3.1. Introduction CHAPTER 3 Transformed Vector Quantization with Orthogonal Polynomials In the previous chapter, a new integer image coding technique based on orthogonal polynomials for monochrome images was proposed. After the proposed transformation the coefficients are scalar quantized and entropy coded in order to obtain a higher compression ratio. This technique proved to be better in the sense that it gives higher PSNR value. However, the quality of the reconstructed image degrades when the quality factor increases due to the scalar quantization step effect. To overcome this problem, in this chapter, a new vector quantization technique has been proposed in the transformed domain. The rationale behind the introduction of vector quantization is that the vector quantization of signal reduces the coding bit rate significantly with good quality of reconstruction picture. The proposed transformed vector quantization exploits the combined features of energy preservation by the proposed transform coding and high compression ratio of the vector quantization. 3. 1. 1 Vector quantization In the current scenario, Vector Quantization [Alle9] has been found to be an efficient data compression technique for speech and image as it provides many attractive features for image compression. A vector quantizer Q of dimension k and size N is mapped from a point in k-dimensional Euclidean space R k, into a finite set C containing N output or reproduction points that exist in the same Euclidean space as the original point. These reproduction points are known as codewords and these set of codewords are called codebook C with N distinct codewords in the set. Thus, the mapping function Q is defined as, Q: R k C... (3.1)
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 36 The rate of the vector quantizer or the number of bits (r) used to express each quantized vector is given by the relation, r = log N / k (3.) This rate equation is very useful as it gives the amount of compression that can be expected in a particular VQ coding scheme. Vector quantization can be considered as a pattern recognizer where an input pattern is approximated by a predetermined set of standard patterns [Robe0]. Experiments have shown that vector quantization produces superior performance over scalar quantization even when the components of the input vectors are statistically independent. Vector quantization exploits the linear and nonlinear dependence among vectors. It also provides flexibility in choosing multi-dimensional quantizer cell shapes and in choosing a desired codebook size. If scalar quantization is extended to k dimensional vectors using N levels, then the codebook would contain N x k codewords. In the case of vector quantization there could be arbitrary partitions with integer number of codewords N. Another advantage of vector quantization over scalar quantization is the fractional value of resolution that is achievable. This is very important for low bit rate applications where low resolution is sufficient. The number of codewords in the codebook decides the quality of the reconstructed vectors. If the number of code words is large, the output vectors would be close to the input vectors. The dimension (i.e the number of elements present in each vector) of the input vectors and code words also play a crucial role in quality of reconstruction. Ideally, the compression performance improves as the vector dimension increases but the tradeoff is the increased coding complexity. Besides dimension, the difficult task in any VQ scheme is the generation of codewords that best represent the input vectors [Lind80, Gray84]. The performance of the quantizer is assessed using a suitable statistical distortion
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 37 measure. The generic statistical distortion measure as applied to vectors is represented as: D d x Q( x )) (3.3) i ( i, i where x i is the input vector and Q(x i ) is the approximation of x i and d ( xi, Q( xi i )) represents the squared Euclidean Distance between the input vector x and its approximation Q x ). ( i 3. Literature survey Vector quantization is a very powerful method for lossy compression of data such as images and speech. The lossy compression scheme can be analyzed using rate distortion theory [Alle9]. In this scheme the decompressed data will not be a replica of the original. Instead, it will be distorted by an amount D. According to Shannon s rate distortion theory [Jude76], vector quantization of signals reduce the coding bit rate significantly when compared to scalar quantization. Vector Quantization takes M number of multi dimensional vectors and reduces their representation to k number of code words, where k < M. The key to Vector Quantization is to construct a good codebook of representative vectors. The most popular method for designing a codebook was proposed by Linde, Buzo and Gray in [Lind80, Gray84]. This method is now commonly referred to as LBG algorithm. In this algorithm, all the training vectors are clustered, using the minimum distortion principle, around trial code vectors. The centroids of these clusters then become the new trial code vectors at the next iteration. This procedure continues until there is no significant change in the total distortion between cluster members and the code vectors around which they are clustered. Then the training vectors are compared with codebook that is generated by LBG algorithm. The result is an index position of codebook with minimum distortion. This algorithm works directly on the image pixels and it uses the full search technique in the encoding process. So it takes longer time to
