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Compression of Magnetic Resonance Images Using Wavelet Packets Aditya Zutshi

Abstract - Discrete Wavelet Transform (DWT) can be efficiently used in image coding applications because of its data reduction capabilities. Unlike the case of Discrete Cosine Transform (DCT) in which basis function is composed of cosine functions, basis of DWT can be composed of any function, called wavelet, of varying frequency and limited duration that satisfy requirements of Multi Resolution Analysis. The wavelet transform often fails to accurately capture the high frequency information, especially at lower bit rates where such information is lost in quantization noise. A technique called wavelet packets has been developed that is better able to represent high frequency information. Wavelet packets are the conventional wavelet transforms in which the details are iteratively filtered. Quantization of coefficients is done using uniform quantizer, which decides the reconstruction error. The algorithm was tested for Magnetic Resonance Images and compression ratio of 15.045: 1 was achieved. The coefficients can also be encoded using Huffman’s coding technique. The Compression ratio between the original image and the data on disk after Huffman’s coding was found out to be 89.77:1. I INTRODUCTION

When we look at images, generally we see connected regions of similar texture and gray level that combine to form objects. If the objects are small in size or low in contrast, we normally examine them at high resolutions; if they are large in size or high in contrast, a course view is all that is required. If both small and large objects – or low and high contrast objects – are present simultaneously, it can be advantageous to study them at several resolutions. This is the fundamental advantage for multiresolution processing and this is where Wavelet Transform comes into

picture. From a mathematical viewpoint, images are two-dimensional arrays of intensity values with locally varying statistics that result from different combinations of abrupt features like edges and contrasting homogeneous regions. An image pyramid is a collection of decreasing resolution images arranged in the shape of a pyramid. The base of the pyramid contains a high-resolution representation of the image being processed; the apex contains a low-resolution approximation. As we move up the pyramid, both size and resolution decrease. Since base level J is size 2j X 2j or N X N where J=log2N, intermediate level j is size 2j X 2j, where 0≤ j ≤ J. Fully populated pyramids are composed of J + 1 resolution levels from 2j X 2j to 20 X 20, but most pyramids are truncated to P + 1 levels, where j = J – P, ...., J -2, J – 1, J and 1 ≤ P ≤ J. That is we normally limit ourselves to P reduced resolution approximations of the original image; a 1X1 or single pixel approximation of a 512X512 image, for example, is of little value. The total number of elements in a P + 1 level pyramid for P>0 is: N2( 1 + 1/(4)1 + 1/(4)2 + ...... + 1/(4)P ) ≤ (4/3) N2 II METHODS A. Subband Coding 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


subbands, which can be reassembled to reconstruct the original without error. Each subband is generated by bandpass filtering the input. Since the bandwidth of the resulting subbands can be downsampled without loss of information. Reconstruction of the original image is accomplished by upsampling, filtering, and summing the individual subbands. The onedimensional filters can be used as twodimensional separable filters for the processing of images.

of the real, square-integrable function φj,k (x) = 2 (j/2) φ (2j x – k) for all j, k Є Z and φ (x) Є L2 (R). Here, k determines the position of φj,k (x) along the x-axis, j determines φj,k (x)’s width – how broad or narrow it is along the x-axis – and 2 (j/2) controls its height or amplitude. Because the shape of φj,k (x) changes with j, φ(x) is called a scaling function. By choosing φ(x) wisely, {φj,k (x)} can be made to span L2 (R), the set of all measurable, square-integrable functions. It is seen that increasing j allows functions with smaller variations or finer detail to be included in the subspace. In multiresolution analysis (MRA), a scaling function is used to create a series of approximations of a function or image, each differing by a factor of 2 from its nearest neighboring approximations. Additional functions, called wavelets are then used to encode the difference in information between adjacent approximations.

Fig. 1. A two-dimensional, four-band filter bank for subband image coding.

B. Wavelet Functions Transform

As can be seen in Figure 1, separable filters are first applied in one-dimension (e.g. vertically) and then in the other direction (e.g. horizontally). Moreover, downsampling is performed in two stages-once before the second filtering operation to reduce the overall number of computations. The resulting filtered outputs, denoted a(m,n), dV(m,n), dH(m,n), dD(m,n) in Figure 1 are called the approximation, vertical detail, horizontal detail, and diagonal detail subbands 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. Considering the set of expansion functions composed of integer translations and binary scalings

A wavelet function Ψ(x) can be defined as that, together with its integer translates and binary scalings, spans the difference between any two adjacent scaling subspaces. A wavelet is a waveform of effectively limited duration that has an average value of zero. Wavelet analysis represents a windowing technique with variablesized regions. Wavelet analysis allows the use of long time intervals where we want more precise low-frequency information, and shorter regions where we want high-frequency information. Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques miss aspects like trends,

and

Wavelet


breakdown points, discontinuities in higher derivatives, and self-similarity.

