3.8 Algorithm development with an interpolated single dimension image
3.10.3 Analysis of two dimensional voting
In order to analyse the tangential tensors in the two dimensional plane the tissue-earth sequence used in the single dimensional analysis is used. The earth moves to the right, while the tissue moves slowly to the left. This motion is suitable for the further analysis as the skewness measure is analysed in the x direction only in the results. The motion vectors will be aligned in the x direction with no y component. This allows easy visual orientation checks of the tangential tensor voting approach. Simulations on the tissue-earth sequence are run for cases 0, 3, 6, 7 and 8 which are the cases relevant to the two dimensional image in a spatio-temporal volume. In these simulations, the scale factor σ = 20, the number of iterations f = 10000 and the number of voters per votee is set at 32. A single ball vote pass is done with no removal or interpolation of data. In order to get an overall image of the effect of the tensor voting, only half the two dimensional image is shown. Figures 3.39 to 3.43
show the saliency of the first eigenvector ˆe1 as well as its projection on the two dimensional x, y
plane. The projection is the orientation of the motion vector. Also shown are the tensor skewness measures in the two dimensional plane. This is referred to as the Tensor Skewness Map or TS map. Both the overall effect and the detail are shown. In the detail, the motion vector orientations can
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(a) Voters around a votee. Voter markers get darker with increasing Euclidean distance from votee. Mo- tion vectors shown as red line through votee.
(b) A small subset of votees indicating motion esti- mate consistency.
Figure 3.35: 3D image volume voter-votee relationship for case 3 encoding of the tensors.
(a) Voters around a votee. Voter markers get darker with increasing Euclidean distance from votee. Mo- tion vectors shown as red line through votee.
(b) A small subset of votees indicating motion esti- mate consistency.
Figure 3.36: 3D image volume voter-votee relationship for case 6 encoding of the tensors.
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(a) Voters around a votee. Voter markers get darker with increasing Euclidean distance from votee. Mo- tion vectors shown as red line through votee.
(b) A small subset of votees indicating motion esti- mate consistency.
Figure 3.37: 3D image volume voter-votee relationship for case 7 encoding of the tensors.
(a) Voters around a votee. Voter markers get darker with increasing Euclidean distance from votee. Mo- tion vectors shown as red line through votee.
(b) A small subset of votees indicating motion esti- mate consistency.
Figure 3.38: 3D image volume voter-votee relationship for case 8 encoding of the tensors.
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(a) Saliency over half the image. (b) Detail of saliency.
(c) TS map over half the image. (d) Detail of TS map.
Figure 3.39: Case 0 tensor voting applied to the tissue earth ideal sequence. be seen which are horizontal due to no y component in the motion.
The case 0 saliency in Figure 3.39 does not contain much structure, as can be seen by the motion vector orientations being inconsistent in places. Even though the motion vector estimation is not good, the TS map still shows structure at the occluding and disoccluding boundaries.
The case 3 saliency in Figure 3.40 shows good saliency, as evidenced by the bright saliency map. There is a section on the left where the saliency breaks down slightly. This is due to an area which has few features and can be expected. The motion vector orientations are consistent in the x direction, and the TS map shows good structure at the occluding and disoccluding boundaries.
The results of case 6 in Figure 3.41 show little improvement over case 3. In the cases of pixel aliasing and noise, there may be an advantage to using case 6.
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(a) Saliency over half the image. (b) Detail of saliency.
(c) TS map over half the image. (d) Detail of TS map.
Figure 3.40: Case 3 tensor voting applied to the tissue earth ideal sequence.
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(a) Saliency over half the image. (b) Detail of saliency.
(c) TS map over half the image. (d) Detail of TS map.
Figure 3.41: Case 6 tensor voting applied to the tissue earth ideal sequence.
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(a) Saliency over half the image. (b) Detail of saliency.
(c) TS map over half the image. (d) Detail of TS map.
Figure 3.42: Case 7 tensor voting applied to the tissue earth ideal sequence.
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(a) Saliency over half the image. (b) Detail of saliency.
(c) TS map over half the image. (d) Detail of TS map.
Figure 3.43: Case 8 tensor voting applied to the tissue earth ideal sequence.
The results of case 7 in Figure 3.42 show a degradation. This is probably due to the high dimen- sionality (N = 42) that causes inconsistencies in calculation. The effects of higher dimensionality on the Monte Carlo analysis are analysed in the next section.
The results of case 8 in Figure 3.43 show consistency with case 3 and case 6. The detail on the skewness also shows the averaging effect of the wider kernel in Figure 3.43(d), where there is a 2 pixel offset to the right. The effect of averaging is not wanted in trying to determine motion boundaries, and as such this case has limited use.
