Walsh transform

The Walsh Transform matrix is obtained from the Hadamard matrix by re-arranging the rows in increasing sign change order.

The Hadamard matrix is given below

$H[8]=\begin{vmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\ 1 & 1 & -1 & - 1 &1 & -1 & -1 & -1 \\ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ 1 & 1 & 1 & 1 & -1 & -1& -1 & -1 \\ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1\\ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \end{vmatrix}$

Rearranging the rows of the Hadamard matrix to get an increasing order of sign changesgives us the Walsh matrix

$H[8]=\begin{vmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\\ 1 & 1 & -1 & - 1 & -1 & -1 & 1 & 1 \\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 & 1 & -1& -1 & 1 \\ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1\\ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \end{vmatrix}$

Plotting each of the rows of the Walsh matrix gives us the Walsh basis functions. Walsh transformation can be calculated using the Matrix Equation

$X[n]=[W(N) x(n)]$

The inverse is given by $x(n)=1/N [W(N)X(n)].$

Wavelet Transform:

Two dimentional Forward Pyramid Decomposition:

1. In wavelet Transform, an image signal is passed through an analysis filter bank followed by decimation operation.
2. The analysis filter bank consists of a LPF and HPF.

3. When the signal passes through these filters,it splits into two bands.

   The LPF which corresponds to an averaging operation,extracts texture information of the signal.

   The HPF,which corresponds to differencing operation,extracts the detail information of the signal.

4. The output filter is decimated by two.
5. A Two Dimentional Transform is accomplished by Row-wise filtering operation using parallel bank of filters followed by Row-wise filtering operation using parallel bamk of filters. Each time the output of filters is decimated by two.
6. This operation splits input image into four bands of data,LL(Low-Low),HL(High-Low),LH(Low-High),HH(High-High).
7. The LL band can be decomposed once again in the same manner , thereby producing even more subbands. This can be done up to any level, thereby resulting in a pyramidal decomposition.

Two Dimensional Inverse Pyramid Reconstruction:

1. In inverse wavelet transform, the decomposed frequency band images LL,LH,HL,HH are first interpolated by factor two and then further passed through a synthesis filter bank.
2. The synthesis filter bank consists of a LPF and HPF.
3. When the decomposed frequency band image signal is passed through these filters,the output is merged into single band image.
4. The reconstructed image is exactly same as original input image.