Keywords: image alignment, registration, visual tracking, surface, rigid, deformable, direct method, efficient, fast, real time, optimization alignement d'images, suivi visuel, surface, rigide, déformable, méthode directe, efficace, rapide, temps réel, optimisation, ESM, second ordre

Image-based deformations are generally used to align images of deformable objecys moving in the 3D space. For the registration of deformable objects, this assumption has shown to give good results. However it is not satisfying for the registration of images of 3D rigid objects as the underlying structure cannot be directly estimated. The general belief is that obtaining the 3D structure directly is difficult. In this article, we propose a parameterization that is well adapted either to align deformable objects or to recover the structure of 3D objects. Furthermore, the formulation leads to an efficient implementation that can considerably reduce the computational load. Experiments with simulated and real data validate the approach for deformable object registration and 3D structure estimation. The computational efficiency is also compared to standard methods.

Introduction

Image registration (or alignment) of rigid and deformable surfaces is an active field of research and has many applications for example in medical imagery, augmented reality or robotics. In this article, we focus on iterative methods that are potentially real-time. We also focus on methods that do not rely on any off-line learning step as for example [START_REF] Gleicher | Projective registration with difference decomposition[END_REF][START_REF] Cootes | Active appearence models[END_REF][START_REF] Black | Eigen-tracking: robust matching and tracking of articulated objects using a view-based reprsentation[END_REF][START_REF] La Cascia | Fast, reliable head tracking under varying illumination: an approach based on registration of textured-mapped 3D models[END_REF][START_REF] Fleet | Design and use of linear models for image motion analysis[END_REF].

Image registration methods can be roughly classified between feature-based and direct methods. Features-based registration generally consists in extracting features in images and then finding the correspondences based on descriptors. These associations can then be used to recover the deformation of the underlying object. The strength of features-based methods reside in the possibility of working on large displacements. The difficulty is to ensure correct associations at a low computational cost. However, recent advances for matching features robustly and efficiently have lead to real-time registration of non-rigid objects [START_REF] Pilet | Real-time non-rigid surface detection[END_REF].

Direct approaches [START_REF] Irani | All about direct methods[END_REF] minimize a similarity measure between a reference template and a region of the image to register warped with appropriate geometric and photometric parameters. The underlying assumption is that the deformations between two views of the surface are small. This will typically be the case in video sequences or after an initialization by feature matching. The main advantage of dense registration is accuracy.

Initial work mainly focused on registering planar rigid surfaces [START_REF] Lucas | An iterative image registration technique with application to stereo vision[END_REF][START_REF] Shum | Construction of panoramic image mosaics with global and local alignment[END_REF] with the iterative Gauss-Newton minimization of the sum of squared differences (SSD) between a reference template and a template in a new image. The same optimization approach can be used for the direct alignment of deformable surfaces [START_REF] Bartoli | Direct estimation of non-rigid registrations[END_REF]. The contribution of this article is in the field of direct parametric methods. We propose a parameterization that is well adapted either to deformable object tracking or to the registration of complex 3D surfaces with the direct recovery of the 3D structure. We also focus on the problem of the efficiency of the registration.

One important step towards real-time applications with fast frame rate has been improving the efficiency of the Gauss-Newton optimization method. Two approaches are possible for building efficient algorithms. The first one is to keep the same convergence rate (the number of iterations needed to obtain the minimum of the similarity measure) while reducing the computational cost per iteration. This can be achieved by pre-computing partially [START_REF] Hager | Efficient region tracking with parametric models of geometry and illumination[END_REF] or completely [START_REF] Baker | Equivalence and efficiency of image alignment algorithms[END_REF] the Jacobian used in the minimization. The main limitation of these approaches is that, contrarily to [START_REF] Lucas | An iterative image registration technique with application to stereo vision[END_REF], they can only be applied to certain warps. Furthermore, Baker et.al. [START_REF] Baker | Lucas-kanade 20 years on: Part 5[END_REF] have shown that in the case of surfaces in the 3D Cartesian space the convergence rate of the inverse compositional algorithm is not equivalent to the convergence rate of [START_REF] Lucas | An iterative image registration technique with application to stereo vision[END_REF]. An alternative approach for building efficient algorithms is to keep the same computational cost per iteration while increasing the convergence rate. This can be achieved for example by using an efficient second-order minimization method [START_REF] Benhimane | Real-time image-based tracking of planes using efficient second-order minimization[END_REF]. This approach has been applied to the estimation of a homography for the visual tracking of planar surfaces. In this paper, we investigate how to extend this approach to the visual tracking of continuous surfaces. We propose a flexible and efficient algorithm that can be used for the alignment of rigid and deformable surfaces. Compared to existing techniques, a great efficiency is obtained by reducing the number of iterations needed to converge to the minimum of the similarity measure. Experiments with simulated and real data validate the technique for the efficient visual tracking of 3D surfaces.

Modeling

Camera and transformation models

We consider a pinhole camera. A 3D point m = (x, y, z, 1) projects onto the image point p = (u, v, 1) as follows:

p ∝ K 0 m ( 1
)
where K is the upper triangular matrix containing the camera intrinsic parameters. The camera displacement is represented by a (4×4) matrix T homeomorphic to SO(3) × R 3 :

T = R t 0 1 (2)
where R ∈ SO3 is the rotation matrix and t ∈ R 3 is the translation vector. The 3D point m is eventually deformed to the point m in the same reference frame. The new 3D point projects into a new image point p = (u , v , 1):

p ∝ K 0 T m (3)
The camera intrinsic parameters K of the new image may be different from the intrinsic parameters of the reference image. We divide the deformation of the 3D point into two parts:

m = 1 α m + β (4)
where, in the first part, α ∈ R + is a scale factor that take into account projective deformations only (i.e. deformations that change the 3D structure of the object in the reference frame but do not change the reference image) and in the second part, β = (β x , β y , 0, 0) ∈ R 2 is a vector that take into account the remaining deformations. If the surface is rigid we obviously have α = 1, β = 0 and m = m. Plugging equations ( 1) and (4) into equation (3) we obtain:

p ∝ H (p + δ) + ρ e (5)
where H = K RK -1 is the homography of the plane at infinity, e = K t is the epipole in the reference image, ρ = α/z is a projective depth and δ = (δ u , δ v , 0) is an image coordinates deformation vector. The homography matrix is invertible (det(H) = 0) and it is defined up to a scale factor. Thus, we can normalize the matrix such that H ∈ SL(3) (det(H) = 1). When the camera is not calibrated, we cannot measure directly the transformation matrix T but the following (4×4) matrix

Q homeomorphic to SL(3) × R 3 : Q = H e 0 1 (6)
When the 3D surface is rigid δ = 0 and ρ is constant. However, the values of ρ are a function of the image coordinates p. Since ρ is unknown, we can let it vary with respect to time and this allows to compute projective deformations without any additional cost. When the deformation of the 3D surface is any we add the unknowns δ. Again, the values of δ are functions of the image coordinates p. Finally, let us remark that when the object stops to deform we can fix ρ and δ once and for all and the matrix Q becomes the only unknown.

