A Novel 9/7 Wavelet Filter banks For Texture Image Coding

This paper proposes a novel 9/7 wavelet filter bank for texture image coding applications based on lifting a 5/3 filter to a 7/5 filter, and then to a 9/7 filter. Moreover, a one-dimensional optimization problem for the above 9/7 filter family is carried out according to the perfect reconstruction (PR) condition of wavelet transforms and wavelet properties. Finally, the optimal control parameter of the 9/7 filter family for image coding applications is determined by statistical analysis of compressibility tests applied on all the images in the Brodatz standard texture image database. Thus, a new 9/7 filter with only rational coefficients is determined. Compared to the design method of Cohen, Daubechies, and Feauveau, the design approach proposed in this paper is simpler and easier to implement. The experimental results show that the overall coding performances of the new 9/7 filter are superior to those of the CDF 9/7 filter banks in the JPEG2000 standard, with a maximum increase of 0.185315 dB at compression ratio 32:1. Therefore, this new 9/7 filter bank can be applied in image coding for texture images as the transform coding kernel.


INTRODUCTION
Wavelets can effectively be used in several domains including image segmentation, image enhancement, feature extraction, image retrieval, and image coding [1][2][3].Although wavelets play an important role in the field of image coding, designing a wavelet kernel for a specific type of image coding, for instance texture images, is still problematic.The CDF 9/7 filter banks of the biorthogonal 9/7 wavelet proposed by Cohen, Daubechies, and Feauveau, is adopted by the JPEG2000 standard as a core algorithm.Although CDF 9/7 has had great impact and has a wide range of applications, its design method is too complicated and its VLSI hardware implementation is too complex.Therefore, this paper proposes new 9/7 filter banks based on Sweldens' lifting scheme [4][5][6].Starting from the relatively simple 5/3 filter, this paper presents lifting to a 7/5 filter and subsequently to a 9/7 filter.Then, a one-dimensional parametric 9/7 filter family is derived, as well as providing the dynamic range of control parameters according to Daubechies regularity criterion.Finally, the 9/7 filter family designed in this paper is applied in an image coding application to Brodatz standard texture image database, where a new 9/7 filter bank with the optimal control parameter is determined by maximizing the PSNR (Peak Signal to Noise Ratio).
For the optimal design of biorthogonal wavelets, Cheng, constructed, based on the lifting algorithm, the compact support of biorthogonal wavelet filters and proposed a parametric expression for 9/7 wavelets [7].In the meantime, Yang designed 9/7 and 7/5 wavelets on the basis of the lifting algorithm [8][9].The lifting algorithm, presented in [7] and [9] adopts the Euclidean algorithm without providing the lifting operator that can directly improve to a 9/7 wavelet.Phoong and Vaidyanathan proposed the biorthogonal wavelet design method [10].Antonini and Daubechies designed a wavelet base function for image compression through utilizing the visual features both in the space and frequency domain [11].Wei and Burrus designed a novel compact support biorthogonal Coifman wavelet in the time domain [12].The filter design methods mentioned in the above literature are based on traditional Fourier transform and do not use the lifting algorithm.To design a biorthogonal wavelet filter with vanishing moments of arbitrary multiplicity Liu proposed a method that solves trigonometric polynomial equations with two variables on the basis of Diophantine equations [13].On the basis of filter optimization and median operation, Quan and Ho proposed an efficient lifting scheme to construct biorthogonal wavelet [14] with better compression www.ijacsa.thesai.orgperformance than the JPEG2000 standard CDF 9/7 wavelet; however, no optimal filter is provided in the literature for certain type of images.At present, there is relative little literature on texture image coding, yet the study of texture image coding is an important branch of image research, where Brodatz standard texture image database is one of the representative research subjects.The wavelet filter designed in this paper achieves better application performance in the coding of texture images.This paper is organized as follows: Section 2 provides the basic theories of lifting scheme.Section 3 proposes a 9/7 wavelet filter design approach based on lifting scheme and the Euclidean algorithm and results in a one-dimensional parametric 9/7 wavelet filter.The range of the control variable of the one-dimensional parametric 9/7 wavelet filter designed in section 3 is determined in section 4. In section 5 this 9/7 wavelet filter is used for the coding of texture images in the Bordatz standard texture image database, an optimal parameter based on the PSNR criterion is determined, and, finally, experimental results for the optimized 9/7 wavelet filter bank are presented and analyzed.Section 5 states the conclusion of this paper and provides implications for future research.

II WAVELET LIFTING SCHEME
This paper focuses on biorthogonal wavelet filters.Let { ( ), ( ), ( ), ( )} h z g z h z g z be a compactly supported filter bank for such a wavelet.For filters () hz and () gz, their polyphase representations are: where new P and new P are the polyphase matrix and the dual polyphase matrix after lifting, respectively.

