Continuous Angular Equality

The bivalent approach to energy field generation produces either incomplete, flawed, or computationally expensive energy fields. The solution to at least one of these difficulties lies in adapting the way in which angular equality is used to score the mid-points about which the boundary point pairs are symmetric.

As it stands, all the mid-points generated by a bivalent point-to-point comparison are either candidates for identification as points o f interest or they are not; approximate mid-points are not possible. Many mid-points may be on the bilateral axis and have a valid contribution, but fail to fall within the arbitrary range over which the criterion o f point selection is apphed and therefore do not qualify for inclusion. As discussed above, broadening the range over which the criteria for mid-point selection is applied gives a continuous and complete, but less well defined axis. Eliminating the range over which the criterion for mid-point selection is applied means every mid-point is suitable for scoring and the result is an even symmetry field throughout the shape. Although energy field quality is affected by the extent of the range over which point selection is made, energy field quality is not beneficially changed by a large variation of these limits.

Energy field quality can be improved if, instead of using arbitrary limits within which points are bivalently scored, an approach is taken in which a symmetry dependent score is applied to all points. All mid-points are the local axis of symmetry for the two points from which they are derived, but only some of these points lie on the bilateral axis. It is reasonable to assume that of all the points that are not on the axis some are closer to being on the bilateral axis than others. It follows that the appropriate scoring of this “closeness” would give a continuous map of a given shape’s symmetry properties. In such a situation, the gaps that existed in the representation of a bilateral axis in an energy field generated from a small number of points using the bivalent method would now have a value. This value would not be as high as the point values on the axis but it would be significant enough to give a continuous, although potentially uneven representation of the bilateral axis. If this technique can effectively compensate for the gaps in the energy field shown in figure 8.2f the need for a large number of boundary points would be reduced. This, and the effect on energy plateauing, wiU be considered further in the examination o f practical examples given below (section 8.1.1.2).

Chapter 8

Energy Fields

8.1.1.1 Calculating the Energy Field (Continuous)

An approach based on a continuous mode of evaluation requires a function e/p^pj) by which a value indicating proximity to a point of symmetry can be attributed to every mid-point. Angular equality is still valid for this purpose. The angles 9j and 6j subtended by the line joining two points, Pi and Pj, on the boundary of an object and the tangent at either point (figure 7.9) can have values between 0 and Æ radians.

F igu re 8 .4 A lo c a l syym m etry. I f 6 i= Oj then p o in ts A a n d Pj are lo c a lly sym m etric ab o u t the m id -p o in t M.

At the point of angular equality dt = Oj, therefore:

and at the point of maximum angular difference (inequality)

\0. — 6 ,\ = K

( 8 . 1 )

(8 2)

Thus, angular equality can be expressed as a value between zero and K which can then be used to score the associated mid-point. For practical purposes these values are normalised over a range between 0 and 1 and scaled for display by multiplication with max (max was chosen to be 255 for display purposes). To ensure values of high angular equality correspond to high values of e. the normalised values are subtracted from one.

r f

1 - V V

- Ojw

Chapter 8

Energy Fields

8.1.1.2 Examples of the Energy Field

The simple symmetrical nature of a rectangle means that it is aptly suited to demonstrate the desirable qualities of any energy field. An example of the continuous energy field of a rectangle produced using the method described above is shown in figure 8.5.

Figure 8.5 A continuous energy fie ld fo r a rectangle (angular e q u a l i t y ) . T h e function ^c(p,<Pi) gives a continuous energy field in which the sub-symm etries an d the refiectional axes o f sym m etry are distinct and o f equal significance.

The smooth, continuous nature of this type of energy field can be clearly seen. Also, the axes of reflectional symmetry and comer derived sub-symmetries are discernible as distinct ridges. The fact that the sub-symmetries are as prominent as the axes of reflectional symmetry indicates that some of the local symmetries not on the bilateral axis are as significant as those that are. This is a problem; local symmetry based artefacts, e.g., energy plateaus, are likely to be as prominent as the bilateral axis. This is apparent in the continuous energy fields equivalent in shape and boundary point number to those generated to illustrate bivalent energy fields (8.5a,b,c). The plateauing effect is distinct in all three instances and masks a portion of the desired axis as well as the short axis of reflectional symmetry in the more detailed cases. In a worse case example (figure 8.6d) a high energy plateau B can be clearly seen across a large portion of the boundary, obscuring a section of desired axis A. It is arguable, on the basis of visual comparison, that the plateauing effect is stronger in continuous energy fields than it is in the equivalent bivalent energy fields (Figure 8.2e,f,g). The effect on the behaviour of active contours is detrimental, as shown in figure 8.7.

The symmetries generated along the shape boundary have been omitted in the 3D view to aid presentation.

Chapter 8

Energy Fields

Figure 8.6 Continuous energy Fields (angular equality). The quality o f the energy field s (a,b,c) is effected by the number o f boundary p oin ts although the bilateral axis is distin ct in all cases. Energy plateauing B is a dominant side effect oFscuring the desired axis A (d).

Figure 8.7 The effect o f energy plateauing. The persistence o f an energy pla tea u through a boundary sequence can cause an active contour to lose contact with the axis o f interest^^.

The thickness o f the active contour in this example has be exaggerated for reasons o f clarity.

Chapter 8

Energy Fields

Although this new approach to energy field generation does not solve the problem of energy plateauing, the bilateral axis is reasonably prominent in all three of the example energy fields. This suggests that a useful energy field can be generated from relatively fewer boundary points using this technique, thus reducing computation time and expense, assuming plateauing can be eliminated. If energy plateauing is to be avoided, it is necessary to distinguish local syimnetries that give rise to the axes of reflectionally symmetry from the equally valid local symmetries that constitute sub-symmetiies and plateaux.