1468 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 9, SEPTEMBER 2009

Real-Time FPGA Processing for High-SpeedOptical Frequency Domain Imaging

Adrien E. Desjardins*, Benjamin J. Vakoc, Melissa J. Suter, Seok-Hyun Yun, Guillermo J. Tearney, andBrett E. Bouma

Abstract—We present a novel algorithm for reconstructing inter-ferograms acquired in optical frequency domain imaging (OFDI).The algorithm was developed specifically for processing in fieldprogrammable gate arrays (FPGAs) and featured the use of a fi-nite-impulse-response (FIR) filter implementation of B-spline in-terpolation for efficiently re-sampling -space. When implementedin FPGAs, the algorithm allowed for real-time processing of inter-ferograms acquired with a high-speed OFDI system at 54 kHz anda sampling rate of 100 MS/s.

Index Terms—B-spline, digital signal processing, optical co-herence tomography (OCT), optical frequency domain imaging(OFDI), real-time processing.

I. INTRODUCTION

O PTICAL frequency domain imaging (OFDI), also knownas swept-source optical coherence tomography (OCT), is

a noninvasive imaging modality that has undergone rapid de-velopment in recent years [1]–[3]. An interferometric method,OFDI measures depth-resolved reflectance of infrared light withmicron-scale spatial resolution. With its high imaging speed,OFDI allows for 3-D image volumes to be acquired across largesurface areas in vivo. As such, it has tremendous potential asa screening tool for identifying pathologies in cardiovascular,retinal, and gastrointestinal tissues [4]–[6].

OFDI images are acquired in the frequency domain, withinterferogram samples acquired successively as the peak laserwavelength is rapidly swept in time [1]–[3]. Frequency domaindetection gives OFDI systems an inherent sensitivity advantageover their time domain counterparts [7]–[9]. By means of a dis-crete Fourier transform (DFT), interferograms are transformedfrom the frequency domain ( -space) to the axial depth do-main. Since most wavelength-swept lasers demonstrated to dateexhibit nonlinear trajectories in -space, direct application ofthe Fourier transform to interferograms acquired with tempo-rally linear sampling results in a loss of axial resolution, espe-cially for frequencies near the Nyquist limit. Although a non-linearly sampled clock that is synchronized to the laser sweep

Manuscript received September 25, 2008; revised February 24, 2009. Firstpublished March 24, 2009; current version published August 26, 2009. Asteriskindicates corresponding author.

*A. E. Desjardins was with the Harvard Biophysics Program, Boston, MA02115 USA (e-mail: adrien.desjardins@post.harvard.edu).

B. J. Vakoc, M. J. Suter, S.-H. Yun, G. J. Tearney, and B. E. Bouma arewith the Wellman Center for Photomedicine, Massachusetts General Hospital,Boston, MA 02114 USA (e-mail: bvakoc@partners.org; msuter@partners.org;syun@partners.org; gtearney@partners.org; bouma@helix.mgh.harvard.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMI.2009.2017740

rate can be used to control the digital acquisition and achievelinear -space sampling [10], this approach is potentially prob-lematic at high ( 10 MS/s) digitization rates and is incompat-ible with OFDI systems incorporating frequency shifters. Analternative is to apply interpolation in the data processing al-gorithm. While this approach imposes additional computationaloverhead, we demonstrate in this letter that a novel algorithm,optimized for field programmable gate array (FPGA) imple-mentation, can yield real-time processing performance at a sam-pling speed of 100 MS/s. This algorithm differs significantlyfrom that utilized for real-time processing of signals acquiredin the time-domain, which did not involve a DFT or -space re-sampling [11]. The algorithm’s performance was evaluated withnumerical simulations and with FPGAs that were integrated intoa high-speed OFDI system.

II. OFDI PROCESSING ALGORITHMS

The “sinc” interpolation method involves convolution of asampled function with the normalized periodic-sinc kernel,also known as the Dirichlet kernel. A method motivated bythe Nyquist-Shannon sampling theorem, periodic-sinc interpo-lation has been used for reconstruction of OFDI images withhigh accuracy. In the study of Yun et al. [1], it consisted of thefollowing steps, in which the FFT was employed to performthe convolution operation.

i) FFT of the interferogram (length ).ii) Zero padding of the result of step i) to increase the length

by a factor of .iii) Inverse FFT of the result of step ii).iv) For each interpolated -space point, computation of a

weighted average of two consecutive -space points inthe output of step iii) (linear interpolation) to produce anoutput of length .

