Non-raster scanning in atomic force microscopy for high-speed … · 2010. 9. 13. · Non-raster scanning in atomic force microscopy for high-speed imaging of biopolymers Peter I

Non-raster scanning in atomic force microscopyfor high-speed imaging of biopolymers

Peter I. Chang and Sean B. Andersson

Abstract Recent advances in actuator design, controller architecture, and systemoptimization have made possible video-rate atomic force microscopy. Despite thisgrand achievement, there are systems with dynamics that occur much faster eventhan video rate. Novel methods are needed to push imaging rates even higher. Herewe present a scheme which is complementary to other high-speed approaches toatomic force: non-raster scanning. Using in real-time the data measured by the tipof the microscope coupled with basic information about the sample, we describe analgorithm that steers the tip of the microscope to remain in the regions of interest.The algorithm, designed for imaging biopolymers and other string-like samples, re-duces overall imaging time not by increasing the speed of scanning but by reducingthe total sampling area.

1 Introduction

Since the invention of atomic force microscopy [3], the technology has been used toincrease our understanding across a wide range of fields, including molecular biol-ogy, medicine, materials science, and nanotechnology. It has been particularly usefulin the field of biology due to its ability to operate in liquid, enabling researchers tostudy systems in their native environments. Despite its wide applicability, the use-fulness of AFM in studying dynamics in systems with nanometer-scale features isseverely hampered by the time scale of the imaging process. Current commercialAFMs typically require times on the order of minutes to produce a single high qual-ity image. The need for higher imaging rates has led to a great deal of work in highspeed AFM.

Peter I. ChangBoston University, Boston, MA 02215, USA, e-mail: [email protected]

Sean B. AnderssonBoston University, Boston, MA 02215, USA e-mail: [email protected]

1

2 Peter I. Chang and Sean B. Andersson

Approaches to high speed AFM can be broadly categorized into two main ap-proaches. The first involves improving the physical hardware, such as using a bal-anced scanner to suppress mechanical resonances [2], using small cantilevers [19],and designing new high speed actuators [8, 18]. The second involves the use ofadvanced control algorithms, including inversion-based feedforward control [20],combined feedback/feedforward design with adaptive control [10], and transient-based signal detection [14] (see also the review in [16]). Combinations of theseschemes have yielded a few research systems that achieve video-rate AFM (see, e.g.the review article [17]).

While achieving video-rate has greatly enhanced the utility of the instrumentand broadened the class of dynamics that can be studied using AFM, there are stilldynamic events that exceed even this temporal resolution. For example, a class ofmotor proteins, the dynien motors, travels inside living cells at a measured rate of1.5 µm/sec [11]. For an AFM at current video-rate (25 frames/sec), in a single framethe dynein would have moved 60 nm. There is therefore a need to bring even bettertemporal resolution to the state of the art AFM systems.

All of the current approaches to high speed AFM view the imaging process asessentially open-loop and seek to move the tip more quickly through the raster-scan pattern. In our local raster-scan scheme, we seek to utilize the measurementsacquired by the AFM tip to adjust the measurement process in real time and re-duce the amount of sampling that needs to be done. The algorithm is designed forbiopolymers and other similar samples which can be modeled as planar curves. Theessential idea is to utilize the measurements to estimate the spatial evolution of thesample and then steer the tip to sample only this region. Overall imaging time isreduced not by increasing the tip speed but rather by decreasing the overall imagingarea.

2 Overview of local raster scanning

As illustrated in the block diagram in Figure 1, the local raster scan algorithm is de-signed as a high-level feedback loop around the AFM system. The algorithm utilizesthe data acquired by the AFM in real time to drive a model of the spatial evolutionof the underlying sample. This model is then used to determine a tip trajectory suchthat the tip is scanned back and forth while moving along the path defined by thesample. As a result, data is acquired only local to the sample and the total imagingtime is reduced purely by eliminating unnecessary measurements. As a guide to thealgorithm, in this section we take around the loop and briefly describe each of theelements in Figure 1.

