wavelets based on bernstein bases image compression

7/29/2019 Wavelets Based on Bernstein Bases IMAGE COMPRESSION

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Published in IET Computer Vision

Received on 3rd September 2009

Revised on 26th April 2010

doi: 10.1049/iet-cvi.2009.0083

ISSN 1751-9632

General framework of the construction of biorthogonalwavelets based on Bernstein bases: theory analysisand application in image compressionX. Yang Y. Shi B. Yang

Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, School of Mathematics and

Systems Science, Beihang University, Beijing 100191, Peoples Republic of China

E-mail: [email protected]

Abstract: The authors present a general framework of the construction of biorthogonal wavelets based on Bernstein bases alongwith theory analysis and application. The presented framework possesses the largest possible regularity, the required vanishingmoments and the passband flatness of frequency response of filters. Based on this concept, the authors establish explicitformulas for filters of biorthogonal wavelets with arbitrary odd lengths. Meanwhile, a new family of parametric filters withsymmetry is constructed. The choice of filter bank in wavelet compression is a critical issue that affects image quality. In thisstudy, an optimal model of FIR aiming at image compression is brought forward, and the optimal finite impulse response (FIR)filters can be obtained correspondingly through sequential quadratic programming (SQP) and genetic algorithm (GA). Theauthors demonstrate the performance of the new family of filters given in this study for image compression with veryencouraging results.

1 Introduction

Biorthogonal wavelets have become powerful tools in signalprocessing, image analysis, communication systems and manyother related fields. Symmetric or antisymmetric compactlysupported wavelets are very desirable in various applications,since they preserve phase properties and also allow symmetric

boundary conditions in wavelet algorithms which normallyperform better. However, there does not exist any real-valuedsymmetric or antisymmetric compactly supported orthogonalwavelet with dyadic dilation except for the Haar wavelet.Many subsequent constructions sought to remedy this by

relaxing some restrictions. Indeed, in [1], symmetry wasobtained at the cost of dropping orthogonality; two compactlysupported dual refinable functions were needed, only one ofwhich could be a spline function. In [2], similar non-orthogonal dual symmetric spline wavelet bases were given,

but only one of them could be compactly supported. As forexamples of [3], symmetric orthogonality and compactsupport were combined at the price of having multi-waveletsfrom a vector multiresolution analysis. In examples of [4],symmetry, orthogonality, interpolatory property and compactsupport were achieved at the cost of using non-dyadicdilations. Since then, the theory of biorthogonal wavelets has

been developed rapidly [520].In this paper, we present a general framework of the

construction of biorthogonal wavelets based on Bernsteinbases along with theory analysis and application. Thepresented framework possesses the largest possible regularity,

the required vanishing moments and the passband flatness offrequency response of filters. We investigate the smoothness

properties of multivariate refinable functions based onBernstein bases in terms of the spectral radius of thecorresponding transition operator restricted to a suitable finite-dimensional invariant subspace and present a generalalgorithm to construct biorthogonal scaling functions fandfwith the largest possible regularity. By using this framework,a new family of parametric filters with symmetry areconstructed. Parameterisations of FIR systems are offundamental importance to the design of filters with special

properties. It should be mentioned that constructing wavelets

satisfying too many nice properties may become verydifficult. The parameterisation-based construction method

provides one way to this goal. For example, it produceswavelets with high-order vanishing moments and goodsmoothness and compactness representation which playimportant roles in image compression. For this reason, the

problem is among attractive problems of wavelet theory andimage processing.

The biorthogonal wavelet filter banks offer outstandingperformance for image compression, but the choice of filterbanks is a critical issue which affects image quality as wellas system design. The regularity and vanishing moments of

biorthogonal wavelets are used in filter evaluation, but theirsuccess at predicting the compression performance is only

partial. Although wavelet filter selection for imagecompression has advanced substantially with thedevelopment of useful metrics and criteria [19], the factor

50 IET Comput. Vis., 2011, Vol. 5, Iss. 1, pp. 5067

& The Institution of Engineering and Technology 2011 doi: 10.1049/iet-cvi.2009.0083

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that dominates the effect of compression is still open to morediscussion. In our study, the new family of wavelet bases

present a unified framework including the widely favouredCDF9/7 [12] wavelet filter. Under the new constructionframework, biorthogonal wavelets boast great regularity andvanishing moments, as well as the flatness of the frequencyresponse of biorthogonal filters. These properties are of greatimportance to image compression. Since the framework

proposed in this paper is generally feasible, it sets the theoryfoundation for transforms aiming at image compression. Inthe meantime, the optimal model of FIR filters aiming at thecharacteristic image is brought forward, and the optimal FIRfilter can be obtained through sequential quadratic

programming (SQP) and genetic algorithm (GA). Wedemonstrate that the performance of the new family of filtersgiven in this paper for image compression is veryencouraging. Finally, in the paper, we assume that thecoefficients of filters discussed are real.

The present paper consists of five parts. Section 2 presents ageneral framework of the construction of biorthogonalwavelets based on Bernstein bases. Section 3 discusses theregularity analysis of biorthogonal wavelets based onBernstein bases. In Section 4, the construction of parametricfilters with symmetry is discussed. In Section 5, the optimalmodel of FIR filters aiming at the characteristics of theimage is brought forward, and the optimal FIR filters areobtained through SQP and GA. In the meantime, results ofour experiments are analysed. In Section 6, we draw theconclusions and introduce some research prospects.

2 General framework of construction ofbiorthogonal wavelets based on Bernsteinbases

In this section, we will provide a general framework of theconstruction of biorthogonal wavelets based on Bernstein

bases.Here we introduce some basic concepts and conclusions.

Define analysis or synthesis low-pass filter as

H(v) = 12

n

hneinv

, H(v) = 12

n

hneinv (1)We assume that only finitely many hn,

hn are non-zero. Definef, f by

f = 1

j=1H(2jv), f =

1

j=1 H(2jv) (2)These infinite products can only converge if

H(0) = H(0) = 1 (3)that is, if

n

hn =

n

hn = 2 (4)If (3) is satisfied, then the infinite products in (2) converge

uniformly and absolutely on compacts. Obviously

f(v) = H v2

f v2

,

f(v) = H v2

f v2

(5)

or, equivalently

f(t) =2

n

hnf(2t n),

f(t) = 2 n

hnf(2t n) (6)at least in the sense of distributions. From [1], fandf havecompact support. We also define the corresponding c andc byc = eiv/2H v

2+ p

f v2

,

c = eiv/2H v2

+ p f v

2

(7)

or, equivalently

c(x) =2

n

(1)n

h1nf(2x n) (8)

c(x) = 2 n

(1)nh1nf(2x n) (9)We now discuss the biorthogonality offandf. Obviously, iffandf defined as (6) are biorthogonal, then we have [14]

H(v)H(v) + H(v+ p)H(v+ p) = 1 (10)Recall that the Bernstein polynomials [21] B

nk(x) =

(nk)xk(1 x)nk, for k = 1, 2, . . . , n, give polynomial

approximations that converge to a continuous function fon the unit interval [0, 1]; we express fapproximations as alinear operator

Bn[f](x) =nk=0

fk

n

Bnk(x)

=nk=0

fk

n

n

k

xk(1 x)nk (11)

The Bernstein form of a general polynomial is expressed by

H1(x) =nk=0

d(k)n

k

x

k(1 x)nk (12)

where d(i) are the Bernstein coefficients.Theorems 1 and 2 indicate that the representation of

decomposing and reconstructing low-pass filters expressed byBernstein polynomials ensures that the coefficients of filtersare symmetric, which is of great significance in image

processing.

