# Wavelet-Based Fluid Motion Estimation.

**ABSTRACT** Based on a wavelet expansion of the velocity field, we present a novel optical flow algorithm dedicated to the estimation of continuous motion fields such as fluid flows. This scale-space representation, associated to a simple gradient-based optimization algorithm, naturally sets up a well-defined multi-resolution analysis framework for the optical flow estimation problem, thus avoiding the common drawbacks of standard multi-resolution schemes. Moreover, wavelet properties enable the design of simple yet efficient high-order regularizers or polynomial approximations associated to a low computational complexity. Accuracy of proposed methods is assessed on challenging sequences of turbulent fluids flows.

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**ABSTRACT:**Expanding on a wavelet basis the solution of an inverse problem provides several advantages. First of all, wavelet bases yield a natural and efficient multireso-lution analysis which allows defining clear optimization strategies on nested subspaces of the solution space. Be-sides, the continuous representation of the solution with wavelets enables analytical calculation of regularization integrals over the spatial domain. By choosing differen-tiable wavelets, accurate high-order derivative regular-izers can be efficiently designed via the basis's mass and stiffness matrices. More importantly, differential constraints on vector solutions, such as the divergence-free constraint in physics, can be nicely handled with biorthogonal wavelet bases. This paper illustrates these advantages in the particular case of fluid flow motion estimation. Numerical results on synthetic and real im-ages of incompressible turbulence show that divergence-free wavelets and high-order regularizers are particu-larly relevant in this context.International Journal of Computer Vision 05/2013; 103(1):80--99. · 3.62 Impact Factor - SourceAvailable from: Cedric Herzet[Show abstract] [Hide abstract]

**ABSTRACT:**Selecting optimal models and hyperparameters is crucial for accurate optical-flow estimation. This paper provides a solution to the problem in a generic Bayesian framework. The method is based on a conditional model linking the image intensity function, the unknown velocity field, hyperparameters, and the prior and likelihood motion models. Inference is performed on each of the three levels of this so-defined hierarchical model by maximization of marginalized a posteriori probability distribution functions. In particular, the first level is used to achieve motion estimation in a classical a posteriori scheme. By marginalizing out the motion variable, the second level enables to infer regularization coefficients and hyperparameters of non-Gaussian M-estimators commonly used in robust statistics. The last level of the hierarchy is used for selection of the likelihood and prior motion models conditioned to the image data. The method is evaluated on image sequences of fluid flows and from the "Middlebury" database. Experiments prove that applying the proposed inference strategy yields better results than manually tuning smoothing parameters or discontinuity preserving cost functions of the state-of-the-art methods.IEEE Transactions on Image Processing 12/2011; 21(4):1437-51. · 3.20 Impact Factor

Page 1

Wavelet-Based Fluid Motion Estimation

Pierre Dérian, Patrick Héas, Cédric Herzet, and Étienne Mémin

INRIA Rennes-Bretagne Atlantique,

Campus universitaire de Beaulieu, 35042 Rennes Cedex, France.

{Pierre.Derian,Patrick.Heas,Cedric.Herzet,Etienne.Memin}@inria.fr

http://irisa.fr/fluminance

Abstract. Based on a wavelet expansion of the velocity field, we present

a novel optical flow algorithm dedicated to the estimation of continuous

motion fields such as fluid flows. This scale-space representation, asso-

ciated to a simple gradient-based optimization algorithm, naturally sets

up a well-defined multi-resolution analysis framework for the optical flow

estimation problem, thus avoiding the common drawbacks of standard

multi-resolution schemes. Moreover, wavelet properties enable the design

of simple yet efficient high-order regularizers or polynomial approxima-

tions associated to a low computational complexity. Accuracy of pro-

posed methods is assessed on challenging sequences of turbulent fluids

flows.

Keywords: Optical flow, continuous fluid motion, wavelet multi-resolution

analysis, high-order regularization, polynomial approximation.

