoretical considerations and initial test,” Int. J. Imaging Syst. Technol. |

13. | P. C. Li, C. J. Cheng, and C. K. Yeh, “On velocity estimation using speckle decorrelation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control |

14. | A. Ahmad, S. G. Adie, E. J. Chaney, U. Sharma, and S. A. Boppart, “Cross-correlation-based image acquisition technique for manually-scanned optical coherence tomography,” Opt. Express |

15. | K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. |

16. | J. K. Barton and S. Stromski, “Flow measurement without phase information in optical coherence tomography images,” Opt. Express |

17. | B. Lau, R. A. McLaughlin, A. Curatolo, R. W. Kirk, D. K. Gerstmann, and D. D. Sampson, “Imaging true 3D endoscopic anatomy by incorporating magnetic tracking with optical coherence tomography: proof-of-principle for airways,” Opt. Express |

18. | J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Speckle in optical coherence tomography,” J. Biomed. Opt. |

19. | J. W. Goodman, |

20. | R. F. Wagner, M. F. Insana, and D. G. Brown, “Statistical properties of radio-frequency and envelope-detected signals with applications to medical ultrasound,” J. Opt. Soc. Am. A |

21. | A. Yariv, |

22. | Y. Liu, Y. Liang, G. Mu, and X. Zhu, “Deconvolution methods for image deblurring in optical coherence tomography,” J. Opt. Soc. Am. A |

23. | P. D. Woolliams, R. A. Ferguson, C. Hart, A. Grimwood, and P. H. Tomlins, “Spatially deconvolved optical coherence tomography,” Appl. Opt. |

24. | K. Zhang and J. U. Kang, “Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT,” Opt. Express |

25. | X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. |

26. | X. Liu and J. U. Kang, “Progress toward inexpensive endoscopic high-resolution common-path OCT,” Proc. SPIE |

27. | J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic functional Fourier domain common path optical coherence tomography for microsurgery,” IEEE J. Sel. Top. Quantum Electron. |

28. | S. Sinch and C. Riviere, “Physiological tremor amplitude during retinal microsurgery,” Proc. 28th IEEE Northeast Bioeng. Conf, 171–172 (2002). |

29. | J. Lee, V. Srinivasan, H. Radhakrishnan, and D. A. Boas, “Motion correction for phase-resolved dynamic optical coherence tomography imaging of rodent cerebral cortex,” Opt. Express |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(120.5800) Instrumentation, measurement, and metrology : Scanners

(170.4500) Medical optics and biotechnology : Optical coherence tomography

(330.4150) Vision, color, and visual optics : Motion detection

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: May 24, 2012

Revised Manuscript: June 23, 2012

Manuscript Accepted: June 29, 2012

Published: July 6, 2012

**Virtual Issues**

Vol. 7, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Xuan Liu, Yong Huang, and Jin U. Kang, "Distortion-free freehand-scanning OCT implemented with real-time scanning speed variance correction," Opt. Express **20**, 16567-16583 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-15-16567

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