Beneficios Posempleo Reconocimiento

The initial attempt in Section 3.3 adjusted the kernel to do tangential voting by making use of a cross product, thus limiting its use to 3D. Using normal tensor voting, the stick votes are normal to the connecting vector between the voter and votee. When a ball vote is simulated, it is implemented as uniform random direction unit vectors at a voter site which then use the tensor voting field to produce a vote vector at the votee site for each voter vector. The ball vote for the normal kernel given in Section 2.3 is shown in Figure 3.5. The surface renditions in the figures do not represent actual voting tokens, but rather give an indication of the shape and structure of the underlying kernel.

The eigenvector aligned to the connecting line between the voter and votee is the eigenvector ˆe3

associated with the smallest eigenvalue λ3. The other more dominant eigenvectors (ˆe1 and ˆe2) are

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(a) Voter (red) and votee (violet) point clouds. (b) Votee detail indicating the directions and strengths (λ1ˆe1, λ2ˆe2, λ3ˆe3) of the spectral decom- position of the second-order tensors from three vot- ers.

(c) Surface rendition of the cloud of votes at the vo- tee.

Figure 3.6: 3D representations of the simulated ball voting process using the normal kernel and three voters in a line.

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tangential mode by stringing voters in a line, the contribution at the voters at the votee can be seen in Figure 3.6. Figure 3.6(b) shows the individual spectral decompositions of the three voters

separately: the only aligned eigenvector directions are ˆe3. The other two eigenvectors (ˆe1 and ˆe2)

do not align themselves due to the circular symmetry. The small magnitude of the last eigenvalue indicates that errors in tensor location will have large effects on the integrity of the voting as the two dominant orthogonal components will adversely affect the result when the spectral decomposition is carried out on the combined second-order tensor.

To change the behavior of the voting, a different voting kernel needs to be formulated that has a dominant direction along the tangential direction, instead of in the normal direction as indicated in Figure 2.10. Graphically, it would simply mean rotating this plot by π/2, but still preserving the desirable radial decay and cone-like properties. This needs to be done without restricting the dimensionality as was done in Section 3.3. The method proposed is to rotate the voter vector direction by π/2. Due to the rotational symmetry we only need to deal with angles and dot products, allowing the formulation to be applicable to N dimensions.

Using the geometry given in Figure 3.7, the vote strength is given in the same form as Equation 2.3.7: VSsticktan(s, κ) = exp

−(s2+ακ2)

σ2 , (3.4.1)

where we define the scalar strength variables as

r = l 2_{| sin(α)|}; b = l 2 cos(α); κ = 1 r; 2θ = π− 2 arccos | sin(α)|; s = 2rθ. (3.4.2)

In order to correctly define the direction such that the vector field is smooth, symmetrical and rotates in the same fashion as in Figure 2.10, we need to define the voter direction. The voter direction is defined as v and will result in a direction given as u. The direction is found by determining the lower center point (defined as the intersection the two lines defined by these vectors), which is a distance b from the voter in the direction of the voter. The direction of u can be found as:

C = rˆv + P; u = Q− C (3.4.3)

and subsequently the unit vector ˆu can be found. The vote strength and direction is plotted in

Figure 3.8. The vector field is smooth and rotated π/2 radians compared to Figure 2.10.

Observing a simulated ball vote for the tangential vote as in Figure 3.9, it is noticed that the

eigenvector aligned with the connecting line between the voter and votee is the eigenvector ˆe1

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P

Q

Figure 3.7: Geometry of 2D tangential stick vote.

associated with the largest eigenvalue λ1. The other eigenvectors (ˆe2 and ˆe3) are normal to the

connecting line but are much smaller that λ1. In Figure 3.9(b) the calculated strengths of the

eigenvalues (λ1, λ2, λ3) are (37,4,4). The absolute values depend on the number of votes received,

so the relationship between the eigenvalues given as λ1 ¿ λ2 ≈ λ3 better defines the geometric

characteristics of the voting. Repeating the experiment where the voters are placed in a line, we obtain results given in Figure 3.10. Figure 3.10(b) shows the individual spectral decompositions of

the three voters separately; the aligned eigenvector directions are ˆe1. The other two eigenvectors (ˆe2

and ˆe3) do not align themselves due to the circular symmetry but are now small in relation to λ1 so

the effects of errors in tensor location will not have large effects on the integrity of the voting. The surface rendition in Figure 3.10(c) has two lobes which is desired structure aligned to the eigenvector ˆ

e1 with the largest eigenvalue λ1.

The graphical results presented in this section describe a novel method of tangential tensor voting as applied to motion trajectories that avoids using higher-order eigenvectors and eigenvalues to determine the motion vector. By using a new voting kernel, the motion trajectory is aligned to the

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20 40 60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200

Figure 3.8: 2D VSsticktan(x, y) showing both the strength and the vector field. Blue indicates low

strength and red indicates high strength.

first eigenvector and eigenvalue which increases stability of the tangential tensor voting process.

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(a) Voter (red) and votee (violet) point clouds. (b) Votee detail indicating the directions and strengths (λ1ˆe1, λ2ˆe2, λ3ˆe3) of the spectral decom- position of the second-order tensors.

(c) Surface rendition of the cloud of votes at the vo- tee.

Figure 3.9: 3D representations of the simulated ball voting process using the tangential kernel.

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(a) Voter (red) and votee (violet) point clouds. (b) Votee detail indicating the directions and strengths (λ1ˆe1, λ2ˆe2, λ3ˆe3) of the spectral decom- position of the second-order tensors from 3 voters.

(c) Surface rendition of the cloud of votes at the vo- tee.

Figure 3.10: 3D representations of the simulated ball voting process using the tangential kernel and three voters in a line.

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