Wavelet Toolbox

The Fast Wavelet Transform (FWT) Algorithm

In 1988, Mallat produced a fast wavelet decomposition and reconstruction algorithm [Mal89]. The Mallat algorithm for discrete wavelet transform (DWT) is, in fact, a classical scheme in the signal processing community, known as a two channel subband coder using conjugate quadrature filters or quadrature mirror filters (QMF).

• The decomposition algorithm starts with signal s, next calculates the coordinates of A₁ and D₁, and then those of A₂ and D₂, and so on.
• The reconstruction algorithm called the inverse discrete wavelet transform (IDWT) starts from the coordinates of A_J and D_J, next calculates the coordinates of A_J-1, and then using the coordinates of A_J-1 and D_J-1 calculates those of A_J-2, and so on.

This section addresses the following topics:

• Filters Used to Calculate the DWT and IDWT
• Algorithms
• Why Does Such an Algorithm Exist?
• One-Dimensional Wavelet Capabilities
• Two-Dimensional Wavelet Capabilities

Details and Approximations

Filters Used to Calculate the DWT and IDWT

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