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PRACTICAL LOW BIT RATE PREDICTIVE IMAGE CODER USING MULTI-RATE

PROCESSING AND ADAPTIVE ENTROPY CODING

Anna N. Kim and Tor A. Ramstad

Department of Telecommunications, NTNU

O.S. Bragstads plass 2B, N-7491 Trondheim, Norway

ABSTRACT

ThemodifiedDifferentialPulseCodedModulation(DPCM)codec

with multi-rate processing has been shown to able to code source

with monotonically decreasing spectrum efficiently at low bit rates

[1]. A practical image coder is designed based on this approach.

Two dimensional DPCM is used along with decimation and inter-

polation to reduce the number of transmitted samples. The deci-

mation rate depends on the signal spectrum and the bit rate. Fur-

ther bit rate reduction is achieved through adaptive entropy cod-

ing. Wiener filter is appended in the decoder for minimizing dis-

tortion caused by quantization noise. The decimation filter can be

implemented using simple IIR filters. The necessary side infor-

mation is low. Simulation results show that the coder is able to

give good compression performance at low bit rates which is su-

perior to conventional DPCM codec and JPEG. Subjective quality

can be as good as JPEG2000. While at very low bit rates the pro-

posed codec is able to retain certain image characteristics better

than JPEG2000.

1. INTRODUCTION

The main advantage of DPCM is its ability to perform efficient

codingwithlowcomplexity. DPCMcodersgenerallyperformwell

at high rates (3-4 bits/pixel), when quantization steps are small and

the optimal predictor is able to minimize the prediction error vari-

ance. However at low bit rate regions, the additive noise model

is no longer applicable. An exact analysis of the system becomes

very involved or even impossible. It is then interesting to modify

the DPCM structure to comply with rate-distortion theory while

at the same time using a rather applicable system model. The as-

sumptions would have to be examined from experimental results.

In [1], we proposed a modified DPCM codec with multi-rate

processing. It is designed for encoding of sources with monotoni-

cally decreasing spectra at low bit rates. Similar to the coder in [2],

our codec was motivated by rate-distortion theory. It incorporates

multi-rate processing and a Wiener receiver filter. Simulation with

Gaussian AR(1) processes show that the decimation and interpola-

tion process are able to provide further rate reduction so the over-

all rate-distortion performance is much closer to the rate-distortion

bound at low bit-rate regions compared to [2].

In this paper we want to verify the potential of the proposed

method in practice by designing an image coder based on this ap-

proach. The goal is to have a practical image coder with good

low-bit rate performance, yet low-complexity implementation.

2. DPCM WITH MULTI-RATE PROCESSING

The rate-distortion theory states that the optimal lossy coding of

a discrete time, continuous Gaussian amplitude source is done by

coding only portions of the source spectrum that is above the noise

floor. This is commonly described as the reverse “Water Filling”

principle. Consider for example a 1st order Gauss-Markov pro-

cess (AR(1)) which has monotonically decreasing spectrum. At

low target bit rates (high distortion), the high frequency portion

of the source spectrum that is below the additive noise should not

contribute to the rate. This is achieved by applying a low-pass filter

to shape the source spectrum prior to encoding with the standard

DPCM.

However, the low-pass filtered source leads to a low-pass in-

put to the quantizer, which is correlated and oversampled. To over-

comethisproblem, weapplydown-samplingtothesourcetolower

the sampling rate before coding and corresponding up-sampling

and interpolation is performed in the decoder. This modification

leads to further overall rate reduction due to the fact that the num-

berofsamplesforcodingafterdown-samplingisreduced. Therate

at the coder output is measured in bits per coded sample. However,

for the overall rate-distortion performance, the rate is measured in

bits per source sample. The actual bit rate is then the obtained rate

divided by the down-sampling rate. A simple uniform quantizer

is used for quantization of the prediction error followed by an en-

tropy coder. We also append a Wiener filter to the decoder. Its role

is to further suppress distortion due to quantization and it does not

affect the rate.

Simulation results showed that the proposed codec was able

to perform very close to the rate-distortion bound for highly cor-

related AR(1) process (correlation coefficient α = 0.9) at low bit

rates. When the codec is not operating at the optimal point, that

is, when there is a mismatch between the low-pass filter cut-off

frequency and the quantization noise level, the Wiener filter was

able to provide significant compensation. More detailed descrip-

tion and discussion of the system can be found in [1].