Non-raster scanning in AFM 3

Fig. 1 Block diagram of the local raster scan control loop. Driven by the data acquired by the AFM,the detector block determines the current position of the sample in the scan. These positions areused by the estimator block to determine the geometric parameters driving the spatial evolutionof the path of the sample. After filtering, these values are fed to the tip trajectory block whichestimates the evolution of the sample and, from that, the desired trajectory of the tip.

2.1 AFM system block

The AFM system block represents a physical AFM. It is abstractly described as con-sisting of two components: one containing the actuators, sensors, and controllers forperforming lateral motion of the sample with respect to the tip and one containingthe cantilever, tip, actuators, sensors, and controllers for performing motion in thevertical direction. From the point of view of the local raster scan algorithm, theAFM system is a black box which is assumed to faithfully follow the desired (x,y)–trajectory while producing (x,y)–measurements and z–measurements containing in-formation about the sample. The details of this block, such as whether the scanner isa frame-in-frame stage or a tube actuator, whether the AFM is operated in constantforce, AC, or alternative sensing mode, and details on the cantilever are unimportantto the basic design of the local raster scan algorithm. (For an overview of AFM froma controls perspective, see [1].)

Such details do, of course, influence the performance of the local raster scan al-gorithm. For example, novel actuator designs and control architectures would allowthe AFM system to follow the desired (x,y)-trajectory with high fidelity at a hightip velocity, given a z-measurement scheme that reflect the sample at those speeds.In addition, details about the content of the z−measurement directly influence thedesign on detector block since the information about the sample may be encoded inthe signal with different ways. Given a detector block, however, the remainder ofthe algorithm is independent of such information.


2.2 Detector block

The local raster scan algorithm is designed to scan the tip along a biopolymer orsimilar sample, moving repeatedly from the substrate up onto the sample and backagain. The role of the detector is to determine the point of transition between sub-strate to sample. There are a variety of ways to do this, depending on the data avail-able. As described in detail in Sec. 3.1, in this work we assume the z– measurementprovides the height of the sample. Combining this data with the (x,y) measurements,the detector essentially builds the local topology of the sample and then uses thattopology to estimate the position at which the transition occurred. This position ispassed to the next block and drives the entire scanning algorithm.

2.3 Estimator block

The desired trajectory of the tip is determined by estimating the future spatial evo-lution of the underlying sample based on the positions provided by the detector. Toachieve this, the sample is modeled as a planar curve whose evolution is determinedby its heading direction, θ , and curvature, κ . The role of the estimator block, de-scribed in Sec. 3.2, is to transform the position values provided by the detector blockinto estimates of the current heading direction and curvature.

2.4 Filter block

Noise in the original AFM measurements propagates into the position values es-timated by the detector. The estimator block then calculates numerical derivativesof that data, thereby amplifying the noise. To compensate for this effect, the rawestimates are run through the filter block. Using a simple dynamic model for theevolution of the curvature and heading direction, described in Sec. 3.3, this blockuses a Kalman filter to produce the optimal estimates of these curve parameters.

2.5 Tip trajectory design block

The final block in the feedback loop transforms the heading and curvature estimatesinto the desired tip trajectory. The block first uses the estimates to propagate themodel of the planar curve forward in space from the last detected position of thesample. The tip trajectory is then specified as a sinusoidal variation along this planarcurve in which the amplitude, A, and spatial frequency, ω , are specified by the user.The result is a smooth curve that on average tracks the underlying sample, leading to


an image of width A and resolution (1/ω). The details on trajectory determinationare provided in Sec. 3.4.

3 Controller details

The details of each of the blocks in Fig. 1 are given in this section. The entire schemebegins with the detection of the sample.

3.1 Sample detection

Consider the motion of the tip across a cylindrical sample of constant diameter d,depicted in Fig. 2. The method of detecting the transition from substrate to sample orsample to substrate in the data is dependent upon the specific application and uponthe data available to the detector. Under the assumption that the AFM is imaging asample in intermittent contact (or tapping) mode, detectors can be designed based ondirect measurements of the cantilever position or on information generated by the zcontroller, including the height, amplitude and phase signals. For example, using thetransient force detection scheme in [15], transitions could be detected within a fewoscillations of the cantilever. Such an approach would allow for detection to occurat tip speeds far higher than the limits imposed by the need for accurate imaging.