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Surface and deformation models

We suppose that the camera observe a continuous textured surface of the 3D Cartesian space. In some cases we may have a simple parametric model of the 3D surface. For example, a planar surface can be simply modeled with three parameters (the scaled normal vector to the plane). We can model the projective depths of a surface point corresponding to each pixel p with the following function:

ρ = f (p; s) (7)
where s is a vector containing the parameters defining ρ. When this model is unknown we propose two methods to approximate it. In both cases, the first step is to select q image points c k (k = {1, ..., q}) called centers. The number of centers q depends on the complexity of the surface we want to approximate. Centers can be dynamically added or removed.

In the first method we suppose that our unknown parameters in the vector s are the projective depths of the surface points that projects onto the centers: s = (ρ 1 , ρ 2 , ..., ρ q ). The remaining unknown projective depths for all pixels in the area of interest are computed by a bilinear or bi-cubic interpolation.

A second method we propose for approximating the surface ρ is to use Radial Basis Functions [START_REF] Carr | Surface interpolation with radial basis functions for medical imaging[END_REF]. We chose for example thin-plate splines and a first degree polynomial. In this case, the projective depth of point p is computed as follows

f (p; s) = γ p + q k=1 λ k φ( p -c k ) (8)
where γ = (γ 1 , γ 2 , γ 3 ) and λ = (λ 1 , ..., λ q ) are unknown parameters and φ(r) = r 2 log(r). To compute the parameters of the RBF we need to impose the side conditions ans the interpolation conditions [START_REF] Carr | Surface interpolation with radial basis functions for medical imaging[END_REF]. Let C = [c 1 , c 2 , ..., c q ] the (3×q) full rank matrix containing all the coordinates of the centers. The side conditions to be added to equation [START_REF] Fleet | Design and use of linear models for image motion analysis[END_REF] are C λ = 0. This implies λ ∈ ker(C ). Let v j (j ∈ {4, ..., q}) be a vector basis of ker(C ) which can be obtained from the SVD of C . Then we have λ = q j=4 γ j v j . In this case, the vector s contains the parameters defining the RBF: s = (γ 1 , γ 2 , ..., γ q ). The interpolation conditions f (c k ; s) = ρ k can be imposed directly if we can measure ρ k or indirectly by minimizing a similarity error in the image (see section3).

When the 3D surface is deformable we need also to approximate the two functions f u and f v defining the deformations of the image coordinates in the reference frame:

δ =   f u (p, s u ) f v (p, s v ) 0   (9)
where s u and s v are the corresponding parameters. The two approximation methods described above can be used to approximate the functions but we have found that RBF provide better results thanks to their regularization properties. In the experiments, we have tested both approximation methods obtaining similar results for rigid surfaces. We have chosen arbitrarily to select the centers either on a regular grid or centered on interest points. When the interest points were well distributed in the area of interest we obtain similar results.

Unified warping function model

We define a set of parameters η (composed of the camera and surface parameters) needed to align the two images of the surface. From equation ( 5) we define a warping function w(•; η) as a function

P 2 → P 2 : p = w(p; η) (10)
We let H be any homography matrix and we consider ρ as a projective depth. Even if it cannot be computed explicitly, we suppose that the inverse warping function p = w -1 (p ; η) exists. For rigid objects, we set δ = 0 and this warping function is able to capture all the possible projective deformations. For example if the surface is a plane then we can set ρ = 0 and the warp reduces to the standard homographic warp. For deformable surfaces we can capture all projective deformations by letting the parameters s vary with respect to time. The generalization of the warping to deal with any deformation is thus straightforward by considering δ varying with time. In the particular case when the surface is rigid and the cameras are calibrated, i.e. K and K are known, we can set η = (T, s) homeomorphic to the Lie group SO(3) × R q . In the general case, we set η = (Q, s) homeomorphic to the Lie group SL(3) × R 3q . The proposed unified warping function is well adapted for the efficient image alignment of both rigid and deformable surfaces. On the other hand, standard image-based warping maps like for example the one used in [START_REF] Bartoli | Direct estimation of non-rigid registrations[END_REF] are not adapted to the efficient alignment of rigid surfaces. Since they use a warping function that is composed by an affine transformation part and a non-linear one, much more parameters are needed to capture all projective transformations. Consider for example the case of the simple alignment of a planar surface that can be done by estimating 8 parameters only (a homography matrix). The affine part (6 parameters) cannot capture the projective deformations, thus the non-linear part is used. More than 2 parameters are needed to capture the remaining deformations. On the other hand, our warping map is able to capture all possible projective deformation using the 8 homography parameters and the non-linear part of the RBF is not needed.

Image alignment

Let us assume that we have acquired two images I and I of the same surface. We make the standard assumption that the changes in intensity are only due to camera (or surface) motion. The standard "brightness constancy assumption" can be reformulated as follows. There exists an optimal η such that the image I can be warped back to exactly match the reference image:

I (w(p; η)) = I(p) (11)
We suppose to have a prediction η of η. In visual tracking applications the prediction can be provided by a filter or simply the parameters computed at the previous image. We search for the unknown η such that η • η = η. The composition law depends on the group we are considering (see section 2.3). We suppose that the increment η is close to the identity element e of the simply connected Lie group. We can thus parametrize η via the exponential map η = η( x), where x are coordinates in INRIA the Lie algebra [START_REF] Warner | Foundations of Differentiable Manifolds and Lie Groups[END_REF]. Finally, we need to solve the following non-linear equations (one equation per pixel):

I (w(p; η • η(x)) = I(p) (12)
This non-linear system of equation is generally over-constrained and it may not have an exact solution in the presence of measure and modeling errors. Instead, a least-squares problem is iteratively solved by extending the efficient second-order minimization proposed by Benhimane and Malis [START_REF] Benhimane | Real-time image-based tracking of planes using efficient second-order minimization[END_REF] to 3D surfaces.

