Fast-updating and nonrepeating Lissajous imagereconstruction method for capturing increased

dynamic information

Christopher L. Hoy,1 Nicholas J. Durr,2 and Adela Ben-Yakar1,2,*1Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA2Department of Biomedical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA

*Corresponding author: ben‐yakar@mail.utexas.edu

Received 29 November 2010; revised 7 February 2011; accepted 8 February 2011;posted 8 February 2011 (Doc. ID 138842); published 23 May 2011

We present a fast-updating Lissajous image reconstruction methodology that uses an increased imageframe rate beyond the pattern repeat rate generally used in conventional Lissajous image reconstructionmethods. The fast display rate provides increased dynamic information and reduced motion blur, as com-pared to conventional Lissajous reconstruction, at the cost of single-frame pixel density. Importantly, thismethod does not discard any information from the conventional Lissajous image reconstruction, andframes from the complete Lissajous pattern can be displayed simultaneously. We present the theoreticalbackground for this image reconstruction methodology along with images and video taken using the al-gorithm in a custom-built miniaturized multiphoton microscopy system. © 2011 Optical Society ofAmericaOCIS codes: 100.3010, 100.2000, 180.5810.

1. Introduction

Lissajous scanning patterns are often employed inlaser scanning systems where the more common ras-ter scanning pattern is impractical. This is often thecase in miniaturized laser scanning microscopesusing microelectromechanical system (MEMS) scan-ning mirrors or piezoelectric fiber scanners for beamscanning [1–5]. In such devices, the frequency of theslow-axis deflection required for a raster scan isusually well below the mechanical resonance fre-quency. Actuation off resonance requires larger driv-ing voltages, and the maximum achievable deflectionangle is often much less than the maximum de-flection achievable on resonance. In such cases, it be-comes desirable to actuate both axes of the scannerat their mechanical resonance to achieve the maxi-mum scanning angles and therefore the largest fieldof view (FOV). Scanning the laser with two orthogo-

nal axes at resonance produces Lissajous scanningpatterns [6].

In laser scanning microscopy applications, theframe rate of a Lissajous-scanned image is tradition-ally taken as the rate at which the Lissajous patternrepeats itself [2,3]. Selection of the driving frequen-cies determines the pattern repeat rate and requiresoptimization to maximize the imaging speed and thenumber of pixels in the image, while minimizing thenumber of unsampled pixels in a single frame.

Here we present a new implementation of Lissa-jous scanning and image reconstruction in whichthe imaging frame rate is significantly faster thanthe Lissajous pattern repeat rate to provide addi-tional dynamic information. While we originally de-veloped the technique for use in our miniaturizedmultiphoton imaging and femtosecond laser micro-surgery probe [4], the technique is broadly applicableto any Lissajous-scanned laser scanning imagingsystem. This concept could also be applied to anyother scanning pattern that unevenly samples theentire FOV during construction of a single frame,

0003-6935/11/162376-07$15.00/0© 2011 Optical Society of America

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such as the rose or propeller scan. Similar methodshave since been demonstrated in a high-speed min-iaturized multiphoton microscope [5] and a handhelddual-axes confocal microscope [7].

This fast-updating image reconstruction methodeffectively subsamples a slower, more densely sam-pled Lissajous scanning pattern and quickly displaysthe partially scanned frames to provide imaging fas-ter than the pattern repeat rate. Because the Lissa-jous pattern continuously scans across the entireFOV while completing one cycle of the Lissajous pat-tern, partially scanned frames contain sufficient in-formation from throughout the FOV to reconstructthe image. Since the scanning pattern is not repeatedin any two subsequent frames, pixels that are un-sampled in one frame may be sampled in the next,which reduces the appearance of unsampled pixelsto the user when viewed at higher frame rates. Nopixel information is discarded compared to the tradi-tional image reconstruction method, meaning thatthe traditional image can still be reconstructed fromthe fast-updating images. We present methods forreal-time display of both the traditional Lissajous-scanned image and a moving average image along-side the higher frame rate images, which provideadditional image detail.

