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Compression of Magnetic Resonance Images Using Wavelet Packets

Self Developed Compression Technique

Aditya Zutshi


Need For Multiresolution Analysis ď Ż

If both small and large objects – and low and high contrast objects – are present simultaneously, it can be advantageous to study them at several resolutions. This is the fundamental advantage for Multi-Resolution processing and this is where Wavelet Transform comes into picture.


Subband Coding ď Ż

Subband coding is an important imaging technique with ties to Multiresolution analysis. In Subband coding, an image is decomposed into a set of band-limited components, called Subband, which can be reassembled to reconstruct the original without error.

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Each Subband is generated by band pass filtering the input. Since the bandwidth of the resulting Subband can be down sampled without loss of information, Reconstruction of the original image is accomplished by up sampling, filtering, and summing the individual Subband. The one-dimensional filters can be used as two-dimensional separable filters for the processing of images.


A two-dimensional, four-band filter bank for Subband image coding


Split Subbands ď Ż The resulting filtered outputs, denoted a(m,n), dV(m,n), dH(m,n), dD(m,n) are called approximation, vertical detail, horizontal detail, and diagonal detail Subband of the image, respectively. One or more of these Subbands can be split into four smaller Subbands, which can be split again, and so on.


Wavelet ď Ż A wavelet is a waveform of effectively limited duration that has an average value of zero.

ď Ż The term wavelet means a small wave. The smallness refers to the condition that this (window) function is of finite length (compactly supported). The wave refers to the condition that this function is oscillatory.


Wavelet Analysis ď Ż Wavelet analysis represents a windowing technique with variable-sized regions. ď Ż Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques miss aspects like trends, breakdown points, discontinuities in higher derivatives, and self-similarity.


CWT – Mathematical Model


Daubechies Wavelet - Mathematical


Daubechies Wavelet - Order 2


Experimentally Designed Algorithm For Compression Technique


Algorithm – Transmitter End  The original image is loaded.  Pre-filtering techniques are applied.  Discrete Wavelet Transform is applied  Coefficients are threshold and stored with much less data on disk requirement


Algorithm – Receiver End  Interpolation.  Inverse Discrete Wavelet Transform.

 Image is Reconstructed.  Morphological Operations are applied.  Image is displayed to the user.


Wavelet Packet Decomposition  Magnetic Resonance Image is decomposed into an Nlevel Binary Tree by Wavelet Packets Decomposition.   The number of possible decompositions is often so large that it is impractical, if not impossible to enumerate or examine them individually.  One reasonable criterion for selecting decomposition is the additive cost function.

 Only if the combined entropy of the offspring is less than the entropy of the parent, the offspring is included in the analysis tree. In this way the best tree of the wavelet packet is constructed from the binary tree.


Best Tree Algorithm Results


Results

Original Image

Compressed & Reconstructed Image


Difference Between Original & Reconstructed Image


Result & Conclusion With this algorithm a compression ratio of 15.045:1 was achieved. The correlation between the original image and the reconstructed compressed image at the receiver end was found out to be 0.9521 which shows a high degree of alikeness between the original image and the reconstructed compressed image. Huffman’s coding technique was also applied on the compressed image and could reduce the data on disk requirement by a ratio of 89.78: 1.


Application of Wavelet Compression Technique ď Ż The most successful commercial application of wavelets to date has been digitizing of fingerprint information for the FBI. At 500 pixels per inch with 256 gray levels, each fingerprint card has about 10e7 bits of data, some compression is required. The standard form of Joint Photographic Experts Group (JPEG) is a Fourier based and unable to provide the 20:1 compression ratios required for this application.


References 

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[1] Sonja Grgic, Kresimir Kers, Mislav Grgic, “Image Compression using wavelets”, IEEE-ISIE 199 – Slovenia. [2] Dominik Engel & Andreas Uhi, “ Adaptive Object – Based Image Compression using wavelet packets”, IEEE Region 8 International Symposium on Video/Image Processing & Multimedia Communications 2002, Croatia. [3] Mohamed A. El-Sharkawy, Christian A. White & Harry Gundnim, “Subband Image Compression using Wavelet Transform & Vector Quantization”, IEEE. [4] Sonja Grgic, Mislav Grgic & Branka Zovko – Cihlar, “ Performance Analysis of Image Compression using Wavelets”, IEEE Transactions on Industrial Electronics, Vol-48, No. 3, 2001. [5] Amir Averbuch, Danny Lazar & Moshe Israeli, “ Image Compression using Wavelet Transform & Multiresolution Decompositio”, IEEE Transaction on Image Processing, Vol. 5, No. 1, 1996. [6] Jose Oliver & Manuel Perez Malumbres, “Fast & Efficient Spatial Scalable Image Compression using Wavelet Lower Trees”, IEEE International Conference of Data Compression (Dec ’03) [7] Agostino Abbate, Casimer M. Decusatis, Pankaj K. Das, “Wavelets & Subbands – Fundamentals & Application


Thank You


Compression of Magnetic Resonance Images Using Wavelet Packets