Efficient second-order minimization

We stack all the n equations ( 12) in a (n × 1) vector y(x). The second-order Taylor expansion of y(x) about x = 0 is:

y(x) = y(0) + J(0)x + 1 2 M(x)x + r T ( x 3 ) (13)
where

M(x) = (x H 1 , x H 2 , ..., x H n ) is a (n × m
) matrix containing the symmetric Hessians matrices and r T ( x 3 ) is a third-order Taylor remainder. Similarly, the first-order Taylor expansion of J(x) about x = 0 is:

J(x) = J(0) + M(x) + R T ( x 2 ) (14)
where R T ( x 2 ) is a second-order (n × m) remainder matrix. Plugging equation ( 14) into equation ( 13) we obtain an exact third order expansion of y(x) without the second order terms:

y(x) = y(0) + 1 2 (J(0) + J(x)) x + r( x 3 ) (15)
indeed, r( x 3 ) = r T ( x 3 ) -R T ( x 2 )x/2 is a third-order remainder. Thus, a second-order approximation of y( x) is:

y( x) ≈ y(0) + 1 2 (J(0) + J( x)) x (16)
Since the system of equation is over-constrained, we solve the following least squares problem:

min y(0) + 1 2 (J(0) + J( x)) x 2 (17)
The Jacobian J(0) can be completely computed from image data. The problem is that in the general case x is needed to compute J( x). In [START_REF] Benhimane | Real-time image-based tracking of planes using efficient second-order minimization[END_REF] the authors use the following property J( x) x = J x where the Jacobian J can be measured from image data directly. However, this property is valid only if the warping map defines a group action on P 2 . When estimating the 3D surface structure, this is not the case. In section 3.3, we show that only a part of the Jacobian needs to be approximated. We found experimentally that this approximation is worthwhile as it improves greatly the convergence rate over any first-order approach. Let J ≈ J( x), the solution of the problem (17) provides the increment:

x = -2 J(0) + J + y(0) (18)
where + denotes the matrix pseudo-inverse. The update of the estimated parameters is η ← η•η( x).

Computation of J(0)

The Jacobian J(0) is defined as follows:

J(0) = ∇ x I(w(p; η • η(x))| 0 (19)
It can be written as the product of 3 Jacobians using the chain derivation rule:

J(0) = J I J w ( η)J x (0) (20)
Indeed, equation ( 19) can be expanded as follow:

J(0) = ∇ x I(w(w -1 (w(p; η • η(x)); η); η) 0 (21)
Using the chain rule for derivatives we obtain:

J(0) = J I ∇ x w -1 (w(p; η • η(x)); η) 0 (22)
The first Jacobian contains the spatial derivatives of the image warped with η:

J I = ∇ q I(w(q; η))| p (23)
Now we have to compute the second Jacobian. In the particular case when the surface is planar it is possible to compute directly w -1 (w(p; η • η(x)); η). In the general case, the function w -1 (•; η) is not easy to find. Thus, we need to expand the Jacobian further:

∇ x w -1 (w(p; η • η(x)); η) 0 = (24)
= ∇ r w -1 (r; η)

w(p; b η) ∇ x w(p; η • η(x))| 0
We don't need to compute the inverse map w -1 (•; η) directly since, by a change of variable:

∇ r w -1 (r; η) w(p; η) = ∇ q w(q; η)| p -1 (25)
the other Jacobian can be written:

∇ x w(p; η • η(x))| 0 = ∇ η w(p; η • η) e ∇ x η(x)| 0 (26)
and finally we have:

J w ( η) = ∇ q w(q; η)| p -1 ∇ η w(p; η • η) e (27)
J x (0) = ∇ x η(x)| 0 ( 28
) INRIA

Computation of J( x)

The Jacobian J( x) is defined as follows:

J( x) = ∇ x I(w(p; η • η(x)))| e x (29)
It can be written as the product of 3 Jacobians using chain derivation rule:

J( x) = J I J w (η)J x ( x) (30)
Indeed, equation ( 29) can be expanded as follows:

J( x) = ∇ x I(w(w -1 (w(p; η • η(x)); η); η) e x (31)
Using the chain rule for derivatives we obtain:

J( x) = J I ∇ x w -1 (w(p; η • η(x)); η) e x ( 32
)
The first Jacobian contains the spatial derivatives of the image warped with η (i.e. the reference image):

J I = ∇ q I(w(q; η))| p (33)
The second Jacobian is expanded again:

∇ x w -1 (w(p; η • η(x)); η) e x = (34)
= ∇ r w -1 (r; η) w(p;η) ∇ x w(p; η • η(x))| e x
Again, the Jacobian of the inverse warp is:

∇ r w -1 (r; η) w(p;η) = ∇ q w(q; η)| p -1 (35)
and the second Jacobian can be written:

∇ x w(p; η • η(x))| e x = (36) = ∇ η w(p; η • η) e ∇ x η -1 • η(x) e x
Finally, we have:

J w (η) = ∇ q w(q; η)| p -1 ∇ η w(p; η • η) e (37)
J x ( x) = ∇ x η -1 • η(x) e x (38)
The Jacobian J w (η) cannot be exactly computed since η is unknown. However, η ≈ η and we can use the Jacobian J w ( η) instead. The last Jacobian J x ( x) verifies the following property J x ( x) x = J x (0) x from the Lie algebra parameterization [START_REF] Benhimane | Real-time image-based tracking of planes using efficient second-order minimization[END_REF]. Finally, J = J I J w ( η)J x (0) and the second-order increment (18) can be written:

x = -2 ( (J I + J I )J w ( η)J x (0) ) + y(0) (39)
The computational cost of the second-order approximation is equivalent to the cost of the Gauss-Newton method (the average of J I and J I is negligible with respect to the computation of the pseudo-inverse).

Experiments

We applied the proposed image registration to the visual tracking of rigid and deformable surfaces. We select a template in the image I 0 and then we align I 1 . To initialize the minimization we suppose initially that the observed surface is a 3D plane parallel to the image plane. The parameters estimated in the alignment of images I 0 and I k are used as a starting point for the alignment of images I 0 and I k+1 . We can obviously use a filter for prediction and smoothing. When the surface is rigid, the parameters s are constant and a simple Kalman filter with constant position should work well. For deformable surfaces an appropriate motion model for the parameters s should be selected.

We have successfully tested our algorithm on several video sequences. The image sequences are available at ftp://ftp-sop.inria.fr/icare/malis/software/ESM/. The brightness constancy assumption is often violated in the sequences. We found it sufficient to normalize the images at each iteration in order to take into account ambient light changes. However, more complex photometric models could easily be included to our approach.