2. Lissajous Image Reconstruction Methods

A. Conventional Lissajous Reconstruction

Steering a laser beam by two orthogonal sinusoidsproduces a Lissajous pattern, where the locationðx; yÞ of the focused laser beam is given by the para-metric equations

xðtÞ ¼ 12X½sinð2πf xtþ ϕxÞ þ 1�; ð1Þ

yðtÞ ¼ 12Y½sinð2πf ytþ ϕyÞ þ 1�: ð2Þ

Here f and ϕ are the driving frequency and phaseshift, respectively, in the x or y directions, and X andY are the maximum extents of the FOV in x and y,respectively. For image reconstruction, Eqs. (1) and(2) describe the pixel location of each data samplewhere X and Y are in units of pixels. The pattern re-peat rate, f P, of a Lissajous pattern is given by

f P ¼ f xnx

¼ f yny

; ð3Þ

where nx and ny are the smallest integer divisors thatsatisfy Eq. (3). Thus, for all real and rational scan-ning frequencies, the corresponding Lissajous pat-tern is stable and will repeat at a fixed rate. Theintegers nx and ny determine the number of cyclesthat occur in either the x or y direction, respectively,before the pattern repeats [2]. Larger values of n

increase the number of lines scanned across theFOV during one completion of the Lissajous pattern.

Within the resonance bandwidth of the scanningdevice, the driving frequencies can be chosen tomaximize the n values that satisfy Eq. (3), therebyincreasing the line density of the image. FromEq. (3), it is clear that an increase in line density in-herently reduces the pattern repeat rate. When theframe rate (f F) is made equal to f P, as is commonlydone, there exists an intrinsic trade-off between sam-pling density and frame rate. Furthermore, becausethe Lissajous pattern is repeated once for each frame,regions of the image left unsampled by a sparseLissajous pattern in one frame remain unsampledin subsequent frames, and this information is lost.

For illustration, Fig. 1 displays MATLAB simula-tions for single Lissajous-scanned frames where f F ¼f P for several scanning frequency combinations andpixel numbers. In these examples, the driving fre-quencies are restricted to the resonant scanningpeaks of the MEMS scanning mirror used in our re-cently developed 9:6mm diameter microsurgeryprobe [8]: f x ¼ 980Hz with an 18Hz bandwidth(FWHM) and f y ¼ 2260Hz with a 100Hz bandwidth.For quantitative comparison, Fig. 2 plots thepercent of pixels sampled at least once and the per-cent of pixels sampled ten times or more for each in-teger frame rate achievable within these resonantbandwidths up to 35 frames per second.

In Fig. 1, pixels that do not get sampled by theLissajous pattern are shown white. To reduce theseunsampled pixels, higher line densities, and subse-quently lower frame rates, can be used, and/or therelative pixel size can be increased. In either solu-tion, pixels toward the outside of the image can be

Fig. 1. (Color online) Conventional Lissajous reconstructions forvarious pattern repeat rates and pixel sizes. The color map repre-sents the number of times a given pixel is sampled before one cycleof the Lissajous pattern completes. For the conventional casewhere f F ¼ f P, this represents the number of times a pixel issampled in each frame. White pixels denote unsampled regionsof the image. Data generated by MATLAB simulation.

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heavily oversampled (e.g., sampled ten or moretimes) in each frame. Moderate resampling of pixelscan be useful in reducing noise if intensity valuesfrom different cycles within the same frame areaveraged; however, significant resampling of pixelsprovides little new spatial information despite the in-creased laser exposure to these oversampled regions.

When selecting the number of pixels for a givenFOV, we need to consider several constraints. Onone hand, as seen in Figs. 1 and 2, image reconstruc-tion with a fewer number of pixels can reduce thenumber of unsampled pixels to achieve a completelyfilled image. On the other hand, for a given FOVandresolution, there is a minimum number of pixels thatcan be used before the image resolution becomes lim-ited by the pixel size. Using the Nyquist criterion, wedesire at least two pixels per resolvable spot in ourimage. Thus, for an example case of a 200 μm ×200 μm FOV and a 1 μm resolution, this criteria re-quires the reconstructed image consist of no fewer

than 400pixels × 400pixels. However, in additionto increasing the likelihood of unsampled pixels,increasing the number of pixels used in image recon-struction also increases the required data acquisitionrate to avoid missing pixels at the center of the im-age. Explicitly, the pixel scanning speed can be ex-pressed by the first derivatives of Eqs. (1) and (2):

_xðtÞ ¼ πf xX ½cosð2πf xtþ ϕxÞ�; ð4Þ

_yðtÞ ¼ πf yY ½cosð2πf ytþ ϕyÞ�: ð5Þ

When both axes are traveling at their maximumvelocity, which occurs at the center of the image,the maximum scanning speed, vmax, in pixels persecond is given by

vmax ¼ πPffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2x þ f 2y

q; ð6Þ

for a square image in which X ¼ Y ¼ P. In the exam-ple case whereP¼ 400, f x ≈ 980Hz, and f y ≈ 2260Hz,vmax ¼ 3:1 × 106 pixels=s. Therefore, the data acquisi-tion rate must equal or exceed 3:1MHz to avoid miss-ing pixels in the center of the image, and thus theimage size introduces hardware constraints inaddition to resolution constraints.

