Fast Wavelet Transform - Inverse DWT

Inverse DWT

Given the coefficient sequence for some M and all the difference sequences, k=M,...,J-1, one computes recursively

s^{(k+1)}_n:=\sum_{k=-N}^N a_k s^{(k)}_{2n-k}+\sum_{k=-N}^N b_k d^{(k)}_{2n-k} or s^{(k+1)}(z)=a(z)\cdot(\uparrow 2)(s^{(k)}(z))+b(z)\cdot(\uparrow 2)(d^{(k)}(z))

for k=J-1,J-2,...,M and all . In the Z-transform notation:

  • The upsampling operator creates zero-filled holes inside a given sequence. That is, every second element of the resulting sequence is an element of the given sequence, every other second element is zero or . This linear operator is, in the Hilbert space, the adjoint to the downsampling operator .

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