1Introduction

Recent years have seen significant progress in signal processing techniques for

fluid motion estimation. The wider availability of image-like data, whether com-

ing from experimental facilities (e.g. particle image velocimetry) or from larger-

scale geophysical study systems such as lidars or meteorological and oceano-

graphical satellites, strongly motivates the development of computer-vision meth-

ods dedicated to their analysis. Correlation-based and variational methods have

proven to be efficient in this context. However, the specific nature of fluid mo-

tion highly complicates the process. Indeed, one has to deal with continuous

fields showing complex structures evolving at high velocities. This is particu-

larly problematic with optical flow methods, where the problem non-linearity

requires to resort to an ad-hoc multi-resolution strategy. Although leading to

good empirical results, this technique is known to have a number of drawbacks.

Moreover, the underdetermined nature of the optical flow estimation problem

imposes to add some prior information about the sought motion field. In many

contributions dealing with rigid-motion estimation, first-order regularization is

considered with success. However, when tackling more challenging problems such

as motion estimation of turbulent fluids, this simple prior turns out to be inade-

quate. Second-order regularizers allowing to enforce physically-sound properties

Page 2

2Pierre Dérian, Patrick Héas, Cédric Herzet, and Étienne Mémin

of the flow are considered [2,4,10,11], but their implementation raises up several

issues.

In this paper, we propose an optical-flow estimation procedure based on

a wavelet expansion of the velocity field. This approach turns out to offer a

nice mathematical framework for multi-resolution estimation algorithms, which

avoids some of the drawbacks of the usual approach. Note that algorithms based

on wavelet expansion of the data [1] or the velocity field [9] have been previously

proposed. However, unlike the algorithm presented hereafter, the computational

complexity of the later seriously limits its application to small images and/or

the estimation of the coarsest motion scales. Moreover, we consider the effective

implementation of high-order regularization schemes, based upon very simple

constraints on the wavelet coefficients at small scales. We finally assess the rel-

evance of proposed methods on challenging image sequences of turbulent fluid

motions. Simulation results prove that the proposed approach outperforms the

most effective state-of-the-art algorithms.

2Optical Flow Background

Optical flow estimation is an ill-posed inverse problem. It consists in estimating

the apparent motion of a 3D scene through image brightness I(x,t) variations in

space x = (x1,x2)T∈ Ω ⊂ R2and time t ∈ R. Optical flow, identified by a 2D

velocity field v(x,t) : Ω × R+?→ R2is the projection on the image plane of the

3D scene velocity. Its estimation involves two main aspects: a data model that

links image data to the velocity field and a regularization scheme to overcome

the ill-posedness.

2.1Non-Linear Data Model

Data models are commonly built upon assumptions about the temporal varia-

tions of the image brightness. The integration of a conservation assumption leads

to the well-known Displaced Frame Difference (DFD) equation, which is studied

in the following. However, the approach remains valid for any other integrated

data model. Let us denote by I0(x) and I1(x) two consecutive image samples

of the continuous sequence I(x,t) which has been discretized in time with a

unit interval. Under rigid motion and stable lighting conditions, v = (v1,v2)T

satisfies the standard DFD equation, which reads:

∀x ∈ Ω, fDFD(I,v) = I1(x + v(x)) − I0(x) = 0 .

The estimated motion field ˆ v is obtained by minimizing a cost function, which

we chose quadratic in the following to clarify the presentation:

(1)

ˆ v = argmin

v

JDFD(I,v) , with JDFD(I,v) =1

2

?

Ω

|fDFD(I,v)|2dx .

(2)

The data model being non-linear w.r.t. the velocity field v, estimation of optical

flow therefore requires a specific optimization approach.

