Learning vector fields by kernels

Learning vector fields by kernels

Rener Castro, Marcos Lage, Fabiano Petronetto,Geovan Tavares, Thomas Lewiner and

Helio Lopes

MAT. 09/07

Cadastro de Pre–publicacao : MAT. 09/07

Tıtulo : Learning vector fields by kernels

Autores : Rener Castro, Marcos Lage, Fabiano Petronetto,Geovan Tavares, Thomas Lewiner and Helio Lopes

Palavras Chave : 1. Vector Field Recon-struction

2. Geometric Modeling

3. Machine Learning 4. Support Vector Regres-sion

Numero de paginas : 7

Data de submissao : Arpil 25th, 2007Endereco eletronico : http://www.mat.puc-rio.br/˜tomlew/vectorfieldsvr_sgp.pdf

Area de Concentracao : ¤Matematica Pura ¥Matematica Aplicada

Ambito : ¤ Nacional ¥ Internacional

Vınculado com a tese de : Rener Castro, Marcos Lage, Fabiano Petronetto, orientada porHelio Lopes, Geovan Tavares

Linha de Pesquisa do Artigo : Computacao Grafica

Projeto de Pesquisa (conforme relatorio CAPES) : Reconstrucao de objetos Geometricos

Projeto de Pesquisa vinculado (orgao financiador) : Edital CNPq UniversalOutras Informacoes Relevantes :

. .

. .

Breve Descricao do Projeto (em case de projeto novo) :

. .

. .

http://www.mat.puc-rio.br/~tomlew/vectorfieldsvr_sgp.pdf

Learning vector fields by kernels

RENER CASTRO, MARCOS LAGE, FABIANO PETRONETTO, GEOVAN TAVARES, THOMAS LEWINER ANDHELIO LOPES

Departament of Matematics — Pontifıcia Universidade Catolica — Rio de Janeiro — Brazilwww.mat.puc--rio.br/˜rener,mlage,fbipetro,tavares,tomlew,lopes.

Abstract. This paper proposes a novel technique for reconstructing a vector field from unstructured samples.Contrarily to surface reconstruction, which search for local and global coherence, vector field reconstructionmust determine a locally differentiable vector field from a very small number of samples. In this work, thisproblem is formulated as a machine-learning problem, training the machine on the samples and evaluating itover the whole domain. Using an adaptation of the Support Vector Regression method for multi-scale analysis, thismethod provides a global, analytical expression for the reconstructed vector field through an efficient non-linearoptimization. Moreover, it separates support vectors at the inhomogeneous regions without structuring the samples.Experiments on artificial and real data show a robust behavior of the proposed technique.Keywords: Vector Field Reconstruction. Geometric Modeling. Machine Learning. Support Vector Regression.

Figure 1: Reconstruction of a 3D vector field: (left) the support vectors of the norm and the first component with wall colorsaccording to the norm, (right) some streamlines with wall colors according to the magnitude and the angle.

1 IntroductionIn the last few years, a lot of attention has been given

to the problem of object reconstruction from sparse sam-ples. Since curves and surfaces are the most common ob-jects used in Graphics, a significant number of methodshave been proposed to solve their reconstruction problem.However, there are still very few reconstruction methods forvector fields, which is the fundamental object in classicalPhysics (velocity fields, force fields, etc.). Moreover, mostof these methods consider that the samples are structuredon a regular grid.

Sampled vector fields generally appear as measurementsof real phenomena, for example using Particle Image Ve-locimetry acquisition devices [12], or as results of physical

Preprint MAT. 09/07, communicated on Arpil 25th, 2007 to the Depart-ment of Mathematics, Pontifıcia Universidade Catolica — Rio de Janeiro,Brazil.

simulations, such as fluid flow simulations [3, 8, 9] or opti-cal flow images [5]. For all these examples, the vector fieldreconstruction problem consists in inferring a differentiablevector field on the whole region of experimentation fromonly a sampling of it (Figure 1). Reconstructing a sampledfield is a necessary step to analyze the field structure identi-fying the existence of vortices and singularities, to improvesimulations incorporating global information in local com-putations and to help interpreting the measured or simulatedfield returning numerical and visual information.

