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# Motion microscopy for visualizing and quantifying small motions

Edited by William H. Press, University of Texas at Austin, Austin, TX, and approved August 22, 2017 (received for review March 5, 2017)

## Significance

Humans have difficulty seeing small motions with amplitudes below a threshold. Although there are optical techniques to visualize small static physical features (e.g., microscopes), visualization of small dynamic motions is extremely difficult. Here, we introduce a visualization tool, the motion microscope, that makes it possible to see and understand important biological and physical modes of motion. The motion microscope amplifies motions in a captured video sequence by rerendering small motions to make them large enough to see and quantifies those motions for analysis. Amplification of these tiny motions involves careful noise analysis to avoid the amplification of spurious signals. In the representative examples presented in this study, the visualizations reveal important motions that are invisible to the naked eye.

## Abstract

Although the human visual system is remarkable at perceiving and interpreting motions, it has limited sensitivity, and we cannot see motions that are smaller than some threshold. Although difficult to visualize, tiny motions below this threshold are important and can reveal physical mechanisms, or be precursors to large motions in the case of mechanical failure. Here, we present a “motion microscope,” a computational tool that quantifies tiny motions in videos and then visualizes them by producing a new video in which the motions are made large enough to see. Three scientific visualizations are shown, spanning macroscopic to nanoscopic length scales. They are the resonant vibrations of a bridge demonstrating simultaneous spatial and temporal modal analysis, micrometer vibrations of a metamaterial demonstrating wave propagation through an elastic matrix with embedded resonating units, and nanometer motions of an extracellular tissue found in the inner ear demonstrating a mechanism of frequency separation in hearing. In these instances, the motion microscope uncovers hidden dynamics over a variety of length scales, leading to the discovery of previously unknown phenomena.

Motion microscopy is a computational technique to visualize and analyze meaningful but small motions. The motion microscope enables the inspection of tiny motions as optical microscopy enables the inspection of tiny forms. We demonstrate its utility in three disparate problems from biology and engineering: visualizing motions used in mammalian hearing, showing vibration modes of structures, and verifying the effectiveness of designed metamaterials.

The motion microscope is based on video magnification (1⇓⇓–4), which processes videos to amplify small motions of any kind in a specified temporal frequency band. We extend the visualization produced by video magnification to scientific and engineering analysis. In addition to visualizing tiny motions, we quantify both the object’s subpixel motions and the errors introduced by camera sensor noise (5). Thus, the user can see the magnified motions and obtain their values, with variances, allowing for both qualitative and quantitative analyses.

The motion microscope characterizes and amplifies tiny local displacements in a video by using spatial local phase. It does this by transforming the captured intensities of each frame’s pixels into a wavelet-like representation where displacements are represented by phase shifts of windowed complex sine waves. The representation is the complex steerable pyramid (6), an overcomplete linear wavelet transform, similar to a spatially localized Fourier transform. The transformed image is a sum of basis functions, approximated by windowed sinusoids (Fig. S1), that are simultaneously localized in spatial location (

To amplify motions, we compute the unwrapped phase difference of each coefficient of the transformed image at time

We estimate motions under the assumption that there is a single, small motion at each spatial location. In this case, each coefficient’s phase difference, *Relation Between Local Phase Differences and Motions*). The reliability of spatial local phase varies across scale and orientations, in direct proportion to the coefficient’s amplitude (e.g., coefficients for basis functions orthogonal to an edge are more reliable than those along it) (Fig. S3 and *Low-Amplitude Coefficients Have Noisy Phase*). We combine information about the motion from multiple orientations by solving a weighted least squares problem with weights equal to the amplitude squared. The result is a 2D motion field. This processing is accurate, and we provide comparisons to other algorithms and sensors (Fig. 1, * Synthetic Validation*, and Figs. S4 and S5).

For a still camera, the sensitivity of the motion microscope is mostly limited by local contrast and camera noise—fluctuations of pixel intensities present in all videos (5). When the video is motion-magnified, this noise can lead to spurious motions, especially at low-contrast edges and textures (Fig. S6). We measure motion noise level by computing the covariance matrix of each estimated motion vector. Estimating this directly from the input video is usually impossible, because it requires observing the motions without noise. We solve this by creating a simulated noisy video with zero motion, replicating a static frame of the input video and adding realistic, independent noise to each frame. We compute the sample covariance of the estimated motion vectors in this simulated video (Fig. S7 and *Noise Model and Creating Synthetic Video*). We show analytically, and via experiments in which the motions in a temporal band are known to be zero, that these covariance matrices are accurate for real videos (*Analytic Justification of Noise Analysis* and Figs. S8 and S9). We also analyze the limits of our technique by comparing to a laser vibrometer and show that, with a Phantom V-10 camera, at a high-contrast edge, the smallest motion we can detect is on the order of 1/100th of a pixel (Fig. 1 and Fig. S4).