Page 3

Wavelet-Based Fluid Motion Estimation3

2.2Classical Multi-resolution Strategy

Indeed, the equations for the inversion are only valid if the solution remains

in the linearity region of the image intensity function. A standard approach for

tackling non-linearity is to rely on an incremental multi-resolution strategy. This

approach consists in choosing some sufficiently coarse low-pass-filtered version

of the images at which the linearity assumption is valid, and to estimate a first

displacement field assumed to correspond to a coarse representation of the mo-

tion. Then, a so-called Gauss-Newton strategy is used by applying successive

linearizations around the current estimate and warping accordingly a represen-

tation of the images of increasing resolution. More explicitly, let us introduce

the following incremental decomposition of the displacement field at resolution1

2j:

vj= ˜ vj+ v?

j

(3)

where v?

2jand ˜ vj??

considered at resolution 2j. In order to respect the Shannon sampling theorem,

the coarse scale data term is derived by a low-pass filtering of the original images

with a kernel2Gj, followed by a subsampling at period 2j. Using (3), at coarse

scale, image Ij(x) and the motion-compensated image˜Ij(x) are then defined as:

?Ij(x) =↓2j ◦ (Gj? I0(x))

jrepresents the unknown incremental displacement field at resolution

i<jPj(v?

scales; Pj(v?

i) is a coarse motion estimate computed at the previous

i) denotes a projection operator which projects v?

ionto the grid

˜Ij(x) =↓2j ◦ (Gj? I1(x + ˜ vj(x))),

(4)

where ↓2j denotes a 2j-periodic subsampling operator. It yields a functional Jj

defined as a linearized version of (1) around ˜ vj(x):

?

Finally, the sought motion estimate ˆ v is given by solving a system of coupled

equations associated to resolutions increasing from 2Cto 2F:

where the finest scale s = 2−Fcorresponds to the pixel whereas the coarsest scale

is noted s = 2−C. In practice, equations in (6) are usually solved independently,

starting from the coarsest to the finest scale. This coarse-to-fine approach has the

OBS

Jj

OBS(Ij,v?

j) =1

2

Ωj

?˜Ij(x)−Ij(x) +v?

j(x) · ∇˜Ij(x)

?2

dx .

(5)

ˆ v = v?

F+ ˜ vF= v?

F+

F−1

?

i=C

PF(v?

i),

v?

j= argmin

v?

Jj

OBS(Ij,v?) ,∀j ∈ {C,··· ,F} .

(6)

1In this paper, we shall use the following convention: indices j ≥ 0 represent the

resolution 2j—contrary to [7]. Corresponding scale is 2−j.

2A Gaussian kernel of variance proportional to 2jis commonly used.

Page 4

4Pierre Dérian, Patrick Héas, Cédric Herzet, and Étienne Mémin

drawback of freezing (i.e. leaving unchanged), at a given scale, all the previous

coarser estimates. Moreover, the major weakness of this strategy is the arbi-

trary approximation of the original functional in (2) by a set of coarse scale data

terms (5), which are defined at different resolutions by a modification of original

input images with (4) and by a linearization of model (1) around the previous

motion estimate. In the next section, we will see that this multi-resolution strat-

egy has a mathematically-sound formulation within the framework of wavelet

representations.

2.3The Aperture Problem and Usual Regularization Schemes

Previously introduced data models remain under-constrained, as they provide for

each time t a single equation for two unknowns (v1,v2) at each spatial location

x = (x1,x2)T. To deal with this under-constrained estimation problem, the most

common setting consists in enforcing some spatial coherence to the solution.

Implicit Regularization The motion field is constrained to be of the form

v = Φ(Θ), where Φ is a function parametrized by Θ (piece-wise polynomial

functions are often used). Implicit regularization schemes penalize discrepancies

from model (1) by minimizing JDFDwith respect to Θ, i.e.

?

Associated to a low-order parametric representation, this simple approach re-

duces drastically the dimension of the problem, hence addressing its under-

constrained nature. However, when spatiotemporal gradients of the images van-

ish, it is impossible to guarantee the existence of an unique solution: this is the

aperture problem.

ˆ v = Φ

argmin

Θ

JDFD(I,Φ(Θ))

?

.