Contrarily to the curve or surface reconstruction prob-lem, vector field reconstructions require only local con-sistency. Moreover, vector field acquisition or simula-tion systems usually produce very sparse sampling. Wethus argue that the vector field reconstruction problemshould be solved in the Machine-Learning context. Kernel-based methods are considered the state of the art inmachine-learning. Among them, the Support Vector Ma-

Rener Castro, Marcos Lage, Fabiano Petronetto, Geovan Tavares, Thomas Lewiner and Helio Lopes 2

chines (SVM) proposed by Vapnik et al. [19] is one ofthe most robust, since they provide a deterministic, an-alytical result from an efficient non-linear optimization.Starting from a cost function which is insensitive to smallerrors, it reduces the learning process to a linearly con-strained quadratic programming problem, guaranteeing aunique and globally optimal solution. Moreover, the solu-tion is a combination of a reduced set of input, the supportvectors, which turns the evaluation of the SVM particularlyefficient.

Related works. Besides the radial-based interpolants andother approximation methods for multivariate functions[16], the problem of vector field reconstruction of unstruc-tured data compared to surface reconstruction methods isnot so present in the literature. Starting with the work ofMussa-Ivaldi [10] that presents a method for 2D vector fieldreconstruction using least squares schemes. In his strategy,the reconstruction is done in two steps. In the first step hereconstructs the rotational-free part of the velocity field, andin the second step, using the residual vector field, he recon-struct the divergence-free part. In this way, he has at the enda reconstructed vector field obtained by the sum of thesetwo parts. Another work for 2D vector field reconstructionfrom sparse samples was done by Lage et al. in [7]. In theirmethod the vector field is reconstructed by adjusting locallya polynomial for each coordinate and then by the use of apartition of unity the global approximation is obtained. Infact, this work is a generalization of the Multi-level Parti-tion of Unity for surface reconstruction proposed by Ohtakeet al. [11].

Two important surface reconstruction methods based onSupport Vector Machines have been proposed in the lastyears. Scholkopf et al. in [15] introduces a surface recon-struction scheme using the so called single class SupportVector Machine method. Steinke et al. in [18] presents amulti-scale method for surface reconstruction based on aSupport Vector Machine for regression. The Support VectorMachine for regression has also been recently used for op-tical flow reconstruction [2]. This paper proposes methodsthat extend these techniques to vector field reconstruction.

Contributions. This work presents a novel method forvector field reconstruction from sparse points/vectors pairs.This new method uses the Support Vector Machine forfunction estimation technique, called the Support VectorRegression (SVR), as the basic tool for learning (see sec-tion 2. The learning machine is trained on the samples andevaluated over the whole domain using an adaptation ofSVR for multi-scale analysis (see section 3). This methodprovides a single analytical expression for the reconstructedvector field on the whole domain. Experiments on artificialand real data in two and three dimensions show a robustbehavior of the proposed technique (see section 4).

2 Support Vector RegressionThe Support Vector Regression (SVR) is a universal

learning machine for solving multidimensional scalar valueprediction and estimation problems. It received a lot ofattention in the machine-learning community since it isvery well grounded on a statistical learning theory, calledthe VC-Theory [20]. Its consistency conditions, its conver-gence, its generalization abilities and its implementation ef-ficiency have been studied by several authors from the lastfour decades (see [20] and [14]). This Section describesthe ε-SVR method used in our method. A more detailedpresentation can be found in [17].

(a) ε-SVR learning problem

Let S be the training set, containing l samples S =(x1, y1), (x2, y2), . . . , (xl, yl), where xi ∈ Rn are theinput data and yi ∈ R are the target values. The SVRmethod first maps the data x ∈ Rn into some a priorichosen Hilbert space F , called the feature space, via anonlinear function φ : Rn → F . In this feature space, theprediction function is formulated by the affine equation:

f(x) = 〈w, φ(x)〉+ b (1)where 〈·, ·〉 denotes the inner product in F , w ∈ F andb ∈ R.

In the ε-SVR method, the problem of learning is to findthe best function, i.e. the value of w and b that minimizesthe following functional:

R(w, b) =1l

l∑

i=1

|yi − f(x)|ε +12〈w,w〉

where |a|ε =

0 if |a| < ε|a| − ε otherwise.

The solution f(x) of this ε-SVR problem minimizes thedeviation from |f(xi)− yi| for i = 1, . . . , l, while being asflat as possible. Observe that the deviation is controlled bythe loss function | · |ε, called ε-insensitive since it considersthe loss equal to zero when the deviation is less than ε. Theflatness is due to the second term of functional R(w, b),which penalizes the size of w.