Fig. 2 Illustration of an AFM tip crossing a cylindrical sample with diameter d. Under the assump-tion that the tip speed, vtip, is slow enough, the height data directly indicates whether the tip is onor off the sample.

In practice, accessing the cantilever data can be challenging. In this work weassume that the goal is to produce a high-quality image. Under this assumption, thetip speed should be kept slow enough such that the height data acquired by the AFMwill accurately reflect the local topology. Because the local raster scan algorithmkeeps the tip in the vicinity of the sample, it can be assumed that the substratearound the sample is relatively flat with the sample itself rising off the base. As aconsequence of these two assumptions, a simple detector can be designed simply by


establishing a threshold value, τh, relative to the local measurements. Heights belowthe threshold are assumed to be on the substrate while those above the threshold areon the sample. In practice, the choice of τh would be informed by a priori knowledgeof the type of sample (e.g., the height of a DNA strand when imaging in liquid isapproximately 1.5 nm [12]) as well as by user experience with typical noise levelsin the imaging process.

Given a means of detecting transitions, one must then decide what part of thesample is to be represented by the planar curve. Consider the schematic of a sampleand tip path shown in Fig. 3. One possibility is to use the centerline of the sampleas the planar curve to be tracked by the local raster scan algorithm and a schemefor doing this has been previously presented in [5]. For samples with varying width,however, trying to estimate the centerline can lead to widely varying tip trajectoriesthat are difficult for the actuators to follow. As an alternative, one can choose to tracka single edge of the polymer. Consider the motion of the tip from the point p[k] inFig. 3 to point p[k +1]. Under the assumption that the tip trajectory fully covers thesample, there is a sequence of down− up− down transitions between sample andsubstrate. Using a state machine to count these transitions, the crossings on one sideof the sample can be isolated.

Fig. 3 Based on the signals available, a detector can be designed to indicate when the tip movesfrom the substrate to sample or from the sample to substrate. Using high-level logic, one can thenchoose to select only transitions on one side of the sample. Shown here are three detected crossingevents, moving from p[k] to p[k + 2]. In order to prevent chattering between on and off due tomeasurement noise at the edge of the sample, detection is disabled for a portion of the trajectory.

Noise in the measurements can trigger false transitions in the state machine andultimately false detections. Consider, for example, a simple threshold detector. Thenoise can cause chattering in the detector as the tip crosses between sample and sub-strate. To avoid this, after a transition event the detector is disabled for a portion ofthe tip trajectory. Typically detection would be re-enabled again at the maximum of


the sinusoidal component of the tip trajectory. In the example shown, the state de-tector would then only count a down-up transition followed by a up-down transitionto track the right-hand edge of the sample.

When the detector determines that a transition event along the desired edge hasoccurred, the position is recorded in p[k] and the value passed along to the estimationblock.

3.2 Estimation of curvature and heading direction

As described in detail in Section 3.4, the path of the underlying sample from p[k]can be estimated based on the heading direction, θ [k], and curvature κ[k] at p[k].The role of the estimator is to determine the values of θ [k], κ[k] from the history ofcrossings.

Fig. 4 Estimating θ [k] and κ[k]. The heading direction is estimated by calculating the angle of thevector connecting the previous transition point, p[k−1] to the current point p[k] while the curvatureis estimated from the radius of the circle passing through the points p[k−2], p[k−1], and p[k].

Consider a sequence of three crossings, p[k−2], p[k−1], and p[k] as illustratedin Fig. 4. The heading direction at crossing k is by definition the angle of the tan-gent vector to the sample path at that point with respect to a fixed frame. It can beestimated geometrically as follows. Define


δ p[k] = p[k]− p[k−1] =(

δx[k]δy[k]

). (1)

The heading direction at p[k] is then estimated to be

θ [k] = arctan2(δy[k],δx[k]). (2)

The curvature at p[k] is estimated from the previous three transitions, p[k], p[k−1] and p[k−2] using Heron’s formula [4]. That is,

κ[k] =±4

√(l−a)(l−b)(l− c)

abc. (3)

Here a, b and c are the Euclidian lengths of the three sides of the triangle defined byp[k− 2], . . . , p[k], and l = (a + b + c)/2 as the semiperimeter of that triangle. Thesign of κ[k] is defined to be positive when the normal vector to the curve pointsinside the circle passing through the three transition points.