Comparison with the ground truth

The proposed approach is tested with a sequence with known ground truth. A video sequence has been simulated by warping an image onto a sphere. The sphere has a radius of 30 cm and its center is initially at 1 meter in front of the camera. In the simulation, the camera is calibrated and we can use η = (T, s) homeomorphic to the Lie group SO(3)×R q . We compare the second-order minimization with the Gauss-Newton method. A (400×400) template is selected (see Figure 1). The centers for the surface approximation are placed on a regular (5×5) grid. To simulate a real-time experiment we fixed the number of iterations of each algorithm to 5. With our unoptimized Matlab code this corresponds to a fixed time of 5 seconds per image (we have 31 parameters to estimate). INRIA Despite the simple spheric structure of the surface, the standard Gauss-Newton fails to register the images since it does not have enough iterations to converge (with 10 iterations/image the minimization works fine). Figure 2 shows the registration results after 40 images. The top row shows that the regular grid is not transformed accordingly to a spheric surface and the bottom row shows that the area of interest is not correctly registered with respect to the reference template. On the other hand, using the efficient second-order minimization the images are correctly registered. The average RMS error for the registration of the 40 images is 4.9 gray-levels (over 256). Figure 3 shows the registration results after 40 images. The last registered area of interest is correctly aligned with the reference template. Thus, we are able to directly recover the surface up to a scale factor. Obviously, the precision of the reconstruction depends on the translation made by the camera. Indeed, if the camera motion is a pure rotation the structure is not observable. However, the image alignment is correctly performed.

Figure 4 displays two 3D views of the reconstructed sphere after computing the unknown scale factor from the ground truth. The mean error on the depths corresponding to all the pixels of the template is 1.7 mm while the standard deviation is 1.4 mm (the true radius of the sphere was 300 mm). In this experiment we track a smooth vault which is painted in still-life deception. We test our algorithm on two video sequences acquired with an uncalibrated camera. The first sequence is composed by 396 images we selected 99 centers placed on a (11×9) regular grid. We estimate 110 parameters. In figure 6 we show 5 images extracted from the sequence. The first image in the top row shows the regular grid. The last image in the top row corresponds to the last image of the sequence. In the bottom row, the first image show the template and the other images are the current images warped in the first frame with the estimated parameters. All the images were correctly registered and the average intensity RMS error over all the sequence was around 6.6 gray levels (over 256). At the end of the sequence we self-calibrated the camera parameters using the information from the registered images. In figure 5, is displayed the surface reconstructed up to a scale factor. The average number of iterations per image was 6. The second experiment is performed on a different sequence containing 214 images of the same vault acquired from a different point of view. The results are shown in figure 7. The centers are now selected on a (11×9) regular grid.

Experiment "Basketball"

In this experiment we track a basketball in a sequence of 694 images acquired with an uncalibrated camera. The surface does not have too much texture. Thus, if we select the centers on a regular grid and we do not use the RBFs we obtain poor results. The best results are obtained by using the RBFs thanks to their regularization properties. We have selected a (4×4) regular grid (see Figure 8). The bottom row shows the area of interest registered with respect to the template. The average RMS error is 13.6 gray-levels (over 256) but there were several changes of illumination and we did not take them into account in the registration. The algorithm is robust enough to handle these changes. The average number of iterations per image is 7.

Experiment "Vase"

In this experiment we track a vase in a sequence of 450 images acquired with an uncalibrated camera. We select the centers on a regular (6×6) grid and we use again RBFs for surface approximation. Figure 9 shows a selection of five images from the total results. The registration is correctly performed for all images. The average RMS error is 13.8 gray-levels (over 256) but there were again several changes of illumination. These changes are visible in the second row of Figure 9 where the area of interest reprojected in the reference frame appears brighter at the end of the sequence. The average number of iterations per image is 6. In this experiment we track a balloon with an uncalibrated camera. The sequence has 1082 images. We select a (262×262) template and the centers are placed on a 4 × 4 grid. In this experiment we do not use RBFs for surface approximations. We use bi-cubic interpolation to approximate the surface given the centers. We found it sufficient to estimate projective deformations only (i.e. we set δ = 0). Thus, we estimate 27 parameters which leads to a fixed time of 3.5 seconds per image (5 iterations/image) with our unoptimized Matlab code. The bottom row of figure 10 shows that all the images of the sequence have been correctly aligned with the reference template despite the strong change in size of the balloon and its deformation. The average intensity error over all the sequence is around 5.6. The deformation of the balloon surface is shows in the top row of figure 10 by the deformation of the regular grid.

Experiment "Magazine"

In this experiment we track a magazine with an uncalibrated camera. The sequence has 400 images. We select a (251×201) template and the centers are placed on a 6 × 5 grid. In this experiment we use again bi-cubic interpolation to approximate the surface given the centers and estimate projective deformations only. Thus, we estimate 31 parameters. The Figure 11 shows that all the images of the sequence have been correctly aligned with the reference template.

Experiment "Corkes"

In this experiment we track a sheet of paper with an uncalibrated camera. The sequence has 1365 images. We select a (251×201) template and the centers are placed on a 6×5 grid. In this experiment we use RBFs for surface interpolation and we estimate all possible deformations (i.e. δ = 0). Thus, we estimate 98 parameters. The Figure 12 shows that all the images of the sequence have been correctly aligned with the reference template. The average intensity error over all the sequence is around 14.5.

Experiment "Newspaper"

In this last experiment, we have tested our algorithm on a sequence used in [START_REF] Bartoli | Direct estimation of non-rigid registrations[END_REF] and available at http://www.lasmea.univ-bpclermont.fr/Personnel/Adrien.Bartoli/Research/NonRigidRegistration/ Data/Newspaper 01/. The sequence has 224 images. We select a (153×115) template and the centers are placed on a 5 × 4 grid. In this experiment we use RBFs for surface interpolation and we estimate all possible deformations (i.e. we set δ = 0). Thus, we estimate 68 parameters. The Figure 13 shows that all the images of the sequence have been correctly aligned with the reference template. The average intensity error over all the sequence is around 12.

Conclusion

In this paper, we have proposed an efficient method for the image alignment of surfaces in the 3D Cartesian space. The same approach can be used both for rigid and deformable surfaces. For rigid surfaces, in the case of a calibrated camera we directly obtain an approximation of the structure of the surfaces. In the uncalibrated case, the registration can be used to self-calibrate the camera and obtain the 3D structure. We have tested two methods for surface approximation but we find that the RBFs provide more stable results in the presence of low-textured and/or deformable surfaces. The main improvement over standard algorithms based on Gauss-Newton minimization is efficiency in registering large deformations between two images. We use an efficient second order approximation and thus we obtain a faster convergence. The algorithm needs fewer iterations to converge while the computational cost per iteration is equivalent to a Gauss-Newton minimization. The proposed algorithm is thus more suitable for fast real-time applications. Further improvements on the computation speed can be achieved by selecting dynamically the centers in order to reduce the number of unknowns. Future work will be dedicated to handle illumination changes and self-occlusions.

