La prueba indiciaria

One o f the first general systems to incorporate perceptual organisation (SCERPO - Spatial Correspondence, Evidential Reasoning and Perceptual Organisation) was developed by Lowe [1985] for recognising 3D objects in monocular grey scale images. Although symmetry does not feature in his implementation, other grouping processes including collinearity, parallelism and proximity of line segment endpoints are central in the organisation of spatial information at the structural level. These same principles o f grouping also feature in more task specific applications of perceptual organisation which involve the segmentation o f specific types of objects with parallel or rectangular components represented in aerial photographs (e.g., buildings and runways [Huertas and Nevatia 1988; Mohan and Nevatia 1989; Denasi, Quaglia et al. 1992]). Another general approach, developed by McCafferty[ 1990], involves the formulation of the grouping tasks as an energy minimisation problem that is solved by simulated annealing. Although McCafferty acknowledges the importance of symmetry, again it is not implemented.

Symmetry does not feature in the examples of computational perceptual organisation mentioned so far. It has been suggested that symmetry is rarely used because it is much harder to detect compared to other grouping properties [McCafferty 1990; Cham and Cipolla 1995]. However, it has been recognised that in order to handle a broader range of edge and region primitives at the structural level it is necessary to utilise the full range o f grouping mechanisms in order to isolate the diversity of shapes and shape components. A successful implementation of such an approach is the CANC2 [Mohan and Nevatia 1992] vision system which uses most grouping principles to produce quality output for high-level analysis, e.g., shape extraction. The techniques employed in this system have been successfully applied in stereo vision systems used

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for object detection and recognition [Mohan and Nevatia 1989; Ylajaaski and Ade 1996]. As expected, according to the Sakar and Boyer classification system, the CANC2 process begins at the signal level with simple edge detection using a Canny edge detector [Canny 1986]. Edge contours are formed from the detected edge components on the basis o f their neighbourhood connectivity. The next stage in the process (primitive level) entails segmentation o f the contours into curves according to a technique developed by Saint-Marc & Medioni [1988]*^. These curves are grouped into contours based on cocurvilinearity, i.e., the continuity and proximity of similarly orientated curves. At the structural level, symmetries are sought between pairs of contours. If appropriate closures exist at both ends of the synunetrical contours, ribbons can be formed. A closure is either another contour, a set of multiple contours, or the end of other symmetries. After the selection of appropriate ribbons, segmentation of the scene into surfaces is possible'^.

3.3.1 CANC2 - Symmetry

Central to the CANC2 process is the detection and use of symmetries. Within the context of the Mohan and Nevatia schema, symmetries are considered to be pairs o f mutually symmetric curves defined by a mapping between the points on the two curves, where the curves are infinitesimally divisible (Figure 3.1a). The axis of symmetry between the two curves is the locus of midpoints o f the straight lines joining a point on one curve to its image on the other (Figure 3.1b). Curved Segments Axis of Symmetry Point Pair (Local Symmetry) = Mid-Point © = Boundary Point

F ig u re 3.1 The sym m etry betw een tw o cu rves can b e d efin ed a s a m a p p in g b etw een p o in ts on the cu rve s (a). The ax is o f sym m etry b etw een the cu rves is the lo cu s o f the m id -p o in ts o f the s tr a ig h t lin es jo in in g a p o in t on one cu rve to its im age on the o th e r (b).

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3.3.2 CANC2 - Symmetry Detection

Detecting the axis o f symmetry is necessary in the evaluation and selection of symmetries and the formation of ribbons. Axis detection is dependent on finding appropriate matches between points on the curves. The simplest way of doing this requires the consideration of all possible matches: a process that entails matching each point on one curve with every other point on the other curve, and then evaluating the match. Such an approach has a computational complexity of 0 ( n \ where n is the number of points on the boundary [Mohan and Nevatia 1989], and would therefore be computationally slow and expensive. The quantification of point-to-point matching can also be problematic since measures such as tangent direction tend not to vary much along smooth curves thus making the localisation of matches difficult.

Mohan and Nevatia avoid these difficulties by localising the positions along the curve at which matches are considered. They define the symmetry axis between two curves as “...the locus of the midpoints o f the lines joining points at equal length ratios along the curves.” [Mohan and Nevatia 1992 pp 623]. Consider figure 3.2. Given that the length of curve is sj

and that o f CD is S2, point x on AB can be mapped to a pointy on CD if and only if

o . i ;

*

,

where a is the length of curve section A X and b is the length o f curve section CY. Using this method, matched pairs of points can be easily found. Determining the midpoint between each pair of points is a trivial task and it follows that the locus of midpoints, the axis of symmetry, can be determined.

c

F igu re 3 .2 The sym m etry ax is betw een tw o cu rves is the locu s o f the m id -p o in ts o f the lin es jo in in g p o in ts a t eq u a l length r a tio s a lo n g curves.

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3.3.3 CANC2 - The Use of Symmetry

Having discovered all possible symmetries, those with a high probability of correspondence with individual object parts are identified. Mohan and Nevatia achieve this with a constraint satisfaction Hopfield network [Mohan and Nevatia 1992 pp 624-626]. The selected symmetries, i.e., pairs of symmetrical curves, can be considered to be two boundaries of a ribbon-like 2D planar shape. If these boundaries are closed by straight lines joining the ends of the curves, a ribbon is formed that can be described in terms of the symmetry axis between the two curves and a sweeping rule. The sweeping rule gives for each point on the axis the corresponding pair of symmetric points on the two curves. Shape description, however, is not the main purpose of ribbon formation in this procedure. The ribbons used in this process, in fact, defy description in this way because they are closed with edge contours detected in the image which are likely to be irregular. These cause edge effects which are not describable by the traditional sweeping rules (figure 3.3). The ribbons are used to segment the surfaces in the scene. They correspond to the surfaces of the objects in the scene; objects are inferred on the basis of the relationship between the ribbons. Axis Symmetrical Curves S tra ig h t Line Closure Edge Contour Closure E n d / Effect F ig u re 3 .3 A R ib b o n ‘d