Theorem 1: Suppose that the low-pass filter of analysis (orsynthesis) is as follows

H1(x) =

N

k

=0

d(k)N

k

x

k(1 x)Nk (13)

H(v) = H sin2 v2

=j=N

j=Nhje

ivj(14)

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where x = sin2(v/2) and z= eiv. Then, we have

hm = hm =Nk=m

[(km)/2]n=0

(1)m ak2k+2n+m

k

2n + m

2n + m

n , m 0 (15)

and

ak =ki=0

(1)kid(i) Ni

N ik i

(16)

Proof: Suppose that

H1(x)=

N

k=0akx

k(17)

we have

H1(x) =Ni=0

d(i)N

i

xiNij=0

(1)j N ij

xj

=Ni=0

Nij=0

d(i)(1)j Ni

N i

j

xi+j

Thus

ak =ki=0

(1)kid(i) Ni

N ik i

(18)

Furthermore, if x = sin2 (v/2), z= eiv, then

H(v) =Nk=0

ak1 cos v

2

k=Nk=0

ak2k

kj=0

(1)j kj

cos

j(v)

=Nk=0

ak2kkj=0

(1)j kj

z+ z12

j=Nk=0

kj=0

jl=0

(1)j ak2k+j

k

j

j

l

z

j2l

Set j2 2l m, then l (j2m)/2. For the sake of convenience, we define the expression given by

k

j m2

=k

j

m

2 , ifj m

2is a postive integer

0, else

(19)

We obtain

hm =Nk=m

kj=m

( 1)j ak2k+j

k

j

jj m

2

=

N

k=0 km

s=0

( 1)m+s ak2k+s+m

k

s+

m m + s

s

2 =Nk=0

[(km)/2]n=0

( 1)m ak2k+2n+m

k

2n + m

2n + m

n

Thus, (15) is proven. ASimilarly, we can obtain Theorem 2.

Theorem 2: Suppose that the low-pass filter of synthesis (oranalysis) is as follows

H1(x)

= N+2K+1

k=0 d(k)N + 2K+ 1

k xk

(1

x)

N+2K+1k

(20)

H(v) = H sin2 v2

=

j=N+2K+1j=N2K1

hjeivj (21)where x = sin2(v/2) and z= eiv. Then, we have

hm =hm = N+2K+1k=m

[(km)/2]n=0

(1)m ak2k+2n+m

k

2n + m 2n + mn , m 0 (22)and

ak = ki=0

(1)kid(i) N + 2K+ 1i

N + 2K+ 1 i

k i

(23)

Theorem 3 reveals that the number of zeros in Bernsteincoefficients determines the vanishing moments of waveletsfunctions. This conclusion greatly reduces the extent of

complexity in analysing the transforming property ofwavelet functions. Meanwhile, the number of ones inBernstein coefficients decides the flatness of frequencyresponse passbands of filters. These two transformingfeatures are crucial to image compression. The benefit ofapplying the Bernstein polynomial in constructing

biorthogonal wavelets is further testified in Section 5.

Theorem 3: If we represent analysis and synthesis low-passfilters with an odd length in Bernstein basis form

H1(x)

= N

k=0d(k)

N

k xk

(1

x)

Nk(24)

H(v) = H1 sin2 v2

(25)



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for x = sin2(v/2) and satisfy as

dkH(v)

dvk|v=p = 0, k = 0, 1, 2, 3, . . . , 2K 1 (26)

dkH(v)

dvk|v=o = dk,0, k = 0, 1, 2, 3, . . . , 2P 1 (27)

where dk,0 is the Kronecker symbol. If and only if the aboveconditions are satisfied, then d(n) is as follows

d(n) =1, 0 n , Parbitrary, P n N K0, N K, n N

(28)where P and K are integral numbers.

Theorem 4: If H(v) =

NN hne

iv and

H(v) =

N+2K+1N2K1hneiv, where hn andhn are shown by Theorems1 and 2. Assume that (10) holds. The correspondingBernstein coefficients d(i) and d(i) satisfy the followingequation

2

mn=2l

Nk=m

[(km)/2]n=0

N+2K+1k=n

[(kn)/2]n=0

(1)n+m

ak2k+2n+m

k

2n + m

2n

+ mn

ak

2k+2n+n

k

2n + n

2n + n

n

= dl0,

0 l N + K (29)

where

ak =ki=0

(1)kid(i) Ni

N ik i

,

ak = ki=0

(1)kid(i) N + 2K+ 1i

N + 2K+ 1 ik

i

Proof: Since

H(v) =NN

hmeimv

, H(v) = N+2K+1N2K1

hneinvwe have

H(v)

H(v) =

2N+2K+1s=2N2K1

mn=s

(hm

hn)e

isv

H(v+ p)H(v+ p) = 2N+2K+1s=2N2K1

mn=s

(hmhn)(1)seisv

From (10), it follows that

H(v)H(v) + H(v+ p)H(v+ p)= 2

(N+K)l=(N+k)

mn=2l

hmhn

ei2lv = 1, 0 l N + K

We immediately obtain (29). ASuppose that the number of vanishing moments of c

defined by (8) is 2k; according to Theorem 3, we have

H(v) = 1 sin2 v2

kNki=0

d(i)N

i

sin2

v

2

i 1 sin2 v

2

Nik= 1 + e

iv

2

2kF(v) (30)

where

F(v) = eikvNki=0

d(i)N

i

sin

2 v

2

i1 sin2 v

2

Nik(31)

Similarly, suppose that the number of vanishing moments ofcdefined by (9) is 2k; we have

H(v) = 1 + e

iv

2

2kF(v) (32)

F(v) = eikv N+2K+1ki=0

d(i) N + 2K+ 1i

sin2

v

2

i 1 sin2 v

2

N+2K+1ik(33)

Lemma 1: Suppose thatF(v) is defined by (31); we have

F(v) = eiv(Nm)m

s=mbse

isv (34)

where

bs =mk=s

[(ks)/2]n=0

ak2k+2n+s

(1)s k2n + s 2n + s

n

(35)

ak =ki=0

( 1)kid(i) Ni

m ik i

(36)

and s . 0, bs = bs.

Proof: From (31), we have

F(v)=

ei(Nm)v

m

i=0d(i)

N

i sin2 v2 i

1 sin2 v2

mi= ei(Nm)vf(v)



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and

f(x) =mi=0

d(i)N

i

xi(1 x)mi

= m

i=0d(i)

N

i xi

mi

j=0

m i

j (1)jxj

=mi=0

d(i)mk=i

(1)ki Ni

m ik i

xk =

mk=0

akxk

where

ak =ki=0

(1)kid(i) Ni

m ik i

Consequently, for z= eiv

f(v) =mk=0

kj=0

ak2k

(1)j kj

z+ z1

2

j

=mk=0

kj=0

ak2k

(1)j kj

jl=0

1

2 j

j

l

z

j2l

Let j2 2l s, we obtain

f(v) =m

s=m

mk=s

[(ks)/2]n=0

ak2k+2n+s

(1)s

k2n + s

2n + s

n

zs

Thus, Lemma 1 is proven. ATheorem 5 will give sufficient conditions of the

biorthogonality ofc,

c.

