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In any kind of matching problem, there are two questions that must be answered:
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How are points selected for matching features that are matched
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In other words, what are the if any, are
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How are the correct matches chosen What constraints, placed on the displacement vectors Three properties guide matching:
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Discreteness, which is a measure of the distinctiveness of individual points Similarity, which is a measure of how closely two points resemble one another Consistency, which is a measure of how well a match conforms to nearby matches The property of discreteness means that features should be isolated points For example, line segments would not make good features since a point can be matched to many points along a line segment Discreteness also minimizes expensive searching by reducing the problem of analyzing image disparities to the problem of matching a finite number of points The set of potential matches form a bipartite graph, and the matching problem is to choose a (partial) covering of this graph As shown in Figure 1411, initially each node can be considered as a match for each node in the other partition Using some criterion, the goal of the correspondence problem is to remove all other connections except one for each node The property of similarity indicates how close two potential matching points are to one another; it is a measure of affinity Similarity could be based on any property of the features that are selected to implement discreteness The property of consistency is implied by the spatial continuity of surfaces in the scene and assumes that the motion is well behaved Consistency allows the obvious matches to improve the analysis of more difficult matches Some points are sufficiently distinct and similar that it is easy to match them; this match can assist in matching nearby points The discrete feature points can be selected using any corner detector or a feature detector One such feature detector is the Moravec interest operator
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Figure 1411: (a) A complete bipartite graph Here each node in group A has a connection with each node in group B Using a characteristic of nodes (points) and some other knowledge, a correspondence algorithm must remove all but one connection for each node, as shown in (b) This operator detects points at which intensity values are varying quickly in at least one direction This operator can be implemented in the following steps: 1 Compute sums of the squares of pixel differences in four directions (horizontal, vertical, and both diagonals) over a 5 x 5 window 2 Compute the minimum value of these variances 3 Suppress all values that are not local maxima 4 Apply a threshold to remove weak feature points Any feature detector can be used in place of the above operator One can use a corner detector or computed curvature values at every point and select high curvature points as features Next one must pair each feature point in the first image with all points in the second image within some maximum distance This will eliminate many connections from the complete bipartite graph The connections removed are those that are between points far away in two images and hence unlikely to be candidate matches Each node ai has position (Xi, Yi) in the first image and a set of possible labels (disparity vectors) The disparity labels are displacement vectors or the undefined disparity, which allows some feature points to remain unmatched
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14 DYNAMIC
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The initial probabilities of a match are computed using a measure of similarity between the feature points in the two images A good measure is the sum of the squares of the pixel differences, Si, in corresponding windows The following approach may be used for assigning these probabilities Let I be a candidate label at a point This label represents a disparity vector at the point First we compute wi(/), which represents the similarity between the point (Xi, Yi) and its potential match at disparity I wi(/) = 1
(I)'
(1415)
where Si(I) is the sum of the squared differences corresponding to label I, and C is some positive constant The probability that this point has undefined disparity is obtained by first defining
p (undefined)
1 - max(wi(/))
(1416)
This probability is determined based on the strength of the most similar point for (Xi, Yi) If there are no strongly similar points, then it is likely that the point has no match in this image The probabilities of the various matches (labels) are (1417) (I')' where pi(/li) is the conditional probability that ai has label I given ai is matchable, and the sum is over all labels I' excluding the "undefined" label The probability estimates are refined using the consistency property and iterative relaxation algorithm In this approach, the labels at each node are strengthened or weakened based on the labels of the neighboring nodes in that iteration The most important property used here is that all disparities should be similar in a given neighborhood Thus, similar disparities of nodes in a neighborhood should strengthen each other and dissimilar ones should be weakened This is accomplished using the following approach Let us consider probability for disparity vectors of all neighbors of ai For each neighbor, sum the probability of labels (disparities) that are close to, or similar to, the disparity of ai:
pi(llz)
Wi(I)
qf(/) =
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