(7)

Explicit Regularization Global regularization schemes in their simplest form

define the estimation problem through the minimization of a functional com-

posed of two terms balanced by a regularization coefficient µ > 0:

J(I,v,µ) = JDFD(I,v) + µJreg(v) .

(8)

Thus, motion estimate ˆ v satisfies ˆ v = argminvJ(v,I,µ). The data term JDFDis

still defined by (2) . The second term, Jreg(the “regularization term”), encourages

the solution to follow some prior smoothness model formalized with function freg:

?

An n-order regularization writes in its simplest form:

Jreg(v) =1

2

Ω

freg(v,x)dx .

(9)

freg(v,x) =

?

i=1,2

?

j=1,2

?????

∂nvi

∂xn

j

(x)

?????

2

.

(10)

Page 5

Wavelet-Based Fluid Motion Estimation5

A first-order regularizer (i.e. n = 1) enforcing weak spatial gradients of the two

components v1 and v2 of the velocity field v is very often used [6]. Second-

order regularizers (i.e. n > 1) have been proposed in the literature in the case of

fluid flows [2,10,11]. However, since motion variables are considered on the pixel

grid, an approximation of continuous spatial derivatives by discrete operators is

required. For regular pixel grids, it is usually done using finite difference schemes.

Nevertheless, it is well known that ensuring stability of the discretization schemes

of high-order regularizer may constitute a difficult problem.

3 Wavelet Formulation

As shown in Sect. 2, the common optical flow estimation approach suffers from

two main drawbacks: the necessary “empirical” multi-resolution approach and

the implementation of efficient regularizations terms. The use of wavelet bases is

a simple answer to both problems. Moreover, it has been shown that a wavelet

expansion is appropriate for representing turbulent flows [3].

3.1Wavelet Decomposition

In order to avoid the limitations of the classical multi-resolution strategy, we

consider the projection of each scalar component v1, v2of the velocity field v onto

multi-resolution approximation spaces exhibited by the wavelet formalism. Let

us introduce briefly this context for real 1D scalar signals. We consider a multi-

resolution approximation of L2(R) as a sequence {Vj}j∈Z of closed subspaces,

so-called approximation spaces, notably verifying3

Vj⊂ Vj+1; lim

j→−∞Vj=

+∞

?

j=−∞

Vj= {0}; lim

j→+∞Vj= Closure

+∞

?

j=−∞

Vj

= L2(R) .

Since approximation spaces are sequentially included within each other, they can

be decomposed: Vj+1= Vj⊕ Wj. Those Wjare the orthogonal complements of

approximation spaces, they are called detail spaces.

Practically, scalar 1D signals being finite, they belong to a given approxima-

tion space according to their resolution, i.e. number of samples. Let w be a 1D

signal of 2F+1samples, then w ∈ VF+1= VC⊕WC⊕WC+1⊕···⊕WF⊂ L2([0,1]),

where 0 ≤ C ≤ F. The projection of w on this multiscale basis writes:

w(x) =

2C−1

?

k=0

?w,φC,k?L2 φC,k(x) +

F

?

j=C

2j−1

?

k=0

?w,ψj,k?L2 ψj,k(k) .

(11)

Here, {φC,k}k and {ψj,k}k are orthonormal bases of VC and Wj, respectively.

They are defined by dilatations and translations4of the so-called scale function

3See [7] for a complete presentation of wavelet bases.

4Written in a general form fj,k(x) = 2j/2f(2jx − k).

Page 6

6Pierre Dérian, Patrick Héas, Cédric Herzet, and Étienne Mémin

φ and its associated wavelet function ψ. Functions φ and ψ verify the following

two-scale relations:

√2

k∈Z

where sequences h[k] = ?φ(x),√2φ(2x−k)? and g[k] = ?ψ(x),√2φ(2x−k)? are

called conjugate mirror filters. Those filters play an important role in the fast

implementation with filter banks of forward and inverse wavelet transform, i.e.

projection on the wavelet basis and reconstruction, from (11) [7]. Finally, the rep-

resentation of a signal projected onto the multiscale wavelet basis is given by the

set of coefficients appearing in (11): aC,k? ?w,φC,k?L2 and dj,k? ?w,ψj,k?L2 are

approximation and detail coefficients, respectively. Those results are extended

to the case of 2D signals, in order to obtain separable multiscale orthonormal

bases of L2([0,1]2).

