Hybrid pyramid image coding and data compression

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HYBRID PYRAMID IMAGE CODING AND DATA COMPRESSION

Soo-Chang Pei and Ing-Ing Yang Department of Electrical Engineering

National Taiwan University Taipei, Taiwan,

ABSTRACT

The hybrid pyramid coding scheme for image data compression are based on two coding techniques: sub-band pyramid and k p l a c i a n pyramid. Source data are first decomposed into 7 bands by sub-band pyramid coding. Only the lowest band is further decomposed by Laplacian pyramid to reduce correlations between pixels. The advantage of hybrid pyramid is its efficiency as well a s compactness. In coding process each component is PCM coded. The computer simulation results show that the quality of reconstructed image is good up to compression ratio of 10 to 1.

INTRODUCTION

The basic idea of pyramid data structure is based on repeating the decomposition and decimation process of the low-pass filtered image [1]-[4]. In sub-band coding scheme input signal is decomposed into narrow bands where each band is then decimated to reduce data size. Fig. 1 shows the ideal location of subband images in frequency domain. To form pyramid structure the lowest band is further decomposed by the same manner. No matter whether the depth o f the sub-band pyramid is, the total number of pixels in the pyramid is equal to that in the original image, thus the data structure itself is very compact. The advantage of such a coding scheme is that the quantization noise generated in a specific band is limited almost to that band in the reconstruction process. To compensate aliasing errors introduced in decomposition process, Woods [51 and Gharavi C61 apply Quadrature Mirror filter (QMF) to sub-band coding of images. Woods and Gharavi have chosen a type 32D and 16A QMF filter coefficients respectively

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In subband coding of image signals the 2-D QMF filters decompose the input signal into 4 bands. Woods repeats the above process to each sub-band signal and produces a tree-typed data structure. Instead of Woods' 4 band decomposition manner, only the lowest band is further decomposed in Gharavi's paper and sub-band

Rep. of China

pyramid is formed in this case. We prefer Gharavi's subband pyramid, since it is observed that except for the lowest band the pixel to pixel correlation of upper band signals is very low. Theoretically, the decompositon process can be continued as far as possible, however in Woods and Gharavi's experiments they repeated the process just once again. In their subband coding system, QMF filtering is done in frequency transform domain, the filtered signals should be transformed back to spatial domain before decimation. It is time consuming to transform data back and forth between two domains, so Woods and Gharavi allowed only two-level subband decomposition. Another reason of their decision is due to imperfect reconstruction. Since the QMF filters they use are approximated by FIR filters, successive filtering will result in error accumula- tion. In their experiments pixels are highly correlated in the lowest band, therefore more elaborate quantization technique such as DPCM are required to reduce quantization error.

The Laplacian pyramid introduced by Burt C31 is efficiently obtained by fast hierarchical convolution. The most pro- nounced advantage of such a code is its reversibility, the original image can be perfectly reconstructed from its Laplacian pyramid representation. But the total number of pixels in Laplacian pyramid is greater than that in the original image by a factor about

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The hybrid pyramid is a compromise between these two coding schemes. Input signal is first decomposed by two level subband pyramid coding scheme, then the lowest band is further decomposed into Laplacian pyramid to reduce pixel to pixel correlations. In subband coding process we choose short length QMF filter so we can apply fast hierachical convolution technique as the same manner as Burt does. The technique allows filtering and decimation to be done simultaneously, which provides computational efficiency. The excess number of pixels in the lowest band Laplacian pyramid is just a little bit when compared with the original image.

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Each component in the hybrid pyramid can be quantized by simple PCM code. The quality of reconstructed image is still good at high data compression ratio. The hybrid pyramid takes the advantage of data compactness and computational effi- ciency, there is some experiment results presented here to show its effectiveness.

The principle of QMF filter can be found in [5]-[7], in our software simula- tion we used a 8-tap filter designated as type 8A QMF filter in [71. It has a normalized transitional bandwith of 0.14 radian with an overall passband ripple of 0.06 dB and stopband attenuation of 31 dB. The same QMF lowpass filter can be used in Laplacian pyramid generation. The encoding and decoding schemes of hybrid pyramid are described briefly in Section 11. Histogram analysis and quantization results are descussed in Section 111. Section IV gives the compu- ter simulation results.

THE HYBRID PYRAMID Encoding

There are two steps to generate hybrid pyramid, the first step is to construct a subband pyramid, and the second step is to generate the Laplacian pyramid o f the lowest band. For the sake of fast computation, the QMF filter we choose is short in length, a type 8A QMF filter is used in this paper. If IfKtf refers to the number o f levels in the subband pyramid, in general we let KQ to reduce accumulated error. Suppose' the source image is represented by the array Go, let kBij be the component located at ij band in level k o f the subband pyramid we define oBoo to be equal to Go. thus: for l<k<K and Osi,jsl

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ksij(m,n)= j:

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k-leoO(2m-p,2n-q)hij(p,q) ( 1 ) where hij is the 2-D QMF filter.

Separability characteristics o f hij can reduce 2-D QMF filtering problem to 1-D filtering, and it can be written as hij(p,q)=hi(p)hj(q), where hi(p) is the type 8A QMF filter. The lowest band o f the subband pyramid is baseband component kBoo.