## Relation Between Local Phase Differences and Motions

Fleet and Jepson have shown that contours of constant phase in image subbands such as those in the complex steerable pyramid approximately track the motion of objects in a video (7). We make a similar phase constancy assumption, in which the following equation relates the phase of the frame at time

Fleet has shown that the spatial gradients of the local phase,

## Noise Model and Creating Synthetic Video

We adopt a signal-dependent noise model, in which each pixel is contaminated with spatially independent Gaussian noise with variance

The noise level function

With *A*, we take a frame *B*).

We use the same simulation to create synthetic videos with which to estimate the covariance matrix of the motion vectors.

We quantify the noise in the motion vectors by estimating their covariance matrices **S21** and Fig. S7 *A*–*C*).

We estimate the motions in *D*) using our technique with spatial smoothing, but without temporal filtering, which we handle in a later step. This results in a set of 2D motion vectors *E*, we show these components, the variances of the horizontal and vertical components of the motion, and their covariance.

The motion

## Analytic Justification of Noise Analysis

We analyze only the case when the amplitudes at a pixel in all subbands are large (

We reproduce the linearization of phase constancy equation (Eq. **S3**) with noise terms added to the phase variations (

The motion estimate **S24**), and our assumption about the amplitudes being large means

We split

From Eq. **S15**, we know that **S26** is that the motion covariance matrix will be linearly proportional to the variance of the sensor noise.

## Results and Discussion

We applied the motion microscope to several problems in biology and engineering. First, we used it to reveal one component of the mechanics of hearing. The mammalian cochlea is a remarkable sensor that can perform high-quality spectral analysis to discriminate as many as 30 frequencies in the interval of a semitone (8). These extraordinary properties of the hearing organ depend on traveling waves of motion that propagate along the cochlear spiral. These wave motions are coupled to the extremely sensitive sensory receptor cells via the tectorial membrane, a gelatinous structure that is 97% water (9).

To better understand the functional role of the tectorial membrane in hearing, we excised segments of the tectorial membrane from a mouse cochlea and stimulated it with audio frequency vibrations (Movie S1 and Fig. 2*A*). Prior work suggested that motions of the tectorial membrane would rapidly decay with distance from the point of stimulation (10). The unprocessed video of the tectorial membrane appeared static, making it difficult to verify this. However, when the motions were amplified 20 times, waves that persisted over hundreds of micrometers were revealed (Movie S1 and Fig. 2 *B*–*E*).

Subpixel motion analysis suggests that these waves play a prominent role in determining the sensitivity and frequency selectivity of hearing (11⇓⇓–14). Magnifying motions has provided new insights into the underlying physical mechanisms of hearing. Ultimately, the motion microscope could be applied to see and interpret the nanoscale motions of a multitude of biological systems.

We also applied the motion microscope to the field of modal analysis, in which a structure’s resonant frequencies and mode shapes are measured to characterize its dynamic behavior (15). Common applications are to validate finite element models and to detect changes or damage in structures (16). Typically, this is done by measuring vibrations at many different locations on the structure in response to a known input excitation. However, approximate measurements can be made under operational conditions assuming broadband excitation (17). Contact accelerometers have been traditionally used for modal analysis, but densely instrumenting a structure can be difficult and tedious, and, for light structures, the accelerometers’ mass can affect the measurement.

The motion microscope offers many advantages over traditional sensors. The structure is unaltered by the measurement, the measurements are spatially dense, and the motion-magnified video allows for easy interpretation of the motions. While only structural motions in the image plane are visible, this can be mitigated by choosing the viewpoint carefully.