The parameter determines the factor by which the inter-ferogram is over-sampled prior to -space resampling. Qualita-tively, it has been observed that little improvement is obtainedfor , and that excellent image quality is achieved for

. A disadvantage of the sinc method is that it is com-putationally expensive: it requires two FFT operations, one realtransform of length and one complex transform of length

, in addition to the linear interpolation in step iv). Real-time processing of images with the sinc method and isbeyond the capabilities of most commercially-available CPUsand field programmable gate arrays (FPGAs), however.

B-splines provide a continuous representation of discretesamples. As such, they are used in a wide variety of imageprocessing applications such as interpolation, rotation, andedge detection. For this OFDI processing application, we

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DESJARDINS et al.: REAL-TIME FPGA PROCESSING FOR HIGH-SPEED OPTICAL FREQUENCY DOMAIN IMAGING 1469

consider a cubic B-spline approximation as an empiricalmodel of the continuous-time interferogram, . With thisnotation, denotes acquisition time normalized by the timeinterval separating adjacent samples, , so that .This approximation can be represented as

(1)

where is an sequence of reals called the B-spline coeffi-cients, and is a centered cubic B-spline [12]–[14]. Interpo-lation of the interferogram in the frequency domain is performedby applying (1) to a set of normalized time points such that theinterferogram is uniformly sampled in -space. This set can beobtained by inverting , the mapping from normalized timeto -space corresponding to the wavelength-swept source. With

, , and denoting the z-transforms of ,, and , respectively, (1) can be written as

(2)

For a given interferogram, the coefficients can be recov-ered with so-called direct IIR cubic B-spline filter .

Vrcelj et al. suggested performing interpolation with filtercoefficients derived from a short finite impulse response (FIR)filter that approximates the IIR cubic B-spline filter [15]. ThisFIR approach to B-spline interpolation is highly desirable fromthe standpoint of interpolating OFDI interferograms, becausesampled data points can be processed sequentially in real-timeas the laser is swept in time. By comparison, sinc interpolationand other methods such as cubic-spline interpolation are non-sequential, necessitating acquisition of the entire interferogrambefore interpolation is performed. The two computational stepsinvolved with FIR B-spline interpolation were as follows.

a) Convolution of the interferogram with a FIR filter that isa five-tap Kaiser windowed truncation of the IIR cubicB-spline filter.

b) Computation of each interpolated point, as a convolutionof the output of step a) with a centered B-spline, as per (1).

The results of [15] motivated the use of a cubic B-spline filterand a Kaiser window. They also motivated the choice of 1.76 forthe Kaiser parameter which controlled the tradeoff between themain-lobe width and the side-lobe area for the Kaiser window.

The FIR B-spline interpolation method was included into acomplete OFDI processing algorithm that included backgroundsubtraction, demodulation, and postprocessing. Backgroundsubtraction was required to remove components of the inter-ferogram that did not derive from the sample. Demodulationwas required in order to compensate for the effects of fre-quency-shifters [16]. The effect of the frequency shifters wasto induce a time-varying phase in the interferogram signal.The resulting frequency shift, , was chosen to be exactlyone quarter of the sampling rate, . Conceptually, digitaldemodulation was performed by multiplying the input signalseparately by for the real part of the outputand for the imaginary part. In practice, because

, the interferogram was multiplied by the se-quences and to generate thereal and imaginary parts, respectively. Digital demodulation

Fig. 1. Flow diagram for the FPGA algorithm. Numbered steps correspond tothose described in the text.

mapped a single frequency to two frequencies, the first beinglower than the original by , and the second higherby the same amount. The latter frequencies were removedwith a symmetric, 40th-order low-pass FIR filter. This filterwas designed to remove frequencies with magnitude greaterthan , preserving the range . Insummary, the following steps comprised the complete B-splinereconstruction algorithm optimized for sequential processing,with an input interferogram of length , as shown schemati-cally in Fig. 1.

1) Background subtraction to produce one real-valued vectorof length .

2) Demodulation with low-pass FIR filtering, as describedabove, to produce two real-valued vectors of length .

3) Interpolation with the B-spline algorithm. This step wasperformed separately for each vector of step b) to producetwo real-valued vectors of length , where is thelargest power of two that satisfies . A power oftwo was chosen for the output length to optimize thespeed of the FFT [17]. The two vectors were then combinedas real and imaginary parts of a complex vector of length

.4) FFT of length , followed by a magnitude-squared

operation.A standard step in OFDI image reconstruction is the multipli-

cation of the interferogram with a window function prior to per-forming the Fourier transform in order to reduce side-lobe arti-facts. In this B-spline reconstruction algorithm, the weights of a

1470 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 9, SEPTEMBER 2009

Hamming window were included into the interpolation weightsas multiplicative factors (step b) of the FIR B-spline interpo-lation method], so that a separate windowing step was not re-quired. The choice of a Hamming window was based on conti-nuity with previous studies; a different window type could alsohave been chosen to achieve different axial point-spread func-tion characteristics.

