where « -,- > denotes the inner product of two functions, 
a;'s are called the representation coefficients. 
By appropriately choosing the basis function v;'s, we intend 
to extract salient information of f(z,y) in the form of the 
coefficients a;'s. A representation of the form (1) is said to 
be a full-information representation if with a;'s computed via 
(2), the equation 
f(v, 9) - Flay as,...,0:01,%2, 00000) (3) 
holds, where F() is a computable function. An example of 
(3) is that when %,’s constitute an orthonormal basis, we 
have a simple reconstruction procedure 
HE) =) a; ¥;(z,9) (4) 
jz1 
A representation of the form (1) and (3) is said to be uniform 
because each representation coefficient a; is defined and com- 
puted with exactly the same simple mathematical form of (2). 
With some contrast to previous image-domain approaches, we 
.do not scrutinize on the explicit interpretation of the repre- 
sentation coefficients, which may highly nonlinearly relate to 
intensity differentials, textures, shading, surface reflectance 
variations, etc. 
For image matching purpose, it is desirable if the representa- 
tion of the form (3) has properties of good dimensional or- 
thogonality, discriminative uniqueness, space-frequency local- 
ity, multiresolution adaptivity, and computational efficiency 
and robustness. For the particular problem of stereo match- 
ing, we may also require the information representation and 
matching strategies to be invariant to translation, rotation, 
scale and partial correspondence between two stereo images. 
Fourier analysis is a classical example of the uniform and 
full-information representation, which, however, is known to 
be very poor in spatial locality. Wavelet analysis is a new 
approach in this sense, as fundamental as Fourier analysis, 
but with inherently good locality in both spatial and frequency 
(scale) domain and a number of other desired properties. 
3 WAVELETS AND COMPLEX WAVELETS 
In this section, we briefly draw the essentials of wavelet theory 
and lay the mathematical foundation of this image matching 
approach. 
3.1 Wavelet Transform and Wavelet Pyramid 
The wavelet transform is a relatively recent development in 
mathematics and signal processing (Grossmann and Morlet, 
1984; Mallat, 1989), as a signal decomposition approach to 
overcome the shortcomings of the window Fourier transform. 
This decomposition is to project the signal f(z) onto a fam- 
ily of functions which are the dilations and translations of a 
unique function (x). The function (x) is called a wavelet 
and the corresponding wavelet family is given by 
Vea(z) = Vsb(s(z —1)),(s,t) e R (5) 
where R denote the set of real numbers, s and à are called the 
scale and translation respectively. Let L^(R) denote the vec- 
tor space of measurable, square-integrable one-dimensional 
functions f(z). The wavelet transform of a function f(x) € 
L?(R) with a given wavelet 9 is defined by 
+ co 
W /f(s,1)- / f(x). i(v)da' z« f(x), vs (2) > (6) 
— Co 
  
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
A wavelet transform as defined in (6) can be interpreted as a 
decomposition of a signal f(z) into a set of frequency chan- 
nels having the same bandwidth on a logarithmic scale. 
Under a certain condition, f(z) can be reconstructed from 
its wavelet transforms W f(s,t). The adaptivity to scale s 
and translation £ leads to its good locality in both frequency 
and spatial domain, a property desired by image matching 
algorithms. 
For a special class of functions, the redundancy of the con- 
tinuous wavelet transform (6) can be cleared by discretizing 
both the scale factor s and the translation 1, 
des E and t — k, with (j, k) € Z° (7) 
where Z denotes the set of integers, the wavelet family 
et 
v27 
is called dyadic discrete wavelets. 
vna)= G=W(E55),  GReZ À) 
In the dyadic scale space of the form (8), let A;f denote 
the approaximation of a given function f(x) at a scale s = 
+. In practice, we only consider a limited number of levels 
27° 
J=0,1,2,...,n, for some n chosen to be the coarsest level, 
corresponding to the smallest scale s = de Let D;f denote 
the difference between two approximations A;-1 f and A;f, 
i.e. 
Djf-2Aj;Af-4Ajf, 1210124.35 (9) 
where Ao is an identity operator. The function f(z) can be 
decomposed as 
f(z)= Ai Dif 
= Aof + D2f + Dif 
=A.f+) Dif (10) 
k=1 
It was proved that a multiresolution analysis can be realized 
by a scaling function ¢(z) and its associated wavelet function 
P(x), 
Too 
Ajf(s)2 9. «f(u)ójs(u) » ó;u(s) (11) 
fons 
Dif(rim $7 «f(uksun(u) m d eit) (12) 
where 
1 r—k 
$5,k(z) = (kF)eZ' (Q3) 
mh 
and #;,x(x) has the form of (8), which relates to ¢; x(z) by. 
$(2w) — e^" H(e-*")ó(w) (14) 
where ¢ denotes the Fourier transform of the function ¢, H is 
the transfer function of ¢, H denotes the complex conjugate 
of H. 
     
can be 
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p=1, 
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For a s 
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The re 
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Let h a 
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f(z,y) 
(z|2 m