'preted as a
iency chan-
scale.
ucted from
' to scale s
1 frequency
e matching
of the con-
discretizing
(7)
family
A (8)
4;f denote
a scale s =
er of levels
arsest level,
D;f denote
f and A;f,
n (9)
(x) can be
(10)
be realized
let function
z) (11)
z) (12)
2 (13)
$;,k (x) by.
(14)
tion 4, H is
x conjugate
For a 2-D image function f(z, y), a multiresolution analysis
can be written as
f(z,y)2 Avxf Fr Diaf t Dif t Dif
= Af + Deaf - D25f - Daosf
+D1,1$f + D12f + Disf
ZA S IDjaf + D,2f + Djsf] (15)
j=1
Each approximation A;f(z,y) and difference component
D; »f(z,y) can be fully characterized with a 2-D scaling func-
tion ®(z,y) and its associated wavelet functions ¥,(z,y),
p=1,23,
Too Too
Aj. m S Sau bv) (16)
kz—oolz-—oo
Too Too
Djpf(z.y)-— NS SE dy pk V, pku(rz,y) (17)
k-—oolz-—oo
where
1,,r—k y-l
$;i(z, y) em 23 ( 2j ^ E, (7, k, l) € Z (18)
1 r—k y-l
dynt = < Fix. vu), 0; 115,4) > (20)
di prt =< Hoy) spring) > (21)
For a separable multiresolution analysis, the scaling function
®(z,y) and wavelet functions ¥,(z,y),p = 1,2,3 can be
written as
S(z,y) = é(z)ó(y) (22)
Vi(z,y) — ó(z)v(v) (23)
Vo»(z,y) — v(z)ó(v) (24)
Vs(z,y) = v(z)v(v) (25)
where ó is a one-dimensional scaling function, v is the 1-D
wavelet function associated with ó. Apparently V, V5, V3
extract the details of the 2-D image function f(z, y) in the
y-axis, x-axis and diagonal directions respectively.
The representation (15) may be vividly called the wavelet
pyramid of an image f(z, y). Given a discrete image f(z, y)
with a limited support s = 1,2, (Mat gy — 1,2)... fy.
the actual procedure for constructing this pyramid involves
computing the coefficients ajx:, djp,k:, Which can be
grouped into four matrices A;, D;p, P = 1,2,3, on each
level 7
A; = (a;,k,t)(n=1,2,. 2g =12,.., 24) (26)
s 2 (27)
2J
Let h and g be the impulse response of the filter ó and v, the
coefficients a; x, and dj pki, p = 1,2,3, can be computed
via an iterative procedure. The wavelet pyramid of image
f(z,y) and its constructing process are illustrated in Fig.1.
(z|2 means dyadic subsampling).
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
A3] Ds,
D,;| D35
A
A, Dy;
Dy, | D,
A; D,,
D, D, 3
f(X,y)
Figure 1: Wavelet pyramid of an image f(z, y) (left) and the
flowchart for the analysis from level j — 1 to level 7 (right)
3.2 Complex Wavelets for Phase-Based Matching
Wavelet pyramid is ideal for scale adaptive image matching as
it has the advantage at the good locality in both spatial and
frequency domain. However, wavelet pyramid of an image is
neither translation-invariant, nor rotation-invariant. At this
stage, let us concentrate on the translation-invariance prob-
lem, while assuming that either the rotation angle y of the
matched image about its principal axis relative to the refer-
ence image is small enough, or two stereo images have been
resampled along the epipolar lines.
The wavelet pyramid of real-valued wavelets is not
translation-invariant implies that the phase information is not
readily represented. In order to explore the phase information
in the image signals and still on multiscales, complex-valued
wavelets are a suitable representation as the translation in
the spatial domain is represented as a rotation in the com-
plex phase domain. This gives rise to the interpolability of the
wavelet transform, yielding the possibility of subpixel match-
ing through the multilevels of the wavelet pyramids.
The complex wavelets used in this work were designed by
Margarey and Kingsbury (1995), first used for motion estima-
tion of video frames. Bergeaud and Mallat (1984) proposed
similar complex wavelets. It should be pointed out that the
similarity distance measures and various matching strategies
to be described in the following sections are not limited to
those particular wavelets used in this work, they may rather
be generally applicable with other well-designed wavelets with
good properties.
For general image matching purpose, we require the wavelet
filter pair (h, g) (impulse response of the scaling and wavelet
function ó and V) to be compactly supported in spatial do-
main, regular (differentiable up to a high order), symmetric
