Direct Estimation of the Wall Shear Rate using Parametric ... - CiteSeerX

1 downloads 0 Views 293KB Size Report
Abstract We present a new optical-flow-based technique to estimate the wall shear rate using a special illumination technique that makes the brightness of par-.
Direct Estimation of the Wall Shear Rate using Parametric Motion Models in 3D Markus Jehle1,2 , Bernd Jähne1,2 , and Ulrich Kertzscher3 1

Interdisciplinary Center for Scientific Computing (IWR), Im Neuenheimer Feld 368, D-69120 Heidelberg, Germany 2 Institute for Environmental Physics (IUP), Im Neuenheimer Feld 229, D-69210 Heidelberg, Germany {Markus.Jehle,Bernd.Jaehne}@iwr.uni.heidelberg.de 3 Labor für Biofluidmechanik, Charité, Universitätsmedizin D-14050 Berlin, Germany [email protected]

Abstract We present a new optical-flow-based technique to estimate the wall shear rate using a special illumination technique that makes the brightness of particles dependent on the distance from the wall. The wall shear rate is derived directly (that means, without previous calculation of the velocity vector field) from two of the components of the velocity gradient tensor which in turn describes the kinematics of fluid flows up to the first order. By incorporating this into a total least squares framework, we can apply a further extension of the structure tensor technique. Results obtained both from synthetical and real data are shown, and reveal a substantial improvement compared to conventional techniques.

1 Introduction Optical-flow [1] based techniques were established as powerful tools in the field of fluid flow analysis in recent years [2]. Using these methods it is possible to evaluate image sequences using continuous tracer, for example concentration [3] or heat [4], or rigid particles [5]. Under certain circumstances optical-flow based techniques are superior to correlation based techniques, such as Particle Image Velocimetry (PIV) [6], which is quite common in experimental fluid mechanics. For a comparative analysis of correlation based techniques and optical-flow based techniques in the field of computer vision see [7]. In this paper we adopt a novel approach based on an extended version of the generalized brightness change constraint equation which was applied to an image sequence recorded in the context of biofluidmechanics. Short explanations of the medical application and of the considered experiment are given in this introduction. The investigation of the flow near the wall of a blood vessel or an artificial organ is of great interest, since a close relationship is known to exist between the characteristics of the flow such as magnitude and direction of the wall shear stress, and biological phenomena such as thrombus formation or atherosclerotic events. The wall shear stress can be considered as the force, which the viscous fluid excerts tangentially on the wallsurface. It plays an important role, since it influences the structure and function of the endothelial cells as well as the behavior of platelets. The measurement of the wall shear

stress is a requirement to our understanding of atherosclerotic events and also for the ability of avoiding thrombus generation in artificial organs. [8] points out, that previous to her work, there existed no technique, which was capable to measure the influence of the flow close to the wall on biological and pathological events. Firstly this is due to the fact, that we deal with instationary flows at curved walls, secondly, it is not sufficient, to conduct pointwise measurements, but it is necessary to yield temporally and spatially resolved 2D-information of the wall shear stress, which has to be extracted taking into account the 3D-nature of the flow field. The method presented here is based on the observation and the digital recording of buoyant, light-reflecting, spherical particles suspended within the fluid. The particles are all exactly 300 µm in diameter. In contrast to conventional 2D-PIV the entire flow near the wall is illuminated from the outside with monochromatic diffuse light, so that all particles near the wall become visible. A dye is added to the fluid, which limits the penetration depth of the light into the flow model according to Beer-Lambert’s law. The intensity respective gray value gp of the light approaching the particle is gp (z) = g0 exp −z/˜ z∗ , where g0 is the light’s intensity before penetrating into the fluid, z is the distance of the particle’s surface from the wall, and z˜∗ is the penetration depth (Figure 1). The light is reflected by the particle, and passes through the distance z again, before approaching the wall with the intensity g(z) = gp (z) exp −z/˜ z∗ = g0 exp −2z/˜ z∗ = g0 exp −z/z∗ ,