Since the estimates of both θ [k] and κ[k] are essentially numerical derivativesof the sample path with respect to arclength, any noise in the measurements of thetransition positions (originally arising from noise in the AFM measurements) will beamplified. To mitigate the effect of this noise, the estimates are sent to the filteringblock.

3.3 Filtering

To filter the measurements of the heading direction and curvature, we model theirspatial evolution using a discrete time stochastic differential equation. Under theassumption of constant curvature between update steps, the heading direction isdriven directly by the curvature according to

θ [k +1] = θ [k]+∆sκ[k]+wθ [k] (4)

where ∆s = 2π/ω is the (nominal) step size along the sample between crossingsand wθ is a Gaussian white noise process. The curvature is assumed to evolve asa random walk, driven by a Gaussian white noise process wκ . Defining a new sys-tem state as X = [ θ κ ]T and adding in a measurement model yields the stochasticsystem

X [k +1] = FX [k]+W [k], (5a)Y [k] = HX [k]+V [k]. (5b)

where

F =(

1 ∆s0 1

), H =

(1 00 1

), W [k] =

(wθ [k]wκ [k]

), V =

(vθ [k]vκ [k]

). (6)


Here V is Gaussian white noise process capturing the noise in the estimator. AKalman filter is applied to this system, yielding the filter equations

X [k +1|k] = FX [k|k]+E[W ], (7a)

Σ [k +1|k] = FΣ [k|k]FT +ΣW , (7b)

K[k +1] = Σ [k +1|k]HT (HΣ [k +1|k]HT +ΣV )−1, (7c)

X [k +1|k +1] = X [k +1|k]+K[k +1](Y [k +1]−HX [k +1|k]), (7d)Σ [k +1|k +1] = (I−K[k +1]H)Σ [k +1|k], (7e)

where X [k|k] is the a posteriori state estimate and Σ [k|k] is the a posteriori errorcovariance.

3.3.1 Comments on the input and measurement noise

The noise process W represents random fluctuations in the heading direction andcurvature of the underlying sample and arises from the physical characteristics ofthat sample. To date, we have utilized an approach in which the mean and varianceof W are taken to be constant and selected in an ad hoc manner. Better performanceis to be expected, however, if the choice of parameters is informed by a priori knowl-edge of the underlying sample. Biopolymers, for example, are known to twist andturn and a variety of structural models have been developed to describe this bend-ing behavior, such as the freely-jointed chain and the worm-like chain [9]. Thesebehaviors are captured by the choice of input noise parameters.

The noise process V captures noise propagated from the AFM measurementsinto the detection of the crossing positions and from there into the estimation ofthe curvature and heading direction. While we currently use an ad hoc selection ofmeasurement noise parameters, the choice of mean and variance based on the initialmeasurement noise is a topic of ongoing research.

3.4 Tip trajectory

3.4.1 Frenet-Serret frame

As discussed in Sec. 2, at the core of the local raster scan algorithm is a model ofthe sample as a planar curve. The evolution in two dimensions of such a curve canbe described using a Frenet-Serret frame approach. This method uses a system ofordinary differential equations, driven by the curvature, that describes the evolutionof the path in space [13]. Denoting the curve as r(·), the equations are


drds

(s) = q1(s), (8a)

dq1

ds(s) = κ(s)q2(s), (8b)

dq2

ds(s) =−κ(s)q1(s), (8c)

where s is arclength along the curve, q1(s) is the tangent vector to the curve at s,q2(s) is the normal vector to the curve at s, and κ(s) is the curvature at s (see Figure5). The heading direction at s, denoted θ(s), is defined to be the direction of thetangent vector at that point.