Whether or not the proposed reconstruction algorithm ismore computationally efficient than others depends largelyon the hardware platform on which it is implemented. It isexpected to perform particularly well in contexts in whichprocessing is performed on the interferogram in real-time as thelaser is swept in time. Indeed, B-spline interpolation imposesonly modest requirements on FPGAs: for each interpolationpoint, a five point convolution and a weighted sum of at mostfour terms is computed, relative to a weighted sum of two termsin the case of linear interpolation. The increase in memoryrequirements required by B-spline interpolation derives onlyfrom a 1.5-fold increase in the number of stored weights and,trivially, three unique values corresponding to the symmetricFIR filter. Both B-spline and linear interpolation are thereforewell suited to real-time processing. In comparison, the sincinterpolation algorithm involves two FFTs, which can only beperformed after an entire interferogram is acquired.

Interferograms from reflectances at different axial positionswere simulated numerically, and corresponding point-spread-functions (PSFs) were obtained by application of the B-splinereconstruction algorithm. The quadratic and cubic chirp coef-ficients that determined the nonlinearity of -space samplingwere measured from the wavelength-swept laser that was incor-porated into the OFDI system. Nonlinear -space sampling wasmodeled as

. The ratios and , which determinethe strength of quadratic and cubic coefficients, were 6.00and 3.12, respectively. Random white noise was applied ad-ditively. The ratio of the interferogram amplitude squared tothe variance of the random white noise was 38 dB, which isa signal power typical of that received by tissue. The resultingPSFs were compared with ones that were obtained from threevariants of the B-spline reconstruction algorithm. Two variantsemployed linear and sinc interpolation, respectively; a third em-ployed no interpolation. With these three variants, all other pro-cessing steps were identical to those of the B-spline reconstruc-tion algorithm. The importance of PSFs is that, speckle notwith-standing, their profiles completely determine the OFDI imagethat is obtained from a particular sample [18]. All numericalsimulations were performed in Matlab (Mathworks).

For interferogram frequencies with magnitudes approachingthe limit of 0.25 (expressed as a fraction of the sampling fre-quency), the performance of B-spline interpolation was foundto be significantly higher than that of linear interpolation andvery similar to that of sinc interpolation. Indeed, the PSF cor-responding to an interferogram frequency of 0.2 obtained withlinear interpolation had a pedestal that was approximately 15 dBabove the noise floor [Fig. 2(a)]. This pedestal, which was notapparent with B-spline interpolation or with sinc interpolation,is a direct result of the discrepancy between the interferogramand its piecewise-linear approximation that was employed in

Fig. 2. Point-spread functions (PSFs) corresponding to simulated interfero-grams at frequencies of 0.1 (a) and 0.2 (b), where values are expressed relativeto the sampling frequency. The PSF shown in each case was obtained by aver-aging across 1000 simulated PSFs. In case (a), all interpolation methods yieldedsimilar results. In case (b), a pedestal (arrow) approximately 15 dB above thenoise floor was apparent with linear interpolation (black), whereas it was absentwith sinc interpolation (blue) and B-spline interpolation (green). In both cases,reconstruction without interpolation (red) resulted in PSFs that were consider-ably broader.

linear interpolation. For an interferogram frequency of 0.1, thepedestal was not apparent with all three interpolation methods[Fig. 2(b)]. With lower interferogram frequencies, interpola-tion inaccuracies tended to be smaller because there was lesssignal variation across consecutive samples. For both simulatedinterferograms, PSFs obtained without interpolation were sig-nificantly broader than those obtained with interpolation. Thesignal-to-noise (SNR) ratio was found to be insensitive to the in-terpolation method: for both interferogram frequencies, it variedby less than 0.2% (in dB) with linear, B-spline, and sinc inter-polation methods.

The B-spline reconstruction algorithm was integrated intotwo FPGAs (Red River, Virtex-2), each of which correspondedto one polarization channel. Each FPGA contained an iden-tical copy of the algorithm, and it received data from a 12-bitanalog-to-digital converter (ADC) that was clocked at MS/s.Operations in the FPGA were performed with 18-bit preci-sion, with the exception of the magnitude-squared operation,which was performed with 36-bit precision and subsequentlydelivered as output with 32 bits. Each ADC-FPGA pair was

