Wavelet Analysis of 3D Brain Data =================================

1. Brain imaging data is complex and difficult to interpret in its raw format. Here we present a new approach for 3D brain data analysis using wavelets - oscillating compactly supported base functions with many nice properties.
2. 1D Wavelet Analysis - a real cortical profile curve (data) is obtained by a coronal anatomical section through the visual cortex of the human brain. This profile is represented in the wavelet space by the following formula where the signal, f(x), is in green and the wavelet-base function in pink. The wavelet coefficients, left-hand-side, are expressed as an inner-product of the signal, f(x), and the wavelet-base functions, Phi_j_k. All wavelet-base functions are related. They are derived from one common function (mother wavelet). There are two parameters describing each wavelet base function: a scaling frequency index "j" and the position index "k both in red. Increasing and decreasing the scaling frequency index "j" shrinks or expands the domain of the base function. And altering the shifting index k changes the position of the wavelet base function Phi.
3. An estimate of the original signal can be obtained by summing/integrating all of the wavelet coefficients against the wavelet-base functions, for each scaling (j) and shifting (k) indices.
4. Here are some examples of various wavelet-base functions illustrating the local and oscillatory properties of the wavelet-signal representation.
5. How about if in the function reconstruction we only add the contributions of SOME wavelets, not all? This example illustrates the intensities of the cortical ribbon profile on the top, and its wavelet coefficients on the bottom. Using a frequency-adaptive filter (depends only on the scaling frequency index "j") we threshold the wavelet coefficients based on their magnitude. Large wavelet coefficients would remain and the smaller coefficients will be ignored (set to zero) in the reconstruction of an estimate for the cortical profile signal. Here we see the actual "wavelet-shrinkage approach" - large wavelet coefficients survive and small ones are zeroed in the thresholding process.
6. Only a small number of the largest wavelet coefficients represent the essence of the signals (usually 1-2%). This is clearly visible by the reconstruction of the cortical profile from the top 1% of the wavelet coefficients. The smooth curve has similar shape as the raw signal except that it does not include the noise present in the data.
7. Validation of the efficiency of this approach is theoretically shown using this formula. The distance between the raw profile, f, and the reconstructed signal, f^, is bounded above. In fact, the wavelet thresholding estimate can not be outperformed by any other estimate across all possible data signals. It is uniformly the best estimator.
8. Here we see a series of 2D axial brain slices through the human brain. We concentrate on one such planar image. The information content of the image is stripped by iterative application of the wavelet filter extracting features one wavelet at a time.
9. A real 3D brain data is shown on a stereotaxic grid. Then the corresponding wavelet coefficients of the volume are displayed. As before, we recursively extract the information-content of the brain volume one wavelet base function at a time.
10. We have used this technique to quantify quality of image registration, to determine the statistically significant differences in functional brain data and to multi-resolution data storage, compression and analysis.