Concept of Multiresolution Analysis

There were a number of orthonormal wavelet basis functions of the form

$ ~ , , ~ ( t ) = 2TrnI2$I (2+Y ^- n ) discovered in 1980s. The theory of multires- olution analysis presented a systematic approach to generate the wavelets [6, 9, 101. The idea of multiresolution analysis is to approximate a function f ( t ) at different levels of resolution.

In multiresolution analysis, we consider two functions: the mother wavelet

$ ( t ) and the scaling function $(t). The dilated (scaled) and translated (shifted) version of the scaling function is given by $m,n(t) = 2-m/2$(2-mt ^- n). For fixed m, the set of scaling functions $m,n(t) are orthonormal. By the linear combinations of the scaling function and its translations we can generate a set of functions

(4.12) The set of all such functions generated by linear combination of the set n

{$m,n(t)} is called the span of the set {$m,n(t)}, denoted by Span{$,,,(t)}.

Now consider V, to be a vector space corresponding to Span{$,,,(t)}. As- suming that the resolution increases with decreasing m, these vector spaces describe successive approximation vector spaces,

. . . c

^V2

c

^VI

c

^VO

c

^V-1

c

V-2

c

^{. .}

.,

each with resolution 2, (i.e., each space

V,+,

is contained in the next resolution space

K).

In multiresolution analysis, the set of subspaces satisfies the following properties:

84 lNTRODUCTlON TO DISCRETE WAVELET TRANSFORM

1. V,+l

c

V,, for all m: This property states that each subspace is con- tained in the next resolution subspace.

2. = L 2 ( R ) : This property indicates that the union of subspaces is dense in the space of square integrable functions L 2 ( R ) ;

R

indicates a set of real numbers (upward completeness property).

3. n V , = 0 (an empty set): This property is called downward cornplete- ness property.

4. f(t) _E VO cf f(2-,t) E V,: Dilating a function from resolution space Vo by a factor of 2, results in the lower resolution space VTL (scale or dilation invariance property).

5. f ( t ) 6 V , ⁺⁺ f ( t - n ) E VO: Combining this with the scale invariance property above, this property states that translating a function in a resolution space does not change the resolution (translation invariance property).

6. There exists a set

{4(t

^- n ) E VO: n is an integer} that forms an or- thonormal basis of VO.

The basic tenet of multiresolution analysis is that whenever the above properties are satisfied, there exists an orthonormal wavelet basis +,,n ( t ) = 2-m/24(2-mt - n ) such that

(4.13) where P3 is the orthogonal projection of ^$J onto

V, ^.

^{For each m,} consider the wavelet functions +m,n(t) span a vector space W,. It is clear from Eq. 4.13 that the wavelet that generates the space W , and the scaling function that generates the space V, are not independent. W, is exactly the orthogonal complement of V, in V,-1. Thus, any function in V,-l can be expressed as the sum of a function in V, and a function in the wavelet space W,.

Symbolically, we can express this as

V,-l =

v,

⁶⁹

w,.

^(4.14)

Since m is arbitrary, Thus,

Continuing in this fashion, we can establish that

v,

^{= VTLS-1 @} Wrn+l. (4.15)

(4.16) Vm-1 = Vm+l CB wm+l CE Wm.

Vm-1 = Vk CB Wk CE

w!$-1

^{@? Wk-2 @? ' . '}

w,

^(4.17)

for any k

>.

^m.

WAVELET TRANSFORMS 85 Thus, if we have a function that belongs to the space Vm-l (i.e., the func- tion can be exactly represented by the scaling function at resolution m ^- l), we can decompose it into a sum of functions starting with lower-resolution approximation followed by a sequence of functions generated by dilations of the wavelet that represent the loss of information in terms of details. Let us consider the representation of an image with fewer and fewer pixels at successive levels of approximation. The wavelet coefficients can then be con- sidered as the additional detail information needed to go from a coarser to a finer approximation. Hence, in each level of decomposition the signal can be decomposed into two parts, one is the coarse approximation of the signal in the lower resolution and the other is the detail information that was lost because of the approximation. The wavelet coefficients derived by Eq. 4.9 or 4.10, therefore, describe the information (detail) lost when going from an approximation of the signal at resolution 2nL-1 to the coarser approximation at resolution 2m.

4.2.3

It is clear from the theory of multiresolution analysis in the previous sec- tion that multiresolution analysis decomposes a signal into two parts ^- one approximation of the original signal from finer to coarser resolution and the other detail information that was lost due to the approximation. This can be mathematically represented as

Implementation by Filters and the Pyramid Algorithm

(4.18)

n n

where fm(t) denotes the value of input function f(t) at resolution 2m, Cm+l,n

is the detail information, and am+l,n is the coarser approximation of the signal at resolution 2m+1. The functions,

4m+l,n

^{and +m+l,n} are the dilation and wavelet basis functions (orthonormal).