We applied the motion microscope to modal analysis by filming the left span of a suspension bridge from 80 m away (Fig. 3*A*). The central span was lowered and impacted the left span. Despite this, the left span looks completely still in the input video (Fig. 3*B*). Two of its modal shapes are revealed in Movie S2 when magnified 400*C* and *D*, we show time slices from the motion-magnified videos, displacements versus time at three points, and the estimated noise standard deviations. We also used accelerometers to measure the motions of the bridge at two of those points (Fig. 3*B*). The motion microscope matches the accelerometers within error bars. In a second example, we show the modal shapes of a pipe after it is struck with a hammer (*Modal Shapes of a Pipe*, Fig. S10, and Movie S3).

In our final example, we used the motion microscope to verify the functioning of elastic metamaterials, artificially structured materials designed to manipulate and control the propagation of elastic waves. They have received much attention (18) because of both their rich physics and their potential applications, which include wave guiding (19), cloaking (20), acoustic imaging (21), and noise reduction (22). Several efforts have been made to experimentally characterize the elastic wave phenomena observed in these systems. However, as the small amplitude of the propagating waves makes it impossible to directly visualize them, the majority of the experimental investigations have focused on capturing the band gaps through the use of accelerometers, which only provide point measurements. Visualizing the mechanical motions everywhere in the metamaterials has only been possible using expensive and highly specialized setups like scanning laser vibrometers (23).

We focus on a metamaterial comprising an elastic matrix with embedded resonating units, which consists of copper cores connected to four elastic beams (24). Even when vibrated, this metamaterial appears stationary, making it difficult to determine if the metamaterial is functioning correctly (Movies S4 and S5). Previously, these miniscule vibrations were measured with two accelerometers (24). This method only provides point measurements, making it difficult to verify the successful attenuation of vibrations. We gain insight and understanding of the system by visually amplifying its motion.

The elastic metamaterial was forced at two frequencies, 50 Hz and 100 Hz, and, in each case, it was filmed at 500 frames per second (FPS) (Fig. 4*A*). The motions in 20-Hz bands around the forcing frequencies were amplified, revealing that the metamaterial functions as expected (24), passing 50-Hz waves and rapidly attenuating 100-Hz waves (Movies S4 and S5). We also compared our results with predictions from a finite element analysis simulation (Fig. 4 *B* and *C*). In Fig. 4*D*, we show heatmaps of the estimated displacement amplitudes overlaid on the motion-magnified frames. We interpolated displacements into textureless regions, which had noisy motion estimates. The agreement between the simulation (Fig. 4*C*) and the motion microscope (Fig. 4*D*) demonstrates the motion microscope’s usefulness in verifying the correct function of the metamaterial.

## Modal Shapes of a Pipe

We made a measurement of a pipe being struck by a hammer, viewed end on by a camera, to capture its radial–circumferential vibration modes. A standard *A*). Fig. S10 *B*–*F* shows frames from the motion-magnified videos for different resonant frequencies showing the mode shapes, a comparison of the quantitatively measured mode shapes with the theoretically derived mode shapes, and the displacement vs. time of the specific frequency band and the estimated noise SD. The tiny modal motions are seen clearly. Obtaining vibration data with traditional sensors with the same spatial density would be extremely difficult, and accelerometers placed on the pipe would alter its resonant frequencies.

This sequence also demonstrates the accuracy of our noise analysis. The noise standard deviations show that the detected motions before impact, when the pipe is stationary, are likely spurious.

## Synthetic Validation

We validate the accuracy of our motion estimation on a synthetic dataset and compare its accuracy to Ncorr, a digital image correlation technique (30) used by mechanical engineers (31). In this experiment, we did not use temporal filtering.

We created a synthetic dataset of frame pairs with known ground-truth motions between them. We took natural images from the frames of real videos (Fig. S5*A*) and warped them according to known motion fields using cubic b-spline interpolation (32). Sample motions fields, shown in Fig. S5*B*, were produced by Gaussian-blurring IID Gaussian random variables. We used Gaussian blurs with SDs, ranging from zero (no filtering) to infinite (a constant motion field). We also varied the RMS amplitude of the motion fields from 0.001 pixels to three pixels. For each set of motion field parameters, we sampled five different motion fields to produce a total of 155 motion fields with different amplitudes and spatial coherence. To test the accuracy of the algorithms rather than their sensitivity to noise, no noise was added to the image pairs.

We ran our motion estimation technique and Ncorr on each image pair. We then computed the mean absolute difference between the estimated and ground-truth motion fields. Then, for each set of motion field parameters, we averaged the mean absolute differences across image pairs and divided the result by the RMS motion amplitude to make the errors comparable over motion sizes. The result is the average relative error as a percentage of RMS motion amplitude (Fig. S5*C*).