The occluding and disoccluding exponential filters are applied with η = 2. The exponential filters are only applied in the x-direction as the motion is only in that direction.
In case 0 the results of the left hand and right hand movement filters and detectors can be seen in Fig- ure 3.44. The difference in detection of occlusion in Figure 3.44(a) and disocclusion in Figure 3.44(e)
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(a) Left hand movement filter results. (b) Detail of left hand movement filter results.
(c) Left hand movement filter detection at 50% of maximum.
(d) Detail of left hand movement detection.
(e) Right hand movement filter results. (f) Detail of right hand movement filter results.
(g) Right hand movement filter detection at 50% of maximum.
(h) Detail of right hand movement detection.
Figure 3.44: Case 0 TS Map passed through left hand and right hand movement filters.
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In case 3 the results of the occluding and disoccluding filters and detectors can be seen in Figure 3.45. When compared to case 0, the filter responses are much clearer in highlighting the occluding and disoccluding boundaries. The occluding and disoccluding detectors perform very well clearly demar- cating the boundaries. A false detection is found on the left due to an ambiguous motion caused by aliasing.
In case 6 the results of the occluding and disoccluding filters and detectors can be seen in Figure 3.46. Case 6 gives very similar results to case 3. Considering that the images are ideal, case 6 is expected to perform better than case 3 in non-ideal (interpolation and noise) conditions. The occluding and disoccluding detectors perform very well, clearly demarcating the boundaries.
In case 7 the results of the occluding and disoccluding filters and detectors can be seen in Figure 3.47. The results of case 7 are not as good as case 3, 6 and 8 due to the high dimensionality. The poor result leads to more analysis later on the Flower Garden natural image sequence to try understand the problems associated with high dimensionality in tangential tensor voting.
In case 8 the results of the occluding and disoccluding filters and detectors can be seen in Figure 3.48. Case 8 gives very similar results to case 3 and 6. The major difference is the effect of the filter width on the clear demarcation of the boundary. The offset on the detections is not only due to the apparent lag of the filter, but the filter seems to be averaging the edge confirming the hypothesis that larger regions of support affect the edge detection capability of the tensor voting framework. In this section several of the different tensor encodings are applied to an ideal tissue earth image sequence. The reason for doing this is to determine at what stage does an increase in tensor encoding dimensionality have little effect on detecting occlusions and disocclusions. The increase in tensor encoding also affects the ability to discriminate edges accurately. From the results given in Figures 3.44 to 3.48, case 6 emerges as most appropriate in terms of detecting occlusions and disocclusions. Case 3 is also able to discriminate well, but may perform poorly under natural conditions.
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(a) Left hand movement filter results. (b) Detail of left hand movement filter results.
(c) Right hand movement filter detection at 50% of maximum.
(d) Detail of right hand movement detection.
(e) Disocclusion filter results. (f) Detail of right hand movement filter results.
(g) Right hand movement filter detection at 50% of maximum.
(h) Detail of right hand movement detection.
Figure 3.45: Case 3 TS Map passed through left hand and right hand movement filters.
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(a) Left hand movement filter results. (b) Detail of left hand movement filter results.
(c) Left hand movement filter detection at 50% of maximum.
(d) Detail of left hand movement detection.
(e) Right hand movement filter results. (f) Detail of right hand movement filter results.
(g) Right hand movement filter detection at 50% of maximum.
(h) Detail of right hand movement detection.
Figure 3.46: Case 6 TS Map passed through left hand and right hand movement filters.
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(a) Left hand movement filter results. (b) Detail of left hand movement filter results.
(c) Left hand movement filter detection at 50% of maximum.
(d) Detail of left hand movement detection.
(e) Right hand movement filter results. (f) Detail of right hand movement filter results.
(g) Right hand movement filter detection at 50% of maximum.
(h) Detail of right hand movement detection.
Figure 3.47: Case 7 TS Map passed through left hand and right hand movement filters.
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(a) Left hand movement filter results. (b) Detail of left hand movement filter results.
(c) Left hand movement filter detection at 50% of maximum.
(d) Detail of left hand movement detection.
(e) Right hand movement filter results. (f) Detail of right hand movement filter results.
(g) Right hand movement filter detection at 50% of maximum.
(h) Detail of right hand movement detection.
Figure 3.48: Case 8 TS Map passed through left hand and right hand movement filters.
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(a) ˆe1 orientation with N = 42 and f = 1000. (b) ˆe1orientation with N = 42 and f = 30000.
Figure 3.49: Twenty Monte Carlo estimations of the first eigenvectors ˆe1orientation from the votee
towards the voter.