Theorem 5: Let H(v) = NN hneinv and H(v) =N+2K+1N2K1hneinv be shown by Theorems 1 and 2. The

number of vanishing moments of corresponding c and care 2L and 2L, respectively (which are determined bythe number of zeros in Bernstein coefficients).Let m N2L and m = N + 2K+ 1 L. Assume that(10) holds. Let c and c be functions defined by (8) and(9). Suppose that Bernstein coefficients d(i) and d(i)

satisfy the following

ms=m

mk=s

[(ks)/2]n=0

ki=0 (1)kid(i)

N

i

m ik i

2k+2n+s

(1)s

k2n + s

2n + sn

, 22L1/2 (37)

(see (38))We obtain the following results:

1. For some arbitrary integer n . 0

B2n = max

2n1

j

=0

F(2 jv)

1/2n

, 22L(1/2) (39)

B2n = max 2n1j=0

F(2jv)

1/2n

, 22L(1/2) (40)

where F(v) andF(v) are defined by (31) and (33).2.

kf, f(t l)l = dl0f(v) C(1 + |v|)1/2af(v) C(1 + |v|)1/2a

where a = 2L 1/2 log2 B2n , a = 2L 1/2 log2B2n .3. The corresponding c andc are biorthogonal wavelets in

L2(R).

Proof: It is easy to find that sequences B2n and B2n aredecreasing

2n1j=0

F(2 jv)

max

2n11j=0

F(2 jv)|max| 2n1

j=2n1F(2 j2n1 22n1v)

= max2n11j=0

F(2 jv)|max|2n11j=0

F(2 jv)

= max2n11j=0

F(2jv)

2

ms=m

m

k=s

[(ks)/2]n=0

ki=0 (1)kid(i) N + 2K+ 1i m ik i 2k+2n+s

(1)s k2n + s

2n + sn

, 2

2 L1/2 (38)



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Hence, B2n B2n1 B1 = max|F(w)|. Similarly, wecan prove B2n B2n1 B1 = max|F(w)|. CombiningLemma 1, (37) and (38), we obtain conclusion (1) by [1, 14].

From [1, 14], we have kf, f(t l)l = dl0, f(v) C(1 + |v|)1/2a, f(v) C(1 + |v|)1/2a; we obtainconclusion (2).

From [1, 14] it is implied that the corresponding cand

c

are biorthogonal wavelets in L2

(R). Hence, the proof of thetheorem is completed. A

3 Regularity analysis of biorthogonalwavelets based on Bernstein bases

In this section, we will provide the regularity analysis ofbiorthogonal wavelets based on Bernstein bases.

The regularity is defined as follows: fora n + b, n [ N,0 b, 1, the Hoder space Ca [14] is defined as the set offunctions which are n times continuously differentiable andsuch that the nth derivative f

(n)satisfies the following

condition

| f(n)(x + h) f(n)(h)| C|h|b, for all x, h

The numbera is called the regularity (exponent) off. It is wellknown [14] that if

R

|f(v)|(1 + |v|)a , 1 (41)then f[ Ca, which means that the regularity of fcan beestimated via the decay of its Fourier transform. The Sobolevregularity of f is studied with the spectral properties of atransfer operator associated with the coefficients of filters

[14]. Transfer operators are defined as follows.F o r a 2p-periodic function u(v) that depends on the

coefficients of filter is fixed and the associated transferoperator Tu acts on 2p-periodic functions according to

Tu(v) = u(v/2)f(v/2) + u(v/2 + p)f(v/2 + p) (42)

Note that if u = Sn[Z fneinv is a trigonometric polynomial,that is, un = 0 if |n| . N, then the finite-dimensional space

Eu = {

|n|N cneinv

: cn [ C} is invariant under theaction of Tu.

One of the main results in the univariate theory is asfollows.

Assume that H(v) can be factorised as H(v) = ((1 +eiv)/2)LF(v) where F(v) is a trigonometric polynomialand define u(v) = |F(v)|2. The Sobolev exponent[14]: s(f) = sup{s: (1 + |v|s)|f|2 dv, +1} satisfies theestimate

s(f) L log r2 log 2

(43)

where r is the spectral radius of Tu restricted to Eu.Define the associated transfer operator T acts on

2p-periodic functions according to

Tf(v) = F v2 2f v

2 + F v

2+ p 2f v

2+ p (44)

where F(v) is defined by (31).

Lemma 2: For any 2p-periodic continuous functions f, g, weconsider the conjugate operator T of T, and have

Tf(v) = 2|F(v)|2f(v) (45)

Proof: In fact, for any 2pperiodic continuous fand g

,Tf, g. = pp

F v2 2f v

2 g(v) dv

+p

pF

v

2+ p

2f v2

+ p

g(v) dv

= 2p/2

p/2|F(v)|2f(v)g(2v) dv

+3p/2p/2

|F(v)|2f(v)g(2v) dv

=2

3p/2

p/2 |F(v)

|2f(v)g(2v) dv

=,f, T

g.

Thus, Lemma 2 is proven. A

Lemma 3: Suppose that

|F(v)|2 =2m

t=2mbte

itv

where

bt = mtk=m

bk+tbk(t. 0), bt = bt

andbt, t= m, . . . , m is defined by (35).

Proof: By Lemma 1, we have

|F(v)|2 =m

j=mbje

ijv m

k=mbke

ikv

=

m

j=m m

k=mbjbke

i(jk)v

Setting j2 k t, for t. 0, we have

|F(v)|2 =2m

t=2m

mtk=m

bk+tbkeikv

Therefore, Lemma 3 is proven. ALemma 4 shows how to evaluate r(T).

Lemma 4:

r(T) 4m mi=m

b2i (46)

where bi is shown by (35).



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Proof: We consider the conjugate operator T. By directcomputation, for any 2p-periodic continuous functions f,we have

Tf(v) = 2|F(v)|2f(v) (47)

The matrix of T is restricted to E= {2m

j=2m cjeijv,

(c2m, c2m+1, . . . , c2m) [ R2m

+1

}, we have

H = (bk2l)l,k=2m,2m+1,...,2m (48)

Notice that the matrix H is shown as follows (see (49))We remark matrix Hc, which consists of all rows and columnsofH except its first and last rows and columns. To estimatethe bounds of the eigenvalues of H, by Lemma 3, we have

|bt| 1

2

mtk=m

b2k+t +

mtj=m

b2k

=1

2 mm+tb2l + mt

j=mb

2k m

i=mb

2i (50)

It is obvious by (49) that b2m is an eigenvalue of H withmultiplicity 2. Hence, the spectral radius ofH is

r(H) = max{|b2m|, r(Hc)} max{|b2m|, Hc1}

max |b2m|,2m1

i=2m+1|bi2j|

2m

m

i

=m

b2i (51)

By Theorem 5 and Lemma 2-4, Theorem 6 is proven. AFor sufficient conditions of the biorthogonality, and the

regularity analysis of f, f by (8), we have the followingresults. By [5, 6] and Lemma 4, Theorem 6 is proven.

