Image compression using discrete orthogonal transforms with the «Noise-like» basis functions Текст научной статьи по специальности «Математика»

IMAGE COMPRESSION USING DISCRETE ORTHOGONAL TRANSFORMS WITH THE «NOISE-LIKE» BASIS FUNCTIONS

V.M. Chernov, A.G. Dmitriyev * Image Processing System Institute of RAS, vche@smr.ru *) Samara Municipal Nayanova University, alex@,gw.mnu.samara.ru

Abstract

The generalization of the discrete orthogonal transforms with the basis functions generated in a pseudorandom way is the subject of the article.

The examples of such transforms application in the field of videoinformation coding in the channels with the high level of «seldom» noise are also given.

1. Introduction

Discrete orthogonal transforms (DOT)

N-1

x (m ) = X x (n ) hm (n ), m = 0,1,..., N -1

n=1

are widely used in the fields of information coding, information transmission and discrete signals processing. Here x(n) is the N-periodic input sequence,

{hm ( n )} m=0 are the basis functions orthogonal for some scalar (or Hermitian) product.

< hu , hv >=Suv

In the real world channels the noise distorts the transmitted signals, especially those with the low numerical values. If the channel characteristics are such that some samples of the transmitted information can be irretrievably lost or strongly distorted regardless of their values then standard DOTs (Fourier, Hartley etc.) can hardly be used for information coding. The standard DOT properties depend on the correlation properties of the processed signal. Therefore the loss of some high amplitude components (for example Fourier components) during the videoinformation transmission results in a strong noise in the restored image. This noise has periodic structure for which the human vision is very sensitive.

Another decision in this case is to code the information with the DOTs having such basis functions hm ( n ) for which the spectral components {x ( m )} are «energetically equal».

The notion of discrete M-transforms with the noiselike discrete functions is introduced in [1]. The applications of such transforms for one- and two-dimensional information coding are considered in [2]-[4].

Such transforms do not lead to energy concentration in a few spectral coefficients do not lower the redundancy attributed to the statistical relations between the elements of the signal to be transformed and efficiently remove insignificant information.

Particularly in the process of compression-decompression of videoinformation after the inverse transformation was applied, the noise affecting the image is less notable for the human vision than in case of Walsh transforms.

The basis functions of such transforms are based on the m-sequences (recurrent sequences of the finite field elements with the maximum periods) The sequences of

this kind are widely used for the pseudorandom numbers generation, cryptography etc [3]-[4].

The M-transforms with the basis functions taking two different values with the (nearly) equal frequencies are considered in the publications mentioned above. In our article we consider the generalized M-transforms with the basis functions taking p different values. The two-dimensional versions of the proposed structures are also discussed.

2. Generalized one-dimensional M-transforms Let Fp be the finite p -element field. Let <p(n) be the r -order recurrent sequence

p(n) = app(n -1) +... + arp(n - r), aj e Fp

(1)

with nontrivial initial values (p(0),...,p(r -1)).

Definition 1 Let N be the period of sequence (1). If N = pr -1 then sequence (1) is called the m-sequence.

Using the slight modification of the corresponding proof [1] the following statement can be proven:

Proposition 1 Let p be prime, N = pr -1. Let the numbers A0,..., Ap-1 satisfy the following relation

Ap-1 - Ao

Ak = -— + Ao, (k = 0,...,p-1)

p -1

Let the functions hm (n) be determined by the following relations

fh0 (n) = Ak if p(n) = k;

\hm (n) = h0 (m + n) if p(n) ^ k. Then there exists the efficiently calculated constants A0 and Ak such that the functions {hm ( n )} m- form

the orthonormal set

N-1

<

hu, hv >= X huhv =du

n=0

Proof. Let us introduce the following notation

Ap-1 - A0

C =—-, hT (n) = h0 (n +t), A0 = A

p -1

Hk ( n ) =

1, if p(n) = k; 0, if p(n) ^ k.

Then

p-1

hT = X(A + kC)Hk (n + T)

k=0

Let us take A0 and Ap-1 that the orthogonality

condition for the set {hm (n)} is held in the form

n-1

< hT, hv >= N-1 X hT (n) hv (n) = SW (3)

n=0

Since the functions hT are obtained from each other

by the cyclic shifts then the sum (3) depends only on (t - v). Therefore we can consider only the case of v = 0 .

Using (2) and (3) for t = 0 we get

n-1 n-1

N = I ( I(A + Ck)Hk (n))2

(4)

n=0 k=0

Since H, (n)H j (n) = Sji then using (4) we obtain

p-1n-1

p-1 n-1

N = A 2 I I H, (n ) + 2 AC I i IH, (n )

i=0 n=0 i=0 n=0

2 P -1 2 N -1 + C 2 I i2 IH, ( n ) =

i = 0 n =1

P -1 P -1

= A2N + 2AC I ipr-1 + C2 I i2pr-1 = i =1 i =1

= A2(pr -1) + ACpr (p-1) +

+ C2pr (p-1)^ 6

Let t * 0. Then we have

p -1 p -1

0 = I I(A + c, )(A + Cj )Sj (t) i=0 j=0

(5)

(6)

Here

N-1

Sj (t) =XH (n)Hj (n + T). (7)

n=0

The calculation of Sij is the most difficult part of

this poof. Using the standard method from number theory we can bring (7) to the trigonometric sum of the special kind.

Let A be the nontrivial character of a field Fpr = Fq additive group. Then

, _ f1, if P = 0;

q_1 V A(cP) = f H

aeFq 1if 0.

(see [4]).