ζ

ζ+ε

+ε

−ε

−ε

0

Figure 2: The ε-insensitive cost function for Support VectorRegression.

Preprint MAT. 09/07, communicated on Arpil 25th, 2007 to the Department of Mathematics, Pontifıcia Universidade Catolica — Rio de Janeiro, Brazil.

3 Learning vector fields by kernels

(b) ε-SVR optimization problem

In the ε-SVR learning method, the values of w and b aredetermined by the following minimization problem:

Minimizew,b12‖w‖2 + P ·∑l

i=1(ξi + ξi)

subject to: ξi, ξi ≥ 0 i = 1, . . . , l(〈w, φ(xi)〉+ b)− yi ≤ ε+ ξi i = 1, . . . , lyi − (〈w, φ(xi)〉+ b) ≤ ε+ ξi i = 1, . . . , l

where P is a constant parameter that penalizes the ε-insensitive errors (Figure 2). The errors occurring if f(xi)is above (resp. below) yi are represented by the ξi (resp. ξi)slack variables.

One can rewrite this optimization problem in its dualform, using of Lagrange multipliers αi, αi:

Maximizeαi,αil∑

i=1

(αi − αi)yi − εl∑

i=1

(αi + αi) −

12

l∑

i=1

l∑

j=1

(αi−αi)(αj−αj)〈φ(xi), φ(xj)〉

subject to:∑li=1(αi − αi) = 0

0 ≤ αi, αi ≤ P(2)

This is a convex quadratic programming problem, thusit has a unique global solution. Let denote the optimalsolution of the dual problem as: α∗ = (α∗1, α

∗2, . . . , α

∗l ),

α∗ = (α∗1, α∗2, . . . , α

∗l ).

The complementary Karush Kuhn-Tucker conditions forthis dual problem at the optimal solution are:

w −∑li=1(α∗i − α∗i )φ(xi) = 0

α∗i (〈w, φ(xi)〉+ b− yi − ε− ξi) = 0 i = 1, . . . , lα∗i (yi − 〈w, φ(xi)〉 − b− ε− ξi) = 0 i = 1, . . . , lα∗i · α∗i = 0, ξi · ξi = 0 i = 1, . . . , l(α∗i − P )ξi = 0, (α∗i − P )ξi = 0 i = 1, . . . , l

(3)These complementary conditions formulas show several

important and suitable properties of the ε-SVR learningmethod.

The first equation means that at the optimal solution w?

for the primal problem is a linear combination of the inputpoints mapped to the feature space. Since

w? =l∑

i=1

(α?i − α?i )φ(xi),

then equation (1) can be rewritten as:

f(x) =l∑

i=1

(α?i − α?i )〈φ(xi), φ(x)〉+ b?. (4)

where b∗ is chosen so that f(xi) − yi = −ε for any i suchthat α?i ∈ (0, P/l).

The other set of equations in (3) say that when α?i and α?iare both equal to zero the scalar function prediction for theinput point xi distances from the target value yi less than ε.The input points xi whose one of the associated α?i or α?idoes not vanish are called the support vectors. The readerwill see in Section 4 the role of the support vectors on theanalysis of the vector field reconstructed by the schemesthat are proposed in this work.

See Cristianini and Shawe-Taylor [6] and Smola andScholkopf [14] for more details about the ε-SVR and otherSVM methods for regression.

(c) Kernel functions

The priori chosen non-linear function φ, that maps theinput point to the feature space, appears on two places: oneis on the objective function of the ε-SVR dual optimizationproblem (2) as 〈φ(xi), φ(xj)〉, and the other is on theprediction function f on equation (4) as 〈φ(xi), φ(x)〉.Notice that in both cases it is sufficient to know how tocompute the inner-product of two points mapped to featurespace by φ. Thus, a suitable and efficient way to do that isthrough the use of the so called Kernel functions.

Kernel functions have been recognized as an importanttool in several numerical analysis applications, includingapproximation, interpolation, meshless method for solvingdifferential equations and in Machine Learning [16].

