(Sunil S.S,Mar Baselious College of Engg;Peermade,Kerala)


With the rapid growth of internet, multimedia amounts now to transfer of large amount of images. Data compression is urgent due to the limitation in the channel band width. The proposed system presents a modified scheme that offers the decompression technique that used Wavelet decomposition coefficients of gray level images. The discrete wavelet transform(DWT) decomposs an image into different bands that vary in spatial frequency, uniform quantization of a single band of coefficient results in artifacts may be defined as uniform quantization noise, Wavelet coefficients are vector quantized by using a well defined codebook. Wavelet coefficients (Haar & Daubechies) are used and their performance are compared. Experiments demonstrate that Haar wavelet transform is better than Daubechies Wavelet transform,the model can be used for adaptive quantization scheme


Compressing an image is significantly different than compressing raw binary data. Of course, general purpose compression programs can be used to compress images, but the result is less than optimal. This is because images have certain statistical properties which can be exploited by encoders specifically designed for them. Also, some of the finer details in the image can be sacrificed for the sake of saving a little more bandwidth or storage space. This also means that lossy compression techniques can be used in this area. The proposed system presents a modified scheme that offers the decompression technique that used Wavelet decomposition coefficients of gray level images. Wavelet Coefficient Image Compression is a lossy Compression. In this method, Wavelet coefficients are vector quantized by using a well defined codebook



The new JPEG-2000 digital image standard and the WSQ (wavelet scalar quantization) method that the FBI uses to compress the fingerprint image. In this context, wavelets can be thought of as the building blocks of images. An image of a forest can be made from the broadest wavelets: a big swath of green for the forest, a splash of blue of the sky. More detailed shaper wavelets can help distinguish one tree from another. Branches and needles can be added to the image with even finer wavelets. Like an individual brush stroke in a painting, each wavelets is not itself an image, but many wavelets together can recreate anything. Unlike a brush stroke in a painting, a wavelet can be made arbitral small: A wavelets has no physical size limitation because it is simply a series of 0s and 1s stored in computer memory.


Wavelet transform is capable of providing the time and frequency information simultaneously. Hence it gives a time-frequency representation of the signal. Wavelets (small waves) are functions defined over a finite interval and having an average value of zero. The basic idea of the wavelet transform is to represent any arbitrary function ƒ(t) as a superposition of a set of such wavelets or basis functions. These basis functions are obtained from a single wave, by dilations or contractions (scaling) and translations (shifts). The discrete wavelet transform of a finite length signal x(n) having N components, for example, is expressed by an N x N matrix similar to the discrete cosine transform .


There are several ways wavelet transforms can decompose a signal into various sub bands. These include uniform decomposition, octave-band decomposition, and adaptive or wavelet-packet decomposition. Out of these, octave-band decomposition is the most widely used. The procedure is as follows: wavelet has two functions “wavelet “and “scaling function”. They are such that there are half the frequencies between them. They act like a low pass filter and a high pass filter. Figure 2-1 shows a typical decomposition scheme. The decomposition of the signal into different frequency bands is simply obtained by successive high pass and low pass filtering of the time domain signal. This filter pair is called the analysis filter pair. First, the low pass filter is applied for each row of data, thereby getting the low frequency components of the row. But since the low pass filter is a half band filter, the output data contains frequencies only in the first half of the original frequency range. By Shannon's Sampling Theorem, they can be sub-sampled by two, so that the output data now contains only half the original number of samples. Now, the high pass filter is applied for the same row of data, and similarly the high pass components are separated. Figure 2-1 Pyramidal Decomposition of an image This is a non-uniform band splitting method that decomposes the lower frequency part into narrower bands and the high-pass output at each level is left without any further decomposition. This procedure is done for all rows. Next, the filtering is done for each column of the intermediate data. The resulting two-dimensional array of coefficients contains four bands of data, each labeled as LL (low-low), HL (high-low), LH (low-high) and HH (high-high). The LL band can be decomposed once again in the same manner, thereby producing even more sub bands. This can be done up to any level, thereby resulting in a pyramidal decomposition as shown in figure 2-1. The LL band is decomposed thrice in figure 2-1. The compression ratios with wavelet-based compression can be up to 300-to-1, depending on the number of iterations. The LL band at the highest level is most important, and the other 'detail' bands are of lesser importance, with the degree of importance decreasing from the top of the pyramid to the bands at the bottom. This can be done to any image.


