Results from the Wavelet Transform Analysis.
6.2. On the performance prediction of the 4-TAP wavelet filters.
O verall, 276 subim ages w ere u sed in this study, 138 abnorm al an d 138
norm al, as described at a previous parag rap h . In this p articu lar experim ent w e
algorithm , into eight decom position levels. In oth er w ords, w e fully m a p p e d
the subim ages from the tim e dom ain into the w avelet transform dom ain. Each
subim age w as analysed 360 tim es, each tim e w ith a different w av elet filter
w hich w as obtained from the E quation 6.4. The variance of the w avelet
transform coefficients w as m easured. As a result for each subim age w e collect
the variance of their w avelet transform coefficients w h e n th ey are analysed
w ith all the 4-tap w avelet filters. These variance values w ere th en norm alised
a n d m a p p e d to the radii Rg of the concentric circles w ith respect to circle
(Equation 6.3), usin g the formula:
/ \ 0 = 3 6 0 °
maxiVflÆ - VarRg
^ ■ ^ 0 — / \ 0 = 3 6 0 ° / \ 0 = 3 6 O °
mâx(VarRg
So, the form ula m aps the variance (VarRg ) values to variable rad ii Rg follow ing
the conventions listed below:
• The max(VarRgy^_^^^ corresponds to the centre of the circle (Eq. 6.3).
• The imniyarRg corresponds to the circum ference of the circle (Eq. 6.3).
• All the other VarRg values corresponds to the interior of the circle (Eq. 6.3).
This m ap p in g efficiently describes the d istribution of the variance of the
w av elet transform coefficients w h en the subim ages are analysed w ith all the
From Figure 6.1. it is clear th at only half of these 4-tap w avelets are unique, the
o th er half can be obtained using the sym m etry about the line Cq=c^. This
co rresponds to reversing the ord er of the coefficients, as sh o w n in Table 6.1.
A ngle 0 C2 ^3
165° -0.129410 0.224144 0.836516 0.482963
285° 0.482963 0.836516 0.224144 -0.129410
Table 6.1: Half of the 4-tap wavelet filter coefficients obtained using the symmetry about the line
Co = C3.
The m axim um values of the norm alised variance of the w avelet transform
coefficients are obtained w h en 0 = 165° or 0 = 285°, w hich corresponds to
D aubechies w avelets, w hile the m inim um values are obtained w h en 0 = 45°.
It sh o u ld also be n o ted th at the locus in Figure 6.1., of the norm alised variance
m atches the locus of the regularity for a b ro ad range of values of 0 a ro u n d the
D aubechies w avelets. H ow ever, they differ slightly aro u n d 0 = 45°, since this
p o in t does n o t determ ine a w avelet orthonorm al analysis for Thus,
p red ictio n of the norm alised variance of the w avelet transform coefficients from
th e regularity locus Eq. 6.6, although quite inform ative, is n o t reliable enough
for a large n u m b er of values of 0 as can be seen from Figure 6.1.
A s w ith the norm alised variance (Eq. 6.5), the sam e eq u ation can be u sed for the
m a x ( - R E G ,
^ ^ 0 ~ / \ 8 = 3 6 0 ° / \ 8 = 3 6 0 ° W - O j
max(/?£G, - mm[REG,
From the equality of Equations 6.5 an d 6 . 6 w e can obtain the p red icted variance
values u sin g the follow ing equation:
VarRa =
/ \ 8=360° r / \ 8=360° 1 / \ 8=360° / _ \ 8=360° 1 max(Varfi,)^ KEGj-min(KEG,)^ j + m3x(REGg\^ -REG,]
(
6.
7)
max(RRG,)nr-nun(REG.)::“/ \ 8 = 3 6 0 ° , / \ 8 = 3 6 0 °
or by su b stitu tin g min(^EGg = -0 .2 7 , an d m a x ^ / ^ E G ^ =- 0. 73,
= max(Vflr/ ? 0 [/?EGg + 0.27] + wix)fyarRQ
[o
.73- REGq ] (6.8)or
+027max(Vbr%)^ +073mn(%y:R^)^ (6.9)
The above form ulas can pro v id e predicted variance values of the w avelet
tran sfo rm coefficients by taking into account the values of the reg u larity of all
the 4-TAP w avelet filters.
H ow ever, from Figure 6.1. it can be n oted th at the big locus (circle-like) curve
of the norm alised variance locus can be approxim ated usin g the lim açon of
Pascal (cardiod) equation Rq = b + a cos(0 - 45°), w hereas the sm all curve (in the
3^^ qu ad ran t) can be approxim ated using the 4-leaved rose eq u ation
Rg = csin(26) [67]. Prediction of the w avelet transform variance usin g 4-TAP
2Rq—
l(h + a cos{6 - 45°)) Pascal's limaçon,
0
2{c sin(20)) 4 - leaved rose,
0
2{b + a cos{6 - 45°)) Pascal's limaçon.
V 0g[O°,16O°) V 0g[16O°,18O°) V 0g[18O°,27O°) V 0 G [270°,290°) V 0 G [290°,360°) (6.10)
From the equality of Equations 6.5 an d 6.10, w e can derive E quation 6.11, by
w hich w e can obtain the predicted variance values u sin g the form ulas:
VcütRq = -2(Z7+aco8(0-45'))(nBx(l^/^)^ -2(è+aoŒ(e-45'))|nB3!(V&^)^ -irir(Mrf^) 0=36QP 0=OP )0=36OP et=OP 0=36OP 0=OP
vee[cp,ieop)
V0^16OP.18O") ve^isop^TOP) V0e[27OP31f’) V0s[29CP,3eCP)0.25 0.5 270 Mean ABNORMAL Mean NORMAL Regularity
Figure 6.1: The perform ance o f the 4-tap wavelet filters, in term s o f th eir mean norm alised V ariance, for the norm al (green), and the abnorm al (blue) m am m ographie images. It is also shown for com parison th eir norm alised regularity (red).