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 38 construct the codebook and each codeword contains every pixel of a block. Due to the enormous size of the code book the search time to find the best match vector in the encoding process increases drastically. The methods available in the literature to alleviate this problem are presented below. Generation of fast codebooks for Vector Quantization of images, based on the features of training vectors has been reported by Hsieh. C.H et. al [Hsie91]. This method uses the good energy compaction capability of the Discrete Cosine Transform and uses certain significant components of the feature space to construct the binary tree. Design of codebook for vector quantization with the discrete cosine transform Coefficients as training vector feature has been reported by Hsieh. C. H. [Hsie9]. In this work, the energy preserving property of the DCT has been used to reduce the dimension of the feature training vector. From these works, research activities on design of transformed vector quantization (TVQ) that combines the features of transform coding and vector quantization have gained popularity. Timo Kaukoranta, Pasi Fanti and Olli Nevalanen [Timo00] have reported a scheme for reducing the number of distance calculations in the LBG algorithm and are included in several fast LBG variants reducing their running time by over 50% on average. A scheme based on Vector Quantization in transformed domain has been reported in [Robe0] by Roberts et al. This scheme uses a fast Kohonen self-organizing neural network algorithm to achieve reduction in codebook construction time and transformed vector quantization to obtain better reconstructed images. Hsien-Wen Tseng et al. [Hsie05] have reported a classified vector quantization (CVQ) in the DCT transform domain. In this method DC coefficients are coded by difference pulse code modulation and only the most important AC coefficients are coded using classified vector quantization (CVQ). These AC coefficients are selected to train the codebook according to the energy packing region of different block classes. Evaluation of TVQ as low bit rate
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 39 image coding has been reported by Clyde Shamers et al. [Clyd04]. This coding technique which is based on the combined use of discrete cosine transform and Vector Quantization eliminates the artifacts generated in JPEG compression. Use of wavelet transformation in the design of TVQ has been reported in [Min05]. Here the relationship between the input vector and codeword, as well as the relationship among code words and characteristics of code words in wavelet domain are utilized. Another scheme for image compression with transform vector quantization of the wavelet coefficients has been reported by Momotaz Begum et al. [Momo03]. This scheme utilizes mean-squared error and variance based selection for good clustering of data vectors in the training space. The two major drawbacks of the LBG algorithm namely, the choice of initial codebook and the huge computational burden have been alleviated by this scheme. Fast search algorithm for vector quantization with multiple triangle inequalities in the wavelet domain has been reported by Chaur H.H and Liu. Y.J. [Chau00]. The multiple triangle inequalities confine a search range using the intersection of search areas generated from several control vectors. Also a systematic way for designing the control vectors is reported. The wavelet transform combined with the partial distance elimination is used to reduce the computational complexity of the distance calculation of vectors. A fast codeword searching algorithm based on mean-variance pyramids of codewords [Lu00] and Hadamard Transformation [Lu00a] have also been found in the literature. Given initial vectors, two design techniques for adaptive orthogonal block transforms based on VQ codebooks are presented in [Cagl98]. Both the techniques start from reference vectors that are adapted to the characteristics of the signal to be coded, while using different methods to create orthogonal bases. The resulting transforms represent a signal coding tool that stands between a pure VQ scheme on one hand and signalindependent, fixed block transformation like discrete cosine transform (DCT) on the other.
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 40 Review of early works on VQ can be found in [Nasr98]. S. Esakkirajan et al. [Esak06] have proposed an image coding scheme based on contourlet transform and multiscale VQ. Recently filter banks approach for VQ has been developed by Brislawn and Wohlberg [Chri06] to overcome obstructions for a class of half-sample symmetric filter banks. They employ lattice vector quantization to ensure symmetry preserving rounding in reversible implementations. Z. Liu et al. [Liu07] reported the use of biorthogonal wavelet filter banks (BWFBs) for image coding with lower computational costs. Here a new class of Integer Wavelet Transforms (IWT) parameterized simply by one parameter, obtained by introducing a free variable to the lifting based factorization of a Deslauriers-Dubuc interpolating filter, is introduced. The exact one-parameter expressions for this class of IWTs are deduced. In this technique, different IWTs are obtained by adjusting the free parameter and several IWTs with binary coefficients are constructed. In this chapter we explore the possibility of introducing a new VQ with the integer transform coding proposed in chapter. This new integer Transformed Vector Quantization takes the advantage of decorrelation and energy compaction properties of Orthogonal Polynomials based Transform coding and the superior rate distortion performance of VQ in the orthogonal polynomials transformed domain. In this work, the energy preserving property of the proposed transformation coding scheme is analyzed to truncate higher frequency components. This truncated sub-image is then subjected to vector quantization for effective codebook design with less complexity in sample space. The important steps involved in the work presented in this chapter are highlighted below. Analysis of the point spread operator introduced in chapter that defines the proposed coding and shows its completeness, for the purpose of perfect reconstruction of the original image by proposing difference operators.