3. LOW BIT-RATE PREDICTIVE IMAGE CODER WITH

ADAPTIVE ENTROPY CODING

We now introduce the practical image coder based on the codec

described in the previous section.

The image coder has essentially the same structure (Figure 1).

The source image is first low-pass filtered and down-sampled two

dimensionally. It is then fed through the DPCM encoder, where

the difference between the down-sampled image and its prediction

is quantized using a uniform quantizer and the quantization indices

are entropy coded. On the decoder side, the received quantization

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Fig. 1. Block Diagram of Image Coder: (a) Encoder (b) De-

coder. EC: Entropy Coder, ED: Entropy Decoder, L Low-

pass Filter, P: Prediction Filter, W: Wiener Filter, r: Decima-

tion/Interpolation Rate

indices first go through entropy decoder and DPCM decoder, fol-

lowed by the Wiener filter. The resulting image is then up-sampled

and interpolated using the same low-pass filter.

A two dimensional DPCM (2D-DPCM) is used in the pro-

posed image coder. We model the down-sampled images as a sep-

arable 1st order Gauss-Markov model which has auto-correlation

function:

R(i,j) = σ2ρ|i|

1ρ|j|

2,

|ρ1| < 1,|ρ2| < 1

(1)

where ρ1 and ρ2 are the one step correlation coefficients in the

row and column direction after decimation and σ2is the variance.

The current pixel value can be calculated by its three surrounding

pixels [3]:

ˆ x(i,j) = ρ1˜ x(i−1,j)+ρ2˜ x(i,j−1)−ρ1ρ2˜ x(i−1,j−1) (2)

The Wiener filter gives an optimal tradeoff in the mean squared

sense between suppression of additive quantization noise and lin-

ear distortion due to filtering. Its frequency response is given by:

W(ωi,ωj) =

Sxx(ωi,ωj)

Sxx(ωi,ωj) + σ2

q,

(3)

whereSxx(ωi,ωj)isthepowerspectraldensityofthedown-sampled

image and σ2

filter is generally not separable even if the point spread function

and the variance covariance functions are [3]. The required auto-

correlationfunctioncanbederivedfromthepredictioncoefficients,

whereas the quantization noise variance that can be approximated

by σ2

We would like to stress that although the over-all image coder

is designed for low-bit rate operation, due to down-sampling, the

DPCM part of the system is in fact operating at fairly high rates

(above 2 bits/pixel)1.The additive noise model is then applicable

and the assumption that the noise and the signal are uncorrelated

is quite accurate when a uniform quantizer is applied. It is safe to

consider that the quantization noise is uncorrelated with the quan-

tizer input. The approximation of σ2

Further rate reduction of the system is obtained through adap-

tive entropy coding. For our codec in Section 2, the prediction

qis the quantization noise variance. The 2-D Wiener

q= ∆2/12, where ∆ is the quantization step.

qis then also valid.

1[4] defines high rate above 2 bits/pixel

error is quantized by the uniform quantizer then coded by one en-

tropy coder. However, images have varying local statistics. When

a fixed predictor is used in the DPCM encoder, the prediction error

image will also have this property. It can be split into small blocks

with size that makes their internal statistics constant. One should

note that the block size must not be too small, in which case a large

amount of side information is required. These blocks can then be

classified into a small number of classes depending on their vari-

ances. After uniform quantization with the same quantizer for the

entire image, an entropy coder can be designed for each class.

A method for optimal selection of class boundaries and en-

tropy coders for each class for a given number of classes was de-

signed in [5],[6]. It was shown that there is an optimal number

of classes for which the gain in increasing the number of classes

is counteracted by the increase of side information needed for in-

forming the receiver about the block classification. The number

of classes, of course, depends on the block size as well. [5] also

showed it is adequate to keep the source splitting class number

M between 4 and 6. For our proposed coder, M = 4 is cho-

sen along with block size of 4x4 pixels. We then use the iteration

algorithm in [5] to find the optimal variance decision levels to de-

termine where to apply the four different entropy coders.