DESJARDINS et al.: REAL-TIME FPGA PROCESSING FOR HIGH-SPEED OPTICAL FREQUENCY DOMAIN IMAGING 1471

contained within a single-board computer (Curtiss-Wright,Champ-AVIII). A custom program written for the single-boardcomputers channeled the outputs of the FPGAs simultaneouslyto a single RAID hard drive array (Apple Xserve) via separatefiber channels, and to a user-interface computer (Apple, G5; 8GB RAM) via gigabit Ethernet. The RAID array received alldata delivered by both single board computers; its write-speedsaccommodated continuous data storage. The coefficients

of the mapping were measured by a) applyingthe Hilbert transform to an interferogram corresponding toreflection from a single depth; b) unwrapping the phase ofthe result from part a); c) fitting a third-order polynomial tothe result of step b). Prior to the start of data acquisition,parameters were uploaded to the FPGA from the user-interfacecomputer. These parameters consisted of background signalsacquired from the ADCs in the absence of an imaging sample,the B-spline interpolation parameters, and a flag indicatingwhether to output raw or processed data. The optical elementsof the OCT system, including the laser and the interferometer,were developed for high-speed cardiovascular imaging [4]. Theaverage power of the laser was 50 mW; its tuning repetitionrate was 54 kHz. For each A-line, the input vector had a lengthof wavelength samples, and the FPGA outputconsisted of samples. Two postprocessing stepswere performed on the user-interface computer: signal powersfrom the two FPGAs were added, and the output was loga-rithmically transformed using a look-up table. The interfacecomputer displayed a subset of the processed frames immedi-ately after FPGA processing; in that sense, real-time displaywas achieved. The display rate of approximately 5 Hz (with1080 A-lines per frame) could be significantly improved withsoftware optimization, however.

For imaging of human skin (finger tip) in vivo, a glasswindow angled at 20 with respect to the incident beam wasplaced beneath the focusing lens to provide mechanical stabi-lization and to reduce specular reflections. In a typical imagereconstructed in real-time by the two FPGAs, the boundarybetween the stratum corneum and the papillary dermis and thatbetween glass and the skin surface were sharply delineated[Fig. 3(a)]. The spiral structure of the sweat ducts within thestratum corneum was clearly resolved. Overall, the imagequality was good, with no prominent image artifacts apparent.The FPGA-processed image compares favorably with those re-constructed offline in software [Fig. 3(b) and (c)], and thereforeit demonstrates the feasibility of accurate real-time processingat sampling frequencies of 100 MS/s with the B-spline recon-struction algorithm.

The advantages and disadvantages of the proposed OFDI re-construction algorithm compared with existing algorithms de-pend on the context in which it is implemented. As demonstratedin this study, its accuracy is significantly higher than algorithmswith linear interpolation; in general however, higher accuracycan be expected from algorithms with sinc interpolation. Therelative performances of different interpolation methods can beexpected to depend on the extent to which the trajectory ofthe laser deviates from linearity in -space, which in turn de-pends on the optical configuration of the laser. The reconstruc-tion speeds achieved with the FPGAs utilized in this study were

Fig. 3. OFDI images of human skin (finger tips). For image (a), processingwas performed in real-time with FPGAs. The arrow indicates the zero-delaypoint at which the reference and sample arms had identical optical lengths. SC:Stratum corneum; PD: papillary dermis; SD: sweat duct. Scale bar: 250 (m;transverse extension: 7 mm. For images (b) and (c), processing of a differentsection of human skin was performed offline in software, without contributionsfrom FPGAs. In image (b), processing was performed without �-space re-sam-pling, with resulting axial blur. In image (c), processing was performed with thesame �-space re-sampling algorithm as implemented in the FPGAs.

comparable to those that could be achieved with optimized soft-ware running on current desktop CPUs. Looking forward, it islikely that FPGA processing of OFDI data will be most valuablefor real-time processing in cases where the tuning repetition rateof the source is faster than that of the system utilized in this study[2], [3].

III. CONCLUSION

A novel OFDI reconstruction algorithm was proposed, andits accuracy was demonstrated with numerical simulations andan FPGA implementation integrated into a high-speed OFDIsystem. By using an FIR implementation of B-spline interpo-lation, reconstruction accuracy was shown to be significantlyhigher than linear interpolation and very similar to sinc inter-polation. With interpolation and demodulation performed se-quentially in time, the reconstruction algorithm was particu-larly well suited to FPGA processing. It allowed for continuous,

1472 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 9, SEPTEMBER 2009

real-time processing in conjunction with sampling frequenciesof 100 MS/s. From a clinical standpoint, the significance of thisalgorithm lies in its ability to make OFDI images available morequickly to clinicians after acquisition, which in turn can lead tofaster diagnostics.

ACKNOWLEDGMENT

The authors are grateful to B. Vrcelj (Caltech DSP Group)and to the anonymous reviewers for their critical feedback.

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