Fig. 2: Graphical representation of “db 2” wavelet The Daubechies wavelet function can mathematically be expressed as:

The one-dimensional transforms are easily extended to two-dimensional functions like images. In twodimensions, a two-dimensional scaling function, φ(x,y), and three twodimensional wavelets, ΨH(x,y), ΨV(x,y), ΨD(x,y), are required. Each is the product of a one-dimensional scaling function φ and the corresponding wavelet Ψ. Excluding products which produce one-dimensional results, like φ(x)Ψ(x), the four remaining products produce the separable scaling function – φ(x,y)= φ(x) φ(y) and separable, “directionally sensitive” wavelets – ΨH(x,y) = Ψ(x) φ(y) ; ΨV(x,y) =φ(x) Ψ(y) ; ΨD(x,y) = Ψ(x) Ψ(y). These wavelets measure functional variations – intensity or gray-level variations for images- along different directions i.e. the horizontal, vertical and the diagonal.

C. Wavelet Packets The fast wavelet transform decomposes a function into a series of logarithmically related frequency bands. The low frequencies are grouped into narrow bands, while the high frequencies are grouped into wider banks. To get a greater control over partitioning of the time-frequency plane, the Fast Wavelet Transform must be generalized to yield a more flexible decomposition – called a Wavelet Packet. The cost of this generalization is an increase in computational complexity.

Fig. 3. Block Diagram of the Wavelet Packet based Compression Scheme utilized for the Compression of Magnetic Resonance Images of Human Brain.


III RESULTS The Magnetic Resonance Image is decomposed into an N-level Binary Tree by Wavelet Packets Decomposition. The single wavelet packet tree presents numerous decomposition options. In fact, the number of possible decompositions is often so large that it is impractical, if not impossible to enumerate or examine them individually. An efficient algorithm for finding optimal decompositions with respect to Image Compression application criteria is designed.

Ε (f) = ∑m,n | f (m,n) | This function measures the entropy or information content of two-dimensional function f. Entropies near 0 indicate functions with little to no information. For each node of the analysis tree, beginning with the root and proceeding level by level to the leaves, computation of the entropy of the node, denoted E P (for parent entropy), and the entropy of its four offspring – denoted EA, EH, EV and ED is done. For two-dimensional wavelet packet decompositions, the parent is a two-dimensional array of approximation or detail coefficients; the offspring are the filtered approximation, horizontal, vertical and diagonal details. 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. The Wavelet Packet best tree decomposition, shown in Figure 6 is calculated by minimizing cost function of wavelet packet tree of Figure 4.

Fig. 4. 2D Coronal MRI Image Slice of Brain

Fig.6: Wavelet Decomposition. Fig. 5: Wavelet Packet Decomposition of the Image One reasonable criterion for selecting decomposition for the compression of the image of Figure 4 is the additive cost function:

Packet

Best

Tree

Thresholding is done to the coefficients of the wavelet packet best tree decomposition. Thus we get the compressed image containing thresholded coefficients of the best tree decomposition. This


algorithm could compress the image by a compression ratio of 15.045:1. At the receiver end, inverse wavelet transform of the coefficients is taken and the image is reconstructed. Various image enhancement techniques, interpolation techniques and morphological operations are applied as an intermediate stage to enhance the quality of the reconstructed image. Figure 7 shows the image reconstructed from the compressed image. The correlation between the original image and the reconstructed compressed image was found out to be 0.9521 which shows a high degree of alikeness between the original image and the reconstructed compressed image.

Fig.7.Reconstructed and Enhanced Compressed Image at the Receiver End. The visible loss in the reconstructed image is negligible. The difference image between the original image and the reconstructed image is shown in Figure 8.

Fig. 8. Difference between Original Image & Reconstructed Image

IV DISCUSSION 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. V 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. VI REFERENCES [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


Compression of Magnetic Resonance Images Using Wavelet Packets