----pixel spacing can be recorded. If t-2d-.cross and t-2d-along represent the pixel spacing corresponding to t3d-across and t3d-along respectively, and p, and P2 represent the change in pixel spacing corresponding to the scale changes s, and 2, then the description of the model line group that can be used to generate the line groups in the image becomes < tm-2d-acrossi tm-2d-alongi tm-local-orienti tm-qlobal-orient7P17P2
>-5.4.3 Selective generation of matching line groups
We now present an algorithm for selectively generating line groups in the image that match a given model line group description Sce the model description places a bound on the tolerable scale and pose changes, the basic strategy is to generate line groups with successively increasing inter-line (both d,o,, and d,,jong) spacing until the upper bound specified in the model description is reached. That is, given a model line group des crip tion < t - 2d- across t - 2d- along t - local - orient t - global - orient Pi P2
the across-the-line spacing allowed between the lines varies from tn- 2d-ao,, PI t tm-2d-across + p2 (and s arly for the along-the-line spacing). The matching
< tacross-low7tacrossitalong-low7talongitlocal-orient)tglobal-orient > with the following interpretation: It is the largest group of non-intersecting line segments such that for each line in the group, there eists no line the group such that da,,,,, < tacross-low
Selection using Parallel-line Groups 192
3. The orientation difference between the lines is no more than a threshold
and the entire group satisfies the assumption that the orientation change between any two lines in the group is no more than a threshold
tglobal-orient-If the observation conditions clude only pose changes, and if all the lines a model group are equi-spaced, then the model group appearing in the n'nage (the visible part of it, that is) is bound to be present in one such augmented groups because its consecutive lines would exhibit an inter-line spacing within the specified range. In the presence of occlusions and other observation conditions, and when the inter-line spacing the model group 'is not uniform, the model group can still be captured in the augmented groups, albeit in a fragmented form. That is, the model group may be partitioned (and possibly merged wth adjacent lines) into several augmented groups. But as long as two adjacent lines in the model group are visible in the image, they can still be captured in one of the augmented groups and this should, in theory, be sufficient for recognition (as the two line end points can provide four features)
Algorithm for generating augmented line groups
The algorit for selectively generating the groups satisfying a model description
< tm-2d-acrossitm-2d-alongitm-local-orientitm-global-orient7PliP2 > proceeds by first
spaced line groups specified by <, across I ta&crossi along7 alongitiocal-orientitglobal-orient >
are successively generated using a modified version of the grouping algorit of Sec-tion 52.2 as follows:
1. Let t across " t-2d-acros - Pi and to along tm - 2d- along - P1
2. For i = to p - pl do
-, 1 . I II II III I MONISM
9 Let t across = + t1 acros I and ti along = + talong'i-1
* The grouping algorithm of Section 52.2 is applied to the line segments with the following modifications to Steps 2a and 2c:
- In Step 2a, an us of neighborhood lying between the rectangles 2ti-1 across x 2(t'-' + ) and 2tcross x 2(ti along a along+ 1), where I is the length of the line, is scanned and all lines passing through this annulus but not through the =er rectangle are retained.
- In Step 2c, in adition to checking for t_global-orient7 each enclosing
group being merged with the current group is checked for violations
against the annulus neighborhood constraint mentioned above.
Analysis
The above algorit makes (p2 - pl) passes over the line segments in generating the matching groups for each model line group. However, the total number of matching groups generated is still lineaz in the number of line segments since each line can belong to at most (P2 - P) groups, one for each line lying in the annulus of neigh-borhood between the outer and inner rectangles of imensions 2t'-' acro x 2 ti-i + )along and 2t' across x 2(t'long + 1). a The above procedure can be repeated for generating matching groups for each model group separately. Alternatively, the aowable inter-line spacings for all model inter-line groups can be pooled together to form ranges of pixel spacing for all the model groups, and the search for matching groups can be clone for each such range using the above algorithm. The time to generate the augmented line groups for an iteration i is still O(n2) (n 'is the number of line segments) since the grouping algorit is the same as before, and Step 2c. ex es each pair of line segments at most once. For the allowed scale changes, the range (p2 - pi) is s enough so that the entire operation of selective group generation can be clone O(kn2) time, where k < n is a constant representing the number of passes over the line segments.
194 Selection using Parallel-line Groups An Example
We now illustrate model-driven selection using paraRel-line groups with an ex-ample. Figure 5.8 shows model line groups being used to perform selection. Here the parallel-lines on the ladder part of the toy fire truck serve as the model line groups por-tion of it occluded. Figures 5.8d-g show the matching augmented closely-spaced line groups obtained using successively increasing line-spacing as specified by the range
< 0270,016,10 > < 2370,016,10 > < 34,070,6710 > and < 45,010,6710 >1 respectively. These matches to the indicated model line groups are shown collectively in Figure 58h. As can be seen from the figure most of the model line groups are captured in the matching groups, although there i's evidence of fragmentation in two of the groups marked and 2 as shown in Figure 5.8b.
Discussion
Model-driven selection using the above algorithm generates enough candidate match groups for a model line group to avoid false negatives under most observation con-ditions. Further, by requiring the groups to have a ' ' inter-line spacing it avoids generating nnecessary false positive matches to model line groups. This can easily happen if a simpler strategy for group generation were used such as generating ordinary (rather than augmented) closely-spaced parallel line groups by successively increasing the line spacing from t2d-across - Pi t t2d-across + P2- Such a scheme would create successively bigger groups (since a group with small line spacing would always satisfy the constraint of a bigger line spacing) that are often more unreliable and ely matches to model groups.