Abstract

This article introduces a method to extract the speed and density of microparticles in real time at several kHz using an asynchronous event-based camera mounted on a full-field optical coherence tomography (FF-OCT) setup. These cameras detect significant amplitude changes, allowing scene-driven acquisitions. They are composed of an array of autonomously operating pixels. Events are triggered when an illuminance change at the pixel level is significant at 1μs time precision. The event-driven FF-OCT algorithm relies on a time-based optical flow computation to operate directly on incoming events and updates the estimation of velocity, direction and density while reducing both computation and data load. We show that for fast moving microparticles in a range of 0.4 – 6.5mm/s, the method performs faster and more efficiently than existing techniques in real time. The target application of this work is to evaluate erythrocyte dynamics at the microvascular level in vivo with a high temporal resolution.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2017 (1)

J. You, A. Li, C. Du, and Y. Pan, “Volumetric Doppler angle correction for ultrahigh-resolution optical coherence Doppler tomography,” Appl. Phys. Lett. 110, 011102 (2017).
[Crossref] [PubMed]

2015 (1)

X. Lagorce, C. Meyer, S. H. Ieng, D. Filliat, and R. Benosman, “Asynchronous event-based multikernel algorithm for high-speed visual features tracking,” IEEE T. Neural Netw. 26, 1710–1720 (2015).

2014 (1)

R. Benosman, C. Clercq, X. Lagorce, S. H. Ieng, and C. Bartolozzi, “Event-based visual flow,” IEEE T. Neural Netw. 25, 407–417 (2014).

2013 (1)

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measure- ment of rbc speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 1707–1710 (2013).
[Crossref] [PubMed]

2012 (6)

E. Meijering, O. Dzyubachyk, and I. Smal, “Methods for cell and particle tracking,” Methods Enzymol. 504, 183–200 (2012).
[Crossref] [PubMed]

R. Benosman, S.-H. Ieng, C. Clercq, C. Bartolozzi, and M. Srinivasan, “Asynchronous frameless event-based optical flow,” Neural Netw. 27, 32–37 (2012).
[Crossref]

Z. Ni, C. Pacoret, R. Benosman, S. Ieng, and S. Regnier, “Asynchronous event-based high speed vision for microparticle tracking,” J. Microsc. 245, 236–244 (2012).
[Crossref]

V. J. Srinivasan, H. Radhakrishnan, E. H. Lo, E. T. Mandeville, J. Y. Jiang, S. Barry, and A. E. Cable, “OCT methods for capillary velocimetry,” Biomed. Opt. Express 3, 612–629 (2012).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20, 22262–22277 (2012).
[Crossref] [PubMed]

Y. Jia, J. C. Morrison, J. Tokayer, O. Tan, L. Lombardi, B. Baumann, C. D. Lu, W. Choi, J. G. Fujimoto, and D. Huang, “Quantitative oct angiography of optic nerve head blood flow,” Biomed. Opt. Express 3, 3127–3137 (2012).
[Crossref] [PubMed]

2011 (2)

S.-L. Chen, Z. Xie, P. L. Carson, X. Wang, and L. J. Guo, “In vivo flow speed measurement of capillaries by photoacoustic correlation spectroscopy,” Opt. Lett. 36, 4017–4019 (2011).
[Crossref] [PubMed]

T. Zhu, R. Cheng, and L. Mao, “Focusing microparticles in a microfluidic channel with ferrofluids,” Microfluid. Nanofluid. 11, 695–701 (2011).
[Crossref]

2006 (1)

2004 (2)

2003 (2)

2002 (1)

Audinat, E.

E. Chaigneau, M. Oheim, E. Audinat, and S. Charpak, “Two-photon imaging of capillary blood flow in olfactory bulb glomeruli,” Proc. Natl. Acad. Sci. 100, 13081–13086 (2003).
[Crossref] [PubMed]

Bajraszewski, T.

Barry, S.

Bartolozzi, C.

R. Benosman, C. Clercq, X. Lagorce, S. H. Ieng, and C. Bartolozzi, “Event-based visual flow,” IEEE T. Neural Netw. 25, 407–417 (2014).