4. SIMULATION RESULTS AND DISCUSSION

Fig. 2. Coding performance at decimation rate of 3/2x3/2, 2x2 and

3x3 comparing with JPEG. Image “Lena”.

In the first experiment we used the 512x512, monochrome

“Lena”imagetoassessthesourcecodingperformanceofourcoder.

Figure 2 shows the results for down-sampling rates3

3x3 compared with JPEG and JPEG2000

DPCM performance at such low bit rate region is unsurprisingly

substantially worse, hence not included in the plot. The collection

of curves of the proposed coder have the same shape as the ones

show in [1]. The convex hull of the curves are the best operation

point of the codec. For each down-sampling rate, there is an op-

timal quantization step ∆. When the quantization noise level is

too low, the low-pass filtering before down-sampling removes too

much of the source spectrum so the rate-distortion performance

2x3

2, 2x2 and

2. The conventional

2JPEG2000 standard used here was implemented in Java through The

JJ2000 project. Source code v5.1 http://jj2000.epfl.ch

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Fig. 3. Coding performance at decimation rate of 2x2. Image

“Lena”.

is further away from the “elbow” of the curve. As the quantiza-

tionnoiseincreases, theperformanceimproves. The“elbow”point

is reached when the quantization noise level intercepts the source

spectrum at the same frequency of the stop-band of low-pass filter.

The performance is again worsened when the quantization noise

becomes too high for the source shaping low-pass filter. In prac-

tice, for a chosen down-sampling rate, the quantization step ∆ for

the ideal operating point can be determined through simple itera-

tion since the quantization noise level is approximated by ∆2/12.

Figure3showstheeffectofadaptiveentropycodingandWiener

filter. As our results in [1] indicated, improvement by Wiener filter

is more pronounced beyond the convex hulls when the quantiza-

tion noise level is higher. The adaptive entropy coding shows re-

duction rate around 0.15 bits/pixel. Figure 4 compares the original

and coded version of “Lena”. The down-sampling rate is limited

to3

due to much removal of the high frequency components. It can

be observed in Figure 2 however, that the three decimation factors

are adequate to cover a large range of bit rates at almost optimal

quality. The common blocking artifact seen in Transform Coding

is not present. The ringing noise is very limited when decimation

rate is low.

Necessary side information includes prediction coefficient (2

bytes), quantization step size (2 bits), decimation rate (2 bits) and

entropy coder location table. The number of bits required for the

first three quantities is negligible when the image size is large.

Information needed for adaptive entropy coding can be coded as

such: in the case of 4 entropy coders and 4x4 block size, the sim-

plest way is to use 2 extra bits per block. which gives 0.125 bpp

extra. With decimation rate r the number of bits spent per source

pixel is then 0.125/r2. The alternative, which is more efficient,

is to entropy code the entropy coder location table since the dis-

tribution of the coders is not uniform. The probability distribution

needed for entropy coding can be modelled as generalized Gaus-

sian distribution and requires 2 additional bytes for the necessary

parameters. The resulting over-all side information is low.

FIR filters are generally used in image processing since they

can be designed with linear phase. For lower complexity, IIR filers

are good alternatives. Phase distortion can be avoided by filtering

the image in the reverse direction during interpolation. As an ex-

2x3

2, 2x2 and 3x3 since higher rates result more blurred images

ample, the low-pass filter for decimation and interpolation rate of

2x2 can be implemented in simple form using an allpass filter and

a delay [7]:

H(z) =1

2

with α < 1 for stability. The transfer function can be realized with

only one multiplication and three additions.

?

z−1+α + z−2

1 + αz−2

?

(4)

Fig. 4. Comparison of original and coded versions of “Lena”.

Top Left: Original. Top Right: Decimation Rate 3/2x3/2 Bit

Rate 0.97 bits/pixel. Bottom Left: Decimation Rate 2x2 Bit Rate

0.55 bits/pixel. Bottom Right: Decimation Rate 3x3 Bit Rate 0.25

bits/pixel.