Theorem 6: Let

H(v) =NKk=0

d(i)N K

k

sin

2k(v/2)(cos

2(Nk)(v/2)

(52)

H(v) = N+2K+1 Kk=0

d(i) N + 2K+ 1 Kk

sin

2k(v/2)

(cos2(N+2K+1k) (v/2)) (53)

Set m N2K, m = N + 2K+ 1 K. Assume that (10)holds. Let f and f be function defined by (6). Suppose

that generalised Bernstein coefficients d(i) andd(i) satisfym

s=m

mk=s

[(ks)/2]n=0

ki=0 (1)kid(i)

N

i

m ik i

2k+2n+s

(1)s

k

2n + s 2n + s

n , 22K1/2 (54)

ms=m

mk=s

[(ks)/2]n=0

ki=0 (1)kid(i) N+ 2K+ 1

i

m ik i

2k+2n+s

(1)s k

2n +s

2n +s

n

, 2

2 K1/2(55)

We obtain the following results:

1. kf, f(t l)l = dl0, f(v) C(1 + |v|)1/2a, f(v) C(1 + |v|)1/2a, where a = 2K 1/2 log2|F(v)|, a =2K 1/2 log2F(v), F(v) andF(v) are defined by (31)and (33), respectively.2. The corresponding c and c are biorthogonal, waveletsin L2(R). c[ Ca andc[ Ca , where a, a are more than2K 1

2log2 4m

mi=m

b2i , 2

K 1

2log2 4

mm

i=mb

2

i (56)

where bi is defined by (35), and

bs = mk=0

[(ks)/2]n=0

a k2k+2n+s

(1)s k2n + s

2n + s

n

a k = ki=0

(1)kid(i) N + 2K+ 1i

m ik i

We now give a general model to construct the biorthogonalscaling functions f and

f with the largest possible

regularity. In fact, this method can be described as aconstrained optimisation problem of finding suitable

Bernstein coefficients {d(i)} and {d(i)}. By (56) of Theorem 6, when log2 4m

mi=m b

2i and log2 4mmi=m b2i

reach a minimum value, the regularity (exponent) of c andc will increase. The optimisation problem can be written asfollows.

Model 1: (the non-linear optimisation model of the largestpossible regularity for biorthogonal wavelet function c

H=

b2m b2m2 b2m4 b2m6 b2m 0 0 00 b2m1 b2m3 b2m5 b2m+1 0 0 00 b2m b2m2 b2m4 b2m+2 b2m 0 00 0 b2m1 b2m3 b2m+3 b2m+1 0 0...

.

.

....

.

.

. ... ... ... 00 0 0 0 0 0 0 b2m

(49)



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defined by (8): Suppose that the objective function is

f(d(0), d(1), . . . , d(N))

=m

s=m

mk=s

[(ks)/2]n=0

ki=0 (1)kid(i)

N

i

m ik i

2k+2n+s

(1)s

k2n + s

2n + s

n

The following Bernstein coefficients ensure that the functionreaches its minimum

minf(d(0), d(1), . . . , d(N)) (57)

Subjected to

2 mn=2l

Nk=m

[(km)/2]n=0

N+2K+1k=n

[(kn)/2]n=0

(1)n+m

ak2k+2n+m

k

2n + m

2n

+ mn

ak2k

+2n+n

k

2n + n

2n + n

n

= dl0, 0 lN+K (58)

ms=m

mk=s

[(ks)/2]n=0

ki=0 (1)kid(i)

N

i

m ik i

2k+2n+s

(1)s

k2n +s

2n +s

n

, 2

2L1/2 (59)

m

s=m

mk=s

[(ks)/2]n=0

ki=0 ( 1)kid(i) N+ 2K+ 1

i

L ik i

2k+2n+s

( 1)sk

2n +s 2n +sn , 22L1/2 (60)Restriction (58) assures that the decomposing and reconstructinglow-pass filters satisfy perfect reconstruction conditions.Restriction (59) and (60) assure the biorthogonality ofcandc.If the above constrained optimisation problem (57) is solved, thelargest possible regularity ofcdefined by (8) can be acquired.

Model 2 (the non-linear optimisation model of the largestpossible regularity for biorthogonal wavelet function cdefined by (9): Suppose that objective function

g(d(0),d(1), . . . ,d(N + 2K+ 1))= m

s=m

mk=s

[(ks)/2]n=0

ki=0 (1)kid(i) N + 2K+ 1

i

m ik i

2k+2n+s

(1)s

k2n + s

2n + s

n

ming(d(0),d(1), . . . ,d(N + 2K+ 1)) (61)Subjected to

2

mn=2l

Nk=m

[(km)/2]n=0

N+2K+1k=n

[(kn)/2]n=0

(1)n+m

ak2k+2n+m

k

2n + m

2n

+ mn

ak

2k+2n+n

k

2n + n

2n + n

n

= dl0,

0

l

N

+K (62)

ms=m

mk=s

[(ks)/2]n=0

ki=0 (1)kid(i)

N

i

m ik i

2k+2n+s

(1)s

k

2n + s

2n + s

n

, 22L1/2 (63)

m

s=m

mk=s

[(ks)/2]n=0

ki=0 (1)kid(i) N + 2K+ 1i L ik i 2k+2n+s

(1)s k2n + s

2n + s

n

, 2

2 L1/2 (64)

Restriction (62) assures that the decomposing andreconstructing low-pass filters satisfy perfect reconstruction

conditions. Restrictions (63) and (64) assure thebiorthogonality of c and c. If the above constrainedoptimisation problem (61) is solved, the largest possibleregularity of biorthogonal wavelet function c defined by(9) can be acquired.

Model 3 (the non-linear optimisation model of thelargest possible regularity for biorthogonal wavelet

functions c and c defined by (8) and (9)): Suppose thatobjective function

w(d(0), d(1), .. ., d(N),d(0),d(1), .. .,d(N+2K+1))= m

s=mmk=s

[(ks)/2]n=0

ki=0 (1)kid(i) Ni m ik i 2k+2n+s



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(1)s k2n+s

2n+s

n

+ m

s=m m

k=s [(ks)/2]

n=0 ki=0 (1)ki

d(i)

N+ 2K+1i

m ik

i

2k+2n+s

(1)s k2n+s

2n+s

n

(65)min w(d(0), d(1), .. ., d(N),d(0),d(1).. .,d(N+2K+1))Subjected to

2

mn=2l

Nk=m

[(km)/2]n=0

N+2K+1k=n

[(kn)/2]n=0

(1)n+m

ak2k+2n+m

k

2n + m

2n + m

n

ak2k

+2n+nk

2n + n

2n

+ nn

= dl0,

0 l N + K (66)

ms=m

mk=s

[(ks)/2]n=0

ki=0 (1)kid(i)

N

i

m ik i

2k+2n+s

( 1)s

k

2n +s

2n +s

n

, 22L1/2

(67)

ms=m

mk=s

[(ks)/2]n=0

ki=0 ( 1)kid(i) N+ 2K+ 1i m ik i 2k+2n+s

(1)s k2n +s 2n +sn

, 22L1/2 (68)

Restriction (66) assures that the decomposing and reconstructinglow-pass filters satisfy perfect reconstruction conditions.