R. Benosman, S.-H. Ieng, C. Clercq, C. Bartolozzi, and M. Srinivasan, “Asynchronous frameless event-based optical flow,” Neural Netw. 27, 32–37 (2012).
[Crossref]

Baumann, B.

Y. Jia, J. C. Morrison, J. Tokayer, O. Tan, L. Lombardi, B. Baumann, C. D. Lu, W. Choi, J. G. Fujimoto, and D. Huang, “Quantitative oct angiography of optic nerve head blood flow,” Biomed. Opt. Express 3, 3127–3137 (2012).
[Crossref] [PubMed]

R. Haindl, A. Wartak, W. Trasischker, S. Holzer, B. Baumann, M. Pircher, C. Vass, and C. K. Hitzenberger, “Total retinal blood flow in healthy and glaucomatous human eyes measured with three beam doppler optical coherence tomography,” OSA Technical Digest (online) (Optical Society of America, 2016), paper TTh1B.2 (2016).

Beaurepaire, E.

Benosman, R.

X. Lagorce, C. Meyer, S. H. Ieng, D. Filliat, and R. Benosman, “Asynchronous event-based multikernel algorithm for high-speed visual features tracking,” IEEE T. Neural Netw. 26, 1710–1720 (2015).

R. Benosman, C. Clercq, X. Lagorce, S. H. Ieng, and C. Bartolozzi, “Event-based visual flow,” IEEE T. Neural Netw. 25, 407–417 (2014).

R. Benosman, S.-H. Ieng, C. Clercq, C. Bartolozzi, and M. Srinivasan, “Asynchronous frameless event-based optical flow,” Neural Netw. 27, 32–37 (2012).
[Crossref]

Z. Ni, C. Pacoret, R. Benosman, S. Ieng, and S. Regnier, “Asynchronous event-based high speed vision for microparticle tracking,” J. Microsc. 245, 236–244 (2012).
[Crossref]

Boas, D. A.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measure- ment of rbc speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 1707–1710 (2013).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20, 22262–22277 (2012).
[Crossref] [PubMed]

Boccara, A.-C.

Boccara, C.

Burnett, M.G.

Cable, A. E.

Carson, P. L.

Chaigneau, E.

E. Chaigneau, M. Oheim, E. Audinat, and S. Charpak, “Two-photon imaging of capillary blood flow in olfactory bulb glomeruli,” Proc. Natl. Acad. Sci. 100, 13081–13086 (2003).
[Crossref] [PubMed]

Charpak, S.

E. Chaigneau, M. Oheim, E. Audinat, and S. Charpak, “Two-photon imaging of capillary blood flow in olfactory bulb glomeruli,” Proc. Natl. Acad. Sci. 100, 13081–13086 (2003).
[Crossref] [PubMed]

Chen, S.-L.

Cheng, R.

T. Zhu, R. Cheng, and L. Mao, “Focusing microparticles in a microfluidic channel with ferrofluids,” Microfluid. Nanofluid. 11, 695–701 (2011).
[Crossref]

Choi, W.

Clercq, C.

R. Benosman, C. Clercq, X. Lagorce, S. H. Ieng, and C. Bartolozzi, “Event-based visual flow,” IEEE T. Neural Netw. 25, 407–417 (2014).

R. Benosman, S.-H. Ieng, C. Clercq, C. Bartolozzi, and M. Srinivasan, “Asynchronous frameless event-based optical flow,” Neural Netw. 27, 32–37 (2012).
[Crossref]

Detre, J.A.

Drexler, W.

Du, C.

J. You, A. Li, C. Du, and Y. Pan, “Volumetric Doppler angle correction for ultrahigh-resolution optical coherence Doppler tomography,” Appl. Phys. Lett. 110, 011102 (2017).
[Crossref] [PubMed]

Dubois, A.

Durduran, T.

Dzyubachyk, O.