Figure 1 :

 1 Figure 1: Case study for testing the proposed approach and comparing the convergence rate of minimization algorithms. The image on the right shows the regular grid defining the area of interest in the first image. The image on the right shows the corresponding template.

Figure 2 :

 2 Figure 2: Visual tracking with the Gauss-Newton minimization. The image on the left shows the transformation of the regular grid after 40 images of the sequence. The registration fails due to the real-time constraints. This is clearly visible in the right image where the warped area of interest in the reference frame is not equal to the reference template.

Figure 3 :

 3 Figure 3: Visual tracking with the second-order minimization. The image on the left shows the transformation of the regular grid after 40 images of the sequence. The image on the right shows that the warped area of interest in the reference frame is equal to the reference template.

Figure 4 :

 4 Figure 4: Two 3D views of the reconstructed surface. The surface lie on a sphere of 300 mm radius.

Figure 5 :

 5 Figure 5: Two 3D views of the reconstructed surface up to scale for the first sequence of the vault.

Figure 6 :

 6 Figure 6: Visual tracking of a vault in a sequence of 396 images acquired with an uncalibrated camera. The top row shows the the (11×9) regular grid used to track the area of interest in the sequence. The bottom row shows the area of interest registered with respect to the template.

Figure 7 :

 7 Figure 7: Visual tracking of a vault in a sequence of 214 images acquired with an uncalibrated camera. The top row shows the (9×9) regular grid used to track the area of interest in the image sequence. The bottom row shows the area of interest registered with respect to the template.

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Figure 8 :

 8 Figure 8: Visual tracking of a basketball in a sequence of 694 images acquired with an uncalibrated camera. The top row shows the (4×4) regular grid used to track the area of interest in the sequence. The bottom row shows the area of interest registered with respect to the template. The average RMS error is 13.6 gray-levels (over 256). The average number of iterations per image is 7.

Figure 9 :

 9 Figure 9: Visual tracking of a vase in a sequence of 450 images acquired with an uncalibrated camera. The top row shows the (6×6) regular grid used to track the area of interest in the sequence.The bottom row shows the area of interest registered with respect to the template. The average RMS error is 13.8 gray-levels (over 256). The average number of iterations per image is 6.

Figure 10 :

 10 Figure 10: Visual tracking of a deformable surface in a sequence of 1082 images acquired with an uncalibrated camera. The top row shows the (4×4) regular grid used to track the area of interest in the sequence. The bottom row shows the area of interest registered with respect to the template. The average intensity error over all the sequence is around 5.6. The average number of iterations per image is 6.

Figure 11 :Figure 12 :

 1112 Figure 11: Visual tracking of a deformable surface in a sequence of 400 images acquired with an uncalibrated camera. The top row shows the (6×5) regular grid used to track the tracked area of interest in the image sequence. The bottom row shows the area of interest registered with respect to the template.

Figure 13 :

 13 Figure 13: Visual tracking of a deformable surface in a sequence of 224 images acquired with an uncalibrated camera. The top row shows the (5×4) regular grid used to track the tracked area of interest in the image sequence. The bottom row shows the area of interest registered with respect to the template. The average intensity error over all the sequence is around 12.

RR n°6089


GRAVITATIONAL-EL ECTROMAGNET IC RES ONANCE OF THE S UN AS ONE OF THE POSS IBLE SOURCES OF AURORAL RADIO EMISSION OF PLANETS IN KILOMETRIC RANGE Annotation

Keywords: universal Planck proportions, auroral rad io emission, gravitational-electro magnetic resonance

Gravitational-electro magnetic resonance of the Sun (GERS) at a frequency of 202.97 KHz may be a secondary source of auroral radio emission in kilo metric range (the auroral kilo met ric rad iation -AKR) of planets having magnetosphere, such as Earth, Saturn, Jupiter, Uranus and Neptune. One of the envelopes of solar wind spectrum can be modulated with electro magnetic signal with a frequency of gravitational-electro magnetic resonance of the Sun. This component of the solar wind at a frequency of 202.97 KHz can also be a driver and a source of modulat ion of rad io emission of the planets in kilo metric range. In the spectrum of radio emission of the planets, except the Solar, it may present the components caused by their own grav itational-electro magnetic resonance, and gravitational-electro magnetic resonance of their satellites.

Introduction

In [START_REF] Gu Rnett | The Earth as a radio source: Terrestrial kilo metric radiat ion[END_REF][START_REF] Kurth | Direct ion-finding measurements of auroral kilo metric radiation[END_REF][START_REF] Kaiser | Terrestrial kilo metric radiation. III -Average spectral properties[END_REF][START_REF] Wu | A theory of terrestrial kilo metric radiat ion[END_REF][START_REF] Benson | Auroral kilo metric radiation: Wave modes, harmonics, and source region electron density structures[END_REF][START_REF] Hayes | Generat ion of ordinary mode auroral kilo metric radiat ion fro m ext raordinary mode waves[END_REF][START_REF] Zarka | Auroral radio emission at the outer planets: Observations and theories[END_REF][START_REF] Louarn | Generation of the auroral kilo metric rad iation in p lasma cavities -I. Experimental study[END_REF][START_REF] Morioka | Fundamental characteristics of fieldaligned auroral accelerat ion derived fro m AKR spectra[END_REF] it is shown, that the AKR is closely connected with the emergence of magnetospheric storms (during magnetospheric sub storms), and its main source and drivers are the following phenomena: gyro magnetic resonance of electrons with an energy of 1 keV (and more) around the magnetic field lines of a planet in the cyclotron the frequency and on the heights, for example the Earth, fro m 1 3 of its radius; plasma of planet's magnetosphere and the solar wind plasma. A KR Main characteristics: general energy, for examp le, for the Earth, can be fro m 10 6 to 10 7 watts [START_REF] Kaiser | Terrestrial kilo metric radiation. III -Average spectral properties[END_REF]; frequency ranges of the spectrum: for Earth 30 KHz -800 KHz, for Jupiter fro m 10 KHz to 1500 KHz, and further in Decameter to 40 MHz, for Saturn fro m several KHz to 1.2 MHz -1.3 MHz, fo r Uran iu m fro m tens of KHz to 850 KHz, for Neptune from 20 KHz (and possibly lower) to 600 KHz (and possibly more); polarization in all cases is mostly circu lar; in all cases there is modulation of frequency of AKR spectrum with the planet rotation and the solar wind [START_REF] Zarka | Auroral radio emission at the outer planets: Observations and theories[END_REF].

