Title: New Clustering Algorithm for Vector Quantization using Rotation of Error Vector
ArXiv ID: 1004.1686
Date: 2010-04-13
Authors: ** 논문에 저자 정보가 명시되어 있지 않음. (추정: 해당 논문은 International Journal of Computer Science and Information Security 2010년 7권 3호에 게재된 것으로 보임) **
📝 Abstract
The paper presents new clustering algorithm. The proposed algorithm gives less distortion as compared to well known Linde Buzo Gray (LBG) algorithm and Kekre's Proportionate Error (KPE) Algorithm. Constant error is added every time to split the clusters in LBG, resulting in formation of cluster in one direction which is 1350 in 2-dimensional case. Because of this reason clustering is inefficient resulting in high MSE in LBG. To overcome this drawback of LBG proportionate error is added to change the cluster orientation in KPE. Though the cluster orientation in KPE is changed its variation is limited to +/- 450 over 1350. The proposed algorithm takes care of this problem by introducing new orientation every time to split the clusters. The proposed method reduces PSNR by 2db to 5db for codebook size 128 to 1024 with respect to LBG.
💡 Deep Analysis
📄 Full Content
World Wide Web Applications have extensively grown since last few decades and it has become requisite tool for education, communication, industry, amusement etc. All these applications are multimedia-based applications consisting of images and videos. Images/videos require enormous volume of data items, creating a serious problem as they need higher channel bandwidth for efficient transmission. Further high degree of redundancies is observed in digital images. Thus the need for image compression arises for resourceful storage and transmission. Image compression is classified into two categories, lossless image compression and lossy image compression technique.
Vector quantization (VQ) is one of the lossy data compression techniques [1], [2] and has been used in number of applications, like pattern recognition [3], speech recognition and face detection [4], [5], image segmentation [6][7][8][9], speech data compression [10], Content Based Image Retrieval (CBIR) [11], [12], Face recognition [13], [14] iris recognition [15], tumor detection in mammography images [29] etc. VQ is a mapping function which maps k-dimensional vector space to a finite set CB = {C 1 , C 2 , C 3 , ..…., C N }. The set CB is called as codebook consisting of N number of codevectors and each codevector C i = {c i1 , c i2 , c i3 , ……, c ik } is of dimension k. Good codebook design leads to reduced distortion in reconstructed image. Codebook can be designed in spatial domain by clustering algorithms [1], [2], [16][17][18][19][20][21][22].
For encoding, image is split in blocks and each block is then converted to the training vector X i = (x i1 , x i2 , ..….., x ik ). The codebook is searched for the nearest codevector Cmin by computing squared Euclidean distance as presented in equation ( 1) between vector X i and all the codevectors of the codebook CB. This method is called as exhaustive search (ES).
Exhaustive Search (ES) method gives the optimal result at the end, but it intensely involves computational complexity. Observing equation (1) to obtain one nearest codevector for a training vector computations required are N Euclidean distance where N is the size of the codebook. So for M image training vectors, will require M*N number of Euclidean distances computations. It is obvious that if the codebook size is increased the distortion will decrease with increase in searching time.
A variety of encoding methods are available in literature: Partial Distortion search (PDS) [23], nearest neighbor search algorithm based on orthonormal transform (OTNNS) [24]. Partial Distortion Elimination (PDE) [25], Kekre’s fast search algorithms [26], [27], [28] etc., are classified as partial search methods. All these algorithms minimize the computational cost needed for VQ encoding keeping the image quality close to Exhaustive search algorithm.
In this section existing codebook generation algorithms i.e. Linde Buzo Gray (LBG) and Kekre’s Proportionate Error (KPE) are discussed.
A. Linde Buzo and Gray(LBG)Algorithm [1], [2] In this algorithm centroid is computed by taking the average as the first codevector for the training set. In Figure 1 two vectors v1 & v2 are generated by using constant error addition to the codevector. Euclidean distances of all the training vectors are computed with vectors v1 & v2 and two clusters are formed based on closest of v1 or v2. This modus operandi is repeated for every cluster. The shortcoming of this algorithm is that the cluster elongation is +135 o to horizontal axis in two dimensional cases resulting in inefficient clustering. [10], [18] Here to generate two vectors v1 & v2 proportionate error is added to the covector. Magnitude of elements of the codevector decides the error ratio. Hereafter the procedure is same as that of LBG. While adding proportionate error a safe guard is also introduced so that neither v1 nor v2 go beyond the training vector space eliminating the disadvantage of the LBG. Figure 2. shows the cluster elongation after adding proportionate error. The codevectors of the two clusters are computed and then both clusters are splitted by adding and subtracting error vector rotated in k-dimensional space at different angle to both the codevector. This modus operandi is repeated for every cluster and every time to split the clusters error ei is added and subtracted from the codevector and two vectors v1 and v2 is generated. Error vector e i is the ith row of the error matrix of dimension k. The error vectors matrix E is given in Equation 2.
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ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 7, No. 3, 2010