The following condition holds for m-sequences N-1

£A(p( n)) = -1.

n=0

Thus

n-1

Sj (t) = q-2 I( iA(a(y(n) -i)))

n=1 ae Fq

x( IA( P(<p( n + t) - j ))) =

PeFq

= q

-2

IA (-ai -pi )>

(a,P)eFq xFq

n-1

x IA(ap(n) + Pp(n + t))

n=0

Since for (a,P) * (0,0) e Fq x Fq the functions

0(cPp( n) = a(p( n) + P<p( n + t) are the m-sequences then

Sj (t) = q-2 N - q-2

IA(-a-p) (8)

(a,P)eFq xFq\{O}

Since

I A(-a)A(-p) = J q

(a ,P )eFq xFq

if i = j = 0 0 , else

then

Sj (t) = q -2 N - q -2 IA(-ai )A(-p )

(a,p)eFq xFq

+ q"2 I A(-ai) + q-2 IA(-p)-q-2 =

aeFq

PeFq

q "2 N - q "2, q - 2 N + q _1 - q7

q - 2 N -1 + 2 q _1 - q

-2

if i, j * 0 if i = 0, j * 0 or i * 0, j = 0 if i = j = 0

Substituting Sj (t) in (6) we get an explicit relation

between A and C. This relation together with (5) brings the system of equations for determining A and C and therefore A0 and A

ip-1.

The examples of the basis function h0 (n) for different p and N are shown on the fig. 1.

Definition 2 The transform (1) with the basis functions {hm (n)} defined in Proposition 1 is called the generalized M-transform (GM-transform).

3. Two-dimensional GM-transforms

The one dimensional M-transforms introduced in the previous section can be used for two-dimensional digital arrays (images) coding after the standard digital image processing methods were applied. (a) The N*N pixel images can be represented by one-dimensional arrays in a number of ways (Fig. 2).

hjn)

a)

h»

b)

Fig. 1. Function Hq (n) for (a) N=26, p=3, r=3, (b) N=24, p=5, r=2

a)

b)

c)

Fig. 2. Different ways of representing two-dimensional N*N array as a one-dimensional N2 - element array.

(b) We can introduce the separable two-dimensional GM-transform by the following relation.

>n2)hm1 (n1)hm2 (n2)

X (m1, m 2 ) =

N-1 N-1

= V Vx (n1-

n1 =0 n 2 = 0

In case (a) the two-dimensional N*N-points GM-transform calculation is reduced to the calculation of one-dimensional N2-points GM-transform. In case (b) the two-dimensional N*N-points GM-transform calculation is reduced to the calculation of N one-dimensional N-points GM-transforms. This calculation is done using the standard "row-column" (cascade) scheme:

N-1 N-1

x(m1 , m2) = V ( V X(n1' n2 )hm1 (n1 ))hm2 (n2 )

n1 = 0 n2=0

4. Fast algorithms for GM-transforms

The main property of GM-transforms is the existence of fast algorithms of their calculation.

Let us show that transforms (1) and (9) can be represented in a form of one- and two-dimensional convolution respectively.

Let n = -n, n = -n, V2 = n2.

The signal x(n) and the functions hm (n) are N-periodic. Thus considering the introduced notation expressions (1) and (9) can be transformed into

N-1

x(m) = ^ x(n)h0 (m - rf) = (x * h)(m)

n=i

x(mi,m2 ) = n-1 n-1

= Z Z x (n , n 2 ) hq (m1 -m)ho(m2-^2)

m =Qn 2=q

(10)

(11)

Array (10) can be calculated in a standard way using the discrete Fourier transform (DFT).

Fig. 3.

The drawback of this scheme is that N = pr -1 can hardly be factored as

N = pC1 •... • pC (12)

The numbers p1 ,... , pt in (12) are primes.

a C ■

The DFT calculation for N = p11 •... • ptJ can be done using Good-Thomas decomposition [6]. According

a ■

to it we have to calculate pjJ -point DFT (J = 1......t ).

There exists efficient FA for all p j .

Another decision in this case is to calculate (10) and (11) using polynomial transform method [5].

The detailed discussion concerning the fast algorithms for calculation of (10) and (11) will be given on presentation.

5. Experimental results

Figures 4b-4d illustrate the reconstructed images after 70 of 256x256 spectral components have been replaced by zeroes for Hartley transform (b), Hadamard transform (c) and GM-transform (d). The original image is depicted on Figure 4a. The «lost» transforms were chosen in a random way.

The more detailed discussion concerning the restoration quality will be given on presentation.

a)

c)

d) I

Figure 4. (a) Original image, (b) Hartley transform, (c) Hadamard transform, (d). GM transform

6. Conclusion In authors' opinion the capabilities of GM-transforms are not limited to the examples given in the

article. It is clear that GM-transforms can be used for signal processing not only in the frequency field but also in the time field. Such problems arise when processing (in particular, when interpolating) non-uniform sampling signals [7]-[8].

Acknowledgement

This work was performed with financial support from the Russian Foundation of Fundamental Investigations (Grant 97-01-00900).

References

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2. H.G. Musmann, P.Pirsch, H.J. Grallert, «Advances in picture coding», IEEE Proc., 1985, vol.73, No 4, pp.523-548.

3. G. Birkhoff, T.C. Bartee, Modern applied algebra, McGraw-Hill, N.Y., 1970.

4. R. Lidl, H. Niderreiter, Finite fields, Addison-Wesley, 1983.

5. Nussbaumer H.J, Fast Fourier Transform and Convolution Algorithms, Springer Verlag,1971.

6. Blahut R.E. Fast Algorithms for Digital Signal Processing, Addison-Wesley, 1985.

7. Bloomfield P., «Spectral analysis with randomly missing observations», J.Roy.Statist.Soc.Ser.B., 1970, vol.32, No 3, pp.369-381.

8. Jones R.N. «Spectral analyses with regularly missed observations», Ann.Math.Stat., 1962, vol.23, No 2, pp.455-461.