Wavelets are functions defined over a finite interval and having an average value of zero.

The basic idea of wavelet is to represent an arbitrary function t as a superposition of a set of such wavelets or basis functions. These basis function or baby wavelets are obtained from single prototype wavelet called the mother wavelet, by dilations or contradictions(scaling) and translations(shifts).

1. Two dimensional Forward Pyramid Decomposition:

• In wavelet Transform, an image signal is passed through an analysis filter bank followed by decimation operation.
• The analysis filter bank consists of a LPF and HPF.
• 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.
• The output filter is decimated by two.
• A Two Dimensional 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.
• This operation splits input image into four bands of data,LL(Low-Low),HL(High-Low),LH(Low-High),HH(High-High).
• 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.

1. Two Dimensional Inverse Pyramid Reconstruction

• 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.
• The synthesis filter bank consists of a LPF and HPF.
• When the decomposed frequency band image signal is passed through these filters,the output is merged into single band image.
• The reconstructed image is exactly same as original input image.