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Investigation of an oversampled filter bank near perfect reconstruction of input image.

Introduction

Multirate filter banks (FB's) are widely used in signal processing for different purposes, including speech processing and image compression, digital audio, statistical signal processing etc. A multirate FB system is composed of a series of filters and downsamplers at the analysis stage and a series of upsamplers and filters at the synthesis stage. Most of the results cited consider called critically decimated FBs, meaning that the decimated factor M is equal to the number of filters K. This has the advantage of giving rise to the same number of samples in the subband domain, as the input signal.[1]. Multirate techniques is used to split the fullband problem into smaller subband problems. If the well researched critically sampled perfect reconstruction filter banks are used to decompose the input signals, the subband signals are contaminated by aliasing which requires adaptive filters in between adjacent bands to compensate for this distortion [2]. The term oversampled refers to filter banks where the sampling factor is less than the number of channels; that is, 1 [less than or equal to] M< K; the term critically sampled refers to the case K = M. The interest in oversampling filter banks is due to some improvements over critically decimated filter banks, such an addition design flexibility, improved frequency selectivity. Experimentation is carried out on filter banks having non integer oversampling ratio M/K, and non uniform bandwidth filters.

Many of the algorithms for processing and analysis of critical sampled filter bank were developed. Our main goal is to develop oversample filter bank with more no. of channels with dyadic sampling factor (oversample factor 2) which gives the output near input image. Harteneck, Stephan Weiss, and Robert W. Stewart [3,4] have developed oversampled filter bank for one dimensional signal. There is open problem to design oversample filter bank without inband aliasing and without amplitude distortion for two dimensional image. Taking a step towards design of oversampled filter bank (OSFB), initially uniform and dyadic filter banks were designed using real valued FIR filters. Here we found that we were not able to achieve perfect reconstruction output because of spectral gap between transition band. Similarly design of three channel and eight channel oversampled filter banks were developed for 2D images [6]. We have developed filter banks using direct form -II transposed structure real valued filters for even order and odd length . We have developed three, seven and ten channel oversampled filter banks with oversample factor 2 and we have achieved output near input image for PSNR(peak signal to noise ratio) approximately 37 for gif images and 33 for tif images . For gif image PSNR for filter bank output near input image is from 30 to 40dB and for tif images PSNR from 28 to 37dB [5,8]. Therefore we are trying to develop FB with histogram matches and also maintain PSNR in above ranges.

In this paper a fast converging and efficient design algorithm for ten channels OSFB's with near perfect reconstruction property which suppresses aliasing in the subbands . In this paper number of channels 14 are selected on trial basis, one can select more number of filters in filter bank. But frequency bands and oversampled factors per channels are most important factors for achiving output near input image. Section 2 will introduce real valued, multirate filter bank. Section 3 highlights implementation and result discussion.

M-Channel Multirate Filter Bank

The generalized M-channel multirate filter bank has the form shown in Fig. 1.

[FIGURE 1 OMITTED]

The output of this system can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

and W represents the phase term, [e.sup.-j2[??]/M] [7]. The alias terms are readily identified as X(z. [W.sup.p]) since they represent identical, but frequency shifted versions of X(z). [G.sub.p](z), then, is the associated gain factor for a given alias term. Thus, the filter bank is free from aliasing only if

[G.sub.p] (z) = 0 for 1 [less than or equal to] p [less than or equal to] (3)

Once again, this alias free system can be represented as a single transfer function of slightly different form than that of the two-channel QMF case.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

The criteria for amplitude and phase distortion are the same. If [G.sub.0](z) is not allpass, the filter bank suffers from amplitude distortion. If [G.sub.0](z) does not have linear phase, the filter bank suffers from phase distortion. Again, when the filter bank is free from aliasing, amplitude distortion, and phase distortion, it is called a PR filter bank. Mchannel banks lend themselves to matrix representations which are useful in the design of specific filters. Fig. 2a and b illustrate the analysis and synthesis filter matrix equations in polyphase form . Fig. 2c illustrates the new M-channel filter bank after simplifying with the noble identities. Notice the two new matrix equations this representation yields.