Alternatively, given the heading direction, the vectors q1(s) and q2(s) can bedefined by

q1(s) =(

cosθ(s)sinθ(s)

), q2(s) =

(−sinθ(s)cosθ(s)

). (9)

With a constant curvature, the path defined by (8) is a circle of radius 1κ

. Notethat our scheme assumes the curvature remains constant between crossings of thetip over the sample and thus the sample is modeled as a sequence of arcs of circles.

Fig. 5 Illustration of the local raster scan AFM tip trajectory. Both the path of the tip (solid black)and sample (dashed red) begin at the origin. The sample curve evolves with a constant curvatureof κ = 0.2 from an initial heading direction of 53o. The tip trajectory is shown over one period andwith an amplitude of A = 1.2 units and a spatial frequency of ω = 0.49 radius/unit. The Frenet-Serret frame of the sample curve is shown at s = 0 and at s = 2π/ω .


3.4.2 Defining the tip trajectory

Given the sample path, the tip trajectory, denoted rtip(s), is designed by adding asinusoidal variation to r(·),

rtip(s) = r(s)±Asin(ωs)q2(s), (10)

where A, in units of length, and ω , in units of 1/length, are user parameters thatdefine the amplitude and spatial frequency respectively, of the tip scanning motion.The choice of positive or negative sign in (10) is alternated at each detection of thetransition between sample and substrate (c.f. Section 3.5).

An example of the pattern produced by our scheme is shown in Figure 5. Theunderlying curve (dashed red) begins at the origin with an initial heading directionof θ = 53o and a constant curvature of κ = 0.2. The tip trajectory (solid black)begins at the origin as well and is illustrated over one cycle for a choice of A = 1.2units and ω = 0.49 radius/unit. Also illustrated is the initial Frenet-Serret frame andthe frame at s = π/(2ω).

The tip trajectory in (10) is specified with respect to the arclength of the (esti-mated) path of the sample. In practice, the trajectory must be specified with respectto time. The relationship between arclength and time is given by

vtipt =∫ s

0

√(1−Aκ sin(ωσ))2 +A2ω2 cos2(ωσ)dσ , (11)

where vtip, the speed of the tip, is assumed to be constant. Given the current valueof t, (11) must be solved for s before the tip trajectory can be determined. Detailscan be found in [6].

3.4.3 User-defined parameters

The two user parameters A and ω influence the shape of rtip. A is the amplitudeof the sinusoidal scanning and is analogous to the scan range in the standard rasterimaging procedure. In the local raster scan, however, this range is defined relative tothe path r(·), ensuring the tip sweeps out an image of width A around the sample. Ingeneral, the size of A should be chosen to ensure that the tip trajectory fully coversthe width of the sample to be imaged. In practice, one typically knows the type ofsample to be imaged. Thus, the choice of A can be informed by a priori knowledgeof the theoretical width of the sample. We note, however, that the algorithm can bemodified to replace a fixed value of A with a detection-based scheme in which itsvalue is determined based on an evolving estimate of the sample width.

The parameter ω defines the spatial frequency of the tip trajectory and is analo-gous to the resolution of the image. Increasing the value of ω decreased the distancebetween crossings of the tip trajectory and the sample path, leading to an increasein resolution in the sample data.


It should be noted that error in the measured transition locations, in the headingdirection, or in the curvature, or a breakdown in the assumptions such as that of con-stant curvature between transition locations can lead to loss of tracking. The choiceof A and ω influence the robustness of the algorithm to such errors. Guidelines onchoosing A and ω to ensure tracking in the face of error in the curvature estimatecan be found in [7] and a more general approach is the subject of ongoing research.

3.5 Summary

Given an initial position of the sample path, r(0), an initial heading direction, θ(0),and a value of the curvature κ , the predicted path of the sample is calculated from(8) under the assumption of constant curvature. Based on this prediction, the tiptrajectory is determined from (10) (with a choice of positive or negative sign in theequation). This trajectory is sent as a reference trajectory to the scanner controllerto drive the tip along this path.