(1)

where an effective penetration depth z∗ = z˜∗ /2 was introduced for convenience. Within the illuminated layer the particles appear more or less bright, depending on their normal distance to the wall: Particles near the wall appear brighter, i. e. have a higher gray value than particles farther away from the wall. The correlation between the gray value of a particle and its distance to the wall, which is expressed in terms of the hypothetical grayvalue g0 of the particle at the wall and z∗ , can be assessed experimentally. If the concentration of the dye, the illumination and the size of the particles are chosen properly, the particles closest to the wall fall within a region where the velocity distribution is considered to be proportional to the wall distance. This permits the calculation of the wall shear stress τw according to Newton’s shear stress formula, using the measured velocity component tangential to the wall u, the normal distance to the wall z and the dynamic viscosity of the fluid η: · ¸ ∆u du ≈η . τw = η dz z=0 ∆z [8] seperates the near-wall flow in several layers by means of gray-value-thresholding. For each layer, which is characterized by a distinct distance from the wall, the motion of the particle can be determined with a conventional PIV algorithm. This results in a vector field u(z) for each layer, from which the wall shear stress can be derived. We present an optical flow-based approach for analyzing image sequences recorded using the technique described above, which leads to following benefits compared to the analysis proposed in [8]:

particle gp=g0 exp z/z* z dyed fluid wall air

g=gp exp z/z*

g0

light

Figure 1. A monochromatic beam of light penetrates the dyed fluid with the intensity g0 , and hits the particle with intensity gp after covering the distance z. After reflecting, it passes through the dye again, and hits the camera sensor with the intensity g. The intensity decrease can be calculated using Beer-Lambert’s law.

– Since PIV is a correlation-based technique, the particle density in image sequences suitable for a PIV-based analysis has to be sufficiently high. For this reason, [8] chooses the width of the distinct layers relatively large, taking into account inaccuracies. We will overcome this by regarding every particle individually. – Since PIV in its simple form is a 2D technique, there is no possibility of estimating out-of-plane motions. In our method brightness changes, i. e. motions perpendicular to the image plane, will be incorporated in the underlying equations. – In order to estimate the wall shear stress [8] has to calculate the velocity vector fields first. Our method delivers the wall shear rate directy, without previous estimation of the vector fields.

2 Estimation of Depth and Velocity We estimate the distance of the particle’s surface from the wall by eliminating z in Beer-Lambert’s law (1): z = z∗ (ln g0 − ln g) . In order to estimate the particle’s velocity, we consider two cases. First we assume that the suspended particles move parallel to the wall, so that z won’t change. The grayvalue then remains constant for all times, and we can apply the brightness change constraint equation (BCCE) to obtain the optical flow: dg/dt = (∇g)T f + gt = 0 .

(2)

The optical flow represents the components of the particle’s velocity parallel to the wall: f = (u, v)T . Secondly if the particles don’t move parallel to the wall, i. e. with z not

constant, the grayvalue will change with time, according to: dg g0 dz 1 dz w =− exp −z/z∗ = − =− g , dt z∗ dt z∗ dt z∗ where the component of the particle’s velocity perpendicular to the wall w = dz/dt is introduced. From this we are able to construct some kind of generalized brightness change constraint equation GBCCE, as proposed in [9]: (∇g)T f + gt = −(w/z∗ )g ,

(3)

which can be written as a scalar product of the data vector d and the parameter vector p: d · pT = (gx , gy , g/z∗ , gt ) · (u, v, w, 1)T = 0 , where gx , gy and gt denote the partial derivatives of the gray values with respect to the spatial and temporal dimensions. To sufficiently constrain the equation system, we assume a constant p over a small spatio-temporal neighborhood, surrounding the location of interest containing n pixels. With the data matrix D = (d1 , . . . , dn )T replacing the data vector, the equation-system can be solved in a total least squares (TLS) sense akin to the structure tensor [7]: kDpk2 = pT D T Dp → min . with pT p = 1 to avoid the trivial solution p = 0. The Eigenvector e = (e1 , e2 , e3 , e4 )T to the smallest eigenvalue of the generalized structure tensor   hgx · gx i hgx · gy i hgx · gi /z∗ hgx · gt i  hgx · gy i hgy · gy i hgy · gi /z∗ hgy · gt i   DT D =   hgx · gi /z∗ hgy · gi /z∗ hg · gi /z∗2 hg · gt i /z∗  hgx · gt i hgy · gt i hg · gt i /z∗ hgt · gt i represents the sought after solution to the problem. In this notation local spatiotemporal averaging using a binomial filter is represented by pointed brackets. In the case of full flow, which means no aperture problem is present, the parameter vector is given by p = 1/e4 (e1 , e2 , e3 )T . The structures (here: particles imaged to circles of diameter smaller than 5 pixels) contain no edges, whose dimensions are greater than the size of the neighborhood which is chosen for velocity estimation (here: 33 × 33 pixels). So the image sequences recorded with the technique described above generally exhibit no aperture problem. Image sequences recorded with the technique described above generally exhibit no aperture problem, so we consider only full flow.