Restrictions (67) and (68) assure the biorthogonality of andc. If the above constrained optimisation problem (65) issolved, the largest possible regularity of andc defined by (8)and (9) can be acquired.

4 Construction of parametric filters ofbiorthogonal wavelets

In this section, we shall give a new familyof filterswith differentlengths and vanishing moments via Theorems 1, 2, 4 and 5.Moreover, we shall analyse the regularity of f and f andshow the largest possible regularity by solving Models 13.

Example 1: Let the lengths of low-pass filters in analysisand synthesis be 9 and 7, respectively (or 7 and 9).Suppose that the vanishing moments of c and c defined

by (8) and (9) are 4 and 2. If (10) is satisfied, then thecorresponding Bernstein coefficients are obtained by (seeequation at the bottom of the page)where b [ R is a parameter.

Note: when d 1.3069, H(v) and H(v) are the widelyfavoured CDF9/7 [12].

Example 2: Let the lengths of low-pass filters in analysis andsynthesis be 9 and 11, respectively (or 11 and 9). Suppose thatthe vanishing moments ofcand

c defined by (8) and (9) are

4 and 2. If (10) is satisfied, then the corresponding Bernsteincoefficients are obtained by (see equation at the bottom of the

page)where c [ R is a parameter.

Example 3: Let the lengths of low-pass filters in analysis andsynthesis be 5 and 11, respectively (or 11 and 5). Suppose thatthe vanishing moments ofcandc defined by (8) and (9) are2 and 4. If (10) is satisfied, then the corresponding Bernsteincoefficients are obtained by (see equation at the bottom of the

page)where b [ R is a parameter.

Example 4: Let the lengths of low-pass filters in analysis andsynthesis be 11 and 13, respectively (or 13 and 11). Supposethat the vanishing moments ofcandc defined by (8) and (9)are 4 and 2. If (10) is satisfied, then the corresponding

d = [1, b, 0, 0] D = 1, 34

b + 74

, 72

b + 72

+ 32

b2,(27/4)b3 + (63/4)b2 (63/4)b + (35/4)

3b + 1 , 0

d = [1, 1, c, 0, 0] D = 1, 1, 35

c + 85

,1

5

108c3 50 288c2 + 393c36c2 + 24c 5 ,

2

5

36c2 96c + 2336c2 + 24c 5 , 0

d= [1, b, 0] D = 1, 25

b + 75

,2

5b2 7

5b + 21

10, 1

10

8b3 35 28b2 + 42b

1 + 2b , 0, 0



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Bernstein coefficients are obtained by (see equation at thebottom of the page)where d[ R is a parameter.

Example 5: Let the lengths of low-pass filters in analysis andsynthesis be 13 and 11, respectively (or 11 and 13). Supposethat the vanishing moments ofcand

c defined by (8) and (9)

are 2 and 2. If (10) is satisfied, then the corresponding Bernstein

coefficientsareobtainedby(seeequationatthebottomofthepage)where e [ R is a parameter.

Example 6: Let the lengths of low-pass filters in analysis andsynthesis be 7 and 13, respectively (or 13 and 7). Suppose thatthe vanishing moments ofcandc defined by (8) and (9) are 2and 4. If (10) is satisfied, then the corresponding Bernsteincoefficientsareobtainedby(seeequationatthebottomofthepage)where c [ R is a parameter.

Example 7: Let the lengths of low-pass filters in analysis andsynthesis be 9 and 15, respectively (or 15 and 9). Suppose that

the vanishing moments ofcandc defined by (8) and (9) are 4and 4. If (10) is satisfied, then the corresponding Bernsteincoefficientsare obtainedby (see equationat thebottomof thepage)where c [ R is a parameter.

Example 8: Let the lengths of low-pass filters in analysis andsynthesis be 13 and 15, respectively (or 15 and 13). Supposethat the vanishing moments ofcandc defined by (8) and (9)are 6 and 2. If (10) is satisfied, then the correspondingBernstein coefficients are obtained by (see equation at the

bottom of the page)where d[ R is a parameter.

Example 9: Let the lengths of low-pass filters in analysis andsynthesis be 7 and 17, respectively (or 17 and 7). Suppose thatthe vanishing moments ofcandc defined by (8) and (9) are 2and 6. If (10) is satisfied, then the corresponding Bernsteincoefficientsare obtainedby (see equationat thebottomof thepage)where c [ R is a parameter.

d = [1, 1, 1, d, 0, 0]

D = 1, 1, 1, (1281/2)d+ 108 + (2715/2)d2 375d3

550d2 355d+ 500d3 + 72 ,200d3 + (2050/3)d2 (854/3)d+ 48

550d2 355d+ 500d3 + 72 ,(250/3)d3 + (850/3)d2 (355/3)d+ 24

550d2 355d+ 500d3 + 72 , 0

d = [1, 1, 1, 1, e, 0]

D = 1, 1, 250e4 + (11 300/3)e3 7590e2 + (17 543/3)e 15341875e3 6050e2 + 5425e 1534 + 625e4 ,

(125/4)e4 + 2000e3 4280e2 + 3380e 9001875e3 6050e2 + 5425e 1534 + 625e4 ,

2275e2 + (5474/3)e + 1050e3 (1480/3)1875e3 6050e2 + 5425e 1534 + 625e4 ,

(875/2)e3 + 770e (5675/6)e2 (634/3)1875e3 6050e2 + 5425e 1534 + 625e4 , 0

d = [1, 1, c, 0] D = 1, 1, 15

c + 65

,(207/10)c 6 (81/10)c2 + (27/20)c3

9c 4 + 9c2 ,(12/5) (12/5)c2 + (39/5)c

9c 4 + 9c2 , 0, 0

d = [1, 1, c, 0, 0] D = 1, 1, 27

c + 97

, 1835

c + 5735

,(648/35)c3 (2232/35)c2 + (3222/35)c (102/7)

24c + 36c2 5 ,24c2 + (198/7)c + (72/7)c3 (18/7)

24c + 36c2

5, 0, 0

d = [1, 1, 1, d, 0, 0, 0] D = 1, 1, 1, 47

d+ 117

,(8000/7)d4 (21 600/7)d3 + (35 880/7)d2 + 154 (8784/7)d

2000d3 590d+ 77 + 1600d2 ,

(8000/7)d3 + (24 400/7)d2 (8360/7)d+ (808/7)2000d3 590d+ 77 + 1600d2 ,

(4000/7)d3 + (10 800/7)d2 (2780/7)d+ (54/7)2000d3 590d+ 77 + 1600d2 , 0

d = [1, 1, c, 0]

D

=1, 1,

3

28

c

+31

28

,

15

56

c

+9

7

,(789/35)c (342/35)c2 (228/35) + (27/14)c3

9c + 9c2

4,

(27/56)c3 (81/28)c2 + (207/28)c (15/7)

9c + 9c2 4 , 0, 0, 0



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Example 10: Let the lengths of low-pass filters in analysis andsynthesis be 7 and 17, respectively (or 17 and 7). Suppose thatthe vanishing moments ofcandc defined by (8) and (9) are4 and 6. If (10) is satisfied, then the corresponding Bernsteincoefficients are obtained by (see equation at the bottom of the

page)where b [ R is a parameter.

