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 sub-images 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.

 

 

 

\begin{figure}
\centerline {
\psfig{figure=/home/pizzato/latex/cap1/dwt_non_stand.eps,angle=0,width=6cm}}\end{figure}

Figure 1: Non-standard 2D-DWT decomposition

The standard solution consists of alternating one decomposition by rows and another one by columns, iterating only on the low-pass sub-image.

 

 

\begin{figure}
\centerline {
\psfig{figure=/home/pizzato/latex/cap1/dwt_stand1.eps,angle=0,width=12.5cm}}\end{figure}

Figure 2: Bank of filters iterated for the 2D-DWT standard

 

 

The resulting decomposition is visible in the figure 2.

 

 

\begin{figure}
\centerline {
\psfig{figure=/home/pizzato/latex/cap1/dwt_stand2.eps,angle=0,width=6cm}}\end{figure}

Figure 3: Standard 2D-DWT 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 sub-images 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 2D-DWT 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 low-pass and high-pass analysis filters are reported in the following table:

 

 

i

Hl(i)

Hh(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

Hl(i)

Hh(i)

0

3/4

1

1

1/4

-1/2

2

-1/8

 

 

 
Is it possible to use filters defined by the user?

 

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