As illustrated in Figs. 1 and 2, it is possible to com-pletely fill images with large numbers of pixels by se-lecting driving frequencies corresponding to a slowlyrepeating Lissajous pattern. However, for use in aclinical imaging system, slow frame rates lead tomotion artifacts, while large oversampling leads tounnecessarily high laser dosages. While the 10Hzupdating pattern with a 256pixel × 256pixel imagesize appears to be a good compromise in our example,approximately 6.5% of the pixels toward the center ofthe image remain unsampled and will be unsampledin every frame, resulting in a small loss of informa-tion. What we desire is to achieve full coverage ofthe FOV, such as the one found with the 1Hz updat-ing pattern, but with the ability to capture dynamicinformation at a faster rate and to use the laser ex-posure more efficiently.

B. Fast-Updating and Nonrepeating LissajousReconstruction

To improve upon this traditional approach to Lissa-jous image reconstruction for clinical imaging appli-cations, we have implemented a new fast-updatingLissajous image reconstruction method where theLissajous pattern does not repeat for each frame.In this method, we drive the scanning device at re-sonant frequencies that can provide a high line den-sity but update the frame at a rate many times fasterthan the pattern repeat rate. Here we simply takeadvantage of the fact that a Lissajous scanning pat-tern repeatedly samples across the entire extent ofthe FOV while acquiring information for a givenframe. Thus, information about the entire FOV canbe extracted from a partially scanned frame.

Fig. 2. Sampling statistics for conventionally reconstructedLissajous images with resonant frequencies at f x ¼ 2260Hz andf y ¼ 980Hz. (a) Percentage of pixels in each frame sampled fora given frame rate. (b) Percentage of pixels sampled tentimes or more in each frame at a given frame rate. The solid,dotted, and dashed curves represent 512 × 512, 256 × 256, and128 × 128 pixel images, respectively. Data generated by MATLABsimulation.

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In our method, the image reconstruction algorithmtracks the pixel location continuously over time,rather than refreshing at the end of each frame.The location for the first pixel of each frame is dic-tated by the Lissajous trajectory and pixel locationat the end of the preceding frame. The frame rate,f F, is then set independently of f P to optimize sin-gle-frame detail and imaging speed (f F > f P). In thismanner, the number of oversampled pixels in a singleframe is greatly reduced, and the frame rate is in-creased for increased temporal resolution. Figure 3compares the percent of pixels sampled once andsampled ten or more times using this method usinga 1Hz Lissajous pattern (f x ¼ 979Hz and f y ¼2263Hz) for frame rates from 1 to 10 frames=s and512pixel × 512pixel image reconstruction. Usingthis method, the scanning pattern for each frameis distinct until the 1Hz Lissajous pattern beginsto repeat. In other words, a 5Hz frame rate will con-sist of a repeating series of five distinct frames. In

Fig. 3, the statistics for each distinct frame areprovided to show frame-to-frame variability.

From Fig. 3 we see that the percentage of pixelssampled per frame in our method decreases with in-creasing frame rate at a similar rate to the conven-tional reconstruction where f F ¼ f P. However, in ourmethod the pixels that are unsampled in a givenframe are generally sampled in the subsequentframe, since the pattern does not repeat for eachframe. Furthermore, due to the increased frame rate,the overall sampled pixel density can appear higheras the frame rate approaches the flicker fusionthreshold of the human visual response [9]. Criti-cally, no information is lost from the conventionalslower-displaying Lissajous image. The informationis simply separated and displayed into shorter timeintervals to capture additional dynamic information.Thus, both the fast-updating, lower-pixel density im-age of our fast-updating algorithm and the slower,high-pixel density image of the traditional algorithmcan be viewed side-by-side during real-time imagereconstruction.

3. Experimental Implementation

We first incorporated the image reconstruction meth-odology described above into the miniaturized 18mmprobe we have designed for combined nonlinear op-tical microscopy and femtosecond laser microsurgery[4]. Here we will present results from themost recent9:6mm diameter probe [8]. This new smaller probeutilizes a two-axis resonant MEMS scanning mirrorwith the resonant frequencies described earlier. Wedrive the mirror at f x ¼ 979Hz and f y ¼ 2263Hz,and reconstruct and display 512pixel × 512pixelframes at a frame rate of 7Hz. These scanning pat-terns correspond to the 1Hz pattern shown in Fig. 1.Based on the measured 1:32 μm two-photon re-solution of our 9:6mm probe, this pattern can pro-vide full coverage at the sample for FOV well over1mm × 1mm. Based on Fig. 3, reconstruction at7 frames=s is an attractive choice, as it offers a 7×enhancement in temporal resolution over the con-ventional reconstruction method, which updates at1Hz for these scanning frequencies. Also, at 7Hz re-construction, each frame samples approximately thesame number of pixels (>76%) with little visibleframe-to-frame variability.