Example 11: Let the lengths of low-pass filters in analysis andsynthesis be 11 and 17, respectively (or 17 and 11). Supposethat the vanishing moments ofcandc defined by (8) and (9)are 4 and 4. If (10) is atisfied, then the correspondingBernstein coefficients are obtained by (see equation at the

bottom of the page)where d[ R is a parameter.

Example 12: Let the lengths of low-pass filters in analysis andsynthesis be 11 and 17, respectively (or 17 and 11). Suppose

that the vanishing moments ofcandc defined by (8) and (9)are 6 and 4. If (10) is satisfied, then the correspondingBernstein coefficients are obtained by (see equation at the

bottom of the page)where c [ R is a parameter.

Example 13: Let the lengths of low-pass filters in analysis andsynthesis be 15 and 17, respectively (or 17 and 15). Suppose

that the vanishing moments ofcandc defined by (8) and (9)are 6 and 2. If (10) is satisfied, then the correspondingBernstein coefficients are obtained by (see equation at the

bottom of the page)where e [ R is a parameter.

Example 14: Let the lengths of low-pass filters analysis andsynthesis be 19 and 25, respectively (or 25, 19). Supposethat the number of vanishing moments of c andc defined

by (8) and (9) are 10 and 4. If (10) is satisfied, then the

d = [1, b, 0, 0]

D =

1, 38

b + 118

,9

28b2 33

28b + 55

28, 27

56b3 + 99

56b2 165

56b + 165

56,

81

70b4 + 99

14b2 297

70b3 99

14b + 33

7,

(243/56)b5 + (891/56)b4 (1485/56)b3 + (1485/56)b2 (495/28)b + (33/4)3b + 1 , 0, 0, 0

d = [1, 1, 1, d, 0, 0]

D = 1, 1, 1, 528

d+ 3328

,(9690/7)d2 (3250/7)d3 (8751/14)d+ 108 + (500/7)d4

550d2 + 500d3 355d+ 72 ,(2875/14)d3 + (20 425/28)d2 (9255/28)d+ 54

550d2 + 500d3 355d+ 72 ,(500/7)d3 (1425/14)d+ (3425/14)d2 + (108/7)

550d2 + 500d3 355d+ 72 , 0, 0

d = [1, 1, c, 0, 0, 0]

D = 1, 1, 514

c + 1914

, 1528

c + 127

,10

7c2 23

7c + 47

14,

(625/7)c4 + (2875/14)c3 (5575/28)c2 + (4005/28)c 63 + 25c + 50c2 ,

(125/7)c3 25c2 + (1105/14)c (289/14)

3 + 25c + 50c2 , 0, 0

d = [1, 1, 1, 1, e, 0, 0, 0]

D = 1, 1, 1, 1, 19 200 + (1 921 535/2)e3 (1 158 213/2)e2 + 167 964e 240 100e4

300 125e4 + 360 150e3 286 895e2 + 99 176e 12 800 ,

(2174375/4)e3 (1 157 625/4)e2 + 82 382e 9600 (300 125/2)e4300 125e4 + 360 150e3 286 895e2 + 99 176e 12 800 ,

85 750e4 + (591 675/2)e3 (309 295/2)e2 + 49 588e 6400300 125e4 + 360 150e3 286 895e2 + 99 176e 12 800

,

(1020425/8)e3 (300 125/8)e4 + 24 794e (277 095/4)e2 3200300 125e4 + 360 150e3 286 895e2 + 99 176e 12 800 , 0



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corresponding Bernstein coefficients are obtained by (seeequation at the bottom of the page)where b [ R is a parameter.

Theorem 7 shows the sufficient conditions of wavelets cand c to be biorthogonal from Examples 1 14, whenBernstein coefficients belong to certain range.

Theorem 7: In the following tables, when Bernsteincoefficients belong to certain ranges, c and c fromExamples 1 14 are biorthogonal wavelets in L2(R).Furthermore, by solving Models 13, we obtain the largest

possible regularities of c and c. The detailed conclusionsare given in the Table 1.

Proof: We will prove the conclusion of Example 1 in Table 1,while the conclusion of other cases can be proven similarly.Suppose that (see equation at the bottom of the page)

Thus

f(b) = 2 2 14

+ 34

b + 12 + 32 b , 8 2

In order to find out the interval thatb belongs to satisfying theabove expression, we draw an image as is shown in Fig. 1.From Fig. 1a, we know b [ B (22.6667, 2.6667).Suppose that (see equation at the bottom of the page)

We have

f(b) = 2 21564 932 b + 964 b2 1

16

(27/4)b3 + (63/4)b2 (63/4)b + (35/4)3b

+1

2532 + 34 b 932 b2 + 38

(27/4)b3 + (63/4)b2 (63/4)b +(35/4)

3b + 1

+ 164 + 932 b 964 b2 1516

(27/4)b3 + (63/4)b2 (63/4)b +(35/4)

3b+

1 +3316 32 b + 916 b2 + 54

(27/4)b3 + (63/4)b2 (63/4)b +(35/4)

3b + 1

, 2

2

From Fig. 1b, we know b [ B = (1.1912, 1.2587). So

d = [1, 1, 1, 1, b, 0, 0, 0, 0, 0]

D = 1, 1, 1, 1, 14

55 b +69

55 , 21

44 b +18

11 , 9

22 b +21

11

(94 517 766/11)b5 (535600674/11)b4 + (959604975/11)b3 (1146 216 393/44)b2+(147464725/44)b 11 7650

18003384b4 + 13 288 212b3 4 608 576b2 + 815 805b 58 825

(756142128/55)b5 (394074072/11)b4 + (2213 447 796/55)b3 (341604144/55)b2+(113142630/11)b 1 977 714

18003384b4 + 13288212b3 4 608 576b2 + 815 805b 58 825

(567106596/55)b5 (81015228/5)b4 + (835299864/55)b3 (212865408/55)b2+(97643826/55)b (2 371 474/11)

18003384b4

+ 13 288 212b3

4 608 576b2

+ 815 805b 58 825 (12002256/11)b

4 + (86516262/11)b3 (62 780 886/11)b2 + (17498835/11)b 162 90518 003 384b4 + 13288212b3 4 608 576b2 + 815 805b 58 825 , 0, 0

f(b) =2

s=2

2k=s

[(ks)/2]n=0

ki=0 (1)kid(i)

4 + 2k+ 1i

2 ik i

2k+2n+s

(1)s k2n + s

2n + s

n

, 22(42)1/2

f(b) = 4s=4

4k=0

[(ks)/2]n=0

ki=0 (1)ki d(i) 5 + 2k+ 1i 4 ik i 2k+2n+s

(1)s k2n + s

2n + sn

, 2

2(54)1/2



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by Theorem 6, when Bernstein coefficients belong tothe set B> B = (2.6667, 2.6667) > (1.1912, 1.2587) =(1.1912, 1.2587), candcare biorthogonal wavelets in L2(R).

