16 Chapter 3 Set Redundancy in Magnetic Resonance Brain Images 3.1 MRI (magnetic resonance imaging) MRI is a technique of measuring physical structure within the human anatomy. Our proposed research focuses on brain structures but will include other image types to demonstrate our techniques. MRI systems produce brain images in cross-sections of a human head. These brain images are acquired by measuring the interaction between pulses of RF (radio frequency) radiation and tissues in a strong magnetic field. Then they are transformed to reconstruct a "3-dimensional" digital image volume. Figure 3.1 depicts the 2 slices comprising an example MR volume with 256*256 pixels in each single slice. Figure 3.1 2 slices of MR brain images comprising an example volume
17 In MRI, several tissue characteristics including the PD (proton density), the longitudinal relaxation time (T1) and the transversal relaxation time (T2) can be measured. The intensity of each voxel of an MR volume image is related to the PD, T1, and T2 of the tissues located at the corresponding anatomical position. The contrast between different tissue types can be controlled at the time of acquisition by varying several MRI parameters including the pulse repetition time (T R ) and echo time (T E ). Choice of these parameters can result in PD-weighted, T1-weighted, or 59, 6 T2-weighted images. As a result, MRI data is inherently multi-spectral. In this proposed research, we use all three image types, that is, PD-weighted, T1- weighted, or T2-weighted images, as the basic test sets for proposed compression method. 3.2 Special Redundancy in a 3-D MR Brain Image Set MRI data contain large quantities of noise, which are uncorrelated from slice to slice. This makes the structure of the cross dependence more complicated than temporal sequences. Yet, a significant amount of redundancy between successive slices of MR data can be found after a close investigation of the structure of MR brain images (Figure 3.1). For instance, they are similar in terms of the shape, pixel intensity at certain anatomical position, and analogous anatomical structures from subjective observation. In addition, statistical analysis was also done to further illustrate the similarity of these images. 3.2.1 Histogram Analysis Histograms for several nonconsecutive MR brain images on Figure 3.1 are presented on Figure 3.2. This shows that more than 5 percent of the pixels have zero intensity, which corresponds to 1 percent black, and all the other intensities are uniformly distributed with a possibility of less than 1 percent.
18.7 Histogram of T2-Weighted Image #1.7 Histogram of T2-Weighted Image #6.6.6.5.5..3..3.2.2.1.1-5 5 1 15 2 25 3-5 5 1 15 2 25 3 (a) Image #1 (b) Image #6.7 Histogram of T2-Weighted Image #11.7 Histogram of T2-Weighted Image #16.6.6.5.5..3..3.2.2.1.1-5 5 1 15 2 25 3-5 5 1 15 2 25 3 (c) Image #11 (d) Image #16 Figure 3.2 Comparable histograms of four nonconsecutive MR brain images (Figure 3.1). 3.2.2 Plot of Wavelet Coefficients A two-level wavelet transform was applied to the same four nonconsecutive MR brain images presented on Figure 3.2. The plots of the coefficients of all decomposed sub-images for all four images are depicted on Figure 3.3.,, V2, D2, H1, V1 and D1 represent level 2 and level 1 decompositions of wavelet transform in average, horizontal, vertical and diagonal directions respectively. The abscissa represents the vector position and the ordinate represents the value of the decomposed coefficients in their corresponding vector position. All four images have very similar (not