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 41 Formations of training vectors with scale quantized high energy transform coefficients. Design of codebook on training vectors as in LBG algorithm. Identification of index values by performing vector quantization on training vectors and entropy coding of the identified indices. Inverse transformation with proposed basis operators after carrying out a simple look-up process in the codebook. 3.3 Proposed orthogonal polynomials based framework for vector quantization 3.3.1 Completeness of the proposed transformation Before presenting the new Vector Quantization in the orthogonal polynomials based transformation domain, we first prove its completeness and the same is presented in this subsection. The point spread operator in equation (.3) described in chapter that defines the linear orthogonal transformation for gray scale images is obtained as M M, where M is computed and the elements are scaled to make them integers as follows. u 0 x0 u1x0 ux0 x1 u1x1 ux1 x u x u x 0 M u0 = u 1 1 1 1 1 0 1 1 1... (3.5) n The set of polynomial operators O ij (0 i, j n-1) can be computed as n t O ij = û i û j where û i is the (i + 1) st column vector of M and is the outer product. In this subsection we prove that the proposed polynomials based difference operators is complete and hence the reconstruction of the image under
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 4 analysis after the said -D transformation is possible in terms of linear combination of basis operators O ij and the transform coefficients. The following symmetric finite differences for estimating partial derivatives at (x, y) position of the gray level image I are analogous to the eight finite difference operators O ij s excluding O 00. I 1 x, y 1 y i1 I x i, y 1 Ix i, y I 1 x, y, x i1 I y 1 x, y I x1, y i Ix1 y i I xi, y 1 Ixi, y Ixi, y 1 i1 I x 1 x, y and so on. I x1, y iix, y i Ix1, y i i1 (3.6) In general, i x y j O ij and i j I i x y j O, I, 0 i, j and i j 0 ij (3.7) where indicates the arrangement in dictionary sequence and (, ) indicates the inner product. Hence, O ij s are symmetric finite difference operators. s are the coefficients of the linear transformations and are defined as follows. = M t I (3.8) where M is the -D point-spread operator defined as M = M M.
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 43 Now we will show that the orthogonal transformation defined in equation (3.8) by the orthogonal system M is complete. We may obtain an orthogonal system H by normalizing M as follows. H = M ( M t M ) -½ Consider the following orthonormal transformations Z = H t I = ( M t M ) ½ M t I = ( M) t M ) -½ Since, H is unitary, I = H Z = M ij O ij (3.9) i0 j0 where = ( M t M ) -1. As per equation (3.9) the image region I can be expressed as a linear combination of the nine basis operators of which O 00 is the local averaging operator and the remaining eight are finite difference operators (equation 3.7). From equation (3.9) we obtain the completeness relation or Bessel's equality as follows., I Z, Z i. e. I I ij Z... (3.10) i0 j0 i0 j0 ij Thus it is proved that the proposed transformation is complete and hence the transformed image can be reconstructed perfectly. In the next section, Vector Quantization on the proposed orthogonal polynomials based transformation coefficients is presented.