Both Ncorr and our method perform best when the motions are spatially coherent (filter standard deviations greater than 10 pixels) with relative errors under 10%. This reflects the fact that both methods assume the motion field is spatially smooth. Across motion sizes, our method performs best for subpixel motions (5% relative error). This is probably because we assume that the motions are small when we linearize the phase constancy equation (Eq. **S3**). Ncorr has twice the relative error (10%) for the same motion fields.

The relative errors reported in Fig. S5*C* are computed over all pixels, including those that are in smooth, textureless regions where it is difficult to estimate the motions. If we restrict the error metric to only take into account pixels at edges and corners, the average relative errors for small (

We generated synthetic images that are slight translations of each other and added Gaussian noise to the frames (Fig. S8*A*). For each translation amount, we compute the motion between the two frames over 4,000 runs. We compute the sample covariance matrix over the runs as a measure of the ground-truth noise level. We also used our noise analysis to estimate the covariance matrix at the points denoted in red.

The off-diagonal term of the covariance matrix should be zero for the synthetic frames in Fig. S8*A*. For both examples, it is within *B*).

The relative errors of the horizontal and vertical variances vs. translation (Fig. S8 *C* and *D*) are less than

## Low-Amplitude Coefficients Have Noisy Phase

Each frame of the input video

Filters

Suppose the observed video

The transformed representation has response

We suppress the indices **S5**, the noiseless and noisy phases are given by*A*–*E*.

For these high-amplitude points, we compute the variance of the phase of a coefficient,**S10**.

When the amplitude is low compared with the noise level (**S10** is not accurate. In this case, *E*, red point).

## Conclusion

Small motions can reveal important dynamics in a system under study, or can foreshadow large-scale motions to come. Motion microscopy facilitates their visualization, and has been demonstrated here for motion amplification factors from 20

## Materials and Methods

### Quantitative Motion Estimation.

For every pixel at location

To increase the signal-to-noise ratio, we assume the motion field is constant in a small window around each pixel. This gives additional constraints from neighboring pixels, weighted by both their amplitude squared and the corresponding value in a smoothing kernel **3**. To handle temporal filtering, we replace the local phase variations

We use a four-orientation complex steerable pyramid specified by Portilla and Simoncelli (25). We use only the two highest-frequency scales of the complex steerable pyramid, for a total of eight subbands. We use a Gaussian spatial smoothing kernel with a SD of 3 pixels and a support of

### Noise Model and Creating Synthetic Video.

We estimate the noise level function (26) of a video. We apply derivative of Gaussian filters to the image in the

### Estimating Covariance Matrices of Motion Vectors.

For an input video

The temporal filter reduces noise and decreases the covariance matrix. Oppenheim and Schafer (27) show that a signal with independent and identically distributed (IID) noise of variance

### Comparison of Our Motion Estimation to a Laser Vibrometer.

We compare the results of our motion estimation algorithm to that of a laser vibrometer, which measures velocity using Doppler shift (28). In the first experiment, a cantilevered beam was shaken by a mechanical shaker at *A*), but we did not use it in this experiment.

We used our motion estimation method to compute the horizontal displacement of the marked, red point on the left side of the accelerometer from the video (Fig. 1*A*). We applied a temporal band-stop filter to remove motions between 67 Hz and 80 Hz that corresponded to camera motions caused by its cooling fan’s rotation. The laser vibrometer signal was integrated using discrete, trapezoidal integration. Before integration, both signals were high-passed above 2.5 Hz to reduce low-frequency noise in the integrated vibrometer signal. The motion signals from each video were manually aligned. For one video (exposure, *B*–*D*). They agree remarkably well, with higher modes well aligned and a correlation of

To show the sensitivity of the motion microscope, we plot the correlation of our motion estimate and the integrated velocities from the laser vibrometer vs. motion size (RMS displacement). Because the motion’s average size varies over time, we divide each video’s motion signal into eight equal pieces and plot the correlations of each piece in each video in Fig. S4 *E* and *F*. For RMS displacements on the order of

As expected, correlation increases with focal length and excitation magnitude, two things that positively correlate with motion size (in pixels) (Fig. S4 *G* and *H*). The correlation also increases with exposure, because videos with lower exposure times are noisier (Fig. S4*I*).