A custom LabVIEW program implements our im-age reconstruction algorithm, shown in Fig. 4. Theprogram drives the scanning mirror and handlesreal-time image reconstruction and display throughtwo data acquisition cards. One high-speed analoginput card (PCI-6115, National Instruments) is usedfor collecting the image data and to drive one axis ofthe MEMS mirror. The other card (PCI-6711,National Instruments) is used to drive the secondaxis of the MEMS mirror. We used the second cardto provide additional analog output channels, be-cause we were limited by the output channels avail-able on our data acquisition card. The two cards areconnected with a real-time system integration

Fig. 3. Sampling statistics for Lissajous images reconstructedusing our fast-updating method with scanning frequencies of f x ¼2260Hzand f y ¼ 979Hzand a 512 × 512 image size. (a) Percentageof pixels in each frame sampled for a given frame rate. (b) Percen-tage of pixels sampled ten times or more in each frame at a givenframe rate. The gray circles denote the values for each frame at agiven frame rate, while the solid line displays the average for eachframe rate. Data generated by MATLAB simulation.

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(RTSI) cable to allow synchronization to a commonclock.

For nonlinear optical imaging, we digitize thephotocurrent from a photomultiplier tube (PMT)(H7422-40, Hamamatsu) at a sampling rate of 4MHzthrough a high-speed preamplifier (DHPCA-100,FEMTO Messtechnik). This sampling rate allows re-construction of up to 512pixel × 512pixel images atthe given scanning frequencies without missed pixelsat the center of the FOV arising from the high scan-ning speed. The collected data are stored in the mem-ory buffer until it is time to display the next frame, asdetermined by the frame rate.

The sinusoidal driving waveforms to the MEMSmirror are generated according to the frequenciesand peak voltage values defined by the user. By vary-ing the peak driving voltage, we can adjust theamplitude of mirror deflection and thus control themagnification of the microscope. For proper syn-chronization, the sampling rate for the generatedwaveforms must be equal to the input sampling rate,which is 4MHz in our case. The program nextdigitizes the driving voltage waveforms into pixel lo-cations for image reconstruction. We have implemen-ted a real-time variable phase delay control, whichallows us to shift the pixel locations along the Lissa-

jous trajectory to compensate for phase delay be-tween mirror driving signal and pixel location. Theresulting array of pixel coordinates is buffered untilall image data have been collected for the currentframe, at which point the image data are placed ina 512 × 512 array according to the correspondingpixel location coordinates and displayed as an image.

For pixels that are sampledmultiple times in a sin-gle frame, the most recent value is used, rather thanaveraging the values. While averaging can help sup-press noise, selective pixel-by-pixel averaging duringimage reconstruction was found to be too computa-tionally intensive.

Proper synchronization is maintained in the pro-gram by triggering data collection and waveformgeneration off the hardware clock. The clocks of thetwo cards are synchronized via the RTSI cable. Main-taining proper synchronization is critical to this tech-nique, as temporal drift may lead to accumulatingartifacts in the image. We have implemented our im-age reconstruction program on a personal computerwith a 2:8GHz quad-core Intel i7-860 processor and3:5GB of random access memory (RAM) runningLabVIEW 2009 on the Windows XP Professionaloperating system. With the current hardware andLabVIEW code, our fast-updating Lissajous image

Fig. 4. Flow chart illustrating our fast-updating Lissajous image reconstruction algorithm. User inputs are shaded gray and located onthe left. User-selectable options are denoted by diamond process blocks. TheMEMS driving signal and the data collection are synchronizedby triggering from the same master clock in the software and by linking the data acquisition cards physically with a RTSI cable.

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reconstruction algorithm is capable of displayingframe rates up to 15Hz without loss of data or depar-ture from the desired frame rate. We believe thisspeed could be increased through further optimiza-tion of the code; however, frames taken at such highspeeds will sample only a small percentage of pixelsper frame.

The program can display in three different imageconstruction modes: (1) the conventional Lissajousimage at 1Hz, (2) our fast-updating Lissajous imageat 7Hz, or (3) a third imaging mode, called “movingaverage,” in which we average every 7Hz image withthe six previous images. The moving average imageprovides all the detail of the 1Hz image, along withnoise suppression from averaging, at the cost of in-troducing mild motion artifacts. By averaging entireframes, we can achieve real-time noise suppressionin a computationally efficient manner.

