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138 3 Image processing

H Ļ2 o H 0 Q H 0 2 o F

L Ļ2 e L 1 L 1 2 e I

(a)

Ļ2 o – H 0 Q H 0 2 o

L C C L
–
Ļ2 e L 1 L 1 2 e

(b)
Figure 3.38 One-dimensional wavelet transform: (a) usual high-pass + low-pass filters followed by odd (↓ 2 o )
and even (↓ 2 e ) downsampling; (b) lifted version, which first selects the odd and even subsequences and then
applies a low-pass prediction stage L and a high-pass correction stage C in an easily reversible manner.

can the corresponding reconstruction filters I and F be computed? Can filters be designed
that all have finite impulse responses? This topic has been the main subject of study in the
wavelet community for over two decades. The answer depends largely on the intended ap-
plication, e.g., whether the wavelets are being used for compression, image analysis (feature
finding), or denoising. Simoncelli and Adelson (1990b) show (in Table 4.1) some good odd-
length quadrature mirror filter (QMF) coefficients that seem to work well in practice.
Since the design of wavelet filters is such a tricky art, is there perhaps a better way? In-
deed, a simpler procedure is to split the signal into its even and odd components and then
perform trivially reversible filtering operations on each sequence to produce what are called
lifted wavelets (Figures 3.38 and 3.39). Sweldens (1996) gives a wonderfully understandable
introduction to the lifting scheme for second-generation wavelets, followed by a comprehen-
sive review (Sweldens 1997).
As Figure 3.38 demonstrates, rather than first filtering the whole input sequence (image)
with high-pass and low-pass filters and then keeping the odd and even sub-sequences, the
lifting scheme first splits the sequence into its even and odd sub-components. Filtering the
even sequence with a low-pass filter L and subtracting the result from the even sequence
is trivially reversible: simply perform the same filtering and then add the result back in.
Furthermore, this operation can be performed in place, resulting in significant space savings.
The same applies to filtering the even sequence with the correction filter C, which is used to
ensure that the even sequence is low-pass. A series of such lifting steps can be used to create
more complex filter responses with low computational cost and guaranteed reversibility.