In 1989, Mallat [6] proposed the multiresolution approach for wavelet de- composition of signals using a pyramidal filter structure of quadrature mirror filter (QMF) pairs. Wavelets developed by Daubechies [9, lo], in terms of discrete-time perfect reconstruction filter banks, correspond to FIR filters. In multiresolution analysis, it can be proven that decomposition of signals using the discrete wavelet transform can be expressed in terms of FIR filters [6, 101 and the all the discussions on multiresolution analysis boils down to the fol- lowing algorithm (Eq. 4.19) for computation of the wavelet coefficients for the signal f(t). For details see the original paper by Mallat [6].

(4.19) where g and h are the high-pass and low-pass filters, gz = (-l)'h-i+l and h, = 2'1's

4(z

- 2)4(2z) c m , n ( f ) dx. = Actually, c k g2n-k a,,,(f) am-l,k(f) are the coefficients charac-

1

a m , n ( f ) = c k h Z n - k a m - l , k ( f )

86 /NTRODUCT/ON TO DlSCRE TE WAVELET TRANSFORM

terizing the projection of the function f(t) in the vector subspace V, (i.e., approximation of the function in resolution 2m), whereas c,,,(f) ^E W, are the wavelet coefficients (detail information) at resolution 2“’. If the input signal f ( t ) is in discrete sampled form, then we can consider these samples as the highest order resolution approximation coefficients ao,,,(f) E Vo and Eq. 4.19 describes the multiresolution subband decomposition algorithm to construct a,,,(f) and c,,,(f) at level m with a low-pass filter h and high- pass filter g from C ~ ~ - . I , ~ ( ~ ) , which were generated at level m- 1. These filters are called the analysis filters. The recursive algorithm to compute DWT in different levels using Eq. 4.19 is popularly called Mallat’s Pyramid Algorithm [6]. Since the synthesis filters h and g have been derived from the orthonormal basis functions q!~ and $J, these filters give exact reconstruction

n n

Most of the orthonormal wavelet basis functions have infinitely supported

$J and accordingly the filters h and g could be with infinitely many taps.

However, for practical and computationally efficient implementation of the DWT for image processing applications, it is desirable to have finite impulse response filters (FIR) with a small number of taps. It is possible to construct such filters by relaxing the orthonormality requirements and using biorthogo- nal basis functions. It should be noted that the wavelet filters are orthogonal when (h’,g’) = (h,g), otherwise it is biorthogonal. In such a case the filters (h’ and g’, called the synthesis filters) for reconstruction of the signal can be different than the analysis filters ( h and g) for decomposition of the signals.

In order to achieve exact reconstruction, we can construct the filters such that it satisfies the relationship of the synthesis filter with the analysis filter [la]

as shown in Eq. 4.21:

gk = (-l)nh-n+i

(4.21)

If (h’,g’) = ( h , g ) , the wavelet filters are called orthogonal, otherwise they are called biorthogonal. The popular (9, 7) wavelet filter adopted in JPEG2000 is one example of such a biorthogonal filter [8]. The signal is still decomposed using Eq. 4.19, but the reconstruction equation is now done using the synthesis filters h‘ and ^g‘ as shown in Eq. 4.22:

n n

(4.22) Let’s summarize the DWT computation here in terms of simple digital FIR filtering. Given the input discrete signal ~ ( n ) (shown as a(0, n) in Figure 4.2),

EXTENSION TO TWO-DIMENSIONAL SIGNALS 87

THREE-LEVEL SIGNAL DECOMPOSITION THREE-LEVEL SIGNAL RECONSTRUCTION Fig. 4.2 Three-level multiresolution wavelet decomposition and reconstruction of sig- nals using pyramidal filter structure.

it is filtered parallelly by a low-pass filter ( h ) and a high-pass filter (9) at each transform level. The two output streams are then subsampled by simply dropping the alternate output samples in each stream to produce the low- pass subband y~ (shown as a(1,n) in Figure 4.2) and high-pass subband Y H

(shown as c(1,n) in Figure 4.2). The above arithmetic computation can be expressed as follows:

TL-1 T H - 1

yL(n) =

c

_i=O ^{h(i)2(2n - i), yH(n) =}

_c

_i=O ^{g(i)z(2n - i) (4.23)}

where r~ and ^TH are the lengths of the low-pass ( h ) and high-pass ( 9 ) filters respectively. Since the low-pass subband a(1,n) is an approximation of the input signal, we can apply the above computation again on a(1, n) to produce the subbands a(2, n ) and c(2, n ) and so on. This multiresolution decomposi- tion approach is shown in Figure 4.2 for three levels of decomposition. During the inverse transform to reconstruct the signal, both a ( 3 , n ) and c(3,n) are first upsampled by inserting zeros between two samples, and then they are filtered by low-pass (h’) and high-pass (9’) filters respectively. These two fil- tered output streams are added together to reconstruct a(2,n) as shown in Figure 4.2. The same continues until we reconstruct the original signal a(0, n ) .

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