The concepts that have been developed for one-dimensional signals can be easily generalized to two-dimensional signals. We consider the case where the two-dimensional scaling function is separable.

Where $\phi$(x) is a one-dimensional function.

Let $\Psi$(x) be its compansion wavelet.

Since the scaling and wavelet functions are separable, each convolution breaks down into one-dimensional convolution on the rows and columns. This is diagrammatically shown in figure 28.

Figure 27

Let the size of the original image, f(x, y) be N X N.

First stage: - Convolve the rows of the image with h(n) and g(n) and discard alternate columns (downsample columns by 2).

Second stage:

• The columns of each of the N/2 X N data are then convolved with h(n) and g(n) and the alternate rows are discarded (downsample rows by 2).
• The result of this entire operation give us four N/2 X N/2 images.
• This decomposition can be carried out recursively.
• The first N/2 matrix is again decomposed into four N / 4 matrices.
• The operation is shown in figure 29.

Figure 28

• Image processing based on wavelet transform:

  1. Applying a 2-D wavelet transform on the image.

  2. Manipulate the values of the wavelet transform.

  3. Perform the inverse wavelet transform to obtain the processed image.

• The original image can be reconstructed from the four N/2 X N/2 quite easily.
• The reconstruction process is shown in figure 30.

Figure 29