3 Estimation of the Wall Shear Rate In the introduction we emphasized that knowledge about the spatially distribution of the wall shear stress is essentially for understanding biofluidmechanics near the wall of a blood vessel or an artificial organ. In this chapter we show a method which delivers

the wall shear rate directly. The wall shear rate is the wall shear stress divided by the dynamic viscosity of the fluid. We derive the wall shear rate by selecting certain components of the velocity gradient tensor at the wall. This object describes the kinematics of the fluid up to first order completely. The velocity gradient tensor may be regarded as a generalization of the concept of the affine parameterization of 2D-optical flow fields, which will be recapitualted briefly in the following: The optical flow f (x, t) may be expanded to a first order Taylor series in the vicinity of (x0 , t0 ) [10]: ¶ ¶ µ µ ∂u/∂t ∂u/∂x ∂u/∂y t ≡ t + Ax + at . x+ f (x, t) ≈ f (x0 , t0 ) + ∂v/∂t ∂v/∂x ∂v/∂y The BCCE (2) supplemented by this parameterization yields the extended brightness change constraint equation (EBCCE): (∇g)T (t + Ax + at) + gt = 0 .

(4)

Geometric transformations of the local neighborhood may be computed from the components of the matrix A. Examples are divergence or vorticity: div(f ) = ∂u/∂x + ∂v/∂y

or

rot(f ) = ∂u/∂v − ∂v/∂x .

In the following we consider 3D physical flow fields. We apply the notation u ≡ (u1 , u2 , u3 )T ≡ (u, v, w)T for the 3D velocity vector at the 3D position x ≡ (x1 , x2 , x3 )T ≡ (x, y, z)T . A flow field u(x) can be extended to a first order Taylor series in the vicinity of (x0 , t0 ): ∂ui ∂ui ui (xj , t) ≈ ui (xj,0 , t0 ) + xj + t . ∂xj ∂t We made use of Einstein’s summation convention, and i and j are defined from 1 to 3. In vector-matrix-notation this reads: u(x, t) ≈ s + Γ x + bt , where s is a 3D-translation, Γ = (γij ) = (∂ui /∂xj ) is the 3×3-velocity gradient tensor which is essentially the Jacobian, and b is a 3D-acceleration. By using an alternative formulation of the GBCCE (3) e T u+gt = 0 , (gx , gy )T ·(u, v)+gt +(w/z∗ )g = (gx , gy , g/z∗ )T ·(u, v, w)+gt = (∇g)

e is an augmented gradient, the 3D-parametrisation can be incorporated into a where ∇ 3D-EBCCE: e T · (s + Γ x + bt) + gt = 0 . (∇g) (5)

From the components of the matrix Γ important physical quantities of the local neighborhood in the flow field can be computed, like – the vorticity vector ωk = ǫijk γij , – the strain rate tensor sij = 1/2(γij + γji ) or

– the dissipation rate ǫ = −2νsij sij = −ν(γij + γji )γij . If we assume that we have pure 2D-flow u = u(x, y), v = v(x, y) and w = 0, (5) reduces to the optical flow-parametrization case (4). In the following we address the case of uniform wall parallel shear flow, i. e. u = u(z), v = v(z) and w = 0. The only non-vanishing components of the velocity gradient tensor γij = ∂ui /∂xj are ∂u ∂u1 = = γ13 ∂z ∂x3

and

∂v ∂u2 = = γ23 . ∂z ∂x3

Therefore the 3D-EBCCE (4) can be rewritten to       x 0 0 γ13 e T · 0 +  0 0 γ23  ·  y  + 0 + gt = 0 , ∇g 00 0 z

(6)

which can be transformed to a scalar product of the data vector d and the parameter vector p after some simple algebraic manupilations: d · pT = (gx , gy , gt ) · (γ13 , γ23 , 1)T = 0 . Starting from this scalar product, we can construct an expanded structure tensor, similar to the way presented in Section 2:   hgx z · gx zi hgx z · gy zi hgx z · gt i T D D =  hgx z · gy zi hgy z · gy zi hgy z · gt i  . hgx z · gt i hgy z · gt i hgt · gt i By performing an eigen-decomposition we obtain an estimation for the parameter vector to p = 1/e3 (e1 , e2 )T = (m13 , m23 )T in the full flow case.

