77.7/ The envelope of epipolar lines in .NET Print 2d Data Matrix barcode in .NET 77.7/ The envelope of epipolar lines

77.7/ The envelope of epipolar lines use vs .net gs1 datamatrix barcode printer tocreate data matrix ecc200 on .net Web application framework distribution. I DataMatrix for .NET n this case, the assumption of a Gaussian distribution of errors is less tenable.

11.11.1 Verification of epipolar line covariance We now present some examples of epipolar line envelopes, confirming and illustrating the theory developed above.

Before doing this, however, a direct verification of the theory will be given, concerning the covariance matrix of epipolar lines. Since the 3 x 3 covariance matrix of a line is not easily understood quantitatively, we consider the variance of the direction of epipolar lines. Given a line 1 = (7i, l2, l-.

i)J, the angle representing its direction is given by 6 = arctan( /i/72). Letting J equal the 1 x 3 Jacobian matrix of the mapping 1 6, one finds the variance of the angle 6 to be > T u . = JEiJ . This result may be verified by simulation, as follows. One considers a pair of images for which point correspondences have been identified.

The fundamental matrix is computed from the point correspondences and the points are then corrected so as to correspond precisely under the epipolar mapping (as described in section 12.3). A set of n of these corrected correspondences are used to compute the covariance matrix of the fundamental matrix F.

Then, for a further set of "test" corrected points x, in the first image, the mean and covariance of the corresponding epipolar line 1 = Fx, are computed, and subsequently the mean and variance of the ^ orientation direction of this line are computed. This gives the theoretical values of these quantities. Next, Monte Carlo simulation is done, in which Gaussian noise is added to the coordinates of the points used to compute F.

Using the computed F, the epipolar lines corresponding to each of the test points are computed, and subsequently their angle, and the deviation of the angle from the mean. This is done many times, and the standard deviation of angle is computed, and finally compared with the theoretical value. The results of this are shown in figure 11.

8 for the statue image pair of figure 11.2(p289). Epipolar envelopes for statue image.

The statue image pair of figure 11.2(j 289) is interesting because of the large depth variation across the image. There are close points (on the statue) and distant points (on the building behind) in close proximity in the images.

The fundamental matrix was computed from several points. A point in the first image (see figure 11.9) was selected and Monte Carlo simulation was used to compute several possible epipolar lines corresponding to a noise level of 0.

5 pixels in each matched point coordinate. To test the theory, the mean and covariance of the epipolar line were next computed theoretically. The 95% envelope of the epipolar lines was computed and drawn in the second image.

The results are shown in figure 11.10 for different numbers of points used to compute F. The 95% envelope for n = 15 corresponds closely to the simulated envelope of the lines.

The results shown in figure 11.10 show the practical importance of computing the epipolar envelopes in point matching. Thus, suppose one is attempting to find a match for the foreground point in figure 11.

9. If the epipolar line is computed from just 10 point matches, then epipolar search is unlikely to succeed, given the width of the. 11 Computation of the Fundamental Matrix F ,", r 0 2 4 6 8 10 12 ECC200 for .NET 14 16 6 8 10 12 14. Point Number Point Number Fig. 11.8.

Comp arison of theoretical and Monte Carlo simulated values of orientation angle ofepipolar lines for 15 test points form the statue image pair offigure 11.2(p289). The horizontal axis represents the point number (1 to 15) and the vertical axis the standard deviation of angle, (a) the results when the epipolar structure (fundamental matrix) is computed from 15 points, (b) the results when 50 point matches are used.

Note : the horizontal axis of these graphs represent discrete points numbered 1 to 15. The graphs are shown as a continuous curve only for visual clarity..

envelope. Even visual .net ECC200 for n 15, the width of the envelope at the level of the correct match is several tens of pixels.

For n = 25, the situation is more favourable. Note that this instability is inherent in the problem, and not the result of any specific algorithm for computing F. An interesting point concerns the location of the narrowest point of the envelope.

In this case, it appears to be close to the correct match position for the background point in figure 11.9. The match for the foreground point (leg of statue) lies far from the narrowest point of the envelope.

Though the precise location of the narrow point of the envelope is not fully understood, it appears that in this case, most points used to the computation of F are on the background building. This biases towards the supposition that other matched points lie close to the plane of the building. The match for a point at significantly different depth is less precisely known.

Matching points close to the epipole - the corridor scene. In the case where the points to be matched are close to the epipole, then the determination of the epipolar line is more unstable, since any uncertainty in the position of the epipole results in uncertainty in the slope of the epipolar line. In addition, as one approaches this unstable position, the linear approximations implicit in the derivation of (11.

14) become less tenable. In particular, the distribution of the epipolar lines deviates from a normal distribution. 11.

12 Image rectification This section gives a method for image rectification, the process of resampling pairs of stereo images taken from widely differing viewpoints in order to produce a pair of "matched epipolar projections". These are projections in which the epipolar lines run parallel with the x-axis and match up between views, and consequently disparities between the images are in the rc-direction only, i.e.

there is no y disparity..
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