E. Meijering, O. Dzyubachyk, and I. Smal, “Methods for cell and particle tracking,” Methods Enzymol. 504, 183–200 (2012).
[Crossref] [PubMed]

Fercher, A.F.

Filliat, D.

X. Lagorce, C. Meyer, S. H. Ieng, D. Filliat, and R. Benosman, “Asynchronous event-based multikernel algorithm for high-speed visual features tracking,” IEEE T. Neural Netw. 26, 1710–1720 (2015).

Fujimoto, J. G.

Greenberg, J.H.

Grieve, K.

Guo, L. J.

Haindl, R.

R. Haindl, A. Wartak, W. Trasischker, S. Holzer, B. Baumann, M. Pircher, C. Vass, and C. K. Hitzenberger, “Total retinal blood flow in healthy and glaucomatous human eyes measured with three beam doppler optical coherence tomography,” OSA Technical Digest (online) (Optical Society of America, 2016), paper TTh1B.2 (2016).

Hitzenberger, C. K.

R. Haindl, A. Wartak, W. Trasischker, S. Holzer, B. Baumann, M. Pircher, C. Vass, and C. K. Hitzenberger, “Total retinal blood flow in healthy and glaucomatous human eyes measured with three beam doppler optical coherence tomography,” OSA Technical Digest (online) (Optical Society of America, 2016), paper TTh1B.2 (2016).

Holzer, S.

R. Haindl, A. Wartak, W. Trasischker, S. Holzer, B. Baumann, M. Pircher, C. Vass, and C. K. Hitzenberger, “Total retinal blood flow in healthy and glaucomatous human eyes measured with three beam doppler optical coherence tomography,” OSA Technical Digest (online) (Optical Society of America, 2016), paper TTh1B.2 (2016).

Huang, D.

Ieng, S.

Z. Ni, C. Pacoret, R. Benosman, S. Ieng, and S. Regnier, “Asynchronous event-based high speed vision for microparticle tracking,” J. Microsc. 245, 236–244 (2012).
[Crossref]

Ieng, S. H.

X. Lagorce, C. Meyer, S. H. Ieng, D. Filliat, and R. Benosman, “Asynchronous event-based multikernel algorithm for high-speed visual features tracking,” IEEE T. Neural Netw. 26, 1710–1720 (2015).

R. Benosman, C. Clercq, X. Lagorce, S. H. Ieng, and C. Bartolozzi, “Event-based visual flow,” IEEE T. Neural Netw. 25, 407–417 (2014).

Ieng, S.-H.

R. Benosman, S.-H. Ieng, C. Clercq, C. Bartolozzi, and M. Srinivasan, “Asynchronous frameless event-based optical flow,” Neural Netw. 27, 32–37 (2012).
[Crossref]

Jia, Y.

Jiang, J. Y.

Lagorce, X.

X. Lagorce, C. Meyer, S. H. Ieng, D. Filliat, and R. Benosman, “Asynchronous event-based multikernel algorithm for high-speed visual features tracking,” IEEE T. Neural Netw. 26, 1710–1720 (2015).

R. Benosman, C. Clercq, X. Lagorce, S. H. Ieng, and C. Bartolozzi, “Event-based visual flow,” IEEE T. Neural Netw. 25, 407–417 (2014).

Lecaque, R.

Lee, J.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measure- ment of rbc speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 1707–1710 (2013).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20, 22262–22277 (2012).
[Crossref] [PubMed]

Leitgeb, R.A.

Lesage, F.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measure- ment of rbc speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 1707–1710 (2013).
[Crossref] [PubMed]

Li, A.

J. You, A. Li, C. Du, and Y. Pan, “Volumetric Doppler angle correction for ultrahigh-resolution optical coherence Doppler tomography,” Appl. Phys. Lett. 110, 011102 (2017).
[Crossref] [PubMed]

Li, P.

Lo, E. H.

Lombardi, L.

Lu, C. D.

Luo, Q.

Mandeville, E. T.

Mao, L.

T. Zhu, R. Cheng, and L. Mao, “Focusing microparticles in a microfluidic channel with ferrofluids,” Microfluid. Nanofluid. 11, 695–701 (2011).
[Crossref]

Matolin, D.