4 Results We apply the analysis presented in Section 3 to synthetically generated and to real acquired image sequences. All sequences are evaluated by using the EBCCE based on the special case of wall parallel shear flow (6). 4.1

Synthetic Data

The following image sequences are generated by providing a uniform, wall-parallel 3D-Flow: u(z) = γ13 z, v = w = 0. The flow is texturized using Gaussian intensity distributions of equal maximum intensity and equal maximum variance, representing the particles. The z−position of the particles is indicated by attenuating the maximum intensity of the Gaussians according to Beer-Lambert’s law (1). The first synthetic image sequence contains particles, which are distributed in such a way, that they never will overlap each other: The particles are arranged in rows; each particle in one row having the same depth, and therefore the same brightness and the

γ 13

γ 23

z 1

0.5

0.55

0.6

0.65

0.7

0.1

0.05

0

0.05

u

0.1

0.6

γ 13

γ 23

z 1

0

0.5

1

1.5

2

1

0.5

0

0.5

1

1

u

Figure 2. Example images of the synthetic image sequences (left), maps of the wall shear rates estimated with our algorithm (center) and velocity profiles, which deliver the ground truth for the wall shear rates (right). The distinct image sequences are described in the text.

same speed (Figure 2, top, left). Here the wall shear rate m13 is exactly 0.6, which is indicated by the profile u(z) (Figure 2, top, right), and the wall shear rate γ23 vanishes. Our algorithm yields wall shear rates, which are displayed in Fig. 2, top, center. The ground thruth is reproduced very well. Slight deviations occur, where there are several particle-rows are adjacent, moving with approximately the same speed. The reason for these deviations is the fact that the spatio-temporal neighborhood is of limited size (in this case 65 × 65 pixels). In the second synthetic image sequence the particles are randomly distributed (Figure 2, bottom, left), moving so that they follows the wall shear rates m13 = 1 and γ23 = 0. The estmated wall shear rates are mapped in Fig. 2, bottom, middle. A a result of the fact, that overlappings may occur, there are regions, where our algorithm produces significant deviations from the ground truth. This is evidence, that our model fails in the presence of multiple motions. 4.2

Real Data

The analyzed image sequence was recorded by [8]. To examine the applicability of the method presented in Section 1 for the investigation of complex flows, a U-shaped channel with a rectangular cross-section and a step was constructed (Figure 3, right). In combination with the bending, the step in the cross-section generates a complex flow which detaches from the wall. Figure 3, left shows a sample image of the recorded sequence. Since the diameter of the spheres is significantly larger than the penetrationdepth, only the particles close to the wall are imaged and no overlapping particles are

γ 13

20

0

γ 23

20

20

0

20

Figure 3. Example image of the recorded image sequence (left), maps of the wall shear rates estimated with our algorithm (center) and geometry of the U-shaped channel with the arrows denoting the direction of the flow (right).