A kernel function K : Rn × Rn → R is defined asfollows:

K(z,w) = 〈φ(z), φ(w)〉.Using this definition, the prediction function in (4) is betterwritten as:

f(x) =l∑

i=1

(α?i − α?i )K(xi,x) + b?. (5)

In fact, kernel functions represent implicitly the mappingφ to the feature space F . For example, consider that zand w are in R2. Also consider the non-linear mappingφ : R2 → R3 as φ(z) = (z2

1 , z22 ,√

2z1z2). Then,

〈φ(z), φ(w)〉 = z21w

21 + z2

2w22 + 2z1z2w1w2 = (〈z,w〉)2.

Thus, in this case to compute the inner-product using φ,one has to evaluate the non-linear mapping at each 2D pointand, after that, compute their inner-product in R3. Insteadof that, one can obtain the same calculus in a more efficientby the use of the following kernel function K(z,w) =(〈z,w〉)2, which computes firstly the inner-product in R2

and then takes the square of it.In conclusion, it is more efficient and more suitable to

choose kernels rather than non-linear mappings φ. How-ever, not all functions K represent an inner-product in thefeature space. The Mercer´s theorem characterizes thesefunctions [20]. Some examples of kernel functions that sat-isfy the Mercer´s conditions are:



3(a): F1, color=magnitude 3(b): F2, color=angle 3(c): R, color=magnitude 3(d): S, color=angle

Figure 3: Comparing the reconstruction from Cartesian (left) and polar (right) coordinate system: the support vectors of the polarcoordinate system detect finer features of the field, while the Cartesian system is more accurate for the same SVR parameters.

– Polynomial kernel [19]: K(z,w) = (1 + 〈z,w〉)d.

– Gaussian kernel [19]: K(z,w) = e−‖z−w‖2

2σ2 .

– Wavelet kernel [21]: K(z,w) =n∏

i=1

h

(zi − wiσ

),

where h(u) = cos(1.75u)e−u22 .

– Fourier kernel [13]: K(z,w) =n∏

i=1

g

(zi − wiσ

),

where g(u) = 1−q2

2(1−2q cos(u))+q2 .

3 The Reconstruction Method(a) Sampled vector fields

A vector field F defined on a subset Ω ⊂ Rn is amap that assigns to each point x ∈ Ω a vector on Rn. Inthe Cartesian coordinate system, the vector field is repre-sented by an ordered n-tuple of scalar functions F(x) =(F1(x), F2(x), . . . , Fn(x)). The function Fi is called the i-th coordinate function of F. A vector field is differentiablewhen all of its n coordinate functions are differentiable.

The aim of this paper is to provide a differentiable vectorfield F : Ω → Rn that approximates the vector field F onthe region Ω by the use of a learning-machine method basedon the ε-SVR. As an input of the reconstruction problem,it is considered a set of l pairs of n dimensional pointsS = (x1,y1), (x2,y2), . . . , (xl,yl) sampled from F,which means that xi ∈ Ω and yi = F(xi). It is assumedthat both xi and yi are sampled on the same basis of theCartesian coordinate system.

(b) Learning 2D vector fields

There are two classical ways to represent a vector v ∈R2. One is using the Cartesian coordinates v = (v1, v2)and the other is using the polar coordinates (r, θ), for r ∈[0,∞) and θ ∈ [0, 2π) (Figure 3). The equality (v1, v2) =‖v‖(cos θ, sin θ) is used to convert one system into theother. Thus, we propose two methods for learning 2D vectorfield F : Ω ⊂ R2 → R2, one for each coordinate system.

Learning in Cartesian coordinates. The first 2D vectorfield learning method determines, from the sampling setS, the prediction function F by learning each coordinatefunction of F. That means that this method solves twoε-SVR machine learning problems, one to find F1 thatapproximates F1 and other to find F2 that approximatesF2. According to this, the approximation for the vector fieldfunction F is obtained by the formula:

F2Dc(x) = (F1(x), F2(x)),

where Fj(x) =l∑

i=1

(α?i,j − α?i,j)K(xi,x) + b?j .

Learning in polar coordinates. The second 2D methoddetermines from the sampling set S the prediction functionF by learning three functions:

R(x) = ‖F(x)‖, C(x) =F1(x)‖F(x)‖ , and S(x) =

F2(x)‖F(x)‖ .

The first represent the norm of the vector F(x), the secondrepresents the cosine of θ and the last the sine of θ. The ap-proximation for these three functions, respectively namedR, C, and S, are also expressed using equation (5).