φ(x) =

?

h[k]φ(2x − k) ; ψ(x) =

√2

?

k∈Z

g[k]φ(2x − k),

(12)

3.2Wavelet Data Term

In this work, the representation of the velocity field v is obtained by the wavelet

decomposition (11) of each component. We denote by Θ1 and Θ2 the sets of

coefficients respectively associated to v1and v2; Θ = (Θ1,Θ2)Tis the set of all

coefficients. Denoting the linear reconstruction operator by Φ for convenience,

we may write

∀x ∈ Ω, v(x) = Φ(x)Θ .

Here the constant coefficients vector Θ is the unknown of our optical flow esti-

mation problem. Replacing v(x) by (13) in DFD data term (1), we obtain

?

and the estimation problem becomes

(13)

JDFD(Θ) =1

2

Ω

[I1(x + Φ(x)Θ) − I0(x)]2dx

(14)

ˆ v = ΦˆΘ ∈ VF+1, whereˆΘ = argmin

Θ

JDFD(Θ) .

(15)

3.3Multiscale Estimation

Unknown coefficients are estimated sequentially from coarsest scale C to a cho-

sen finest one L (with C ≤ L ≤ F) using a gradient-descent algorithm. At each

scale j, all coefficients from scales C to j are estimated. Coefficients previously

estimated at coarser approximation spaces are used to initialize the gradient

descent; this strategy enables the update of the latter coarser coefficients while

estimating “new” details at current scale j. In other words, solution ˆ v is sequen-

tially sought within higher resolution spaces: VC⊂ VC+1⊂ ··· ⊂ VL. This way,

the projection of the current solution ˆ v ∈ Vj onto every coarser space Vpwith

C ≤ p < j is constantly updated, contrary to the standard incremental approach

(Sect. 2.2). The use of wavelet bases thus leads to a “natural” and well-defined

multi-resolution framework. At each refinement level, minimization of functional

Page 7

Wavelet-Based Fluid Motion Estimation7

JDFDis efficiently achieved with a gradient-based quasi-Newton algorithm (L-

BFGS) [8], to seek the optimumˆΘ. For any coefficient θi,p∈ Θi⊂ Θ,

∂JDFD

∂θi,p

(Θ) =

?∂I1

∂xi(· + Φ(·)Θ)[I1(· + Φ(·)Θ) − I0(·)],Φp

?

L2([0,1]2)

(16)

where Φpis the wavelet basis atom related to θi,p. As a consequence, components

of the spatial gradient of the data-term functional (14) are simply given by the

coefficients of the wavelet decomposition of the two terms

∂I1

∂xi(x + Φ(x)Θ)[I1(x + Φ(x)Θ) − I0(x)] , i = 1,2 ,

on the considered wavelet basis. It is easy to see that the proposed coarse-to-

fine estimation strategy enables to capture large displacements: at large scales,

the decomposition of (3.3) is obtained by convolutions with the atoms of the

wavelet basis having the largest support. Note that conversely to the algorithm

proposed in [9], the low-complexity of gradient computation via fast wavelet

transform does not restrict motion estimation to large scales and/or images of

small size.

4 Regularizations

4.1Wavelet Properties

Wavelet-based regularizers which are described in the following are based upon

wavelet properties such as polynomial reproduction, differentiation and interpo-

lation. Those aspects are linked to the notion of vanishing moments (VM). A

wavelet ψ(x) ∈ L2(R) has n VM if :

?

R

x?ψ(x)dx = 0, for 0 ≤ ? < n .