recorded. Therefore, the problem of multiple-motions does not occur in this situation. Though not having a uniform flow, we applied (6) since it is evident that the components of the velocity gradient tensor containing the derivatives w. r. t. z are large compared to the components containing the derivatives w. r. t. x and y. The components m13 and m23 of the wall shear rate are mapped in Fig. 3, center. Since the examined flow is stationary, the shear rate is averaged over 300 frames. The gaps in the otherwise dense flow field indicate the spots, where the confidence measure, which is provided by the structure-tensor-technique, was to low for providing a reliable result. Figure 4 displays the magnitude of the wall shear rate, obtained with different techniques, and compares the methods to each other. To establish some kind of “ground truth” [8] computed the flow numerically with the solver FLUENT6, which is shown in Fig. 4, top, left. The analysis of the flow using the PIV-technique and subsequent derivation of the wall shear rate, as carried out by [8] is shown in Fig. 4, top, center. Our result is mapped in Fig. 4, top, right. In order to compare the techniques with each other, we filled the gaps by means of interpolation, and afterwards smoothed the result using a 2D-anisotropic diffusion. Our optical-flow-based method provides a dense, highly resolved vector field of the wall shear rate, which is capable of estimating this value at positions where the PIV-method fails. The flow-detachment in the lower left corner can, for instance, be reproduced very well. Both techniques, optical flow and PIV, show a deficit of the wall shear rate in the upper left corner, and a surplus in the upper right corner, compared to the analysis provided by computational fluid dynamics. These systematical deviations may occur as a result of the fact, that the particles, which are of a comparable large size, cannot follow the fluid ideally, or influence the fluid. To provide a measure of how much the results of the experimental methods are apart from the numerical solution, we added up the magnitudes of the differences on each pixel. Optical flow resulted in about 10% better results, than PIV, when the CFD-solution was regarded as the “ground truth”. Besides that, the optical flow analysis yielded a much better spatial resolution, and the area, where a reliable estimate of the wall shear rate is possible, is about 30% greater compared to the area, obtained using the PIV analysis.

CFD

10

PIV

20

30

10

CFD−PIV

−10

0

OF

20

30

10

CFD−OF

10

−10

0

20

30

OF−PIV

10

−10

0

10

Figure 4. Top: Wall shear rates determined by computational fluid dynamics (left), PIV-technique (center) and our optical-flow-based method (right). Bottom: Pointwise Differences between CFD and PIV (left), CFD and optical flow (center), optical flow and PIV (right).

5 Conclusion

A novel technique is presented for the direct estimation of the wall shear stress from particle-based image sequences. We propose an extension of the BCCE so that estimation can be done of the particle’s velocity perpendicular to the image plane, using an exponential brightness change model, and also so that a direct analysis of the components of the strain rate tensor such as the wall shear rate is possible. Both synthetical and real experiments demonstrate the feasibility of the technique in good agreement with the ground truth. Though in this paper we addressed stationary wall-parallel flows only, our method may be extended to instationary, full 3D-flows in principle. In order to solve these challenges we are currently investigating a convection-driven free-surface flow. Furthermore we will solve the problem cuased by the restriction of using spheres of exactly the same size, so that smaller and less expensive particles may be used, by means of illuminating with light conisting of two wavelengths [11].

Acknowledgements. We gratefully acknowledge the support by the priority program 1147 of the German Research Foundation.

References [1] Horn, B.K.P., Schunk, B.G.: Determining optical flow. Artificial Intelligence 17 (1981) 185–204 [2] Jehle, M., Klar, M., Jähne, B.: Optical-flow based velocity analysis. In Tropea, C., Foss, J., Yarin, A., eds.: Springer Handbook of Experimental Fluid Dynamics. (in preparation) [3] Corpetti, T., Memin, E., Perez, P.: Dense estimation of fluid flows. IEEE Trans. on Pattern Analysis and Machine Intelligence 24(3) (2002) 365–380 [4] Garbe, C., Spies, H., Jähne, B.: Estimation of surface flow and net heat flux from infrared image sequences. Journal of Mathematical Imaging and Vision 19 (2003) 159–174 [5] Ruhnau, P., Kohlberger, T., Schnörr, C., Nobach, H.: Variational optical flow estimation for particle image velocimetry. Experiments in Fluids 38 (2005) 21–32 [6] Adrian, R.J.: Particle-imaging techniques for experimental fluid mechanics. Annu. Rev. Fluid Mech. 23 (1991) 261–304 [7] Jähne, B., Haussecker, H., Geissler, P., eds.: Handbook of Computer Vision and Applications. Academic Press, San Diego, CA, USA (1999) [8] Debaene, P.: Neuartige Messmethode zur zeitlichen und örtlichen Erfassung der wandnahen Strömung in der Biofluidmechanik. Phd thesis, TU Berlin (2005) [9] Haussecker, H.W., Fleet, D.J.: Computing optical flow with physical models of brightness variation. PAMI 23(6) (2001) 661–673 [10] Fleet, D.J.: Measurement of Image Velocity. Kluwer Academic Publishers, Dordrecht, Netherlands (1992) [11] Jehle, M., Jähne, B.: A novel method for spatiotemporal analysis of flows within the water-side viscous boundary layer. In: 12th International Symposium of Flow Visualisation, Göttingen, Germany (2006)