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 44 3. 4. Proposed vector quantization In this section, a new Transformed Vector Quantization scheme that facilitates the image coding using Orthogonal Polynomials is proposed. This proposed scheme combines both transform coding and VQ technique. The advantage of combining both the proposed transformation and VQ is that, when a linear transform is applied to the vector signal, the information is compacted into a subset of the vector components. In the frequency domain, the high energy components are concentrated in the low frequency region. This means that the transformed vector components in the high frequency regions have very little information and so these low energy components can be entirely discarded. The procedure involved in the proposed transformed vector quantization is presented hereunder. 3.4.1 Formation of training vector The proposed TVQ starts by portioning the original image of size (R x C) into non-overlapping sub-blocks of size (n x n) and mapping them to the frequency domain by applying the proposed orthogonal polynomials based transformation as described in section... The resultant transform coefficients {i, j = 0, 1,,, n-1} are subjected to scale quantization using the default quantization table of JPEG baseline system. The aim of using the scale quantization is to obtain a suboptimal VQ codebook with reduced reconstruction. As the proposed orthogonal polynomials based transform and DCT based JPEG are both unitary, without loss of generosity, the default quantization table of JPEG is utilized here. The scale quantized transform coefficients are then arranged into 1-D zig-zag sequence to form a k-dimensional transformed input vector Y and it is mapped into a p-dimensional (p < k) training vector T by considering only the energy preserving components due to the proposed
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 45 orthogonal polynomials transformation. The energy preservation by the proposed transform is extracted as follows. The energy preserving property of the proposed transformation is based on the estimates of the variances Z i, j s corresponding to the mean squared amplitude responses of the basis, difference operators O i,j. These variances are computed as Z i, j W W i, i i, j j, j (3.11) t where W M M and M M I t The F-ratio test [Fish87] is then conducted on the variances Z i, j s to identify the significant responses towards signal compared to a threshold. The fact that a variance passes the test implies that considerable energy is compacted in the transform coefficients the F-ratio test on every energy can be discarded. corresponding to that variance. After applying Z i, j the insignificant responses that do not contain much The aforementioned process is repeated to form the training vectors of all the partitioned sub-blocks and the codebook is designed as described in the following section using the training vectors. 3.4. Codebook design The selection of the initial set of codewords is a very tricky problem in any VQ design. A variety of techniques are available in the literature for the initial selection of codewords [Alle9]. The proposed technique uses the splitting technique for choosing the initial set of codewords. In the splitting technique, the
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 46 initial codeword C 0 is chosen by taking the centroid of the entire training vectors T. Then, this codeword is split into two, namely C 0 + ε and C 0 - ε, where ε is any Euclidean norm and indicates the optimization precision. This process of iteratively splitting each codeword into two continues until the desired number of codewords of the codebook is obtained. These codewords do not qualify as final codewords for quantizing the input vectors as they do not satisfy the necessary conditions of optimality. However these can be used as the initial codewords. To optimize the codewords, the proposed technique makes use of Linde Buzo Gray (LBG) Algorithm [Lind80]. Here the training vectors T are clustered by computing the minimum distortion of the training vectors against the initial codewords C. The centroids of the clusters thus formed become the new codewords for the next iteration. This procedure continues until there is no significant change in the total distortion between cluster members and the codewords around which they are clustered. The final set of code vectors obtained constitutes the codebook. The steps involved in the design of codebook in the proposed transform domain are presented below: Step 1. Initialize the initial codeword C 0 with the mean of the entire set of training vectors Z and perturbation value ε to a fixed value. Step. Initialize iteration number n to zero and distortion D -1 to. Step 3. Form the desired number of codewords for the new codebook by splitting each codeword into two using the binary splitting operation. Step 4. Optimize the new codebook formed in step 3 using the centroid condition. Step 5. Compute r = (D n-1 - D n)/ D n. where D n-1 distortion before optimization and D n is the distortion after optimization. Step 6. Repeat steps 3 through 5 until r ε.
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 47 Then the training vectors T are compared with codebook, and index positions of code vectors that give minimum distortion, are identified. These index values are entropy coded as in JPEG baseline system and are transmitted to the receiver. 3.4.3 Reconstruction The receiver decodes the received bits to get the index values. It then initiates the lookup process in order to get the p-dimensional transformed coefficients vector from the codebook which is identical to the one at the sender. Then (k p) additional components with value zero are appended to the vectors, producing the k-dimensional vectors Yˆ and scale dequantization is performed on the elements of Yˆ to get the transformed coefficients. Finally these coefficients are subjected to inverse transform with the help of basis functions of the proposed orthogonal polynomials as described in section.3 to get back the decompressed image. 3.4.4 Time minimization The goal of the proposed TVQ scheme is threefold. First, the proposed scheme tries to minimize the time taken for construction of the codebook. The second goal is to reduce the size of the codebook. Thirdly, the proposed scheme aims to reduce the time consumption for the encoding process. To perceive how these goals are achieved, let us consider a k-dimensional input vector X with a resolution of r-bits per component constituting a total bit allocation of r x k bits. Normally in VQ, the codebook size would be N = rk. With the proposed TVQ, using the same r-bits the maximum possible codebook size is reduced to N = rp, which can be of smaller magnitude since p < k. Also, the time taken for constructing the codebook is reduced drastically, as the proposed scheme uses only p-dimension vectors instead of k-dimension vectors. In a generic VQ, the
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 48 number of comparisons required during the encoding process is N x k whereas the proposed scheme requires only N x p comparisons. Hence, it is evident that the proposed TVQ technique consumes less time and takes less storage for image coding. Alternatively, for the same time and space, the resolution or codebook size can be increased to obtain better performance. The steps involved in the proposed TVQ technique, are presented below. 3. 4. 4. TVQ Algorithm Input: Gray-level image of size (R x C). [ ] denotes the matrix and the suffix denotes the elements of the matrix. Let [I] be the (n x n) non-overlapping image region (block) extracted from the image. Begin Step 1. Divide the given input image into number of non-overlapping image regions [I] of size (n x n). Step. Repeat the steps 3 to 7 for all the image regions. Step 3. Compute the orthogonal polynomials based transform coefficients [] as described in section... Step 4. Apply scale quantization on the transform coefficients []. Step 5. Arrange the scale quantized [] in 1-D zig-zag sequence. Step 6. Truncate the low energy components from the scale quantized [] based on the energy preserving property of the proposed Orthogonal Polynomials based transformation as described in section 3.4.1. Step 7. Form a vector T using the truncated low frequency coefficients. Step 8. Design the codebook with LBG algorithm on the vectors T as described in section 3.4..