19 identical) plots for level 2 average decomposition, which is the coarse approximation of the original image at a high scale, as shown on each subplot. Deviations from the average approximation of horizontal, vertical and diagonal directions are also similar to each other. 2 5 2 5-2 2 V2 6 5-5 2 D2 6 2-2 2 V2 6 5-5 2 D2 6 2-5 2 H1 6 2-2 2 V1 6 2-5 2 H1 6 2-2 2 V1 6 2-2.5 D1 1 1.5 2 5 x 1-2.5 1 1.5 2 x 1-2.5 D1 1 1.5 2 5 x 1-2.5 1 1.5 2 x 1-5.5 1 1.5 2 x 1-5.5 1 1.5 2 x 1 (a) Image #1 (b) Image #6 2 5 2 5-2 2 V2 6 5-5 2 D2 6 2-2 2 V2 6 5-5 2 D2 6 2-5 2 H1 6 2-2 2 V1 6 2-5 2 H1 6 1-2 2 V1 6 1-2.5 D1 1 1.5 2 5 x 1-2.5 1 1.5 2 x 1-1.5 D1 1 1.5 2 5 x 1-1.5 1 1.5 2 x 1-5.5 1 1.5 2 x 1-5.5 1 1.5 2 x 1 (c) Image #11...(d) Image #16 Figure 3.3 Coefficients of decomposed sub-images for a 2-level wavelet transform 3.2.3 Feature Vector Table 3.1 numerically lists the mean and variance for all decomposed sub-images presented on Figure 3.3. The corresponding vectors (MA 2, VA 2, MH 2, VH 2, MV 2, VV 2, MD 2, VD 2, MH 1, VH 1, MV 1, VV 1, MD 1, VD 1 ) derived from the first row of Table 3.1 are defined as feature vectors in our research since both mean and variance are good statistical tools to represent the general characteristics of this population. Here, MA 2,
2 MH 2, MV 2, MD 2, MH 1, MV 1 and MD 1 represent a mean of level 2 and level 1 decompositions of wavelet transform in average, horizontal, vertical and diagonal direction respectively, whereas VA 2, VH 2, VV 2, VD 2, VH 1, VV 1 and VD 1 represent the corresponding variances. There are only slight deviations for feature vectors of four MR brain images. MA 2 VA 2 MH 2 VH 2 MV 2 VV 2 MD 2 VD 2 MH 1 VH 1 MV 1 VV 1 MD 1 VD 1 #1 229.51 277.52.9 1.93 -.37 38.5.2 2.3 -.6 12.89 -. 9.78.3 2.87 #6 256.9 313.32.3 6.7.26.61 -.11 19.69.5 12.78.2 1.9 -.3 3. #11 255.57 32.37.11 7.8 -.13 37.13 -.5 21.3-13.9 -.3 11.28 -.8 3.38.3 #16 2.11 296..3 38.9.8 37.5 -.5 17.73 -.1 1.52.3 9.79.6 2.78 3.2. Entropy Table 3.1 Feature vector for four nonconsecutive MR brain images The entropy for original four nonconsecutive MR brain images and their corresponding wavelet transform images are shown in Table 3.2. The entropy of an image is a measure of the amount of information an image contains, and it is also used as a measure for the compressibility of the image (lower entropy means better compressibility). The comparable entropy values for all four original MR brain images illustrate that each contains almost an equal amount of information. This holds true for the wavelet transform entropy. Image Original Entropy WT Entropy Image #1.551 3.921 Image #6.568 3.8518 Image #11.3363 3.696 Image #16 3.5683 3.111 3.2.5 Correlation Table 3.2: Original entropy and wavelet transform entropy for four nonconsecutive MR brain images in testing image sets Finally, the correlation was determined (Table 3.3) between these same four nonconsecutive MR brain images. The existence of statistical correlation between two images can be verified graphically with a scatter plot of pixel values, and numerically by calculating the correlation coefficient. It was observed in the four
21 nonconsecutive MR brain images, which should have less similarity than consecutive images, that the correlation was between.6 and.7. This signifies similarities among all four images and even more similarity with neighboring MR brain images. 3.3 Summary Correlation Image #1 Image #6 Image #11 Image #16 Image #1 1.7311.7663.6272 Image #6.7311 1.7953.638 Image #11.7663.7953 1.7213 Image #16.6272.638.7213 1 Table 3.3 Correlation between all four nonconsecutive MR brain images Based on these statistical analyses, we can conclude that the redundancies in a 3-D MR brain image set can be summarized as follows: Similar pixel intensities in the same areas, Similar edge distributions, Analogous distributions of features, Comparable histograms (Figure 3.2), Identical sub-image coefficients of each decomposition direction and comparable feature vectors after applying transforms such as wavelet transform. (Figure 3.3 and Table 3.1), Comparable entropy for both original images and wavelet transformed images respectively (Table 3.2), High correlation (Table 3.3). The special redundancies in a 3-D MR brain image set may be expanded to any other 3-D medical image set. We will choose a 3-D knee medical image set to further demonstrate the prevalence of these redundancies.
22 It has been proven 61 that entropy of the image set will decrease when the redundancies of the same image set increases. Therefore, we will utilize the set redundancy in a 3-D medical image set to further the compression.