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福彩号码查询 www.edasl.tw DESIGN OF LINEAR PHASE COSINE MODULATED FILTER BANKS FOR SUBBAND IMAGE COMPRESSION
J. E. Odegard, R. A. Gopinath, C. S. Burrus Department of Electrical and Computer Engineering, Rice University, Houston, TX-77251

ABSTRACT
This paper proposes to investigate a new class of M -band linear phase modulated lter banks. In particular we propose to investigate di erent methods for designing the prototype lter based on perceptual measures such as reduced image artifacts as well as more traditional measures such as coding gain and mean square di erence. Image compression at low bit rates typically introduces image artifacts annoying to the human observer. One can classify the various types of compression artifacts in to three categories: blurring, blocking and ringing. Each of these artifacts will have a di erent visual impression on the observer and typically blurring will be the most acceptable distortion; blocking and ringing artifacts tends to cause more visual interference to the observer. Blocking is typically found in transform coding based algorithms (e.g., the JPEG standard which is based on the DCT) while ringing is typically found in subband image coders and is due to the coarse quantization of the high frequency bins.

De ne

1.1. Modulated Filter Banks 1 p2 i 2 f0; M g ki =
1 =
8 M > 2 > M ?1 < J = > M2 2 ? > M2 : 2

and

otherwise, M ? 1 Type 1 FB M ? 2 Type 2 FB

(1) (2)

Type 1 FB, M even Type 1 FB, M odd (3) Type 1 FB, M even Type 1 FB, M odd Given the above de nitions we can now de ne the new MFBs. For even (i.e., Type 2 and M is even or Type 1 M is odd) a DCT/DST I, 2M channel FB is obtained from the prototype lters h(n), and g(n) as 5]:
hi (n) = ki h(n) cos M i(n ? ) ; i = 0; : : : ; M

2

(4a) (4b) (4c)

hM +i (n) = h(n ? M ) sin

M i(n ? 2 ) ; i = 1; : : : ; M ? 1

1. M-BAND AND 2M-BAND DCT/DST MODULATED FILTER BANKS
The M channel maximally decimated modulated lter bank (MFB) has undergone an extensive study over the last few years 4]. Until recently only a class of MFBs referred to as a Class B MFBs 4, 2] were known. In these MFBs it can be shown that linear phase is impossible to achieve. Notice however that linear phase paraunitary lter banks have been designed 6]. This paper proposes to use the new Class A MFBs for even M rst constructed in 1]. The design of a MFB typically requires the design of one or two prototype lters which result in a lter bank where each lter does not have linear phase. The recent discovery of this new class (Class A) modulated lter banks was rst proposed by Lin and Vaidyanathan 5]. These results were further extended to cover a broader class of MFB's 1]. The novelty of this new class of MFB's was that both analysis and synthesis lters would have linear phase. Since linear phase is well know to be a desirable property for image coding applications with coarsely quantized subbands it is expected that this new class of lter banks will have good subband coding properties.
Submitted to ICIP, Austin, Texas, USA - '94

(4d) For odd (i.e., Type 1 and M is even or Type 2 M odd) a DCT/DST II, 2M channel FB is obtained from the prototype lter h(n) as 1]
hi (n) = ki h(n) cos M i(n ? ) ; i = 0; : : : ; M ? 1 (5a)

gi (n) = ki g(n) cos M i(n + 2 ) ; i = 0; : : : ; M gM +i (n) = ?g(n + M ) sin M i(n + 2 ) ; i = 1; : : : ; M ? 1

2

(5b) gi (n) = ki g(n) cos M i(n + 2 ) ; i = 0; : : : ; M ? 1 (5c) gM +i (n) = ?ki g(n + M ) sin M i(n + 2 ) ; i = 1; : : : ; M (5d) Finally we also have the DCT/DST III/IV based M channel FBs 2, 4]
hi (n) = h(n) cos M

hM +i (n) = ki h(n ? M ) sin M i(n ? 2 ) ; i = 1; : : : ; M

1 (i + 2 )(n ? 2 ) ; i = 0; : : : ; M ? 1: (6a)