Gravi tational -electromagnetic resonance of the Sun as one of the possible sources of AKR

In [START_REF] Timkov | Planck universal proportions. Grav itation alelectro magnetic resonance[END_REF][START_REF] Timkov | Planck universal proportions. Grav itationalelectro magnetic resonance[END_REF] it is proposed and experimentally proved the law of "Universal Planck proportions." According to this law, in the observable universe, any body with mass m, creates a gravitational field, which bends the surrounding space with a radius of curvature S (in fact, S -is the length of a gravitational wave) and introduces time delay in signal propagation t dm into this space. Characteristics of the body m, S and t dm are interconnected with universal Planck proportions [START_REF] Timkov | Planck universal proportions. Grav itation alelectro magnetic resonance[END_REF][START_REF] Timkov | Planck universal proportions. Grav itationalelectro magnetic resonance[END_REF] , and for two bodies with weight m 1 and m 2 , length of a gravitational wave S 1 and S 2 , the time delay t dm1 and t dm2 , at a distance R from each other, the law of gravity is given by [START_REF] Timkov | Planck universal proportions. Grav itation alelectro magnetic resonance[END_REF][START_REF] Timkov | Planck universal proportions. Grav itationalelectro magnetic resonance[END_REF]:

. 2 2 1 2 2 2 1 2 2 1 R t t c F R S S F R m m G F dm dm p p    (3)
Based on the data about the mass of astronomical objects [12] and universal Plank proportions (1) there were calculated length value of a gravitational wave and frequency for thus: Earth, Moon, Venus, Mars, Jupiter, Saturn, Sun, Uranus and Neptune: It is experimentally confirmed the existence of gravitational-electro magnetic resonance of the Earth (GERE) at a frequency of 67.6 GHz [START_REF] Timkov | Planck universal proportions. Grav itation alelectro magnetic resonance[END_REF][START_REF] Timkov | Planck universal proportions. Grav itationalelectro magnetic resonance[END_REF]. For examp le, Fig. 1 of [START_REF] Timkov | Planck universal proportions. Grav itation alelectro magnetic resonance[END_REF][START_REF] Timkov | Planck universal proportions. Grav itationalelectro magnetic resonance[END_REF] shows one of the experimental charts, which confirms the existence of gravitational-electro magnetic resonance of the Earth at frequency 67.6 GHz. Fig. 1. Graph GERE. The dependence of signal power on frequency in the frequency range from 65.7 GHz to 68.6 GHz with 0.1 GHz at the output of the measuring channel. The signal power at the output of the generator is 4,5 mW [START_REF] Timkov | Planck universal proportions. Grav itation alelectro magnetic resonance[END_REF][START_REF] Timkov | Planck universal proportions. Grav itationalelectro magnetic resonance[END_REF].

The injection of electrons during the magnetospheric storm w ith an energy of about 1 keV and higher fro m the plasma reg ion of the magnetosphere of the planet in its auroral region followed by a reflection of some of them because of mirror effect of convergent geomagnetic field leads to resonance and amp lificat ion o f electro magnetic waves at the cyclotron frequency [START_REF] Wu | A theory of terrestrial kilo metric radiat ion[END_REF]. Due to the spread of energies of the electrons and the convergence of the magnetic field, it is formed many natural resonance chambers (the effect of the set of natural masers at the cyclotron frequen cy of electrons), which in its turn leads to multip le resonance in the frequency range from a few KHz to 1.5 MHz. So me of these resonant frequencies and their harmonics are close to the frequency of gravitational-electro magnetic resonance of the Sun 202.97 KHz. Under the influence of the gravitational field of the Sun and envelope of the spectrum of the solar wind, which is modulated by the gravitational-electro magnetic resonance of the Sun, there is a new resonance at frequencies close to the frequency of 202.97 KHz. A distinctive feature of this resonance is that the gravitational-electromagnetic resonance of the Sun is always present, and so it is a relatively stable and slightly damped. While resonance at other frequencies, usually has no permanent recha rge of energy and therefore is periodic and damped.

For example, in [START_REF] Taubenschuss | Linear predict ion studies for the solar wind and Saturn kilo met ric rad iat ion[END_REF] it is presented the results of experiments conducted on board of the Cassini spacecraft studying the effect of radio waves, the waves in the plasma and of the solar wind at AKR Saturn (AKR for Saturn -is SKR). These experiments are the Radio and Plasma Wave Science experiment (RPWS) [START_REF] Gurnett | The Cassini radio and plasma wave investigation[END_REF], the Dual Technique Magnetometer (MA G) [START_REF] Dougherty | The Cassini magnetic field investigation[END_REF] and the Cassini Plasma Spectro meter (CA PS) [START_REF] Young | Cassini Plasma Spectrometer Investigation[END_REF]. At Fig. 2a AKR of Uranus and Neptune, as seen from their graphs in Fig. 4 are similar to each other. Despite the fact that the frequency of GERS is in the local maxima of the graphs, the impact of GERS on AKR of Uranus and Neptune is minimal. This is due to the considerable distance of the planets from the Sun and the refore a significant reduction in its gravitational potential and a decrease in the density of the solar wind at the location of Uranus and Neptune.

Influence of GERS on the magnetosphere of Jupiter and consequently on his AKR is lower than for examp le in the AKR of Earth and Saturn (see. Fig. 4). Th is is due to the fact that the gravitational potential of Jupiter in AKR is much higher than gravitational potential of the Sun and planet's magnetic field is so strong that the effect of modulation of electro magnetic signals using AKR solar wind does not have a dominant influence on the characteristics of AKR, co mpared with other physical processes.

The frequency of gravitational-electro magnetic resonance of Jupiter (GERJ) 212.7 MHz is in the area, which is called synchrotron radio waves (synchrotron radiation), or abbreviated JSR. In the frequency range fro m about 100 M Hz to about 4 GHz, Jovian synchrotron radiation (hereafter referred as JSR) is emitted fro m the relat ivistic electrons, which is a non-thermal and incoherent radiation. JSR has a flat spectrum wh ich is mainly in the decimeter (DIM) range. Fig. 5 [START_REF] Watanabe | Develop ment of the observation system for the Jovian synchrotron radiation using an aperture synthesis array[END_REF] shows a frequency spectrum of the power of radio emission of Jupiter (in comparison with the spectrum of the Earth's AKR), where in the radio DIM (JSR) it is marked the frequency 212.7 M Hz. As can be seen from the graph, the power spectrum of the radio emission at a frequency of 212.7 MHz is in the area of global maximu m of JSR. Fig. 5. Spectra of Jov ian magnetospheric radiations. The power flu x is normalized to constant distance. The spectrum of the Earth's is also shown as a comparison with Jupiter [START_REF] Watanabe | Develop ment of the observation system for the Jovian synchrotron radiation using an aperture synthesis array[END_REF].