A(z) = E([z.sup.M]-d(z) (5)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

where A and S represent the analysis and synthesis matrices, respectively, and d represents the delay chain vector. The bulk of the material covered in this paper will concentrate on the specific case where M=2. Fig. 3 illustrates the associated polyphase matrices for this case.

[FIGURE 2 OMITTED]

It can be shown that the system illustrated in Fig. 2c exhibits perfect reconstruction

if

K(z) E(z) = [cz.sup.[??] I (7)

or, more generally, if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

[FIGURE 3 OMITTED]

for some integer, r, between 0 and M-1, some integer, m, and some non-zero constant, c. If one of these conditions holds, y(n) = c x(n-[n.sub.0]) where [n.sub.0] = M m + r + M - 1 regardless of whether the system is FIR or IIR. Subsequently, if Equation 8 is

satisfied, det R(z) det E(z) = [cz.sup.k] (9)

for some constant, c, and some integer, k. If the analysis and synthesis filters are FIR, then their coefficient matrices and determinants are FIR. Thus, every FIR PR system must then satisfy

det E(z) = [az.sup.k0] and det R (z) = [bz.sup.-k1] (10)

for constant, a, b, k0, and k1. By using above design steps filter bank for 14 channels are selected and applied for 2D images(i.e. tif and gif images). The analysis and synthesis filters are designed by using above technique. Filters of filter bank implemented in one dimensional form and then transformed into 2D form for 2D image application. Downsampling and upsampling of image is carried out by using bilinear interpolation technique. Clearly, the concept of multirate filtering relies on the two processes that effectively alter the sampling rate, decimation and expansion. Decimation or downsampling by a factor of M essentially means retaining every Mth sample of a given sequence. Decimation by a factor of M can be mathematically defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

or equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

Expansion or upsampling by a factor of M essentially means inserting M-1 zeros between each sample of a given sequence. Expansion by a factor of M can be mathematically defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

or equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

Downsample algorithm reduces the size of image by downsample factor and upsample algorithm restores original size of image.

Oversampled FIR Filter Bank

The design of oversampled filter bank implemented by using real valued FIR filters with odd length, even order and proper frequency bands which significantly reduced inband aliasing and amplitude distortion . Design of filter bank with proper oversample factor, which gives the output near input image. After selection of proper oversample factor[i.e.2 after testing through software], filter bank designed with different filter orders . DCT image compression technique applied for analysis of output image in terms of peak signal to noise ratio [PSNR] and histogram of compressed input and output image . Proposed filter bank consists of fourteen channels, without overlapping of frequency bands. Oversampled factor M selected on trial and algorithm tested for proper oversampled factor and proper frequency bands which gives output near input image. Proper frequency bands means frequency bands and design parameter i.e. order of filter selected likewise, which pass the signal without overlapping and also without loss of signal at transition gap.

Fig.4 shows fourteen channel oversampled filter bank. This filter bank applied for 2D image. MATLAB 7.1 software with signal processing and image processing toolboxes applied for implementation.

[FIGURE 4 OMITTED]
Channel selection for 14-channel FIR FB are,

CH1 = 0.001 to 0.12 X pi rad/sample
CH2 = 0.12 to 0.14 X pi rad/sample
CH3 = 0.14 to 0.26 X pi rad/sample
CH4 = 0.26 to 0.28 X pi rad/sample
CH5 = 0.28 to 0.4 X pi rad/sample
CH6 = 0.4 to 0.42X pi rad/sample
CH7 = 0.42 to 0.54X pi rad/sample
CH8 = 0.54 to 0.56X pi rad/sample
CH9= 0.56 to 0.68 X pi rad/sample
CH10 = 0.68 to 0.7 X pi rad/sample
CH11 = 0.7 to 0.82 X pi rad/sample
CH12 = 0.82 to 0.84X pi rad/sample
CH13 = 0.84 to 0.96 X pi rad/sample
CH14 = 0.96 to 0.9999 X pi rad/sample