(17)

Wavelets as polynomial approximations From (17), a wavelet with n VM

is hence orthogonal to any polynomial of degree n−1. Consequently, piece-wise5

polynomials of degree n − 1 belonging to VF+1 are exactly described in VF,

since the atoms of the basis that belong to its orthogonal complement WF have

vanishing coefficients.

Wavelets as Differential Operators Given a signal w ∈ Cn, the behavior of

its small scales coefficients resulting from an n-VM wavelet decomposition can

be related to its nthderivative [7]:

lim

j→∞

?w(x),ψj,k(x)?

2−j(n+1

2)

∝∂nw(x)

∂xn

.

(18)

This result can be extended to the case of 2D signals.

5On the support of {ψF,k}.

Page 8

8Pierre Dérian, Patrick Héas, Cédric Herzet, and Étienne Mémin

Wavelet-Based Interpolation A multiscale interpolation is the orthogonal

projection of a signal w estimated at a given resolution 2jonto the next finer

approximation space Vj+1:

PVj+1w

?

2(p +1

2)

?

=

+∞

?

k=−∞

w(2k)ϕj+1(p − k + 1/2) .

(19)

The interpolation function ϕ is defined as the autocorrelation of the scaling

function: ϕ = φ ?ˇφ, where ? and ˇ · denote respectively convolution and time-

reverse6operators. It can be shown that ϕ interpolates exactly polynomials of

order n if and only if the wavelet associated to scaling function φ has n + 1

VM [7]. This linear interpolation operator PVj+1is also implemented with filter

banks using filter hi, where hi[n] =?h ?ˇh?[2n + 1] and h is defined in (12).

4.2Polynomial Approximation on a Truncated Basis

As seen in Sect. 2.3, a first way to overcome the under-constrained nature of

the optical flow estimation problem consists in reducing the number of un-

knowns through a parametric formulation of the velocity field. Using the pro-

posed wavelet formulation (15), this can be easily achieved by estimating the ve-

locity field on a truncated wavelet basis. This means that the solution ˆ v belongs

to a lower-resolution space VL⊂ VF+1and therefore is a piece-wise polynomial

of order n − 1 in VL+1. Details coefficients associated to non-estimated small

details scales (Wjwith L ≤ j ≤ F) are thus not estimated, but set to zero.

ˆ v = ΦˆΘ ∈ VL, L < F + 1, whereˆΘ = argmin

Θ

JDFD(Θ) .

(20)

Since the basis truncation reduces the number of unknowns, it is theoretically

possible to estimate detail coefficients up to penultimate scale F −1, i.e. ˆ v ∈ VF.

Practically, it is impossible due to the aperture problem.

4.3High-Order Regularization

It has been previously mentioned that smallest scale coefficients might be inter-

preted as the signal’s nthderivative (Sect. 4.1), with n number of VM of the

considered wavelet. The penalization of small scale coefficients’ amplitude thus

enables to control the amplitude of the derivative of the estimated signal. How-

ever, due to the dyadic structure of the discrete wavelet decomposition, only a

“piecewise” control is possible. In order to control the derivative at junctions of

those dyadic blocks, interpolated signal ˜ v of the velocity field v on a shifted 2D

grid is considered. Small scale coefficients {˜ΘF} and {ΘF} of both ˜ v and v are

penalized. “Interpolated coefficients”˜Θ are expressed as a linear combination of

Θ through wavelet inverse and forward transformations (Φ, Φ−1= ΦT, resp.)

6More explicitly,ˇf : t ?→ˇf(t) = f(−t).

Page 9

Wavelet-Based Fluid Motion Estimation9

and interpolation:˜Θ =

term

Jreg(Θ) =1

?ΦT◦ PVF+1◦ Φ?Θ. We finally get the regularization

2?˜ΘF?2and ∇Jreg(Θ) = ΘF+

2?ΘF?2+1

?

ΦT◦ PT

VF+1◦ Φ

?