When a detection of a transition is made, new values of θ and κ are generated andfiltered. These values, as well as the new measurement of the position of the sample,are passed to the trajectory determination block. These measurements are used tocorrect the current estimate of the predicted sample path, leading to an instant jumpboth in the value of r(s) (to the new transition position, p[k]) and to the value of θ(s)(to the new estimate θ [k]). From (10), these discontinuities lead to a discontinuity inthe designed tip trajectory. To minimize the size of this discontinuity, upon detectionof a transition the arclength parameter is reset to 0 and the sign in (10) is switched.This is summarized in the following algorithm.

Algorithm 1 (Local raster-scan)

0. Initialize k = 0.1. Set s = 0.2. Set the initial conditions r(0) = p[k], θ(0) = θ [k], κ = κ[k].3. Predict the sample path r(·) and the desired tip trajectory rtip(·) from (8) and

(10), respectively.4. Steer the tip according to rtip(·).5. Monitor the measured z data until a transition is detected.6. Set p[k +1] to the detected transition position,7. Estimate θ [k + 1] and κ[k + 1] from (2) and (3), respectively. Filter using the

Kalman filter in (7) to produce θ [k +1] and κ[k +1].8. k← k +1. Go to 1.

We note that in practice the scheme would be initialized either based on a pre-liminary, low-resolution raster-scan, or in an automatic fashion. Auto-initializationcould proceed by the following approach. A raster pattern is followed until a tran-sition is detected. The tip is then scanned along a small circle to find a secondtransition. This pair of transition locations defines both the initial position p[0] andthe initial heading direction θ [0]. The initial curvature is set to 0.


4 Simulation experiments

To illustrate the algorithm we performed simulation experiments using data acquiredfrom a standard raster-scan image of DNA. A solution of λ−DNA was purchasedfrom New England BioLabs (Ipswich,MA, USA) and diluted to a concentration of1.25 µg/mL in 5 mM NICl2 and 50 mM HEPES buffer, adjusted to a pH of 6.8.A small droplet was placed on freshly cleaved mica and incubated for five min-utes. The mica surface was then gently rinsed with deinoized water to wash awayunbounded DNA and, finally, dried in air.

Imaging was performed in air using an Asylum Research MFP 3-D with anOlympus AC-240 cantilever operating in intermittent contact mode. The line scanrate was set to 1 Hz, the image resolution to 256 by 256 pixels, and the image areato a square with 400 nm sides. The resulting height image is shown in Fig. 6. Notethat for the given imaging parameters, the tip speed is approximately 800 nm/secand that the image took 4 minutes and 12 seconds to acquire. The measured heightof the DNA of approximately 0.8 nm is within the expected range (see, e.g. [12]).The image also contains residual salt islands with heights of about 0.1 nm.

Fig. 6 Height retrace image of λ−DNA sample acquired in air using raster scanning. Image sizeis 400 by 400 nm and has a 256 by 256 resolution. Total scan time is 4 minutes and 12 seconds.Measured DNA height is 0.8 nm.

The local raster scan algorithm was implemented in Matlab and executed usingthe measurements from the raster scan image in Fig. 6 as the z−measurement outputfrom the AFM block (Fig. 1). Overlaid on Fig. 6 is a box illustrating the region overwhich the local raster scan algorithm was applied. The scan was begun from the


upper right corner and proceeded along the DNA until the sharp turn at the middle ofthe left edge of the box. Note that due to the limited resolution in the original rasterdata, the local raster algorithm was unable to accommodate the large curvature atthe sharp turn. In practice, the spatial frequency ω could be set sufficiently large forthis turn to be tracked.

In the simulations, a simple threshold detector at a height of 0.36 nm was usedto detect sample-substrate transitions. The tip velocity was set to 12.5 µm/sec.Through trial and error, the input and measurement noise parameters in the model(5) were set to be zero mean with covariances of

Σw =(

0.0685 00 0.0009

), Σv =

(1.10 0

0 0.250

)(12)

Note that the discrete nature of the raster scan image led to increased measurementnoise in the detection of the crossing positions and therefore in the measurement ofthe curvature and heading direction.

