PROCESO BÁSICO O AGREGADORES DE VALOR
NORMAS, LEYES, REGLAMENTOS, ACUERDOS, POLÍTICAS, RESOLUCIONES Y PROCEDIMIENTOS PARA LA EJECUCIÓN DEL PROCESO DE ÍNFIMA CUANTÍA,
1. OBJETIVO / ALCANCE Solicitar el Sistema de información utilizado.
The matching procedure has been tested on dozens of dierent image sequences. The astronaut and lab image sequences, shown previously in Figures 5-6 and 5-7, were chosen to illustrate several of the issues which have been raised in this chapter.
In applying the matching procedure to the astronaut sequence, the edge map from the left image was divided into 400 blocks whose centers were regularly spaced in a 2020 array on the pixel grid. Each block measured 2424 pixels, and the search in the right image was conducted over an area of 120120 pixels surrounding the coordinates of the center of each block. The correspondences which were found are numbered sequentially and their locations are shown superimposed on the edge maps in Figure 6-2. The numbered locations of the correspondences are also displayed by themselves directly below the edge maps to aid the reader in nding them.
Of the 400 blocks, 135 passed all the tests and generated acceptable matches. The quality of the matches which did pass the tests can be seen to be quite good. They are all correct to within possible oset error caused by approximating the correspondence at the center of the area covered by the block in the second image. A distinguishing feature of the astronaut images is the absence of repeating patterns. In fact the edge maps have almost the appearance of random dot images, and as a result, few blocks were rejected for not producing a well localized minimum. The vast majority of those which were rejected failed either the threshold test (6.19) or did not have the edge density required to pass the test of (6.20).
The second sequence, composed of images taken in our laboratory, is a very dierent situation. Many of the objects in the scene, i.e., the bookshelves, workstation monitors, and tripods, have long linear features for which it is impossible to nd the correct match with any certainty using a windowing method. There are also regular repeating patterns, such as the supports on the bookshelves and the drawer handles, which generate multiple candidate matches when more than one instance is included in the search window. In addition, the motion, which includes a ^
z
axis rotation, complicates things even more by introducing a relative tilt in the edge patterns.The matching procedure was executed on these images by dividing the left image into 900 blocks (in a 3030 array), each measuring 2424 pixels. The search was conducted over an area of 200 (horizontal) 60 (vertical) pixels. Of the 900 blocks, 49 produced
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Figure 6-2: Binary edge maps of astronaut sequence with correspondence points found by the matching procedure.
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Figure 6-3: Binary edge maps of real motion sequence (lab scene) with correspondence points found by the matching procedure.
acceptable matches according to the dierent tests in the procedure. These are shown in Figure 6-3.
As in the astronaut sequence, most of these matches are very good. However, the proportion of blocks generating acceptable matches is much lower, and there are some obvious errors, such as points 13, 20, 32, and 46. The rst three of these points are simply weak matches that passed the localization test only because they had a single minimum marginally below threshold, while the other minima were marginally above threshold. The last mismatched point, #46, demonstrates a dierent, but frequently encountered problem.
This point lies near the border of the left image on the lower right corner of the workstation monitor which is not in the eld of view in the second image. If the full monitor had been visible in the second image, the localization test would have rejected the match since there are several positions where a low score could be obtained. Instead, however, the wrong match was accepted.
Lowering the detection threshold is not a good solution for removing the marginal cases which slip past the localization test. This has the eect only of reducing the total number of matches, without changing the fact that the threshold can still fall in between the minima as in the cases above. In fact, there is not a simple solution at this level for removing the bad matches which escape detection without compromising the generality of the procedure. In the next chapter we will see how these false matches aect the computed motion estimates.
Solving the Motion Equations
As previously discussed in Section 2.2, the basic procedure for computing general camera motion, or relative orientation, given a set of point correspondencesf(
r
i;
`i)g,i
= 1;:::;N
, is to nd the rotation and baseline direction which minimize the sum of squared errorsS
=XNi=1
2i (7:
1)where
i is the triple product given in equation (2.34) as i =R
`i(r
ib
) (7:
2)Since the measurements of the locations of the correspondence points are not always equally reliable, it is often appropriate to dene
S
as the weighted sumS
=XNi=1
w
i2i (7:
3)where the weightsf
w
ig, 0w
i1, reect the relative condences in the data.The methods presented in Section 2.2 for minimizing (7.3) were developed to be exe-cuted on powerful digital computers where memory and power consumption limitations are not signicant constraints. In this chapter a simplied algorithm which is much more suit-able for implementation on low-level hardware, such as a programmsuit-able microcontroller, is presented. The algorithm, which is based on an adaptation of Horn's second method [4],
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is also an iterative nonlinear constrained minimization procedure. However, it breaks up the problem by alternating between updating the rotation and baseline, and in doing so, considerably reduces the size and complexity of the operations. The largest matrix which must be handled is 44, and the most complex operation at each iteration is solving a 33 eigenvalue-eigenvector problem.
It is not sucient to present a method for computing camera motion without discussing problems of stability. There are several well known and analyzed cases in which the mini-mization problem is numerically unstable and which allow multiple solutions for the motion parameters. If we are going to build a robust system, we must be able to recognize and avoid these cases. In the next chapter, I will derive analytically the conditions for the function