Figure 5 shows an image acquired using our fast-updating Lissajous scanning algorithm [Fig. 5(a)], amoving average image [Fig. 5(b)], along with a frameacquired at 1Hz showing the traditional Lissajousscanning algorithm [Fig. 5(c)]. As seen in Fig. 5(a),the frame updating at 7Hz with the fast-updatingLissajous image reconstruction algorithm providesinformation from the entire FOV; however, the FOVis only sparsely sampled compared to the conven-tional Lissajous pattern shown in Fig. 5(c). Note that,because the optical focusing conditions at the sampleremain unchanged across all reconstruction meth-ods, the resolution is unaffected in Figs. 5(a)–5(c)although the image is more sparsely sampled inFig. 5(a). On the other hand, the frame in Fig. 5(a)

contains the desired fast dynamic information be-cause the frames refresh seven times faster. Thisaspect can only be truly appreciated when viewingthe frames streaming at their respective rates, andMedia 1, Media 2, and Media 3 show video takenusing our 7Hz frame rate fast-updating algorithm,the moving average algorithm, and finally the tradi-tional algorithm.

In Fig. 6(a), the movement of the beads can be fol-lowed easily with little or no noticeable motion arti-facts, depending on the speed. While the percent ofpixels sampled in each individual frame in Fig. 6(a)is reduced compared to the traditional Lissajous im-age in Fig. 6(c), individual beads can still clearly bedistinguished and show no change in fundamentalresolution (i.e., the size of a single bead). Use of themoving average algorithm in Fig. 6(b) providessuperior detail over the traditional image due to thenoise reduction arising from averaging, though at thecost of the introduction of motion artifacts comparedto the raw 7Hz images. Because the moving averageimages still update at 7Hz, however, the user is stillable to identify and react to the occurrence of fast dy-namic events faster than in the traditional 1Hzimage.

4. Conclusion

We have presented a simple fast-updating imagereconstruction algorithm for use with Lissajous-scanned imaging systems. This modification is

Fig. 5. Two-photon fluorescence images of 1 μm fluorescent beadsdeposited on glass. (a) Raw frame using the fast-updating imagereconstruction algorithm updating at 7Hz. (b) Frame taken usingthe moving average algorithm, averaging seven raw frames andupdating at 7Hz. (c) Frame taken using the conventional Lissajousimage reconstruction methodology, updating at the pattern repeatrate of 1Hz. Contrast has been increased on the moving averageimage to match that of the unaveraged images. Scale bars are10 μm.

Fig. 6. Images from real-time videos (Media 1, Media 2, Media 3)of 1 μm fluorescent beads dried on a glass microscope slide andtranslated manually at varying speeds. The videos demonstratethe effect of each image reconstruction algorithm. (a) Our fast-updating algorithm updating at 7Hz. (b) Moving average algo-rithm updating at 7Hz displaying a seven frame moving average.(c) Conventional 1Hz image reconstruction. Note that the conven-tional algorithm video contains motion artifacts not present in thefast-updating algorithm video. The image reconstruction size wasreduced here to 128pixels × 128pixels to demonstrate the densersampling that can be achieved with small FOV. Contrast has beenadjusted on the moving average image to match that of the una-veraged images. Scale bars are 5 μm.

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particularly useful in resonantly scanned lasermicroscopy systems, wherein the choice of scanningfrequencies is restricted to the bandwidth of the re-sonant peaks. Our method provides an alternative tousing the pattern repeat rate as the frame rate, thusenabling dynamic visualization of samples withhigher temporal resolution. This method is simple toimplement and results in no loss in information fromthe conventional Lissajous-scanned image, as theconventional image can be displayed simultaneously.Using our method, the trade-off between single-frame pixel density and frame rate can be chosenfreely by the user, whereas only a limited numberof practical options may be available using the con-ventional method with narrow-bandwidth resonantscanning devices. In addition to the miniaturizedmultiphoton imaging system shown here, this fast-updating image reconstruction algorithm can bebroadly applied for any Lissajous-based laser scan-ning imaging system in which greater temporal reso-lution is desired.

The authors acknowledge support from theNational Science Foundation under grants BES-0548673, CBET-1014953, and Career Award CBET-0846868 as well as a grant from the Texas IgnitionFund by the University of Texas Board of Regents.Also, the authors would like to thank Olav Solgaardfor supplying the MEMS scanning mirror used in theminiaturized microscope and Pengyuan Chen forLabVIEW assistance.

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