If uniform frequency bands used, means all the passband ranges of equal size of all channels of filter bank, distortion occurs at the output image. Therefore above frequency bands are selected for fourteen channels. One can select more or less number of channels, but frequency bands selection accordingly to number of channels. Filter bank designed using direct form-II, type-I, real valued FIR filters. Design of filters per channels in one dimensional form, then it is transformed into two dimensional (2D) form for 2D image processing. 2D Lena (Gif format), Barba (Tif format) and Cameraman (Tif format) images are applied for proposed fourteen channel oversampled FB analysis. As per the above analysis for 14-channel F

Fig.5 shows normalized frequency response of 14-channels for filter order 46. Critically sampled filter bank means number of channels equal to oversample factor i.e.14. Resultant of this FB is as shown in Fig.6b.Fig.6a shows input Lena image. Similarly this FB designed with oversampled factor 12, 8, 4 and 2. Downsampler and upsampler algorithm developed by linear interpolation technique for images . The resultant of oversampled FB's are shown in Fig. 6c, 6d, 6e and 6f respectively for filter order 46 where perfect reconstruction occurs at oversampled factor 2.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Similarly FB with oversample factor 2 applied for Barba (tif) image. Fig.7.a shows input Barba image and Fig.7.b shows output Barba image for 14-channel FB with 46 filter orders for all filters.

Transform coding is simply the compression of the images in the frequency domain. Transform coefficients are used to maximize compression. For lossless compression, the coefficients must not allow for the loss of any information. The DCT is fast. It can be quickly calculated and is best for images with smooth edges like photos with human subjects. The DCT coefficients are all real numbers unlike the Fourier Transform. The Inverse Discrete Cosine Transform (IDCT) can be used to retrieve the image from its transform representation. Therefore for analysis of FB for oversample factor 2, with filter orders 46 for all filters, output image of FB near input image and histogram of input and output images matches.

Fig..8 a shows compressed Input and output image of 14-channel FB with Histogram for 46 filter orders of all filters for Barba image . Fig..8.b shows Input and output Lena image for 46 filter orders of all filters.

Here results of compressed image of each oversampled filter bank compared with input image. It is most easily defined via the mean squared error (MSE) which for two mxn monochrome images I and K where one of the images is considered a noisy approximation of the other is defined as:

MSE = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Comparison of all the results of compressed and segmented output of filter banks with input image in terms of MSE is as shown in table1. After image compression all the results are applied for peak signal to noise ratio [PSNR] estimation. The PSNR is most commonly used as a measure of quality of reconstruction in image compression. It is most easily defined via the mean squared error (MSE) The PSNR is defined as:

PSNR = 10 * [log.sub.10] ([MAX.sup.2.sub.I])/MSE) = 20 * [log.sub.10] ([MAX.sub.I]/[[square root (MSE)]]

Here, MAXI is the maximum pixel value of the image. Comparison of all the results of filter banks in terms of PSNRs for different filter orders with oversample factor 2 is as shown in table1. PSNR and MSE caluculated with respect to input compressed image.