From Fig. 1c, when b 1.1913 [ (1.1912, 1.2587),function f(b) reaches a minimum. So the best parameter(37) of Model 1 is 1.1913.

From Fig. 1d, when b 1.217 [ (1.1912, 1.2587),function f(b) reaches a minimum. So the best parameter(38) of Model 2 is 1.217.

From Fig. 1e, when b 1.217 [ (1.1912, 1.2587), functionf(b) +

f(b) reaches a minimum. So the best parameter (39) of

Model 3 is 1.217. The proof of the theorem is completed. A

5 Construction of optimised biorthogonalwavelets for image compression

Most applications of wavelet bases exploit their ability toefficiently approximate particular classes of functions with afew non-zero wavelet coefficients. The design of c musttherefore be optimised to produce a maximum number ofwavelet coefficients k f, cj,nl that are close to zero. It is wellknown that wavelets can provide sparse and, hence, efficientrepresentations of smooth functions. This property isespecially advantageous to image coding. However, in thevicinity of an image discontinuity, large wavelet coefficients

can be generated. When these coefficients are quantised orlost, a Gibbs-like phenomenon can be observed in thereconstructed image. For example, artefacts marked bysoftness, ringings, halos and colour bleeding can be seen along

the edges. Although regularity and vanishing moments aresometimes used in filter evaluation, their success at predictingcompression performances are only partial. Many issuesrelated to the choice of the filter bank for image compressionremain unresolved. Wavelet filter evaluation for imagecompression, particularly for remote-sensing image is still anopen question. In this section, we present the optimised modelfor image compression and the corresponding solving method.

Taking into account all the factors, for instance, thevanishing moments and the regularity, we propose a brandnew method of designing optimal filters for application inthe compression of images. We can draw the conclusion

that the filters proper for image compression should becharacterised in the following aspects:

1. The regularity of biorthogonal wavelets functions.2. Vanishing moments of biorthogonal wavelets functions.3. The flatness of frequency response of low-pass filters ofanalysis and synthesis.4. The concentration of low-frequency energy. In thefrequency domain, the energy of the entire discrete wavelettransform (DWT) image should concentrate on the low-frequency sub-band.5. The concentration of high-frequency energy. The energyof the high-frequency sub-bands should engross in a small

portion of the coefficients.

The energy in the frequency domain can be presented bythe sum of the square power of all sub-band coefficients of

Fig. 1 Regularity analysis

a function f(b) 82

around zero

b functionf(b) 4 2 around b 1.5c function f(b) around b 1.2d function

f(b) around b 1.2

e function

f(b) + f(b) around b 1.2

Table 1 Largest possible regularities ofcand c by Models 13

Filter pair Set Best parameter of

Model 1

Best parameter of

Model 2

Best parameter of

Model 3

Example 1 (1.1912, 1.2587) 1.1913 1.217 1.217

Example 2 (0.8091, 1.3333) 0.8092 1.251 0.8092

Example 3 (0.6218, 1.2) 0.6219 1.185 1

Example 4 (0.4424, 0.81) 0.4425 0.8 0.4425

Example 5 (0.0828, 0.1778) 0.1286 0.1777 0.1285Example 6 (0.1926, 0.286)< (0.3381, 0.6667) 0.3382 0.3432 0.3432

Example 7 (0.6058, 1.3333) 0.6059 1.3332 0.8333

Example 8 (0.8366, 1.6) 0.8367 1.2268 0.8367

Example 9 (20.0064, 0.3321) < (0.3393, 0.6667) 0.3319 0.6666 0.3320

Example 10 (0.9947, 1.67) 0.9948 1.5078 1.5078

Example 11 (20.0168, 0.8) 0.0666 0.7 0.198

Example 12 (0.9870, 1.6847) 0.9871 1.3262 1.3262

Example 13 (0.4483, 0.9143) 0.4485 0.7406 0.4484

Example 14 (0.8750, 2.0110) 0.8751 1.6412 0.8751



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the DWT image. Thus, an objective function is introduced byus as follows. If the preponderant amount of energy isconcentrated in the low-frequency part, the objectivefunction can reach its minimum value, which is defined as

minf= min

j,k,m

3i=1 (D

ij,k,m)

2

j,k,m ((Cj,k,m)2 +3i

=1 (D

ij,k,m)

2)

(69)

where Cj,k,m are coefficients of a low-frequency sub-band

(LL), Di

j,k,m, i 1, 2, 3 are coefficients of high-frequencysub-bands (LH, HL, HH).

Based on the five aspects mentioned above, the followingalgorithm is suggested. The following steps can be taken inorder to obtain filters which have a better effect oncompression of the image.

Algorithm 1 (algorithm of the optimal filters):

Step 1: Utilising the filters obtained from Examples 114, acoefficient matrix can be formulated after DWT.Step 2: Solving the non-linear optimisation models (69) bySQP [23], we obtain the initialisation value of

parameterisation of filters. The filters of Examples 1 14 bythe parameterisations of filters can be obtained.Step 3: Set the group of solutions obtained by Step 2 as theoriginal population. Finess function is defined by PSNRof reconstruction of image. Searching range of

parameterisations of filters is restricted by Theorem 7.Step 4: Calculate the peak signal to noise ratio (PSNR) valueof the reconstructed image. If it converges, then obtain theoptimal solution, otherwise go to Step 5.Step 5: We use the GA [24] to acquire the new

parameterisation of filters. The filters of Examples 1 14 by

new parameterisations of filters can be obtained. Utilising

the new filters compress wavelet coefficients by the bit-plane prediction coding method, go to Step 4.

Since the framework proposed in this paper ensures that thebiorthogonal wavelets have good regularity, vanishingmoments and flatness of the frequency response, the firstthree requirements mentioned above are satisfied. The

purpose of construction of parametric filters is to achieve(69) by adjusting the parameters. By applying Examples 1

14 and Algorithm 1, we obtain a new family of transformsBBW17/7, BBW15/13, BBW17/11, BBW17/15 and BBW25/19 as shown in Table 2.

As Fig. 2 indicates, the passband of the frequency responseof synthesis and analysis low-pass filters of BBW15/13,BBW17/7, BBW17/11 and BBW17/15 display betterflatness than that of CDF9/7, which testifies the efficiencyof Algorithm 1.