To form a Laplacian Pyramid o f kBoo we can follow Burt's method [l] C31 by substitute kBoo for initial image. The only difference is replaced Burt's kernel function W(m,n) by lowpass QMF filter hoo(m,n). Fig.2 is the block diagram of hybrid pyramid coding scheme.

p=o q=o

Decod inp;

The reconstruction of Laplacian

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pyramid can be found in C31. We only deal with the recovery of subband pyr.amid. Suppose the reconstruction task o f Lapla- cian pyramid is fulfilled, in subband pyramid level K, there are 4 sub-images band : kBoo, kBol, kBlo and kB11.

To recover the baseband component k-1Boo in pyramid level K-1, the decimated signals are interpolated to the same size as k-1Boo and filtered by a similar set of QMF filters before summing together. Let k,lBij be the interpolated and filter- ed version of kBi,j, then

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k,lBlj=4 e

x

kBij(-,-)hiJ(p,q) (2) p=o q=o 2 2

Only terms for which and

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are integers are included in thig sum. The recover baseband image k-1Boo can be written as

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k-lh'(m,n)= E e k,lBij(m,n)

i=o j=o (3)

Starting at k-1200, k-lBol, k-1Blo and k-1B11, we can repeat the above process until o$oo is obtained. The little dif- ference between o%oo and Go is invisible to the human eyes.

HISTOGRAM ANALYSIS AND PCM QUANTIZATION The histogram analysis of a n image is useful for the decision o f quantiza- tion. The uniform quantized 'range, the number o f quantum level L and the size of quantum' step all should be decided before quantization. By the distribution of image histogram, one can make a good choice of these values. We study the histograms of the hybrid pyramid of a source image "Girl".

It is observed that most o f the pixels in subband pyramid are concentrated about the origin, thus we may choose fewer quantization levels to encode them.

Since the dispersive histograms of the lower level Laplacian pyramid, the quantizer would be finer than that o f upper levels. The reduced dimension make it without detriment to compression ratio. In hybrid pyramid, pixel to pixel correlations are largely reduced, so we can apply simple PCM quantizations with different number o f quantum levels and quantum steps to each component.

EXPERIMENTS

In the experiment of computer simula- tion, a source image lfGirlf8 is decomposed into 2-level subband pyramid and 4-level Laplacian pyramid, the total number o f bands is 10. The lowest band in the Laplacian pyramid is a low-pass component

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of dimension 8 by 8. It contains the general brightness area and since pixel to pixel correlation of this subimage is high, it will not be applied to quantizer and remains unchanged. As mentioned earlier, the small size o f this component makes little contribution to average bit rate. The quantization data o f the Girl's hybrid pyramid are listed on Table 1.

pp.1287-1288, Oct. 1986.

[ 6 1 H. Gharavi and A. Tabatabai, "Sub-band Coding of Monochrone and Color Images,If IEEE Trans. Circuits Systems, Vo1.35, [ 7 ] R.E. Crochiere and L . R . Rabiner, "Mil- tirate Digital Signal Processing," Prentice-Hall Inc., 1983, Appendix No.2, pp.207-214, 1988.

7.1, pp.401-404.

Since the diagonal component of the Table 1 . The quantization data used in the subband images contain least information, hybrid pyramid of image "Girl". . . -

ub-band no. of quantum Laplacian no. of quantum yramid levels pyramid levels the quantizer may be chosen as coarse as

possible. In Laplacian pyramid we use finer quantizer to reduce quantization error of the lower frequency subimages. We compute the image data compression ratio

through quantization alone. This may be 1B11 specified a s the average number of bits

represent the quantized hybrid pyramid 2B11 struture. Let there be N2 nodes in the

original image, Level 1 will have 3 subband images with nodes i n each component. If the number o f quantum levels of subimages in Level 1 are all equal to 3 , their contribution to average bit rate is the sum of their sample density multi- plying by their entropy which is given by (3x~x10g,3)/Nz or 1.19 bits per pixel. As the same manner, we can get the estimated average bit rate from 2 bits/pixel to 0.7 bit/pixel in our experiments. Since the original image is 8 bits/pixel, the compression ratio is defined a s the ratio o f 8 to average bit rate, we get the compression ratios from 3.8 to 10.7 in our experiments. Further data compression can be achieved by variable-length codeword such a s Huffman code. The experiment results are shown in Fig.3. Fig.)(c) is obtain from (b) by eliminating Level 1 subimages. The simulation results show the effectiveness o f hybrid pyramid coding scheme, in addition to the computational efficiency make it suitable for image data compression.

REFERENCES

[l] P.J. Burt, "Fast Filter Transforms for Image Processing," Comput. Graphics Image Processing, Vo1.16, pp.20-51, 1981.

[ 2 ] P.J. Burt, C. Yen and X . X u , "Local Correlation Measures for Motion Ana- lysis a Comparative Study," IEEE Conf. pp.269-274, 1982.

C31 P.J. Burt and E.H. Adelson, "The Lap- lacian Pyramid As A Compact Image Code," IEEE Trans. Communication COM- [41 E.H. Adelson and P.J. Burt, ItImage

Data Compression with the Laplacian Pyramid," IEEE Conf., pp.218-223, 1981. C5l J.W. Woods and S.D. O'Nell, "Sub-band Coding o f Images,1f IEEE Trans. Acoust. Speech, Signal Processing Vol.ASSP-34,

Hybrid pyramid image coding and data compression

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Experiment results.