The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
proposed method will be explained. The fourth chapter will 
present the experimental results. 
2. THE FRAMELET TRANSFORM 
As it is well known, except for the Haar filterbank, two-band 
finite impulse response (FIR) orthogonal filterbanks do not 
V j =Span{f(2 j t-k)} 
k 
W i j = Span{tp‘(2 J t-k)}, i = 1,...,N-1 
k 
with 
allow for symmetry. In addition, imposition of orthogonality 
for the two-band FIR filterbanks requires relatively long filter 
support for such properties as a high level of smoothness in the 
resulting scaling function and wavelets, as well as a high 
and the corresponding scaling function and wavelets satisfy the 
following multiresolution equations: 
approximation order. Symmetry and orthogonality can both be 
obtained if the filterbanks have more than two bands. 
Furthermore, due to the critical sampling, orthogonal filters 
suffer a pronounced lack of shift invariance, though the 
desirable properties can be achieved through the design of tight 
frame filterbanks, of which orthogonal filters are a special case. 
№ = ^K(.k)<t>(2t-k) 
k 
ip‘(t) = s[2£/?,.(k)<f>(2t-k), i = \,...,N-l. 
k 
In contrast to orthogonal filters, tight frame filters have a level 
of redundancy that allows for the approximate shift invariance 
Now, the frame bound A takes on the value 
behavior caused by the dense time-scale plane. Besides 
producing symmetry, the tight frame filterbanks are shorter and 
result in smoother scaling and wavelet functions. Before 
proceeding further, let me describe the basic concepts of frame 
theory and oversampled filterbanks 
1 N ~ l 
^ i=0 
2.1 A Tight Wavelet Frame 
2.2 Oversampled Filterbanks 
A set of functions jy/in a square integrable space, 
L 2 (□ ) , is called a frame if there exist A > 0, B < oo so that, for 
any function / £ Z, 2 (D ), 
The frame condition can be expressed in terms of oversampled 
polyphase filters. Given a set of N filters, we define them in 
terms of their polyphase components: 
S||/||\ (1) 
¿=1 /60 ieO 
Hfz) = H j0 (z 2 ) + z-'H.fz 2 ), i = 0,..., N -1, 
where 
= 0,1. 
where A and B are known as frame bounds, 
{f,g) := J f(*)g(x) dx , and ||/|f := (/,/). 
Now define the polyphase analysis matrix as 
The special case of A - B is known as tight frame. In a tight 
frame we have, for all / ei 2 (0), 
ZEl|(/>'< 2 '-*))f='<|/f 
1=1 /60 *e0 
' H oA z ) H 0 fz) ' 
H XQ (z) Hyi(z) 
H (z) = . . 
K H N ., 0 (z) H N _ xx (z)^ 
which implies 
If we now define the signal X(z) in terms of its polyphase 
components, we then have 
/=1 jé0 £eQ 
x(z) = (X 0 (z) X x (z)) T , 
In order to have a fast wavelet frame (or framelet) transform, 
multiresolution analysis is generally used to derive tight 
wavelet frames from scaling functions. 
where the equation X t (z), / = 0,1 is defined in terms of the 
time domain signal, x(ri), as follows: 
Now, we obtain the following spaces: