b-spline filter banks for very low bit rate coding - Semantic Scholar

B-SPLINE FILTER BANKS FOR VERY LOW BIT RATE CODING Sergio de Faria M. Ghanbari Dept. of Electronic Systems Engineering { University of Essex Wivenhoe Park { Colchester CO4 3SQ { England Tel: +44 1206 872448 Fax: +44 1206 872900 E-mails: [email protected], [email protected]

ABSTRACT In this paper we report on the use of B-Spline lters in perfect reconstruction lter banks. For each polynomial order the analysis and synthesis FIR lters are evaluated, considering the error introduced by lter coecient truncation, coding quantization and the distortion introduced by the lters themselves. It is shown that B-Spline lters exhibit higher energy concentration in the lower frequency band than other known lter kernels. This property can be exploited for very low bit rate image coding where the higher frequency bands might be very coarsely quantized. For a two channel subband coding system, the reconstructed image from the low frequency channel of the B-Spline coder is compared with other lter banks reported in the literature. The simulation results show that B-Spline lters achieve better reconstruction quality of images than their counterparts.

1. INTRODUCTION Subband coding [1] is an important development for compression of speech and video signals. In this technique, an input signal is decomposed into a set of frequency bands by their proper analysis lter banks. In the reconstruction process, these bands are decoded, interpolated, ltered (synthesis) and added together to represent a replica of the original signal. For perfect and alias free reconstruction of the signal, several design procedures for the lter banks are known [2, 3]. However, for very low bit rate coding applications, where higher bands are coarsely quantized, lters which concentrate most of the energy in the lowest band are the favorites. In this paper we propose a two-channel scheme where the analysis and synthesis lter banks use B-Spline FIR lters. As a measure of the energy concentration in the lowest band, the reconstructed signal from this band is compared against commonly used lters [4, 5]. Application of this method is the four band decomposition of images for low bit rate coding, and the reconstruction of the image only from the lowest band.  His work has been supported in part by Instituto Politecnico de Leiria and Programa CIENCIA (JNICT), Portugal, under grant BD/CIENCIA/2673/93.

2. B-SPLINES

Polynomials are widely used in the approximation theory and numerical analysis to approximate smooth functions. However, coarse approximation for polynomials of orders greater than 3 or 4 may result in wild oscillations. Fortunately, a class of piecewise polynomials can be free of oscillation, when the entire domain is segmented. B-Splines basis functions have such a property and are commonly used to represent curves and surfaces [6]. Let X (u) be the parametric position vector along a curve, as a function of the parameter u, then B-Spline curve is de ned by X (u)

=

n+1 X

Yi Bi;k (u) i=1 umin  u < umax 2  k  n + 1

where Yi are the position vectors of the n+1 polygon vertices and the Bi;k are the normalized B-Spline basis functions. These polygon vertices are a set of data values which represent the original curve by approximation. The piecewise polynomial curve is formed by a group of consecutive polynomial segments. The joints between these segments form a set of values known as knot vectors. The most important properties of the B-Spline basis functions are:  the continuity (C ) of the surface in each knot vector is two

   

less than the B-Spline order (k), C k;2 . the B-Spline order of the curve is restricted to the number of control vertices. each control vertex in uences  k2 segments of the polygon. B-Spline basis functions always assume non-negative values. the curve is invariant with respect to an ane transformation. This implies that any transformation applied to the curve is the same as if it is applied to the control vertices.

Evaluation of the control vertices of a B-Spline curve or surface for interpolation, however, requires the solution of a large system of equations [7]. Additionally if the interpolation order is larger than one, these points are determined only by approximation, which means that the signal can not be totally recovered. For this reason, we approximate the B-Spline functions by lters and study their behavior in a perfect reconstruction subband coding system. Discrete B-Splines: These functions can be derived using a signal processing interpolation method [8], where a function is interpolated by analytic functions which are a

Mk

M

1

0 (z ) where B1k is the discrete signal at the B-Spline knots, BM is the rectangular pulse of width M , the interpolation order, and k is the B-Spline order. The evaluation of the original digital signal, FM , using the polygon vertices (or coecients), Y , is then

n

FM (z ) = BM (z )Y (z

M );

and the coecients can be obtained by convolving the data n (z ));1 . points with (BM

3. BIORTHOGONAL B-SPLINE FILTER BANKS

A two-channel (M = 2) subband coding system uses biorthogonal lter banks, where the synthesis FIR lter, G0 (z ), and the analysis lter, H0 (z ), are symmetric. While for B-Spline FIR lter banks, G0 (z ) can be determined by Eqn. 1, H0 (z ) is only obtained by approximation, as the number of data points is larger than the number of coecients. This overspeci ed problem leads to a mean value solution, which is represented by an in nite-duration impulse response lter. An FIR lter is obtained by discarding the less signi cant coecients. Using two linear phase FIR lters for the subband coding implementation, then lters have to meet certain conditions, such that the system achieves perfect reconstruction (PR) and a zero overall delay [11]. The corresponding high frequency band lters are obtained using, H1 (z ) =

;z d; G (;z ) 2

1

0.002 PRE_7(z) PRE_5(z) PRE_3(z)

0.0015

0.001

0.0005

0

-0.0005

-0.001

-0.0015

-0.002 0

0.5

1

1.5 2 Frequency [radians]

2.5

3

Figure 1. Reconstruction error of odd order Bspline lters.

deviations in the amplitudes related to the inaccuracy of the truncated values, lters with odd order, which corresponds to even degree polynomials, have a larger reconstruction error at the lower and higher frequencies, and a smaller error in the midrange frequencies. On the other hand, even order 0.002 PRE_8(z) PRE_6(z) PRE_4(z) PRE_2(z)

0.0015

0

and

0.001

0.0005

G1 (z ) =

;z d; (2

1)

;

H0 ( z );

where 2d ; 1 is the resultant delay.

3.1. Coecients truncation error

As the coecients strongly drop to zero the H0 (z ) lter can be truncated without introducing a signi cant error in the reconstructed signal. This distortion error is expressed as, (2)

Let us take the coecients of a lter with absolute values larger than 10;4 , and assume that other coecients have a small in uence in the signal reconstruction. The magnitude response of these lters, for B-Spline orders between 2 and 8 are shown in Fig. 1 and 2. With the exception of the 2nd order B-Spline lters, odd and even order B-Splines have a dierent type of reconstruction error. Apart from a small

Magnitude response [dB]

M

3.2. Generalization of the error criteria for other B-Spline orders

Magnitude response [dB]

piecewise polynomial of a xed degree. Based on the normalized k-order B-Spline for n + 2 uniformly spaced knots [9], Unser et al obtained the discrete B-Splines representation in the z-transform form [10], 1 B k (z )(B 0 (z ))n+1 ; k B (z ) = (1)

"PRE

= j2 ; [H0 (z )G0(z ) + H1 (z )G1(z )]j

which is the dierence between the reconstructed signal and PR. This error can be approximated as a function of  and the lter length, n, n; Pre (n)  (3) =  #;( 4 ) where  stands for the PR error due to the maximum truncation, (n = ) and # = 1:95 for a B-Spline order of 4. Hence for a given error threshold of , the length of the lter, n, can be chosen such that Pre (n)  .

0

-0.0005

-0.001

-0.0015

-0.002 0

0.5

1

1.5 2 Frequency [radians]

2.5

3

Figure 2. Reconstruction error of even order Bspline lters.

B-Spline lters present a larger reconstruction error at the midrange frequencies than at the lower and higher frequencies. In both cases the error at the midrange frequencies is similar and approximately equal to 7:5  10;4 , but at the lower and higher frequencies the error for even B-Splines is almost 2  10;4 and for odd B-Splines is 10 times larger ( 2  10;3). Evaluation of the reconstructed error, as given in Eqn. 3 can be generalized for the complete set of B-Spline orders,

from k = 2 to k = 8, with a good accuracy by



k

(4)

Pre (n) = k #(k)

;

; n;
4

k

H0 (z )G0 (z ) = 2

where ;k ;

#(k) = 1:4 + 30 e

and n is the length of the lter, the error  for the maximum k 2 3 4 5 6 7 8 k

0.05 0.43 0.36 0.76 0.43 1.91 0.95

Table 1. Error for the maximum truncation.

truncation, is shown in the Tab. 1, and (5)



k

= 5 7

; if ; if

where the rst term corresponds to the error in the signal and the second term is the aliasing eect. Using the PR condition

k=even; k=odd:

"(z ) = "PRE (z ) + "ALI (z ) + "COD (z ):

where the "PRE (z ) is substituted from Eqn. 2, the aliasing error, "ALI is 1 "ALI = [H0 (;z )G0(z ) + H1 (;z )G1(z )] 2 and the coding error, "COD , is "COD (z ) = G0 (z )E0 (z

2

) + G1 (z )E1 (z 2):

If the subband coding system is used under the conditions of PR, in the absence of coding errors, the aliasing eect is then canceled. Obviously, when the PR relationships between lters change, then there is no aliasing cancelation. Similarly, when B-Splines are used to approximate curves or surfaces in computer aided geometry, only the low frequency band is used, rstly to perform the inverse operation and obtain the control vertices, and secondly to reconstruct the surface. In the signal processing applications, these operations are carried out by the H0 (z ) analysis and the G0 (z ) synthesis lters. This means that high frequency channel lters, H1 (z ) = G1 (z ) = 0. Because of this, in addition to "PRE (z ) and "COD (z ), we have "ALI (z ) 6= 0. This aliasing error, represented by 12 H0 (;z )G0 (z ) , has a large energy in the midrange frequencies and decreases as the B-Spline order increases. For k = 2, the error has high energy in a wide band of the spectrum, which is related to the large transition passband exhibited by the B-Spline of order 2.