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 49 Step 9. Perform VQ operation as explained in section 3.1.1 and obtain the index values. Step 10. The index values are subjected to entropy coding as in JPEG and the coded value is transmitted to the receiver through channel. Step 11. At the receiving end, decode the index values and form the truncated code words using index values as a table look-up process. Step 1. Obtain all the n coefficients by substituting zero values in truncated high frequency coefficients. Step 13. These n coefficients are then subjected to the scale dequantization to form an approximation to the original transform coefficients []. Step 14. Reconstruct the input image region [I] using the polynomial basis functions as discussed section.3. Step 15. Repeat the steps 11 to 14 until all the image regions [I] are reconstructed. End. The above said proposed algorithm is presented in diagrammatic way in Figure 3. 1.
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 50 Proposed Transformation Scale Quantizer Choose the Energy Preserving Coefficients Original Image 4 x 4 blocks Generate codebook with LBG Nearest Neighbor principle XXX XXX Symbol Encoder Index Value 1001 Code book Channel 1001 Symbol Decoder Index Value Table Lookup Inverse Transformation Scale Dequantizer Add zero values to the decoded code words to compensate 16 coefficients Reconstructed Image Figure 3. 1: Flow diagram of the proposed TVQ technique
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 51 3. 5 Experiments and results The proposed orthogonal polynomials based Transformed Vector Quantization has been experimented with 000 test images, having different low level primitives. For illustration two test images viz, Lena and Pepper, both of size (56 x 56) with gray scale values in the range (0 55) are shown in figures 3.(a) and 3.(b) respectively. The input images are partitioned into various non-overlapping sub-images of size (4 x 4). We then apply the proposed orthogonal polynomials based transformation on each of these image blocks and obtain the transform coefficients. All these s of each block are then subjected to scale quantization. The resulting coefficients are re-ordered to 1-D zig-zag sequence and a subset of corresponding s due to energy compaction of the proposed orthogonal polynomials based transformation are extracted as described in section 3.4.1. These resulting frequency coefficients are treated as a vector T i with dimensionality six and the experiment is repeated for all the subimages to form a set of vectors T = {T i, i = 1,,, k} where i represents the sub-images and k represents total number of sub-images. Then the codebook is designed on truncated scale quantized transform coefficients with LBG algorithm as described in section 3.4.. The vectors and the codebook thus generated are subjected to Vector Quantization to obtain the index value for each vector T i corresponding to the sub-images under analysis. These index values are subjected to entropy coding and are transmitted to the receiver side. In the decompression process, these index values are decoded and are used to generate the approximated truncated transform coefficients, with the help of the codebook that were generated in the earlier stage. These 1-D transform coefficients are reordered to the original -D array after compensating zeros to the truncated high frequency components. Then the decompressed original image is obtained with the orthogonal polynomials basis functions as described in section.3.