A more detailed structure of JSR spectrum shown in Fig. 6 [START_REF] Shiho | Studies on variation characteristics of the Jovian synchrotron radiation[END_REF]. Fig. 6. JSR spectrum fro m 74 MHz to 8 GHz measured in Ju ly 1994 (blue circles) and September 1998 (red circles). The red and blue solid lines are the results of a model simu lation [START_REF] Shiho | Studies on variation characteristics of the Jovian synchrotron radiation[END_REF]. Fig. 6 as well as Fig. 5 confirms that the frequency GERJ is within range of global maximu m of spectrum diagram JSR. Based on this, it can be assumed that GERJ may be one of the secondary sources JSR.

GERS has the greatest influence on the Earth's AKR (AKR Earth also called terrestrial KR, or TKR) and AKR Saturn -SKR.

Nature, sources and parameters of TKR are studied, for examp le, in [START_REF] Gu Rnett | The Earth as a radio source: Terrestrial kilo metric radiat ion[END_REF][START_REF] Kurth | Direct ion-finding measurements of auroral kilo metric radiation[END_REF][START_REF] Zarka | Auroral radio emission at the outer planets: Observations and theories[END_REF][START_REF] Hiroshi | Su mmary on Plasma Wave Emissions Observed by JIKIKEN -Preliminary Report for the In itial Phase of the Observation Results[END_REF][START_REF] Bale | Observation of topside ionospheric MF/HF rad io emission from space[END_REF][START_REF] James | Seasonal and solar cycle dynamics of the auroral kilo metric rad iation source region[END_REF][START_REF] Parrot | Propagation characteristics of auroral kilo met ric radiat ion observed by the M EM O experiment on Interball 2[END_REF]. In [START_REF] Hiroshi | Su mmary on Plasma Wave Emissions Observed by JIKIKEN -Preliminary Report for the In itial Phase of the Observation Results[END_REF] it is presented the results of experiments on TKR research, carried on a space probe JIKIKEN (EXOS-B). The authors note that the peak of the power spectrum of signals TKR is in the range from 100 KHz to 300 KHz, and the amplification of electro magnetic waves, associated with the acceleration of charged particles, occurs in magneto-Earth at height of 1.5 to 2.15 of its radius. In the experiment, Energy Spectrum of Particles (ESP) it was studied spectra of electrons and protons with energies from 10 eV to 20 keV. Fig. 7 of [START_REF] Hiroshi | Su mmary on Plasma Wave Emissions Observed by JIKIKEN -Preliminary Report for the In itial Phase of the Observation Results[END_REF] shows a typical dynamic spectrogram of electrons in the TKR, obtained under the ESP. Fig. 7. The energetic part icle spectrum measured by ESP [START_REF] Hiroshi | Su mmary on Plasma Wave Emissions Observed by JIKIKEN -Preliminary Report for the In itial Phase of the Observation Results[END_REF]. Fig. 7 shows that at a frequency of about 200 KHz resonance is observed, which is much less intense t han, for example, resonances of about 150KHz -175 KHz, 120 KHz, 80 KHz, 40 KHz. Fig. 8 of [START_REF] Hiroshi | Su mmary on Plasma Wave Emissions Observed by JIKIKEN -Preliminary Report for the In itial Phase of the Observation Results[END_REF] shows the dynamic range of TKR. Fig. 8. Dynamic spectra of the terrestrial kilo metric radiation [START_REF] Hiroshi | Su mmary on Plasma Wave Emissions Observed by JIKIKEN -Preliminary Report for the In itial Phase of the Observation Results[END_REF].

On the spectrum of TKR in Fig. 8 it is clearly seen almost continuous line of signal intensity TKR at a frequency of approximately 202.97 KHz. At the same time, at other frequencies the signal intensity TKR is usually intermittent nature. The presence of a solid line at the frequency of 202.97 KHz says about the relative stability of the signal source at that frequency. The nature of the spectra in Fig. 7 and Fig. 8 confirms the earlier suggestion that the cyclotron maser mechanism is the primary source of TKR. Further, the primary resonance frequencies near 202.97 KHz is captured by electromagnetic gravitationally resonance of the Sun and as the gravity of the Sun is always present, there is the effect of gravityelectro magnetic generator on frequency 202.97 KHz with paging sign al at the cyclotron frequency (or its harmonics).

In [START_REF] James | Seasonal and solar cycle dynamics of the auroral kilo metric rad iation source region[END_REF] it is presented the results of some years of research on board of the Imager for Magnetopause -to-Aurora Glob Exp loration (IMA GE) and Polar spacecraft. They were studied the seasonal and solar cycle variations in the spectrum of auroral kilo metric radiation (A KR), influence on the spectrum of AKR, dynamics of seasonal and solar cycles. It is also investigated the dependence of the averaged spectra of AKR as a function of d ipole t ilt angle magnetic field of the Earth. For example, in Fig. 9 [21] is a graph of this dependence. As can be seen from Fig. 10 AKR spectrum signals at a frequency of 202. 97 KHz are in the area of local maxima, and for the positive slope of the dipole with a min imu m of solar activity on the data from Polaris in the area of global maximu m.

In [START_REF] Bale | Observation of topside ionospheric MF/HF rad io emission from space[END_REF] it is presented experimental studies of MF / HF ionospheric radio emission of the Earth's magnetosphere at altitudes between 4 and 7 radiuses of the Earth, obtained by satellite Wind. During the experiment, it was found sporadic intense radiation at frequencies of about 1.8 MHz and more stable, but also less intensive radiation at frequencies of about 4.4 M Hz. Fig. 11 [START_REF] Bale | Observation of topside ionospheric MF/HF rad io emission from space[END_REF] shows graphs of power spectrum of AKR signals and the ranges MF / HF, obtained during the mission Wind. In a part related to the frequency range AKR in the graph in Fig. 11, it can be seen that the frequency 202.97 KHz is in the region of maximu m rad iation. On the falling part of the chart it can also be seen the zone of its AKR correction towards increasing signal power at frequencies of about 400 KHz, 500 KHz, 600 KHz and 800 KHz. If we exclude the frequency of 500 KHz, we can assume that, as in the formation of secondary sources AKR, except GERS signal at the fundamental frequency 202.97 KHz, and its harmon ics 2, 3, 4 are involved. And in format ion of high intensity radiation in the frequency range of VHF and HF it is involved respectively: in the frequency range of about 1.8 M Hz -9 th harmonic GERS, and in the frequency region of about 4.4 M Hz -4.6 MHz -22 and 23 GERS harmonic.

In the graph in Fig.

