What is the Discrete
Wavelet Transform (DWT)?
Which
advantages does the DWT have compared to the DCT?
Can the wavelet
transform be reversible?
Is it
possible to use filters defined by the user?
What is the Discrete Wavelet Transform
(DWT)?
DWT
is the discrete variant of the wavelet transform.
Wavelet transform
represents a valid alternative to the cosine transform used in standard JPEG.
The DWT of images is a transform based on the tree
structure with D levels that can be implemented by using an appropriate bank of
filters.
Essentially it is possible to follow two strategies
that differ from each other basically because of the criterion used to extract
strings of image samples to be elaborated by the bank of filters.
The first solution, definitely not very used,
consists of generating the string by queuing image lines and then executing a
decomposition on D levels; after this operation, we generate D strings by
queuing the columns from the found subimages and another decomposition for
each string is applied. The resulting decomposition, in the simplified version
extended up to the third level, is shown in figure 1.

Figure 1: Nonstandard 2DDWT decomposition 
The standard solution consists of alternating one decomposition by rows and
another one by columns, iterating only on the lowpass subimage.

Figure 2: Bank of filters iterated for the 2DDWT standard 
The resulting decomposition is visible in the figure 2.

Figure
3: Standard 2DDWT
decomposition

Let us observe the typical denomination of the subbands: by reading
their name from left to right, l or h letters tell us, we can see
which filter was used for the analysis; with this convention the subimages
with two letters correspond to the first level of decomposition, the ones with
4 letters to the second level and so on. The subband of approximation results
to be identified only by ‘/’. There is a constant ratio (equal to 4)
also between a subband and the subband of the previous level.
Which advantages does the DWT have compared to the
DCT?
The main advantage is that the cosine transform previously carries out a
division into squared blocks, while the 2DDWT works in its totality.
Moreover
the decomposition into subbands gives a higher flexibility in terms of
scalability in resolution and distortion.
Can the wavelet transform be
reversible?
It can be both reversible and irreversible.
The irreversible transform, generally used in JPEG2000 makes us of the Daubechies
9/7 filter with real values. The coefficients of lowpass and highpass analysis
filters are reported in the following table:
i 
H_{l}(i) 
H_{h}(i) 
0 
0.6029490182 
1.1150870524 
± 1 
0.2668641184 
0.5912717631 
± 2 
0.0782232665 
0.0575435262 
± 3 
0.0168641184 
0.0912717631 
± 4 
0.02674875741 

The reversible transform is generally carried out by a 5/3 filter with
complex values. This filter peculiarity is that it produces a complete and
lossless reconstruction during the synthesis. Filter coefficient values are
reported in the following table
i 
H_{l}(i) 
H_{h}(i) 
0 
3/4 
1 
± 1 
1/4 
1/2 
± 2 
1/8 

This option is not available
in the Part I of the standard, but it is present in the extensions of Part II.