III 9/7 WAVELET FILTER DESIGN BASED ON THE LIFTING SCHEME
A. Wavelet Lifting from 5/3 to 7/5 filter If the filters of a 5/3 wavelet are given by: their polyphase representations are, as follows: Applying the Euclidean algorithm to may give in two steps the following quotients i q and remainders ( 1, 2) Given these quotients the polyphase matrix () Pz can be factorized: where K is a constant scale factor.
Let lifting operator () sz be: where  is a free parameter.The polyphose matrix () Pz for the 7/5 filter can now be given by: After lifting we obtain the new 7/5 filter coefficients: ) Wavelet Lifting from 7/5 to 9/7 filter If the filter of the 7/5 wavelet filter is

() h z h z h z h z h h z h z h z
, their polyphase representations are given by:  ( ) 0 , ( ) ( ) , ( ) 0 The corresponding polyphase matrix factorization is:  , according to (10) we get the following matrix for () ) / ( 3) ) The filters () hz and () gz now follow from the polyphase form: www.ijacsa.thesai.org ) ) With the lifting operator with free parameter  given by the following equation the new polyphase matrix is obtained as follows: According to the perfect reconstruction condition of 9/7 wavelet transform, wavelet properties, and normalizing condition, the above coefficients can be expressed in the form of a one-dimensional function: Based on equation ( 14), the range of t can be determined as [0.78,1.85]t  . If t is a known number, the filter coefficients can be easily be determined using equation ( 14 14) is determined by adopting Daubechies' theorem.We have the following equations: where i ze   , () F  and () Q  are both trigonometric polynomial related with control variable.Concurrently, we get the following equations: the filter coefficients can be obtained according to equation ( 14), thus the new 9/7 wavelet filter can also be determined.

V EXPERIMENT AND ANALYSIS
Taking the above one-dimensional parameterized 9/7 wavelet filter family as the coding kernel, adopting EBCOT (Embedded Block Coding with Optimized Truncation) coding [15], and applying this system to the image coding of Brodatz standard texture image database, it is found, by analysis of the experiment statistics in Fig. 1 and Fig. 2, that the control variable at the optimal PSNR is 1.2050 t  . Therefore, our new 9/7 wavelet filter banks for texture image coding applications is known as given in Table1.
The 111 images in Brodatz standard texture image database are all tested based on the above image coding system.The results show that comparing the coding system of JPEG2000 with the coding system proposed in this paper, when compression rate is 4:1, the average PSNR value of the 111 texture image in Brodatz is only 0.0077dB lower than that of JPEG2000, with the PSNR of 49 images higher than that of JPEG200 at the average height of 0.0373dB, while the PSNR of 62 images are lower than that of JPEG2000 at the average amount of 0.0433dB.
When the compression rate are 8:1, 16:1, 32:1, 64:1, and 128:1, respectively, the results show in Table 2.The objective comparison of experimental statistic of maximum PSNR under different compression ratio for test images in the Brodatz standard texture image database are shown in Table 3.
The subjective comparison of compression performances between our new 9/7 filter and CDF 9/7 for test images D29, D43, D84 and D103 in the Brodatz standard texture image database under compression ratio 32:1 are shown in Figure 3, Figure 4 and Figure 5.
The images in Fig. 4 are the reconstructed images of the above four images through the compression experiments by JPEG2000 standard CDF 9/7 filter at the compression ratio of 32:1.
The images in Fig. 5 are the reconstructed images of same four images from the Brodatz standard texture image database, but through the compression experiments by the new 9/7 filter designed in this paper at the compression ratio of 32:1.www.ijacsa.thesai.orgFrom comparison of Fig. 4 and Fig. 5, the resulting subjective visual quality of reconstructed images using the new 9/7 filter is concluded to be as good as the quality resulting from using the CDF 9/7 filter.

VI. CONCLUSION
Compared with CDF 9/7 wavelet filter, the new 9/7 wavelet filter designed in this paper is much easier to be constructed and more favorable in hardware implementation.The results show that under high compression ratio (low bit rate), the overall coding performance of the new 9/7 wavelet filter is better than that of the JPEG2000 CDF9/7 wavelet filter, therefore, the 9/7 wavelet filter designed in this paper is very effective in image coding for texture image.

For
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Figure 1 .
Figure 1.The PSNR value of test image D29 at the control variable's varying range of 1.000000-1.800000under each compression ratio in the Brodatz standard texture image database

TABLE I
The PSNR value of image D110 at the control variable's varying range of 1.000000-1.800000under each compression ratio in the Brodatz standard texture image database

TABLE II .
COMPARISON OF THE COMPRESSION PERFORMANCES BETWEEN NEW 9/7 AND CDF 9/7 FILTER (PSNR/DB)