### Filming Bridge Sequence.

The bridge was filmed with a monochrome Point Gray Grasshopper3 camera (model GS3-U3-23S6M-C) at 30 FPS with a resolution of 800

The accelerometer data were doubly integrated using trapezoidal integration to displacement. In Fig. 3 *C* and *D*, both the motion microscope displacement and the doubly integrated acceleration were band-passed with a first-order band-pass Butterworth filter with the specified parameters.

### Motion Field Interpolation.

In textureless regions, it may not be possible to estimate the motion at all, and, at one-dimensional structures like edges, the motion field will only be accurate in the direction perpendicular to the edge. These inaccuracies are reflected in the motion covariance matrix. We show how to interpolate the motion field from accurate regions to inaccurate regions, assuming that adjacent pixels have similar motions.

We minimize the following objective function:

In Fig. 4*D*, we produce the color overlays by applying the above processing to the estimated motion field with

### Finite Element Analysis of Acoustic Metamaterial.

We use Abaqus/Standard (29), a commercial finite-element analyzer, to simulate the metamaterial’s response to forcing. We constructed a 2D model with 37,660 nodes and 11,809 eight-node plane strain quadrilateral elements (Abaqus element type CPE8H). We modeled the rubber as Neo-Hookean, with shear modulus

### Validation of Noise Analysis with Real Video Data.

We took a video of an accelerometer attached to a beam (Fig. S9*A*). We used the accelerometer to verify that the beam had no motions between 600 Hz and 700 Hz (Fig. S9*B*). We then estimated the in-band motions from a video of the beam. Because the beam is stationary in this band, these motions are entirely due to noise, and their temporal sample covariance gives us a ground-truth measure of the noise level (Fig. S9*C*). We used our simulation with a signal-dependent noise model to estimate the covariance matrix from the first frame of the video, the specific parameters of which are shown in Fig. S9*D*. The resulting covariance matrices closely match the ground truth (Fig. S9 *E* and *F*), showing that our simulation can accurately estimate noise level and error bars.

We also verify that the signal-dependent noise model performs better than the simpler constant variance noise model, in which noise is IID. The result of the constant noise model simulation produced results that are much less accurate than the signal-dependent noise model (Fig. S9 *G* and *H*).

In Fig. S9, we only show the component of the covariance matrix corresponding to the direction of least variance, and only at points corresponding to edges or corners.

## Using the Motion Microscope and Limitations

The motion microscope works best when the camera is stable, such as when mounted on a tripod. If the camera is handheld or the tripod is not sturdy, the motion microscope may only detect camera motions instead of subject motions. The easiest way to solve this problem is to stabilize the camera. If the camera motions are caused by some periodic source, such as camera fans, that occur at a temporal frequency that the subject is not moving at, they can be removed with a band-stop filter.

If adjacent individual pixels have different motions, it is unlikely that the motion microscope will be able to discern or differentiate them. The motion microscope assumes motions are locally constant, and it won’t be able to properly process videos with high spatial frequency motions. The best way to solve this problem is to increase the resolving power or optical zoom of the imaging system. If noise is not a concern, reducing the width of the spatial Gaussian used to smooth the motions may also help.

It is not possible for the motion microscope to quantify the movement of textureless regions or the full 2D movement of 1D edges. However, the motion covariance matrices produced by the motion microscope will alert the user as to when this is happening. The visualization will still be reasonable, as it is not possible to tell if a textureless region was motion-amplified in the wrong way.

Saturated, pure white pixels may pose a problem for the motion microscope. Slight color changes are the underlying signal used to detect tiny motions. For pixels with clipped intensities, this signal will be missing, and it may not be possible to detect motion. The best ways to mitigate this problem are to reduce the exposure time, change the position of lights, or use a camera with a higher dynamic range.

The motion microscope only quantifies lateral, 2D motions in the imaging plane. Motions of the subject toward or away from the camera will not be properly reported. For a reasonable camera, these 2D motions can be multiplied by a constant to convert from pixels to millimeters or other units of interest. However, if the lens has severe distortion, the constant will vary across the scene. If quantification of the motions is important, the images will need to be lens distortion-corrected before the motion microscope is applied.

The motion microscope visualization is unable to push pixels beyond the support of the basis functions of the steerable pyramid. In practice, this results in a maximum amplified motion of around four pixels. If the amplification is high enough to push an image feature past this, ringing artifacts may occur.