Although VQ is the best way of quantizing and compressing images. It has a major drawback in the amount of computations during the search for optimum code vector in encoding [4,5]. This complexity can be reduced by using an efficient codebook design and wavelet based tree structure. We take multiple stage discrete wavelet transform of code words and use them in both search and design processes. Accordingly, our codebook consists of a table, which includes only the wavelet coefficients.


Every row or column is arranged and assigned in the following manner: S0 D0 S1 D1 S2 D2 S3 D3… For every row or column: Repeat until end of row or column: D0=D0+D0-S0-S0 S0=S0+(2*D0/8) Di = Di+Di –Si-Si+1 Si = Si + (( Di-1+ Di )/8)


(The Haar wavelet function) Let be defined by (5) Define as (6) Define the vector space Wj (7) where denotes the linear span.


The first stage sub band decomposition is done through the two-dimensional wavelet transform on the image. In this project both Haar and Daubechies family of basis functions for DWT, which is widely used in image compression [6] are used for Image sub band decomposition. After the decomposition process, we have four different energy bands at hand: low-low (LL), low-high (LH), high-low (HL), and high-high (HH) parts. The least energy containing and the most redundant band is HH, which we have simply ignored, and actually treated as Gaussian noise [7]. The LH and HL bands also exhibit the characteristics of a high frequency signal. Due to the reasons stated above, one should use quantization for compressing these bands unless they will simply be dismissed as well.. After discarding the HH band and coding the HL and LH sidebands, we further decompose the LL band. This does not bring together much computational load, because the size of the LL band is one fourth of the original image. At this point, the LL band is replaced by the following bands: low-low-low-low (LLL), low-low-high-low (LHL), low-low-high-high (LHH), and low-low-low-high (LLH). LHH band contains significant amount of energy and can not be discarded. We apply the same quantization scheme described earlier in this section to all three high frequency bands of the second level transform, namely LLH, LHL and LHH. After deciding on the codebook size, one has to determine the VQ search algorithm. In this paper, we employed the wavelet tree structured vector quantization (WTSVQ), where during the construction of the tree structure, the vectors are combined according to the Euclidean distance between the LL bands of their wavelet transforms. Thus, the LL band of the code vector becomes the representative for that vector. In fact, WTSVQ can be interpreted as a modified version of the classified VQ scheme. This approach reduces the computational complexity as we work with m/2xn/2 vectors rather than mxn. In fact, for further compression quantized coefficients can be entropy coded. Since we are mainly concerned with VQ and its use in sub band coding, we do not consider scalar entropy coding in this paper. The above algorithm is reversely done in the time of decompression.


(a) (b) (c) (d) (a) Original Image (b) First Level Decomposed Image (c) Second Level Decomposed image (d) Decompressed Image Fig4.1 Image Transform Using Haar wavelet Transform (a) (b) (c) (d) (a) Original Image (b) First Level Decomposed Image (c) Second Level Decomposed image (d) Decompressed Image Fig4.2 Image Transform Using Dubechis wavelet Transform Where I(x, y) is the original image, I'(x, y) is the approximated version (which is actually the decompressed image) and M, N are the dimensions of the images. A lower value for MSE means lesser error, and as seen from the inverse relation between the MSE and PSNR, this translates to a high value of PSNR. Logically, a higher value of PSNR is good because it means that the ratio of Signal to Noise is higher. Here, the 'signal' is the original image, and the 'noise' is the error in reconstruction. So, if you find a compression scheme having a lower MSE (and a high PSNR), you can recognize that it is a better one. For Haar Wavelet Transform, MSE = 468.7633 PSNR = 24.6621 For Daubechies Wavelet Transform, MSE = 877.2814 PSNR = 21.5285 As the PSNR value of HAAR Wavelet Transform is higher , it is the better one than Daubechies Wavelet Transform.

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