12 [START_REF] Parrot | Propagation characteristics of auroral kilo met ric radiat ion observed by the M EM O experiment on Interball 2[END_REF] it is shown fine structure of the Earth's AKR signal, obtained during the experiment MEM O on the spacecraft Interball 2. The presence of higher harmonics GERS in the spectrum AKR says that the GERS is stable and stationary secondary source of AKR.

According to the experimental studies Cassini and Voyager missions, according to [START_REF] Cecconi | SKR polarizat ion and source localization with the Cassini/RPWS/HFR instrument: First results[END_REF][START_REF] Menietti | The influence of Titan on Saturn kilo metric radiation[END_REF] frequency AKR range of Saturn is in the range from several KHz to 1.2 MHz -1.3 M Hz with a peak of signal at a frequency of about 200 KHz. That also confirms the assumption: GERS can be a secondary AKR source of Saturn at a frequency of 202.97 KHz.

Conclusion

Gravitational-electro magnetic resonance of the Sun (GERS) at a frequency of 202.97 KHz can be one of the secondary AKR sources of planets, which have magnetosphere, first of all, Earth and Saturn. Gravitational electro magnetic resonance of Jupiter (GERJ) on frequency 212.7 MHz may be one of the secondary sources of JSR.

If you measure the wavelength of enveloping signal of the electromagnetic spectrum fro m any remote object of the observed Universe, it can determine the mass of the object, based on the law "Un iversal Plank proportion".

  (spectrum was obtained in the experiment Cas sini -RPWS in the time interval 08/ 19/ 2004 -08/21/2004, that is, day of year DOY 232.5-234.0) and Fig. 3a (look at the application) (DOY 224.0-240.0) [13] present the dynamic spectra, which was obtained during the experiment Cassini -RPWS. In the figures it is clearly seen almost continuous line of high values of the power spectral density of electro magnetic signals at frequencies close to the frequency of 202.97 KHz. At the same time, at other frequencies the spectral power density of the electromagnetic signal is of an intermittent nature. At Fig. 2b in the experiment Cassini -RPWS presents research of the Stokes parameter (Stokes parameter S = total intensity), and Fig. 2c and Fig. 2d present studies of the degre e of polarization of the signal, respectively, the circular 2c and line 2d. All three figures show the line corresponding to the presence of gravitational-electro magnetic resonance of the Sun.

Fig. 2 .

 2 Fig.2. Cassini-RPWS dynamic spectra for (a) the Stokes parameter S (= total intensity), (b) the degree of circu lar polarizat ion dc, (c) the degree of polarization d and (d) the degree of linear polarization d L as a result of the Direction-Finding computations for the time period DOY 232.5-234.0, 2004 [13].

Fig. 3 .

 3 Fig. 3. (a) The RPWS dynamic spectrum, (b) the integrated SKR intensity profile, (c) p rofiles for the SW ram p ressure (solid) and bulk velocity (dotted), (d) profiles for the interplanetary magnetic field strength (solid) and its y -co mponent (dotted) in KSM -coordinates and (e) the profile of the reconnection voltage at the dayside magnetopause of Saturn during DOY 224-240, 2004 [13].Gravitational-electro magnetic resonance of the Sun has varying degrees of influence on the AKR p lanets. For co mparison,

Fig. 4 .

 4 Fig. 4. Co mparat ive spectra of auroral radio emissions of 5 planets [7].

Fig. 9 .

 9 Fig. 9. The average spectra of AKR as a function of dipole tilt angle fro m observations by the IMAGE/ RPI instrument [21]. Distribution of the spectrum signals AKR is follo wing: fo r negative inclination angles of the dipole fro m 80 KHz to 500 KHz with a peak power of about 260 KHz, for positive angles of inclination of the dipole fro m 60 KHz to 250 KHz with a peak power of about 150 KHz. Noteworthy is almost unnaturally straight line on the boundary between the two ranges of the spectrum intensity signals (-16 and -16.5 units) AKR in the area of about 200 KHz for negative inclination angles the dipole. This straight line means that for maintain ing constant power of AKR signal at about -16 units, at frequencies of about 200 KHz, the tilt range of the dipole from 0 degrees to -(27-28) degrees it is used stationary process of paging signal at one and the same frequency -200 KHz and with the same intensity. It can be assumed that this process -is the gravitational-electro magnetic resonance of the Sun. For co mparing, Fig. 10 [21] shows graphs of averaged spectra of AKR signals for different inclination angles of the dipole of magnetic field of the Earth and the varying intensity of solar activity, obtained fro m IMA GE and Polar.

Fig. 10 .

 10 Fig.10. Co mparison of the average spectra over the same dipole t ilt ranges where the emission peak is approximately constant for both Polar/PWI (blue) and IMA GE/ RPI (red) for positive (top) and negative (bottom) dipole t ilt angle.[21].

Fig . 11 .

 11 Fig . 11. Po wer spectra during the MF (top) and HF (bottom) events from the RA D1 (0.02-1.04MHz) and RAD2 (1.075-14M Hz) receivers. The MF emission in the top panel peaked at f ≈ 1.8 MHz has bandwidth δf/f ≈ 0.14 HF emission is peaked near f HF ≈ 4.5 MHz and has a similar FWHM bandwidth [20].

Fig. 12 .

 12 Fig. 12. AKR event recorded by MEM O on January 28, 1997, with an electric sensor. The spectrogram starts at 1952:20 UT and ends at 2124:58 UT.[22]. In Fig. 12 under division of 202.64 KHz it is clearly seen almos t solid line of secondary AKR source at a frequency of 202.97 kHz by GERS. Also during the experiment POLRAD mission Interball 2 it was obtained AKR spectra of Earth in the range of 4 KHz -1 MHz. In Fig. 13 [23] in addition to the fundamental frequency of 202.97 KHz GERS it is shown second and third harmonics at frequencies of about 406 KHz and 609 KHz.

Fig. 13 .

 13 Fig.13. An examp le of the AKR recorded with Interball 2 by the POLRAD radio -spectro-polarimeter [23].

Name Weight, kg Length of a gravi tational wave, m Frequency, GHz

	Earth	5.9722 x 10 24	0.00443474	67.6
	Moon	7.3477 x 10 22	0.000054547302	5495.94
	Venus	4.8673 x 10 24	0.0036143131	82.95
	Mars	6.4169 x 10 23	0.00047451718	631.78
	Jupiter	1.8981 x 10 27	1.40948454472	0.2127
	Saturn	5.6832 x 10 26	0.42201429314	0.7104
	Sun	1.989 x 10 30	1477.036	2,0297 x 10 -4
	Uranus	8.68 x 10 25	0.0645	4.65
	Neptune	1.02 x 10 26	0.0757	3.958