Perpendicularlity

The continuous energy field discussed above is a map of symmetry scores attributed to all mid­ points between boundary point pairs Pi r„,pj~„, based on the degree of angular equality that exists between the angles subtended by the tangents to Pi,„ and pj,„ and the line L„ joining them (figure 8.8). Instances of high local symmetry are attiibuted a high score. This is the case regardless of the significance of any larger syimnetry to which the high scoring local symmetries may contribute. Thus, the local symmetries A,B,C, in the sub-syimnetry S2 derived at the comer of a rectangle are as valid as the a local symmetries D,E,F on the bilateral axis SI and are attributed the same score (figure 8.8). If the bilateral axis is to be attributed greater significance to distinguish it from other sub-symmetries and energy plateaus, a propeity unique to the local symmetries which cause the desired bilateral axis has to be taken into account in the scoring process. P..5 P..3 'i+3 'i+2

cc

Figure 8.8 L ocal symmetries. To emphasise the axes o f rejlection al sym m etry it is n ecessary to distinguish the loca l sym m etries D ,E a n d F fro m A.B an d C.

Chapter 8__________________________________________

Energy Fields

Consider the six local symmetries, ^ to F, in figure 8.8. There are two properties of the local symmetries, D,E,F, that lie on the bilateral axis Sj that distinguish them from the others:

1. The perpendicular bisectors of the lines L„ connecting point pairs locally symmetric about the bilateral axis pass through the centroid CC of the rectangle. The equivalent lines connecting point pairs across the comers do not;

2. The angles 0i+n and 0j+n for local symmetries on the axes of reflectional symmetry are right angles.

It is reasonable to assume that a mode of scoring based on either or both of these properties could be used in conjunction with angular equality to highlight the desired axis. Experiments with a centroid based approached revealed fimdamental flaws. This property can be expressed as an continuous angular relationship similar to that described for angular equality. However, to avoid spurious and inappropriate scoring it is necessary to know the position of the centroid in the symmetrical form of the shape. For the banana shape it is necessary to know the location of the centroid in its symmetrical equivalent, i.e., a straight banana. The location of this presupposes knowledge of the bilateral axis before it is found. This is self defeating and therefore this approach was not considered fiuther. The second of these properties is potentially more useful for current purposes. However, before considering its use in conjunction with angular equality it is worth considering in its own right. Like equality, this is an angle related property and therefore it is reasonable to assume that it can be expressed as a continuous function ep(pi,pj) and described in as continuous energy field.

8.1.2.1 Calculating the Energy Field (Perpendicularity)

For any angle 6i(0 <9i<7t) the deviation d of ^ from the perpendicular can be expressed as:

d = \e,~4,

(

,

)

The value of d will be between 0 and y radians. Given the two angles ^ ,6j subtended by the

tangents to the two points pi,pj and the line L joining them, the mid-point o f L can be attributed a value given by the sum of the respective deviations. The resulting value will between 0 and k

and will be a direct indication of the degree of symmetry between pi and pj. If both angles are equal to % the resulting score will be 0. Assuming the need for normalisation and scaling, etc. as discussed in section 8.1.1.1, equation 8.3 can be adapted so ep(0j,0j) is given by:

Chapter 8

Energy Fields

1 -

l ^ , - î | + k - î

7 1

/ /

8.1.2.2 Examples of the Energy Field

The axes of reflectional symmetry dominate and all sub-symmetries are absent in an energy field for a rectangle (figure 8.9) derived from continuous perpendicularity values. More specifically, in energy fields of this type generated for the banana shape, the axes of relational symmetry are clearly visible (figure S.10a,b,c), the short axis of reflectional symmetry is missing and no plateauing effects can be seen (figure S.lOd). This is the case regardless of the three boundary sample sizes considered.

Figure 8.9 A continuous energy fie ld f o r a rectangle (perpendicularity). The function Cp(pi,pf gives a continuous energy fie ld in which the reflectional axes o f sym m etry are distin ct and all sub-symm etries are absent.

Chapter 8

Energy Fields

Figure 8.10 Continuous energy field s (p-type). The quality o f continuous p -typ e energy field s is minimally affected by boun daiy p oin t sam ple size (a,b,c)- Energy p lateau in g is not a problem fh) but axis terminations are indistinct..

It is conceivable that p-type^^ energy fields are superior to the energy fields discussed so far and the need for a combination of energy fields is unwarranted. However, two side- effects severely limit its use. P-type energy fields are fundamentally dependent on parallelism between the boundary elements (more specifically between the tangents at the points under consideration on the boundary). Angles 9, and 6j can only ever both be equal to ^ if the tangents at the points from which they are derived are parallel. In situations where this is not the case the resulting energy field is poor. Consider an equilateral triangle. Such a triangle has three axes of reflectional symmetry that are clearly seen in an e-type energy field (figure 8.1 la). However, the parallel dependant p-type energy field does not indicate this fact; distinct energy ridges corresponding to the axes of reflectional symmetry are not obvious (figure 8.11c). The dependence on boundary segment parallelism is a considerable weakness and the use of this type of energy field alone would limit its general application. This phenomenon accounts for the absence of the short axis in the p-type energy field of the banana shape. The portion of the short axis of the bilateral axis internal to the shape is derived from the points on the outer edge

For ease o f discussion angular equality based energy fields will be referred to as e-type energy fields and perpendicularity based energy fields will be referred to as p-type energy fields.

Chapter 8

Energy Fields

of the shape. Sinee there are no points on this edge that have parallel tangents, the mid-points on the short axis will not get a significant vote and will not become visible.

The second limiting factor is concerned with the nature of the ends of the desired axis as it appears in the energy fields. The points of termination of the bilateral axis in the banana shape are less distinct in p-type energy fields compared with those in an e-type energy field. This can be clearly seen in the simple shape shown in figure 8.11. The termination of the bilateral axis as it appears in p-type energy field is dispersed and ill defined (figure 8.1 Id). This is not the case for the e-type energy field (figure 8.1 lb). The main disadvantage with this effect is an increased possibility of misalignment between the active contour and the point of axial intersection with the boundary.

P arallelism

(c)

T erm ination P o in ts

(b)

Figure 8.11 Energy f ie ld weaknesses. Axis termination is p o o r (d) and energy f ie ld quality is affected when boundary segm ents are non p a ra llel (c) in p -typ e energy fields. This is not the case f o r e-type energy field s (a,h).

Chapter 8________________________________________________Energy Fields