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Image Coding Using Wavelet Transform, Vector Quantization, and Zerotrees

Image Coding Using Wavelet Transform, Vector Quantization, and Zerotrees. L. J. Wang. Contents. Introduction Algorithms Methodology Results. I. Introduction.

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Image Coding Using Wavelet Transform, Vector Quantization, and Zerotrees

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  1. Image Coding Using Wavelet Transform, Vector Quantization, and Zerotrees L. J. Wang

  2. Contents • Introduction • Algorithms • Methodology • Results

  3. I. Introduction • Image storage and transmission pose an important problem to the development of intelligent communication systems due to memory and bandwidth requirements. • Many different image compression techniques have been devised during the last few decades. • Although lossless or reversible schemes are very desirable, the achieved compression ratios are relatively low, which makes necessary the use of lossy (irreversible) schemes, allowing some distortion in the reconstructed images.

  4. The efficiency of a coder can be defined as the image quality for a given bit rate and, for a lossy method, is generally increased at the cost of computational complexity. • Wavelet transform separates the information of the image at different scales and orientations, without changing the image size. • Vector Quantization (VQ) of these transformed images gives a good compression, without the blocking effects usually found in images coded using VQ in the spatial domain.

  5. The embedded zerotree wavelet (EZW) algorithm introduced by Shapiro in December 1993, is a relatively simple technique that is based on the wavelet transform, followed by the successful prediction of insignificant coefficients across scales due to the self-similarity inherent in transformed images. • The EZW addresses the two-fold problem of obtaining the best image quality for a given rate and accomplishing this task in an embedded fashion, that is, in such a way that all encodings of the same image at lower bit rates are contained in the beginning of the bit stream for the target bit rate.

  6. II. Algorithms2.1 Wavelet Transform

  7. COLUMNS ROWS Image corresponding to the low resolution level m h 2 h 2 Initial image corresponding to the resolution level m-1 g 2 Detail images corresponding to the resolution level m h 2 g 2 g 2 h Convolve with low-pass filter 2 Downsampling or decimation g Convolve with high-pass filter Fig. 1: One stage in a multiscale image decomposition.

  8. Step 1 Step 2 Step 3 Fig. 2: One level of a wavelet decomposition in 3 steps.

  9. Fig. 3: Image decomposition.

  10. COLUMNS ROWS Image corresponding to the low resolution level m 2  2 Reconstructed image resolution level m-1 2 Detail images resolution level m  2  2 2 X Convolve with filter X 2 Upsampling Fig. 4: One stage in a multiscale image reconstruction.

  11. 2.2 Vector Quantization (VQ) Fig. 5: Encoding/decoding scheme using wavelet and VQ.

  12. Fig. 6: Multiresolution codebook.

  13. Fig. 7: Subimages bit rate allocation: example of a bit allocation for a total bit rate of 1 bpp and for the 256 by 256 Lena image.

  14. 2.2 Enbedded Zerotree Wavelet (EZW) Fig. 8: Parent-child dependencies of subbands.

  15. Fig. 9: Scanning order of the subbands for encoding a significance map.

  16. Fig. 10: Flow chart for encoding a coefficient of the significance map.

  17. Fig. 11: Example of 3-scale wavelet transform of an 8 by 8 image.

  18. III. Methodology • The proposed procedure has four steps: (1) Obtaining the Wavelet Transform of the image. (2) Each subband in the transform is independently coded using VQ, except for the lowest frequency band, on which scalar quantization to eight bits is used. In this preliminary work, all subbands are coded using a 256-vector codebook, which gives the same compression for the three subbands of each scale.

  19. (3) The VQ coded subbands are subjected to an information elimination procedure based on the EZW. We define a significant vector as one whose distance to the origin is greater than a selectable threshold, T. Subbands are scanned as in Fig. 9, maintaining two lists: the significance map (SM), and the significance vectors (SV). The SM has positional information for each subband. For the lower frequency subbands, the SM has one to one correspondence with the subband vectors, with three possible values: significant vector, isolated zero or zerotree root.

  20. For the higher frequency subbands, the SM includes only those vectors that do not belong to a zerotree. Vectors in the highest frequency subbands have no children, and can only be significant or insignificant, requiring only two symbols. When a vector in any scale is significant, its index is added to the significant vector list. Only one pass is made through the image, as the vector indexes saved represent the full precision of the significant vectors.

  21. In this procedure, T is fixed and controls the quality of the reconstructed image. A file is generated which contains both the SM and SV lists, and the scaler-coded lowest frequency image. (4) Finally, this file is further compressed using arithmetic coding.

  22. IV. Results

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