C. Posch, D. Matolin, and R. Wohlgenannt, “High-dr frame-free pwm imaging with asynchronous aer intensity encoding and focal-plane temporal redundancy suppression,” in Proceedings of 2010 IEEE International Symposium on Circuits and Systems (ISCAS,2010), pp. 2430–2433.
[Crossref]

Meijering, E.

E. Meijering, O. Dzyubachyk, and I. Smal, “Methods for cell and particle tracking,” Methods Enzymol. 504, 183–200 (2012).
[Crossref] [PubMed]

Meyer, C.

X. Lagorce, C. Meyer, S. H. Ieng, D. Filliat, and R. Benosman, “Asynchronous event-based multikernel algorithm for high-speed visual features tracking,” IEEE T. Neural Netw. 26, 1710–1720 (2015).

Moneron, G.

Morrison, J. C.

Ni, S.

Ni, Z.

Z. Ni, C. Pacoret, R. Benosman, S. Ieng, and S. Regnier, “Asynchronous event-based high speed vision for microparticle tracking,” J. Microsc. 245, 236–244 (2012).
[Crossref]

Oheim, M.

E. Chaigneau, M. Oheim, E. Audinat, and S. Charpak, “Two-photon imaging of capillary blood flow in olfactory bulb glomeruli,” Proc. Natl. Acad. Sci. 100, 13081–13086 (2003).
[Crossref] [PubMed]

Pacoret, C.

Z. Ni, C. Pacoret, R. Benosman, S. Ieng, and S. Regnier, “Asynchronous event-based high speed vision for microparticle tracking,” J. Microsc. 245, 236–244 (2012).
[Crossref]

Pan, Y.

J. You, A. Li, C. Du, and Y. Pan, “Volumetric Doppler angle correction for ultrahigh-resolution optical coherence Doppler tomography,” Appl. Phys. Lett. 110, 011102 (2017).
[Crossref] [PubMed]

Pircher, M.

R. Haindl, A. Wartak, W. Trasischker, S. Holzer, B. Baumann, M. Pircher, C. Vass, and C. K. Hitzenberger, “Total retinal blood flow in healthy and glaucomatous human eyes measured with three beam doppler optical coherence tomography,” OSA Technical Digest (online) (Optical Society of America, 2016), paper TTh1B.2 (2016).

Posch, C.

C. Posch, D. Matolin, and R. Wohlgenannt, “High-dr frame-free pwm imaging with asynchronous aer intensity encoding and focal-plane temporal redundancy suppression,” in Proceedings of 2010 IEEE International Symposium on Circuits and Systems (ISCAS,2010), pp. 2430–2433.
[Crossref]

Radhakrishnan, H.

Regnier, S.

Z. Ni, C. Pacoret, R. Benosman, S. Ieng, and S. Regnier, “Asynchronous event-based high speed vision for microparticle tracking,” J. Microsc. 245, 236–244 (2012).
[Crossref]

Schmetterer, L.

Smal, I.

E. Meijering, O. Dzyubachyk, and I. Smal, “Methods for cell and particle tracking,” Methods Enzymol. 504, 183–200 (2012).
[Crossref] [PubMed]

Srinivasan, M.

R. Benosman, S.-H. Ieng, C. Clercq, C. Bartolozzi, and M. Srinivasan, “Asynchronous frameless event-based optical flow,” Neural Netw. 27, 32–37 (2012).
[Crossref]

Srinivasan, V. J.

Tan, O.

Tokayer, J.

Trasischker, W.

R. Haindl, A. Wartak, W. Trasischker, S. Holzer, B. Baumann, M. Pircher, C. Vass, and C. K. Hitzenberger, “Total retinal blood flow in healthy and glaucomatous human eyes measured with three beam doppler optical coherence tomography,” OSA Technical Digest (online) (Optical Society of America, 2016), paper TTh1B.2 (2016).

Vabre, L.

Vass, C.