1

(i + 1 )(n + 2 ) ; i = 0; : : : ; M ? 1: 2 (6b) A DCT/DST I/II 2M lter banks will be referred to as a class 'A' modulated lter and a DCT/DST III/IV M as a class 'B' modulated lter bank. Most results in the literature pertains to Type 1, Class B modulated FB's. Notice furthermore that as with the Class B MFBs, g(n) can be obtained form h(n) by a shift or designed individually. It can be shown that if the prototype lter h(n) has linear phase then for a class A modulated lter bank it follows that the lters are linear phase. This is not true for the class B modulated lter banks and in fact it has been proved that class B modulated lter banks can not have linear phase 1].
gi (n) = g(n) cos M

The authors wishe to thank Prof. Vaidyanathan for discussions regarding this new class of linear phase modulated lter banks. Supported for this research is provided by ARPA, BNR and TI 1] R. A. Gopinath. Modulated lter banks - a general theory. Technical Report CML TR94-01, Computational Mathematics Laboratory, Rice University, 1994. 2] R. A. Gopinath and C. S. Burrus. Theory of modulated lter banks and modulated wavelet tight frames. In Proc. Int. Conf. Acoust., Speech, Signal Processing, Minneapolis, MN, 1993. IEEE. 3] N. Jayant, J. Johnston, and R. Safranek. Signal compression based on models of human perception. Proc. IEEE, 81(10):1385{1422, October 1993. 4] R. D. Koilpillai and P. P. Vaidyanathan. Cosinemodulated FIR lter banks satisfying perfect reconstruction. IEEE Trans. SP, 40(4):770{783, April 1992. 5] Y-P. Lin and P. P. Vaidyanathan. Linear phase cosine modulated maximally decimated lter banks with perfect reconstruction. Technical report, California Institute of Technology, 1993. 6] A. K. Soman, P. P. Vaidyanathan, and T. Q. Nguyen. Linear phase paraunitary lter banks: Theory, factorizations and designs. IEEE Trans. SP, 41(12):3480{ 3496, December 1993.

Acknowledgments

REFERENCES

Over last few years there has been a considerable e ort in improving the image quality for low bit rate (less that 0.25 bpp) still image compression. One of the areas of interest is to design subband coders which are optimal with respect to minimizing visual distortion. Low bit rate image coders tends to exhibit considerable visual distortion. It is believed that to obtain high quality image coders at low bit rates the coders needs to incorporate perceptual coding criteria. For an excellent review of signal compression based on models of human perception see 3]. To achieve low bit rates with a subband coder one traditionally has to use long lters. The down side to this is that long lter results in a considerable ringing artifact around image discontinuities. To compensate for this there are currently two direct approaches: (i) use shorter lter lengths which results in less ringing but introduces more blurring, (ii) use a space varying lter bank which use long lters for smooth image regions and short lter for image discontinuities. In this investigation we will be exploring both these methods in parallel. In Fig. 1 we show a region of the Lenna image which has been subband ltered with a Type 1, Class B MFB, quantized (coarsely) with a uniform quantizer and reconstructed. Notice the non-desirable texture like noise that has been introduced. In Fig. 2 we show the same region of the Lenna but now with the new Type 1, Class A MFB, quantized with the same uniform quantizer and reconstructed. Notice the smoothness in this image compared to the previous image. In this research we will investigate the advantages of using this new class of M -band linear phase DCT/DST Type IIV modulated lter banks. Our research will in particular consider the the e ect of varying M , investigate di erent methods for designing \optimal" (in terms of coding gain, reduced subband coding artifacts, perceptual measures etc.) prototype lters. We will also investigate the trade o involved in choosing lter lengths. In particular we will investigate the possible advantages or disadvantages of using space varying lter banks as opposed to a xed lter bank with short lter lengths to deal with image discontinuities. 2

2. DESIGN OF OPTIMAL PROTOTYPE FILTERS FOR IMAGE COMPRESSION

3. ONGOING AND FUTURE RESEARCH

20 40 60 80 100 120 140 160 180 200 50 100 150 200

Figure 1. Subband ltered, quantized and reconstructed Lenna image using a Type 1 Class B MFB (M=4, lter length N=2Mk=16)

20 40 60 80 100 120 140 160 180 200 50 100 150 200

Figure 2. Subband ltered, quantized and reconstructed Lenna image using a Type 1, Class A MFB (M=8, lter length N=2Mk=16)

3


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