## Acknowledgments

We thank Professor Erin Bell and Travis Adams at University of New Hampshire and New Hampshire Department of Transportation for their assistance with filming the Portsmouth lift bridge. This work was supported, in part, by Shell Research, Quanta Computer, National Science Foundation Grants CGV-1111415 and CGV-1122374, and National Institutes of Health Grant R01-DC00238.

## Footnotes

↵

^{1}Present address: Google Research, Google Inc. Mountain View, CA 94043.- ↵
^{2}To whom correspondence should be addressed. Email: billf{at}mit.edu.

Author contributions: N.W., J.G.C., J.B.S., D.W., M.R., R.G., D.M.F., O.B., S.H.K., K.B., F.D., and W.T.F. designed research; N.W., J.G.C., J.B.S., D.W., R.G., P.W., S.S., S.H.K., and W.T.F. performed research; N.W., J.G.C., J.B.S., and D.W. analyzed data; and N.W., J.G.C., J.B.S., D.W., R.G., D.M.F., O.B., P.W., S.S., S.H.K., K.B., F.D., and W.T.F. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1703715114/-/DCSupplemental.

- Copyright © 2017 the Author(s). Published by PNAS.

This is an open access article distributed under the PNAS license.

## References

- ↵.
- Liu C,
- Torralba A,
- Freeman WT,
- Durand F,
- Adelson EH

- ↵.
- Wu HY, et al.

- ↵.
- Wadhwa N,
- Rubinstein M,
- Durand F,
- Freeman WT

- ↵.
- Wadhwa N,
- Rubinstein M,
- Durand F,
- Freeman WT

- ↵.
- Nakamura J

- ↵.
- Simoncelli EP,
- Freeman WT

- ↵.
- Fleet DJ,
- Jepson AD

- ↵.
- Dallos P,
- Fay RR

*The Cochlea*, Springer Handbook of Auditory Research (Springer Science, New York), Vol 8. - ↵
- ↵
- ↵.
- Sellon JB,
- Farrahi S,
- Ghaffari R,
- Freeman DM

- ↵.
- Ghaffari R,
- Aranyosi AJ,
- Freeman DM

- ↵
- ↵
- ↵.
- Ewins DJ

*Modal Testing: Theory and Practice*, Engineering Dynamics Series (Res Stud, Baldock, UK) Vol 6. - ↵
- ↵.
- Hermans L,
- van der Auweraer H

- ↵
- ↵.
- Khelif A,
- Choujaa A,
- Benchabane S,
- Djafari-Rouhani B,
- Laude V

- ↵.
- Cummer S,
- Schurig D

- ↵.
- Spadoni A,
- Daraio C

- ↵
- ↵.
- Jeong S,
- Ruzzene M

- ↵
- ↵
- ↵.
- Liu C,
- Freeman WT,
- Szeliski R,
- Kang SB

- ↵.
- Oppenheim AV,
- Schafer RW

- ↵.
- Durst F,
- Melling A,
- Whitelaw JH

*Principles and Practice of Laser-Doppler Anemometry*. NASA STI/Recon Technical Report A (NASA, Washington, DC), vol 76. - ↵Hibbett, Karlsson, Sorensen (1998).
*ABAQUS/Standard: User’s Manual*(Hibbitt, Karlsson & Sorensen, Pawtucket, RI) Vol 1. - ↵.
- Blaber J,
- Adair B,
- Antoniou A

- ↵.
- Xu J,
- Moussawi A,
- Gras R,
- Lubineau G

- ↵.
- Unser M

- ↵.
- Fleet DJ

- ↵.
- Hasinoff SW,
- Durand F,
- Freeman WT

- ↵.
- Lucas BD,
- Kanade T

- ↵
- ↵.
- Wadhwa N, et al.

- .
- Wachel JC,
- Morton SJ

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- Abstract
- Relation Between Local Phase Differences and Motions
- Noise Model and Creating Synthetic Video
- Analytic Justification of Noise Analysis
- Results and Discussion
- Modal Shapes of a Pipe
- Synthetic Validation
- Low-Amplitude Coefficients Have Noisy Phase
- Conclusion
- Materials and Methods
- Using the Motion Microscope and Limitations
- Acknowledgments
- Footnotes
- References

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