[FIGURE 7a OMITTED]

[FIGURE 7b OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Comparison of results of oversample filter bank with oversample factor 2 for different filter orders, in terms of PSNR are as shown in table1 and in fig. 9. PSNR caluculated with respect to input compressed image. In this proposed 14 channel oversampled FB with oversample factor 2, we achieved output near input image for filter order 46 and PSNR approximately 36 dB for Lena image . Similarly this system is applied for Barba (tif) and cameraman (tif) images and perfect reconstruction is achieved at PSNR approximately 32 dB for filter order 46 which is shown in fig. 9 with ellipse indication. Histogram which gives information in terms of intensity values of input and output images. Fig. 8 shows that for filter order 46 histogram of output images matches with histogram of input images. This algorithim is tested and applied for all filter orders which are shown in table 1. As per as filter order increases above 46, transistion gap becomes smaller and some frequency components are not passed at the output and as per as filter order decreases overlapping of frequency bands at transistion gap occurs i.e. aliasing. Therefore histogram reduces for more filter orders and splits for lower filter orders. For specific filter order and for proper selection of frequency bands, we achieved output near input image. Selection of frequency bands as per the number of channels and filter orders with proper subsampled factor are important factors to get output near input image.

Conclusion

In this paper proposed oversampled real valued, even order filters, filter banks have been designed. To investigate oversampled FIR filter bank output, which gives resultant near input image, different oversampled factors are applied. For oversampled factors 12, 8 and 4 we achieved distorted output. For oversample factor 2, we achieve the resultant of oversampled FB near input image. For design of this FB's, design of filters with proper passband and proper filter order means there should not be aliasing and signal loss at transition gap . To analyze output of FB, DCT image compression technique is used. one can apply this FB for any type of images.

References

[1] Daubechies, "Orthonormal bases of compactly supported wavelets," Communications on Pure and Applied Mathematics, vol. 41, pp. 909-996, 1988.

[2] M.Harteneck, S. Weiss, and R.W. Stewart. "Design of Near Perfect Reconstruction Oversampled Filter Banks for Subband Adaptive Filters", IEEE Transactions on Circuits and Systems | II: Analog and Digital Signal Processing, 1999.

[3] Harteneck M, Paez-Borallo JM, and Stewart RW "A Filterbank Design for Oversampled Filter Banks without Aliasing in the Subbands." Proc. TFTS'97, Warwick, UK, 1997 .

[4] M. Harteneck and R.W. Stewart, "An Oversampled Filter Bank with Different Analysis and Synthesis Filters for the Use with Adaptive Filters", Proc. Asilomar Conf. Signals, Systems and Computers, vol. 2, pp. 1274-1278, Nov.1997.

[5] Nikolay Polyak ans William A. Pearlman "A New Flexible Bi-Orthogonal Filter Design for Multiresolution filterbanks with application to image compression", IEEE Tran. Signal processing vol.48 no.8 Aug.2000.

[6] Mrs.S.R.Chougule and Dr. R.S. Patil," Design of Oversampled Filter Banks for 2-D Image" Published in ICSCN International Conference Proc.Anna University Madras 6th Jan 08.

[7] P.P.Vaidyanathan "Multirate systems and filter banks"(.Pearson Education 1993).

[8] Nikolay Polyak ans William A. Pearlman,Fello, "A New Flexible BiOrthogonal Filter Design for Multiresolution filterbanks with application to image compression" IEEE Tran. Signal processing vol.48 no.8 Aug.2000.

S.R. Chougule

Asst. Professor, BharatiVidyapeethC.O.E.Kolhapur

E-mail: sr.chougule@rediffmail.com

R. S. Patil

Principal, J.J. Magdum C.O.E. Jaysingpur

E-mail: prekha46@yahoo.com
Table 1

Sr.   FILTER       PSNR[dB]           PSNR[dB]
No.   ORDER    Lena (gif image)   Barba (tif image)

1.      20          50.63              47.59
2.      30          44.88              39.98
3.      40          39.54              34.7
4.      46          36.70              32.54
5.      60          27.98              29.32
6.      80          26.49              28.54

Sr.   PSNR[dB] Cameraman
No.       (tif image)

1.           41.97
2.           38.99
3.           34.48
4.           32.78
5.           28.46
6.           27.39
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Author:Chougule, S.R.; Patil, R.S.
Publication:International Journal of Applied Engineering Research
Article Type:Report
Date:Dec 1, 2008
Words:2871
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