R. Haindl, A. Wartak, W. Trasischker, S. Holzer, B. Baumann, M. Pircher, C. Vass, and C. K. Hitzenberger, “Total retinal blood flow in healthy and glaucomatous human eyes measured with three beam doppler optical coherence tomography,” OSA Technical Digest (online) (Optical Society of America, 2016), paper TTh1B.2 (2016).

Wang, J.

Wang, X.

Wartak, A.

R. Haindl, A. Wartak, W. Trasischker, S. Holzer, B. Baumann, M. Pircher, C. Vass, and C. K. Hitzenberger, “Total retinal blood flow in healthy and glaucomatous human eyes measured with three beam doppler optical coherence tomography,” OSA Technical Digest (online) (Optical Society of America, 2016), paper TTh1B.2 (2016).

Wohlgenannt, R.

C. Posch, D. Matolin, and R. Wohlgenannt, “High-dr frame-free pwm imaging with asynchronous aer intensity encoding and focal-plane temporal redundancy suppression,” in Proceedings of 2010 IEEE International Symposium on Circuits and Systems (ISCAS,2010), pp. 2430–2433.
[Crossref]

Wu, W.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measure- ment of rbc speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 1707–1710 (2013).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20, 22262–22277 (2012).
[Crossref] [PubMed]

Xie, Z.

Yodh, A. G.

You, J.

J. You, A. Li, C. Du, and Y. Pan, “Volumetric Doppler angle correction for ultrahigh-resolution optical coherence Doppler tomography,” Appl. Phys. Lett. 110, 011102 (2017).
[Crossref] [PubMed]

Yu, G.

Zawadzki, R.J.

Zeng, S.

Zhang, L.

Zhou, C.

Zhu, B.

Zhu, T.

T. Zhu, R. Cheng, and L. Mao, “Focusing microparticles in a microfluidic channel with ferrofluids,” Microfluid. Nanofluid. 11, 695–701 (2011).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. You, A. Li, C. Du, and Y. Pan, “Volumetric Doppler angle correction for ultrahigh-resolution optical coherence Doppler tomography,” Appl. Phys. Lett. 110, 011102 (2017).
[Crossref] [PubMed]

Biomed. Opt. Express (2)

IEEE T. Neural Netw. (2)

X. Lagorce, C. Meyer, S. H. Ieng, D. Filliat, and R. Benosman, “Asynchronous event-based multikernel algorithm for high-speed visual features tracking,” IEEE T. Neural Netw. 26, 1710–1720 (2015).

R. Benosman, C. Clercq, X. Lagorce, S. H. Ieng, and C. Bartolozzi, “Event-based visual flow,” IEEE T. Neural Netw. 25, 407–417 (2014).

J. Cereb. Blood Flow Metab. (1)

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measure- ment of rbc speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33, 1707–1710 (2013).
[Crossref] [PubMed]

J. Microsc. (1)

Z. Ni, C. Pacoret, R. Benosman, S. Ieng, and S. Regnier, “Asynchronous event-based high speed vision for microparticle tracking,” J. Microsc. 245, 236–244 (2012).
[Crossref]

Methods Enzymol. (1)

E. Meijering, O. Dzyubachyk, and I. Smal, “Methods for cell and particle tracking,” Methods Enzymol. 504, 183–200 (2012).
[Crossref] [PubMed]

Microfluid. Nanofluid. (1)

T. Zhu, R. Cheng, and L. Mao, “Focusing microparticles in a microfluidic channel with ferrofluids,” Microfluid. Nanofluid. 11, 695–701 (2011).
[Crossref]

Neural Netw. (1)

R. Benosman, S.-H. Ieng, C. Clercq, C. Bartolozzi, and M. Srinivasan, “Asynchronous frameless event-based optical flow,” Neural Netw. 27, 32–37 (2012).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Proc. Natl. Acad. Sci. (1)

E. Chaigneau, M. Oheim, E. Audinat, and S. Charpak, “Two-photon imaging of capillary blood flow in olfactory bulb glomeruli,” Proc. Natl. Acad. Sci. 100, 13081–13086 (2003).
[Crossref] [PubMed]