ItisknownthatthelatestimagecompressionstandardJPEG2000

is able to offer high quality performance at low bit rates. In terms

ofpeak-to-peaksignal-to-noise-ratio(PSNR),JPEG2000hasclearly

an advantage (Figure 2). However in terms of subjective quality,

thedifferenceisbarelynoticeable(Figure5). Herewecomparethe

subjective quality by processing the 512x512, monochrome im-

age “Peppers”. In Figure 6 one can see that at very low bit rates,

JPEG2000 coder although has better edge sharpness, it “smears

out” the object surface. For our proposed codec, the sharpness

is compromised due to decimation but the retained texture of the

objects in the image is clearly closer to the original.

Finally, we would like to comment on the complexity of the

proposed codec. JPEG2000 requires at least 5 multiplications per

pixel to achieve sufficient wavelet transform decomposition plus

additional complexity from the context based arithmetic coder [8].

For our coder, 3 multiplications per pixel are needed for prediction

and variance calculation for adaptive entropy coding. The Wiener

filter can be implemented using FIR filters to lower computational

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complexity [3]. Again since this is operated on decimated image,

theactualmultiplicationperpixelisreduceddependingonthedec-

imation rate. As stated earlier decimation filters can be realized in

very simple IIR filter forms so the overall complexity remains low.

5. CONCLUSIONS

In this paper, we proposed a practical image coder based on our

modified DPCM codec with multi-rate processing. Simulation re-

sults show that the proposed coder provides great improvement

over conventional DPCM coders and is able to outperform JPEG

coders at low bit rates. In addition to the rate reduction by re-

ducing the number of coded sample through down-sampling, we

adopted the adaptive entropy coding scheme which provides fur-

ther improvements of the source coding performance. The key

feature of the image coder is its relatively low complexity. The

low-pass decimation and interpolation filters can be implemented

using very simple IIR filters. Very few down-sampling rates are

needed to maintain the coder gain close to the ones provided by

the optimal operating points. The inclusion of the Wiener filter

further ensures less degradation when the coder is operating away

from the ideal operation points. The adaptive entropy coding is a

small and simple addition to the original codec but gives substan-

tial rate gain. The necessary side information to be transmit to the

receiver is little and can be efficiently coded. Our image coder is

shown to be a good alternative for coding images with monotoni-

cally decreasing spectra at low bit rates.

6. REFERENCES

[1] A.N. Kim and T.A. Ramstad, “Improving the rate distortion

performance of dpcm,” in Proceedings of Int. Symposium on

Signal Proc. and its Applications (ISSPA), 2003.

[2] O. Guleryuz and M. T. Orchard, “On the dpcm compression

of gaussian autoregressive sequences,” IEEE Transactions on

Information Theory, vol. 47, no. 3, pp. 945–956, 2001.

[3] A. K. Jain, Fundamentals of Digital Image Processing, Infor-

mation and System Science. Prentice-Hall, Englewood Cliffs,

New Jersy, U.S.A., 1st edition, 1989.

[4] N.S. Jayant and P. Noll, Digital Coding of Waveforms - Prin-

ciples and Applications to Speech and Video, Signal Process-

ing. Prentice-Hall, Englewood Cliffs, New Jersy, U.S.A., 1st

edition, 1984.

[5] J.M. Lervik and T.A. Ramstad, “Optimality of multiple en-

tropy coder systems for nonstationary sources modelled by a

mixture distribution,” in Proceedings of Int. Conf. on Acous-

tics, Speech and Signal Proc. (ICASSP), 1996.

[6] J.M. Lervik, Subband Image Communication Over Digital

Transparent And Analog Waveform Channels, PhD Disserta-

tion, Department of Telecommunications, NTNU, Trondheim,

Norway, 1996.

[7] N. Viswanadham R.N. Madan and R.L. Kashyap, Systems and

Signal Processing, Oxford and IBH, New Delhi, India, 1991.

[8] D.S. Taubman and M.W. Marcellin, JPEG2000: Image Com-

pression Fundamentals, Standards and Practice, Kluwar Aca-

demic Publishers, 2002.

Fig. 5. Subjective quality comparison of JPEG2000 and the pro-

posed codec. Top: Original. Bottom left: Proposed codec. Dec-

imation rate 2x2, Bit rate 0.42bpp; Bottom Right: JPEG2000 Bit

rate 0.42bpp

Fig. 6. Comparison of JPEG2000 and the proposed codec. Left:

Proposed codec. Decimation rate 2x2, Bit rate 0.27bpp; Right:

JPEG2000 Bit rate 0.27bpp