˜ΘF (21)

The gradient in (21) is a linear form which can be efficiently computed using the

recursive filter banks presented in Sect. 3.1 and 4.1. The addition of the regu-

larization term (21) therefore does not increase significantly the computational

burden. Supplementing (15), the estimation problem becomes:

ˆ v = ΦˆΘ ∈ VF+1, whereˆΘ = argmin

Θ

JDFD(Θ) + µJreg(Θ) .

(22)

5Results

Daubechies wavelets have been chosen since they have a minimum support size

for a given number of VM [7]. Daubechies wavelet with n VM will be referred

to as Dnhereafter. Wavelet transform is implemented with periodic boundary

conditions.

5.1 Synthetic PIV Sequence

(a)(b)(c)(d)(e)

(f) (g)(h)(i)(j)

Fig.1. Sample synthetic PIV image (1a) with below the vorticity of the underlying

reference velocity field (1f). End-point error maps on velocity field estimations for a

polynomial approximation (upper row) and high-order regularization (lower row) with

Dn wavelets are presented, i.e. polynomial (resp. derivative) order of n − 1 (resp. n),

for D1 (1b, 1g), D2 (1c, 1h), D3 (1d, 1i) and D10 (1e, 1j).

The first data set used for evaluation is a synthetic sequence of Particle

Imagery Velocimetry (PIV) images of size 256 × 256 pixels, representing small

particles (of radius below 4 pixels) advected by a 2D periodic forced turbulent

Page 10

10Pierre Dérian, Patrick Héas, Cédric Herzet, and Étienne Mémin

flow. The dynamic of the fluid flow is given by numerical simulation of 2D Navier-

Stokes equations at Re = 3000, using the vorticity conservation equation and

the Lagrangian equation for non-heavy particles transported by the flow (sim-

ulation details can be found in [5]). This simulated flow has a null-divergence

by construction. An image of the PIV sequence is displayed in Fig. 1a together

with its associated ground truth motion vorticity (1f). Estimated velocity field

evaluation is based on the Root Mean Square end-point Error (RMSE)7.

When the true velocity field is decomposed on a Daubechies wavelet with

a number of VM higher than 3, the velocity field reconstructed with the p = 6

coarser scales (out of 8) carries out more than 99,95% of the total kinetic energy.

Those 6 scales are represented with only 12.5% of atoms from the full wavelet

basis. Moreover, when n is chosen high enough (> 7), the reconstruction error

stabilizes around 0,013.

Motion Estimation From the previous analysis of the true velocity fields, it

seems that 6 detail scales out of 8 should give an accurate representation of the

motion in terms of kinetic energy, as long as the chosen number of VM is high

enough. Figure 2a (black curve) shows RMS errors computed on an estimated

velocity field with truncated Daubechies wavelet bases (Sect. 4.2) having differ-

ent VM n, i.e. with a polynomial approximation of order n − 1. As expected,

RMSE converges rapidly towards a median value of 0.0613 when n increases.

Figure 1 (upper row) shows corresponding end-point error maps for motion esti-

mated with wavelets bases D1, D2, D3and D10. Although errors effectively lower

when higher VM wavelets are employed, artifacts due to high-amplitude errors

on 6thscale coefficients (small white “dots”) and on coarse coefficients (white

straight “lines”) remain clearly visible. With the proposed high-order regulariza-

tion scheme (Sect. 4.3), all scales are estimated, which should highly improve

results with n ≤ 3 VM, i.e. for penalization of derivatives of order lower than 3.

This is confirmed on Fig. 2a (red dashed curve), with a reduction of 35% and

30% of the RMS obtained with D1and D2wavelets bases, respectively, whereas

the diminution observed using D10wavelet basis is of 10% at best. At the same

time, derivative penalization eliminates most of the artifacts observed on esti-

mates with truncated bases, which is displayed on the lower row of Fig. 1. Note

also that there are less differences between estimations with different VM, in

comparison to the previous case.