A question that arises now is: why is it better to learn thecosine of θ and the sine of θ instead of the argument angleθ itself? The answer is: because doing that it is avoided thediscontinuity of the argument θ on values very close to zeroor to 2π. However, with this strategy it is not possible toguarantee that C2(x) + S2(x) = 1. To correct this fact,the following adjustment is proposed: the predicted point(C(x), S(x)) is projected on the unit circle by a simplenormalization procedure. Notice that the projected pointhas the same argument of the original. According to this,the adjusted point is:

(C(x), S(x)) =(C(x), S(x))√C2(x) + S2(x)

.



Figure 4: The effect of the ε parameter: decreasing ε (from left to right) fits the reconstruction closer to the input data, but harmsthe smoothness of the reconstructed field.

As a conclusion, the approximation for the vector fieldfunction F using this second strategy is obtained by:

F2Dp(x) = (R(x)C(x), R(x)S(x)).

(c) Learning 3D vector fields

The learning methods proposed to 2D vector field recon-struction are easily generalized to 3D as following.

Learning in Cartesian coordinates. The approximationmethod for a 3D vector field function F : Ω ∈ R3 → R3

using Cartesian coordinates learns from the sampling set Seach coordinate function individually. Thus, at the end it isobtained the following formula for the estimator:

F3Dc(x) = (F1(x), F2(x), F3(x)),

where Fj(x) =l∑

i=1

(α?i,j − α?i,j)K(xi,x) + b?j .

Learning in spherical coordinates. In spherical coor-dinates, a vector v = (v1, v2, v3) ∈ R3 is repre-sented by the triple (r, θ, γ), for r ∈ [0,∞), θ ∈[0, 2π) and γ ∈ [0, π). The equality (v1, v2, v3) =‖v‖(cos θ sin γ, sin θ sin γ, cos γ) is used to convert fromthe Cartesian to the spherical coordinate system and vice-versa.

Similarly to the polar coordinates in 2D, the approxima-tion method for a 3D vector field function F : Ω ∈ R3 →R3 learns from the sampling set S the functions:

R(x) = ‖F(x)‖, C(x) =F3(x)‖F(x)‖ ,

CS(x) =F1(x)‖F(x)‖ , and SS(x) =

F2(x)‖F(x)‖ .

R(x) represents the norm of the vector F(x), C(x) rep-resents the cosine of γ, CS(x) represents the cosine of θtimes the sine of γ, and, finally, SS(x) represents the sineof θ times the sine of γ. The approximation for these fourfunctions, respectively named R, C, CS and SS, are alsoexpressed using equation (5).

In order to adjust the prediction of C, CS, and SS tosatisfy the identity

CS2(x) + SS2(x) + C2(x) = 1

the following adjustment is used:

(CS(x), SS(x), C(x)) =(CS(x), SS(x), C(x))√C2(x) + CS

2(x) + SS

2(x)

.

As a result, the approximation for the 3D vector fieldfunction F using the spherical coordinate strategy is ob-tained by:

F3Ds(x) = (R(x)CS(x), R(x)SS(x), R(x)C(x)).

(d) The parameters for the ε-SVR regression

For all methods discussed above the user has to choosethe following parameters:

– The value of the tolerance ε.– The value of penalizing constant P .– The kernel function K and its corresponding param-

eters. For example, the Gaussian kernel is parameter-ized by σ.

Large ε generally results in a very smooth approximationand a small number of support vectors. Small ε results onbetter approximations since it will use almost all points(Figure 4). Small P generates smooth reconstructions thatapproximate loosely the training data. For large P , theapproximation fits the vector field very close to the trainingset, but this may harm the prediction elsewhere.

Usually, Gaussian kernel is a very nice choice. As sug-gest by Steinke et al. [18] a good initial choice of its pa-rameter σ is the half of the diameter of the bounding box ofthe points. In order to have a control of these parameters,the coordinates of the input vectors xi and the coordinatesof the target values yi are all normalized, that means theirvalue is subtracted by the mean and the result is divided



5(a): Amplitude at single scales = 4.

5(b): Amplitude in multi-scales = 1..4.

5(c): Phase in multi-scale s =1..4.

0.0001

0.001

0.01

0.1

1

10

0 1 2 3 4 5 6 7

x multiscaley multiscale

x singlescaley singlescale

5(d): Comparison of the mean absolute er-ror at different scale.