Other (2)

C. Posch, D. Matolin, and R. Wohlgenannt, “High-dr frame-free pwm imaging with asynchronous aer intensity encoding and focal-plane temporal redundancy suppression,” in Proceedings of 2010 IEEE International Symposium on Circuits and Systems (ISCAS,2010), pp. 2430–2433.
[Crossref]

R. Haindl, A. Wartak, W. Trasischker, S. Holzer, B. Baumann, M. Pircher, C. Vass, and C. K. Hitzenberger, “Total retinal blood flow in healthy and glaucomatous human eyes measured with three beam doppler optical coherence tomography,” OSA Technical Digest (online) (Optical Society of America, 2016), paper TTh1B.2 (2016).

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Figures (11)

Fig. 1
Fig. 1

Left: event-based encoding of visual information. A change of the logarithmic intensity generates ON and OFF events if the absolute change in log(I) is superior to τ. Right: the ATIS camera.

Fig. 2
Fig. 2

3D visualisation of event generated over time by the movement of small particles (grey circles moving from right to left). The vertical axis is the time, the other two are the xy plane. The blue dots are both positive and negative events.

Fig. 3
Fig. 3

Schematic representation of the FF-OCT microscope with a microfluidic chip and syringe pump. This setup will be used for all our experiments.

Fig. 4
Fig. 4

Optical flow for particles moving from right to left over a time window of 15ms. Both axes on the left image are the x and y plane. The black and white dots represent negative and positive events respectively. On the right are two zooms on particles with their corresponding optical flows.

Fig. 5
Fig. 5

Speed of microparticles according to the flow at 50μm on the left and 100μm on the right. In blue, the theoretical speed according to the rate of the syringe pump, in red the estimated speed. The shaded areas on both curves correspond to the measurement error. Results divert from the groundtruth above 6.5ml/h at 50μm and above 4.5ml/h at 100μm.

Fig. 6
Fig. 6

Map of the optical flow around an air bubble (grey circle). The two axes are the x and y plane. Each arrow represents the mean velocity of a particle at a given time accumulated over a time window of two minutes. The color bar represents the angle of the optical flow vector from 90 degrees to −90 degrees.

Fig. 7
Fig. 7

On the left, the mean angle value of the optical flow inside each square. On the right, the mean norm of the optical flow inside each square.

Fig. 8
Fig. 8

On the left, Gaussian blob tracking for microparticles over time window of 20ms. The axes are the x and y plane. The black and white dots represent negative and positive events respectively. On the right are zooms on two particles. The blue circles represent the active blobs which are tracking microspheres.

Fig. 9
Fig. 9

Density of -particles measured for different concentrations. In blue is the measured density of particles per ml. The red lines represent the error to the true value.

Fig. 10
Fig. 10

Error regarding the reference value for the optical flow (blue line) and the particle density (red line) depending on the imaging depth. For depths greater than 120μm this error exceeds 10% of the reference value.

Fig. 11
Fig. 11

Top: in blue is the number of event per bin for the experimental data at 4ml/h. Bottom: in red is the computational ratio r. The maximum value is 0.45 and the mean value is 0.23 indicating a real time computation.

Equations (8)

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Σ e : 2 p Σ e ( p ) = t
Σ e ( p + Δ p ) = Σ e ( p ) + Σ e T Δ p + ( Δ p )
Σ e x ( x , y 0 ) = Σ e | y = y 0 d x ( x ) = 1 v x ( x , y 0 ) Σ e y ( x 0 , y ) = Σ e | x = x 0 d y ( y ) = 1 v y ( x 0 , y )
Σ e = ( 1 v x , 1 v y ) T .
P i ( u ) = 1 2 π | β i | 1 2 e 1 2 ( p μ i ) T β 1 ( p μ i )
μ = α μ + ( 1 α ) p
A i ( t ) = { A i ( t Δ t ) e Δ t δ + P i ( p ) , if e ( p , t ) tracker i A i ( t Δ t ) e Δ t δ , otherwise .
v avg = Flow ChannelSection

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