(a) A = 12 nm, ω = 0.625 rad/nm (b) A = 12 nm, ω = 1.256 radn/nm

(c) A = 15, nm ω = 0.625 rad/nm (d) A = 15 nm, ω = 1.256 rad/nm

Fig. 7 Simulation results: local raster scan on a DNA sample. All units are in nm. The top imagesused an amplitude of A = 12 nm; bottom images used A = 15 nm. Left images used ω = 0.625rad/nm; right images used ω = 1.26 rad/nm. The resulting trajectory of the tip is superimposed onthe original raster scan image for comparison.


To ensure that a region surrounding the DNA was sampled by the tip and toillustrate the effect of the parameter A, the amplitude was set to either A = 12 nm orA = 15 nm, both substantially larger than the DNA width in the image. To illustratethe effect of changing the spatial frequency, scans were performed with ω = 0.628rad/nm and ω = 1.26 rad/nm, corresponding to nominal step sizes along the DNA of5 nm and 2.5 nm, respectively. The results of four scans with these different choicesof A and ω are shown in Fig. 7(d) by superimposing the resulting tip trajectory on theoriginal raster image. Note that the algorithm did not have access to the full imagebut was simply sampling from the raster-scan data set to drive the local raster-scantrajectory.

In Fig. 8(a,b) we show the raw and filtered estimates of the heading directionand curvature along the scan trajectory of Fig. 7(d). The noise on the measurementof κ is clearly much larger than in θ , due in part to the higher-order numericalderivative employed in transforming the crossing positions into curvature estimatesas compared to heading direction estimates. The discrete nature of the underlyingdata set increased the noise somewhat with respect to what would be expected in atrue continuous sample.

(a) θ filter result (b) κ filter result

Fig. 8 Results of estimation and filtering of the heading direction and curvature measurementsalong the trajectory of the simulated run shown in Fig. 7(d). The large noise on κ relative to thatof θ is indicative both of the numerical derivative nature of the estimator as well as of the discretenature of the underlying data set used in the simulations.

The simulation lacked explicit models of the dynamics, capturing them roughlythrough the use of an actual raster-image. As a result, a direct comparison of thetime to capture the data between the raster scan image and the non-raster approachis not useful. One can assume, however, that since the same machine would be usedfor both scans, the tip speed would be the same. Comparing the total path lengthtraveled by the tip, then, is a good estimate of the relative performance. Considerthe raster scan image, limited to the boxed area in Fig. 6. The reduced scan area is233 nm× 307 nm with a resolution of 143× 197 pixels. Since the tip must performa trace and retrace in each line, the total distance traveled is 87.9 µm. The


One way to show efficiency of local raster scanning is to directly compare thetotal scan time for local raster scanning to raster scanning. However, this comparisonwould not have ruled out the difference in tip traveling velocity, thus we compare thetotal scan length for both algorithm instead. Among the four examples shown for thenon-raster, the choice with the largest values of A and ω , shown in Fig. 7(d), led tothe maximum distance traveled, with a total value of only 7.17 µm. The result is anorder of magnitude reduction in total path length and therefore an (expected) orderof magnitude reduction in imaging time. It should be noted that this comparisonholds true regardless of the hardware; should a high-speed AFM be used that allowsfor much higher tip speeds while still producing high-quality images, the same orderof magnitude reduction would be achieved by the non-raster approach.

5 Conclusion and future work

We have presented a non-raster scanning algorithm for AFM systems designed toimage biopolymers and other string-like samples. The algorithm reduces the overallscanned area by automatically tracking the sample and reduces the total scan timeby scanning the area of interest only. The essential concept is to close the high-levelloop in the AFM and, as a result, the method is complementary to current high-speedAFM schemes. The scheme was illustrated through a simulation involving true dataas captured by a raster-scan image.

With the foundation in place, we are pursuing several lines of inquiry in additionto physical implementation. Among the most important to the application of themethod are the development of effective techniques for choosing the parameters inthe Kalman filter and methods for choosing the values of A and ω so as to ensurethe entire sample is imaged (under appropriate assumptions).

Acknowledgements This work was supported in part by the National Science Foundation undergrant no. CMMI-0845742.

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Non-raster scanning in atomic force microscopy for high-speed … · 2010. 9. 13. · Non-raster scanning in atomic force microscopy for high-speed imaging of biopolymers Peter I