Figure 5: Single-scale regression versus multi-scale regression: several scales at the same time improves the reconstruction.

by the standard deviation. After the prediction the valuesare transformed back using the inverse process. Using thisstrategy, an initial suggestion is to set P = 1 and ε = 0.01.

Another option to deduce good parameters is to separatethe samples into a training set, used to compute the vec-tor field F and a validation set used to tune the SVR. Thiscross-validation can be easily implemented by pre-selectingvalues for each SVR parameter and perform a regression foreach combination of these values. The parameters generat-ing the best prediction on the validation set are retained.

(e) Multi-scale ε-SVR regression

Steinke et al. in [18] propose the use of kernels of differ-ent sizes in order to improve the surface reconstruction, forexample to interpolate across holes. Their scheme providesan approximation with enough variability to capture the de-tails while guaranteeing good results in large distances. Ituses a coarse-to-fine approximation. In the first scale, it cap-tures basically the sign of the scalar function and on thesubsequent scale levels it approximates the residual errors.Since they are using a kernel that is a radial basis function,at each level the scale is divided by two. The points thathave the desired error tolerance are not considered on thenext level learning procedure. The final function is givenby the sum of the functions obtained at each level.

Following their ideas, an adaptation of this multi-scalemethod is proposed to improve the approximation resultsof the vector field reconstruction. The scheme is basicallythe same, it differs in the following points: First, instead ofdiscarding points xi that have the desired error tolerance. Inthe proposed scheme, the residual vector is set to zero if thedistance from yi to F(xi) is less than ε. Second, the ε pa-rameter is also divided by two at each level, correspondingto the physical paradigm that higher frequencies generallyhave lower amplitudes.

4 Implementation and ResultsWe implemented the proposed vector field reconstruc-

tion method using a home-made SVR. We use Sequential

Figure 6: Mean absolute error distribution over 112 samplingof the same analytical field: the estimation of F1 have similarbehaviors independently of the sampling, provided it has thesame average density.

Minimal Optimization (SMO) for the quadratic optimiza-tion problem, based on the open libSVM [1, 4]. All the SVRinputs, including the errors in the multiscale regression, arenormalized to a normal distribution before being processed.In our implementation, the regression is not (yet) optimizedfor consecutive runs such as multiscale regression.

Results. We tested our reconstruction on different fields:synthetic, such as the ones of Figures 1 and 3, or real datafrom PIV measures [12], such as Figures 4, 5, 8 and 7.

On synthetic fields, the vector field is reconstructed froma random, unstructured sampling and the error is measuredby averaging the absolute difference at points on a regu-lar grid. We can observe on Figure 6 that, for a reasonablesampling density, the prediction error is well behaved withrelation to a particular sampling.

On real fields, the data is often given on a regular grid,and the error is measured by the prediction error on thetraining set. In this context, the multiscale approach im-proves on single-scale prediction (Figure 5).

The role of the support vectors. The support vectors tendto fit features in the reconstruction, as can be seen on Fig-ure 3. This can be useful for concise representations ofcomplex fields, such as the one of Figure 1. Moreover, thisgives a previous analysis of the field before an eventual rel-



atively long-lasting evaluation. The number of these sup-port vectors heavily depends on the parameters of the SVR,in particular ε and P . This fact can be useful for a multi-resolution analysis of these support vectors, such as the oneimplicitly performed in the multiscale context.

Limitations. The main limitation of the proposed methodis the execution time on huge data. Since the reconstructionis global, it requires a global optimization which harmsits efficiency. This could be partially solved by factoringresults for repetitive regression, or by using local solutionsfor good initial guess of the SMO quadratic solver.

Figure 7: Reconstruction of a real 3D velocity field capturedby PIV: magnitude (left) and phase (right).

5 ConclusionsThis paper proposes to solve the vector field reconstruc-

tion problem in a Machine Learning context. Using the con-secrated Support Vector Regression scheme, we achievefaithful reconstruction on synthetic and real data. More-over, the reconstruction is statistically stable with respectto a specific sampling. A multiscale variation of the methodimproves its robustness on real data.

With this approach, the reconstructed field is globaland analytical. This suits for vector field analysis involv-ing derivatives, which can be directly calculated from thederivatives of the kernel. Moreover, support vectors appearclose to the field features, complementing the analysis.

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Figure 8: Cartesian (left) vs. polar (right) reconstruction: thecolor maps the cosine of the field phase, with the correspond-ing support vector for ε = 0.5.


Learning vector fields by kernels

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