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 5 (a) Figure 3.: Original test images considered for proposed TVQ (b) (a) (b) Figure 3.3: Results of proposed TVQ when bpp = 0.5
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 53 Table 3.1: PSNR values obtained with proposed TVQ, DCT based TVQ and the JPEG baseline for different bpps. Bit rate (bpp) 0.5 0.0 0.18 0.16 0.14 0.1 Proposed TVQ (Dimensionality 6) DCT based TVQ (Dimensionality 6) JPEG baseline system Lena Pepper Lena Pepper Lena Pepper 33.09 30.14 9.51 9.0 8.67 7.64 33. 30.90 9.80 9.47 8.9 8.03 3.49 9.71 8.64 8.14 7.4 6.04 3.53 9.88 9.0 8.44 7.49 6.53 31.6 9.47 8.8 7.0 4.99 1.48 31.86 9.81 8.69 7.6 5.9 1.58 The bit per pixel (bpp) scheme is used to estimate the transmission bit rate. The performance of the proposed TVQ scheme is measured with the standard measure Peak-Signal-to-Noise-Ratio (PSNR) as described in section.5 with the proposed TVQ. We obtain PSNR values of 33.09dB and 33.dB for a bit rate of 0.5 for the input images 3.(a) and 3.(b) respectively and the corresponding resulting images are shown in figures 3.3(a) and 3.3(b) respectively. The experiment is repeated by varying the bpp for all the 000 images and the results for the Lena and Pepper images are presented in table 3.1. In order to measure the efficiency of the proposed orthogonal polynomials based TVQ, we conduct experiments with discrete cosine transform based transformed vector quantization. For this comparison, the proposed orthogonal polynomials based transformation is replaced with discrete cosine transformation and the other steps are kept unaltered. The experiments are conducted for different bpp and the corresponding PSNR values obtained are incorporated in the same table 3.1, for both the input images and the corresponding results for 0.5 bpp are shown in figure 3.4(a) and 3.4(b) respectively. The proposed
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 54 transformed vector quantization algorithm is also compared with the international standard JPEG type compression algorithm where discrete cosine transformation and scale quantization are used. For this experiment normalization and quantization arrays are scaled in JPEG algorithm, to adjust the compression ratio to the desired level. The experiments are carried out for varying bit rates for different images and the corresponding results are incorporated in the same table 3.1. The results of JPEG algorithm for the input images shown in figures 3.(a) and 3.(b) when the bpp is 0.5 are shown in figures 3.5(a) and 3.5(b) respectively. The graphs of PSNR vs. bpp for Lena and Pepper images are plotted and the same are shown in figure 3.6(a) and 3.6(b) respectively. (a) (b) Figure 3.4: Results of DCT based TVQ when bpp = 0.5
PSNR (db) CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 55 (a) (b) Figure 3.5: Results of JPEG when bpp=0.5 Lena 34 3 30 8 6 4 Proposed TVQ DCT based TVQ JPEG 0.1.14.16.18..5 Bit rate (bpp) (a)
PSNR (db) CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 56 Pepper 34 3 30 8 6 4 Proposed TVQ DCT based TVQ JPEG 0.1.14.16.18..5 Bit rate (bpp) (b) Figure 3.6: Bit rate versus PSNR comparison of the proposed TVQ with DCT based TVQ and JPEG From table 3.1 and figures 3.3, 3.4, 3.5 and 3.6, it is evident that the proposed TVQ outperforms JPEG and discrete cosine transformation based TVQ. From these outputs, it is clear that the proposed scheme gives higher PSNR value with a reasonably good reconstruction quality. It can also be observed that the quality obtained with JPEG image coding is found to have degradation at low bit rates. But the proposed Transformed Vector Quantization produces a stable quality of picture even at low bit rates. [Observe that the PSNR value that ranges from 31.6dB to 1.48dB and 31.86 to 1.58dB for Lena and Pepper images respectively for JPEG scheme (from table 3.1)]. 3.6 Conclusion In this chapter a new transformed vector quantization based on orthogonal polynomials has been proposed for -D gray scale images. This technique combines the features of both transform coding and vector quantization. The proposed transform coding is based on a set of orthogonal polynomials. The code book is designed with LBG algorithm that utilizes only few transformed
CHAPTER 3. TVQ WITH ORTHOGONAL POLYNOMIALS 57 coefficients due to the proposed transformation. Training vectors are then formed as a subset from the image data in frequency domain and is compared with the code book, to result in the index position of the code book and sent to the decoder after entropy coding. The decoder has the code book identical to the encoder and decoding mechanism is a simple table look-up process with additional null values added to the high frequency samples. These coefficients are subjected to inverse transform with the help of basis functions of the proposed orthogonal polynomials transformation to get back the decompressed image. The performance of the proposed scheme is measured with standard PSNR value and is compared with DCT based TVQ and JPEG type algorithms. However, the encoder phase of the proposed VQ scheme uses the full search algorithm for finding the best match vectors and it leads to increase in searching time. To overcome this problem, a binary tree based codebook design is presented in the next chapter.