4.1. Reconstruction error from the low frequency channel

The general form for the reconstructed signal of a subband coding system is given by, (7) X^ (z ) = 21 [H0 (z )G0(z )X (z ) + H0 (;z )G0(z )X (;z )] ;

0

0.7 E_8(z) E_7(z) E_6(z) E_5(z) E_4(z) E_3(z) E_2(z)

0.6

0.5 Magnitude response

(6)

0

and substituting in Eqn. 7, we obtain the error for the low frequency band, (8) "l (z ) = H0 (2;z ) [G0 (;z )X (z ) ; G0 (z )X (;z )] : For a rst order Gauss Markov signal X (z ) with a autocorrelation coecient of 0.95, the magnitude response of the error is mainly observed at the high frequencies, except for the second order lter which is also present at lower frequencies. As the order of B-Spline increases, the errors in

4. SUBBAND CODING ERRORS

The output error of a subband coding system can be represented by,

; H (;z )G (;z )

0.4

0.3

0.2

0.1

0 0

0.5

1

1.5 2 Frequency [radians]

2.5

3

Figure 3. Output error at the; low frequency channel (coecients truncation: (10 ). 4

the lower frequencies become smaller, as shown in Fig. 3.

5. EXPERIMENTS

We carried out some experiments to compare the performance of B-Spline biorthogonal lters with respect to some known lters reported in the literature namely: the QMF 16TAP(A) [4], near PR; the biorthogonal LeGall [5], PR and the linear phase biorthogonal wavelet Edu, PR [12]. The simulations were performed using the LENA image of size 512512 pixels, with 8 bits/pixel of resolution. In order to obtain PR, the images were symmetrically extended at the borders and properly delayed. It is assumed "COD = 0 and low, horizontal, vertical and diagonal bands are referred as Si0 , Si1 , Si2 and Si3 , respectively. First the performance of B-Spline lters banks were assessed for various conditions, such as:  the eect of coecient's truncation on the cubic Bspline lters bank,  order of B-Splines in decomposing images into bands,  reconstruction of images without quantization of the subimages, evaluating the magnitude distortion due to ltering,  reconstruction of the image just from the low frequency band to evaluate its energy compaction.

It was observed that truncation of lter coecients to a degree of 10;3 (Length=41) does not introduce signi cant error. With such accuracy the performance of B-Spline lters of orders 3 to 8 were evaluated. It was noticed that odd order B-Spline lters have lower energy compaction ability than even order lters. For lters of order larger than k = 6, the improvement in the compaction is not signi cant. To compare the energy compaction of B-Spline with other kernel lters, we reconstructed the LENA picture just from the lowest band under all the lters. Tab. 2 shows the PSNR(dB) of the reconstructed image, where Len(H0(z )) Filter Len(H0 (z)) Len(G0 (z)) Si0 Cubic B-Spline 41 9 35.77428 16TAP(A) 16 16 35.52514 LeGall 5 3 34.47334 Edu 11 5 35.45363

Table 2. Reconstruction quality using dierent lters.

and Len(G0(z )) are the length of the low pass lters analysis and synthesis respectively. The B-Spline lter has almost 1dB improvement over the short kernel lter of [5] and is marginally better than [4, 12]. The introduced error (Eqn. 8) in the reconstructed image of a Gauss Markov process type with a correlation coecient of 0.95 is shown in Fig. 4. 0.9 E_Edu E_4(z) E_16TAP(A) E_LeGall

0.8 0.7

Magnitude

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5 2 Frequency [radians]

2.5

3

Figure 4. Single band reconstruction error. 6. CONCLUSIONS

B-Spline FIR lters were designed for the subband coding applications that achieve near perfect reconstruction. The introduced error from dierent orders of B-Spline lters were analyzed and a relationship between the size of the truncated analysis lter of dierent B-Spline orders and the introduced error in the reconstruction was established. Furthermore, the possibility of encoding only the low frequency subimage was considered, as well as, the ability of the B-Spline lters to concentrate the energy of the picture in this subimage. The results show that B-Spline lter banks are capable of reconstructing images using only information from the low frequency subimage, with better quality than other known lters reported in the literature. Hence these type of lters can be used for very low bit rate coding.

REFERENCES

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