Comparison with State-of-the-Art Estimators Figure 2 is a comparison

of RMS errors obtained on the synthetic PIV sequence with the proposed high-

order regularizer and various state-of-the-art estimators, after a null-divergence

projection. Our wavelet-based estimator clearly outperforms other methods.

7Ground-truth velocity fields being given on a shifted grid (by 1/2 pixel) by the

numerical simulation, they have been interpolated in order to compute accurate

RMSE on the pixel grid.

Page 11

Wavelet-Based Fluid Motion Estimation 11

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 2 3 4 5 6 7 8 9 10

RMS error (pixel)

Daubechie wavelet's vanishing moments

(a)

0.04

0.06

0.08

0.1

0.12

0.14

20 30 40 50 60 70 80 90

RMS error (pixel)

time sample

(b)

Fig.2. Left: Comparison of RMS errors on velocity fields estimated from a pair of the

synthetic PIV sequence, with the proposed methods and Daubechies wavelets with 1

to 10 VM, i.e. for polynomial approximation (resp. derivative regularization) of order

0 to 9 (resp. 1 to 10). RMSE obtained with the polynomial approximation (6 scales

out of 8, dashed line) and with derivatives regularization (best case, solid line). Right:

Comparison of RMS errors on a sequence of velocity fields estimated with proposed

regularization (thick solid) and with state-of-the-art methods: correlation (thin solid),

first order regularization [6] (long-dashed), div-curl regularization [10] (dashed), self-

similar regularization [4] (dotted).

5.2Real PIV Sequence

This data set consists in 128 PIV pictures of a transversal view of a planar con-

comitant jet flow, of size 1024 × 1024 pixels. The flow has a “top-hat” velocity

profile and is poorly turbulent, but shows two high-shear regions featuring de-

velopment of Kelvin-Helmholtz instabilities. Motion is estimated with proposed

wavelet-based estimator (22), using the following settings: 2 VM and derivatives

penalization with factor µ = 107. Figure 3 presents a PIV image of the sequence

and streamlines of an estimated velocity field along with two consecutive vorticity

maps. A qualitative evaluation of the presented motion field shows a remarkably

good agreement with the physics of concomitant jets. A very good temporal

coherence is also observed, although no prior dynamic model is considered (i.e.

successive pairs of images are processed independently).

6

An optical flow estimation algorithm dedicated to continuous motion has been

introduced. The choice of the wavelet formalism sets-up a well-defined multi-

resolution framework that avoids most drawbacks of such usual approaches. Be-

ing associated to a gradient-based quasi-Newton optimization method, its low

complexity makes possible the estimation of the full range of scales composing

the motion. Moreover, high numbers of vanishing moments enable to truncate

the wavelet basis without increasing the error of the polynomial reconstruction,

thus significantly reducing the number of unknowns and the problem complexity.

A high-order regularization scheme, involving small scale coefficients penaliza-

tion, highly enhances estimation results and generally helps reducing errors by

removing noise of the solution, as emphasized by experiments on a synthetic

PIV sequence. Application to a real PIV sequence shows the capability of the

estimation method to reconstruct accurately vortices of large amplitude.

Conclusion

Page 12

12Pierre Dérian, Patrick Héas, Cédric Herzet, and Étienne Mémin

(a) (b)(c)(d)

Fig.3. Sample estimated motion fields from 2D planar jet PIV dataset: detail of input

PIV image (3a), streamlines (3b) and vorticity (3c). Figure 3d is the vorticity field

corresponding to motion estimated at the next time step. Three different areas are

visible: at the output of the jet (top of the field), shear regions begin to oscillate

slowly. The middle region clearly shows the development of vortices characteristic of

the Kelvin-Helmholtz instability. Finally, in the lower part of the field, structure of

vortices collapse due to their tri-dimensionalization.

Acknowledgments

The authors acknowledge the support of the French Agence Nationale de la

Recherche (ANR), under grant MSDAG (ANR-08-SYSC-014) "Multiscale Data

Assimilation for Geophysics".

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