In order to evaluate energy compaction of the transforms,we define the energy compaction ability as follows

Ecomp(LL) = EM/ETotal, M = LL (70)

where ELL is the energy of the sub-band LL and ETotal is theenergy of the whole image. We analyse the energycompaction ability BBW17/11 and CDF9/7 for city,factory, Toulouse and San Francisco. In Fig. 3, the verticalaxis represents the energy percentage of the lowest-frequency sub-band. The energy percentage of each levelusing BBW17/11 is higher than the corresponding levelusing CDF9/7, which means BBW17/11 has betterenergy compaction ability as shown in Fig. 3. In waveletconstruct, we hope to construct wavelet bases with bigger

Ecomp(LL).We explore the applicability of the newly designed

transforms to still images, in particular remote sensing

images (see Fig. 4). The new transforms and CDF9/7 are

Table 2 Best filters from Examples 114

Filters Parameter H HBBW (15/13) 1.1071 0.7909514153, 0.4143203796,

2 0.0628834756, 20.0690533966,

0.0251533902, 0.0082864076,

2 0.0041922317

0.8505266620, 0.3778139152,

2 0.0791084712, 20.0275108192,

0.0078159446, 0.0034614712,

2 0.0004174138, 20.0002111766

BBW (17/7) 1.4363 0.8227982894, 0.4266704415,

2 0.0578457541, 20.0731170509

0.7769929600, 0.4008823089,

2 0.0784015690, 20.0651822134,

0.0531282040, 0.0190696179,

2 0.0112071569, 20.0012163228,0.0015374324

BBW (17/11) 1.4452 0.7808889544, 0.4266174085,

2 0.0491881155, 20.0874989399,

0.0122970289, 0.0144349220

0.7980748793, 0.4152788416,

2 0.0829711757, 20.0723271361,

0.0528941607, 0.0118257160,

2 0.0168438682, 20.00122403010,

0.0014368342

BBW (17/15) 0.6557 0.786343827, 0.4130474234,

2 0.0594277847, 20.0667620754,

0.0237711139, 0.0070134514,

2 0.0039618523, 0.0002545912

0.8551401110, 0.3791884132,

2 0.0828812880, 20.0296087901,

0.0094685634, 0.0043723636,

2 0.0005783263, 20.0003985961,

2 0.0000256140

BBW (25/19) 1.7566 0.7791080241, 0.4354390791,

2 0.0576009944, 20.1095644608,

0.0288004972, 0.0347566893,

2 0.0082287135, 20.0079698474,

0.0010285892, 0.0008919304

0.8212708854, 0.3928295002,2 0.0770085626,

20.0364590962, 0.0213472550, 20.0019923591,

2 0.0029960949, 20.0016882230,0.0023670737,

0.0009164388, 2 0.0008375691,

20.0000528700,0.0000458457



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incorporated into context-based entropy coding in order tocompare their performance. Here each image isdecomposed up to six scales with the wavelet transform.We carry out a series of experiments to evaluate thesuitability of BBW17/7, BBW17/11, BBW17/15 and

BBW25/19 for image compression. Table 3 shows thatPSNR results at the rate of compression is 1:16. PSNR ofreconstructed images has a little increase, and BBW17/11has the best performance. The detailed comparison ofPSNR is listed in the following Table 3. BBW17/11 and

Fig. 2 CDF9/7 and BBW 15/13 for low-pass filters of analysis

a CDF9/7 and BBW15/13, BBW17/7, BBW17/11, BBW17/15 for low-pass filters of analysisb CDF9/7 and BBW15/13 for low-pass filters of synthesisc CDF9/7 and BBW17/7 for low-pass filters of analysis

d CDF9/7 and BBW17/7 for low-pass filters of synthesise CDF9/7 and BBW17/11 for low-pass filters of analysisf CDF9/7 and BBW17/11 for low-pass filters of synthesisg CDF9/7 and BBW17/15 for low-pass filters of analysish CDF9/7 and BBW17/15 for low-pass filters of synthesis

Fig. 3 Energy percentage comparison between CDF9/7 and BBW17/11

a Level 1b Level 2c Level 3d Level 4



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CDF9/7 have approximately same time expenditure andcomputational complexity for the compression programmeof the image.

Figs. 5 and 6 are reconstructed images of city and Barb by

applying BBW15/13, BBW17/7, BBW17/11, BBW17/15and BBW25/19, respectively. As can been seen from thefollowing reconstructed images, the contour on the housesand edges of the road produced by BBW15/13, BBW17/7,

BBW17/11, BBW17/15, BBW15/13 and BBW25/19 areless distorted by ringing effects and the abundance intextures of remote images are restored perfectly.

6 Conclusions and future research directions

The most significant research in this paper can be concludedas follows: we present a general framework of construction of

Fig. 4 Original remote-sensing imagesa Beijingb Airportc Brusselsd Toulousee Cityf Factoryg San Francisco

Table 3 PSNR of the compressed images

Test images CDF9/7 BBW15/13 BBW17/7 BBW17/11 BBW17/15 BBW25/19

city 23.93 24.07 24.08 24.16 24.07 24.11

factory 21.52 21.65 21.63 21.79 21.66 21.69

Toulouse 26.50 26.52 26.43 26.66 26.54 26.62airport 23.39 23.42 23.40 23.47 23.42 23.41

San Francisco 24.67 24.68 24.67 24.82 24.68 24.78

Brussels 25.62 25.69 25.62 25.78 25.70 25.69

Beijing 25.86 26.95 26.91 27.01 26.96 26.87

Gold Hill 33.41 33.45 33.38 33.52 33.48 33.40

barb 33.58 34.03 34.29 34.65 34.02 34.46



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biorthogonal wavelets based on Bernstein bases along withtheory analysis and application. The presented framework

possesses the largest possible regularities and the requiredvanishing moments. We propose an efficient technique thatgenerates a wide range of new biorthogonal symmetricwavelet transforms. The realisation of the construction of

arbitrary odd lengths filters is achieved. By this framework,a new family of parametric biorthogonal wavelet filters with

symmetry are constructed. The parameterisation is a goodstrategy to obtain biorthogonal wavelet transforms withmore attractive features. In this paper, an optimal model ofFIR aiming at image compression is brought forward. We

Fig. 5 Reconstructed images of city by applying

a CDF9/7b BBW15/13c BBW17/7d BBW17/11e BBW17/15f BBW25/19

Fig. 6 Reconstructed images of BARB by applying

a CDF9/7b BBW15/13c BBW17/7d BBW17/11e BBW17/15f BBW25/19



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optimise the presented parametric biorthogonal wavelet filterswith the optimal model and obtain BBW17/11 with excellentcompression potential. We demonstrate the performance ofthe new family of filters given in this paper for imagecompression with very encouraging results.

Motivated by the observed performance improvement, thefollowing research areas have been identified to be promising.We will research the construction of contourlet transform

[25], an efficient directional multiresolution imagerepresentation. One of the advantages of the contourlettransform is that the basis function is anisotropic, whichovercomes the shortcoming of the basis function of wavelettransform which has vertical and horizontal directions only.Therefore the contourlet transform can represent 2-Dsingular edges of images more efficiently. We will present anew framework of the construction of contourlet transform

by presented Bernstein biorthogonal wavelet filters in thefuture. Based on this, we will research remote-sensingimages fusion, compression and feature extractiontechniques based on the new contourlet transform.

7 Acknowledgments

The authors would like to thank the Associate Editor andanonymous reviewers for their constructive comments andsuggestions, which have lead to a significantly improvedmanuscript. This research was supported by the National

Natural Science Foundation of China under grant 60775018and National Key Basic Research Program (973) of Chinaunder grant 2009CB724001.

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