Second-order tangential voting

The votee points are made up of the central portion of frame 25. The sections close to the left and right hand edges are not used to prevent any biasing due to edge effects. The candidates for voters constitute all the other frames except for frame 25. In order to keep the problem tractable and implementable, we limit the number of voters per votee to 32. This also fits in neatly with the parallel computing architecture. These voters are selected to be:

• The closest Euclidean points to the votee, and

• The points that fall within a cone of interest with a cone slope of 5 pixels/frame.

The first-order tensors are made up of tn = (i, k, Ri, Gi, Bi), which indicates that one dimensional

position, frame, and colour are all used in the data. The components of the first-order tensor are chosen to be as close as possible to the available raw data and not use inferred parameters such as pixel velocities as this requires some form of flow estimation. The reduction of the 5D space to lower

dimension subspaces is not consistently possible due to the changing Ri, Gi, Bi values. Tangential

ball voting using the kernel described in Equation 3.4.2 is used with no alpha weighting (α = 1) and a scale factor of σ = 10. The desired geometric feature is the feature that has a single tangential

component and the rest as normal components. The saliency is therefore described by λ1− λ2 and

the direction is given by ˆe1.

To demonstrate the sensitivity of the tensor voting algorithm on a non-uniform random number

generator on the unit 5-sphere S5_{, the standard uniform random number generator described in}

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(a) frame 1. (b) frame 25. (c) frame 50.

(d) Scan line 100 extracted and plotted over columns and frames.

Figure 3.14: Single dimension movement of a synthetic tissue earth image sequence.

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(a) Voter 1 (closest Euclidian voter). (b) Voter 17.

(c) Voter 32 (furthest Euclidian voter). (d) Cumulative effect of all voters.

Figure 3.15: Tangential ball vote process (using tensor representation tn= (i, k, Ri, Gi, Bi)) on the

votee marked as black disk with a non-uniform random number generator on the unit 5-sphere S5_.

The voters are marked as blue circles, and the projection of the first two eigenvectors weighted with their respective eigenvalues are shown as red and green lines emanating from the votee respectively. Section 1.5.5 is used with projection onto the unit sphere in Figure 3.15. It can be seen that the

first eigenvector ˆe3 has a bias in direction as it should be more aligned to the voters.

By replacing the non-uniform random number generator on the unit 5-sphere S5 _{with a uniform}

generator based on the Gaussian method described in Section 1.5.5 the results in Figure 3.16 look far better aligned, verifying the importance of correct random number generation being applied to the tensor voting problem.

The final result of 32 voters is also tested for the monochrome colour tensor tn= (i, k, Yi), the YCbCr

colour space tensor tn= (i, k, Yi, Cbi, Cri) and the CIELAB colour space tensor tn= (i, k, Li, Ai, Bi)

in relation to the RGB colour space tensor tn= (i, k, Ri, Gi, Bi) on an individual votee level as shown

in Figure 3.17. The analysis is done at this early stage to highlight the inclusion of voters in an

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(a) Voter 1 (closest Euclidian voter). (b) Voter 17.

(c) Voter 32 (furthest Euclidian voter). (d) Cumulative effect of all voters.

Figure 3.16: Tangential ball vote process (using tensor representation tn= (i, k, Ri, Gi, Bi)) on the

votee marked as black disk with a uniform random number generator on the unit 5-sphere S5_{. The}

voters are marked as blue circles, and the projection of the first two eigenvectors weighted with their respective eigenvalues are shown as red and green lines emanating from the votee respectively.

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(a) Monochrome colour space with tensor tn = (i, k, Yi).

(b) YCbCr colour space with tensor t_{n =}

(i, k, Yi, Cbi, Cri).

(c) CIELAB colour space with tensor tn =

(i, k, Li, Ai, Bi).

(d) RGB colour space with tensor t_{n =}

(i, k, Ri, Gi, Bi).

Figure 3.17: Tangential ball vote process using several colour spaces on the votee marked as black disk. The voters are marked as blue circles, and the projection of the first two eigenvectors weighted with their respective eigenvalues are shown as red and green lines emanating from the votee respec- tively.

individual voting process that are noticeably incorrect. Care is taken that the scale of all the colour spaces remains similar to the RGB case to eliminate biases in favour of any single colour space. From the distribution of voters, it seems that the initial motivation to use the RGB colour space is valid.

Expanding on the single votee analysis on the one dimensional image as given in Figure 3.14(d), tangential ball voting is applied to more of the votees on line 25, resulting in Figure 3.18. In the

figure, the saliency λ1− λ2 is normalised to span [0, 1] such that strong saliency is represented by

white and low saliency is represented as black. The projection of the first eigenvector ˆe1 onto the

first two dimensions (x, z) is normalised and displayed as a directionless bar indicating the frame-

to-frame movement of colour, which is equivalent to vx. A measure of correctness is visually seen by

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the alignment to the apparent motion in the underlying image. Where the saliency is strong (white dots), the orientations seem in good agreement with the motion. Weak saliency (darker dots) often includes definite errors in orientation. By censoring low saliency votees, a better sparse estimate of orientation is observed in Figure 3.18(d). An important feature of this vote is how close correct votes are obtained to a moving boundary as can be seen on the right hand motion boundary. The left hand motion boundary does not fare well due to the near-homogeneous colour at the boundary. The motion estimates within the moving objects can be strengthened by formulating a dimension-

11 tensor of the form tn= (i, k, Ri, Gi, Bi, Ri−1, Gi−1, Bi−1, Ri+1, Gi+1, Bi+1) (case 1). The tokens

used for the dimension-11 tensor consist of the adjacent (left hand and right hand) pixel colour values. Choosing the adjacent pixels is based on trying to use the video frame data directly instead of using inferred values such as pixel velocities. The tokens are not orthogonal and an effort to try and find an orthogonal space using techniques such as Principal Components Analysis (PCA) or Karhunen Loeve Transform (KLT) would not be successful due to the varying correlation of the tokens over the video frame. The fact that the tensor is not orthogonal can have an effect on the basis of tensor fields being invalid. The differing tensor forms are enumerated as different cases later in the thesis given in Table 3.2. From Figure 3.19, the orientations are good within the boundaries of the moving object. The right hand motion boundary shows high saliency which is expected due to the general high contrast in this region, but the orientations are erroneous. The left hand motion boundary displays bleeding in that the orientation outside the boundary complies with the motion inside the boundary. This is due to the homogeneous colour on the one side of the boundary not contributing in a directional way to the tensor vote, allowing the i + 1 pixels to dominate.