## One Two and Three Dimensional Wavelet Transform

We use the 1D case to discuss the concept of multiresolution analysis. Consider the discrete signal fm at level m, which can be decomposed into the m + 1 level by convolving it with the h (low pass) filter to form a smooth signal fm+1 and g (high pass) filter to form a detailed signal 1, respectively, as shown in Fig. 1. This can be represented with the following equations using the pyramidal algorithm suggested by Mallat [2]:

fm +1(«) = £ h(2n — k)fm(k) k fmn + 1(n) = (2n — k)fm(k)

Here, fm+1 is the smooth signal and 1; is the detailed signal at the resolution level m + 1. The total number of discrete points in fm is equal to that of the sum of fm+1 and f4+ j. For this reason, both fm+1 and 1 have to be sampled at every other data point after the operation described in Eq. (5). The same process can be further applied to fm 1, creating the detailed and smooth signal at the next resolution level, until the desired level is reached.

Figure 2 depicts the components resulting from three levels of decompositions of the signal f,. The horizontal axis indicates the total number of discrete points of the original signal, and the vertical axis is the level, m, of the decomposition. At the resolution level m = 3, the signal is composed of the detailed signals of the resolution levels f\, f'2, and f'3 plus one smooth signal f3. Signals at each level can be compressed by quantization and encoding methods to achieve the required compression ratio. Accumulation of these compressed signals at all levels can be used to reconstruct the original signal f,.

In the case of the 2D transform, the first level will result in four components, the ^-direction and the y-direction (see Fig. 4, left and middle). Figure 3 shows a two-level wavelet decomposition of a head MR image.

Three-dimensional wavelet transforms can be computed by extension of the 1D and 2D pyramidal algorithm, since the multidimensional wavelet transform can be formulated to be separable. One level of the decomposition process from fm to fm+1 is shown in Fig. 4. First, each line in the ^-direction of the 3D image data set is convolved with filters h and g, followed by subsampling every other voxel in the ^-direction to form the smooth and detailed data lines. The resulting voxels are convolved with h and g in the y-direction, followed with subsampling in the y-direction. Finally, the same procedure is applied to the z-direction. The resulting signal has eight components. Since h is a low-pass filter, only one component, fm 1, contains all low frequency information. The rest of the seven components convolve at least once with the high-pass filter g, and therefore contain the detailed signals 1 in different directions.

The same process can be repeated for the low-frequency signal fm 1, to form the next level of wavelet transform, and so forth, until the desired level is reached.

FIGURE 1 Decomposition of a signal fm into a smooth fm+ 1 and a detailed signal f^+1.

FIGURE 2 A three-level wavelet decomposition of a signal.

FIGURE 1 Decomposition of a signal fm into a smooth fm+ 1 and a detailed signal f^+1.

FIGURE 2 A three-level wavelet decomposition of a signal.

FIGURE 3 Wavelet decomposition of a MR image, (a) Original; (b) one-level decomposition; (c) two-level decomposition. In each level, the left upper corner shows the smooth image; the other three quadrants are the detailed images.