Rob Reeves 
directly from the DCT of the original sequence. This and related properties have been reported previously (Reeves, 1999 
Reeves and Kubik, 1998), and are summarized below. ME 
We define the forward discrete transform and its inverse as 
N—1 
G(m) = T{g(n)} = X g(n)fa(m) i 
n=0 
and 
N-1 
g(n) = T HG (m)} = S G (m)rg (n) () 
m=0 
where f,,(m) represents the forward transform kernel, r,, (1) represents the reverse transform kernel, and n and m are 
integers from 0 to N — 1. The type-2 DCT (Rao and Yip, 1990) is defined by 
v2 
falm) = ran) = A cos((2n + 1)mx/2N) 0 
with c(m) — A form = 0 and c(m) = 1 otherwise. We assume that the sequence g(n) is derived by sampling, 
in accordance with the Nyquist criterion, a band-limited continuous signal g(x) at points x = n,n = 0...N — 1, wi 
sampling interval taken as one without loss of generality. By considering a half-range cosine series expansion of g(x) 
around x = — 1, we can show that, 
N—1 
glx) = So ra (z)G(m) (4) 
m=0 
This expression is equivalent to the reverse discrete transform, except that it gives the continuous band-limited function 
g(x) instead of sequence g(n), and the discrete variable n has been replaced by continuous x in the reverse transfom 
kernel. If g(z) satisfies Dirichlet’s conditions, it’s derivative can be computed by a term by term differentiation of its 
Fourier series (James et al., 1993). Differentiating both sides of Equation 4 gives 
d p N—1 7 
1,79 = 2 Glmirm(a) = 2. G(m) rm (a) 6 
Adopting the notation ¢'(n) to mean the sampled derivative of g(x) at z = n, and r^, (n) to refer to the sampled deriva 
of Taiz) at T = n gives 
N-1 
gn) — S G (m)r;, (n) (6 
m=0 
It follows from Equation 1 that 
N—1 N—1 
Tig} = M Gp fot.) 0 
p=0 n=0 
where p is used as an additional index for the transform coefficients. Equation 7 represents a simple linear transfom 
which computes T | 9 (n)) from the values of 7'{g(n)}. This property can be extended similarly to the second and higher 
derivatives, and to any linear operation on g(z), including the terms involving derivatives required to formulate the les 
squares problem in the DCT domain. This single linear transform is equivalent to reconstructing the continuous signi. 
differentiating it, sampling it, and taking the DCT. The extension to two dimensions is straightforward. 
3 LEAST SQUARES MATCHING 
Our work is based on the formulation of Ackerman (Ackermann, 1984), who uses an affine transformation to model the 
transformation of left image patch to right image patch as follows, 
gi(z,y) = ho + h1g(ao + a1x + a2y, bo + biz + bay) + M (x,y) 
and N 
g2(x,y) = g(x,y) + na(x,y) 0 
  
762 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
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