Noiselets are functions which gives the worst case behavior for the Haar wavelet packet analysis. In other words, noiselets are totally incompressible by the Haar wavelet packet analysis. Like the canonical and Fourier bases, which have an incoherent property, noiselets are perfectly incoherent with the Haar basis. In addition, they have a fast algorithm for implementation, making them useful as a sampling basis for signals that are sparse in the Haar domain.
Definition The mother bases function χ ( x ) {\displaystyle \chi (x)} is defined as:
χ ( x ) = { 1 x ∈ [ 0 , 1 ) 0 otherwise {\displaystyle \chi (x)={\begin{cases}1&x\in [0,1)\\0&{\text{otherwise}}\end{cases}}}
The family of noislets is constructed recursively as follows:
f 1 ( x ) = χ ( x ) f 2 n ( x ) = ( 1 − i ) f n ( 2 x ) + ( 1 + i ) f n ( 2 x − 1 ) f 2 n + 1 ( x ) = ( 1 + i ) f n ( 2 x ) + ( 1 − i ) f n ( 2 x − 1 ) {\displaystyle {\begin{alignedat}{2}f_{1}(x)&=\chi (x)\\f_{2n}(x)&=(1-i)f_{n}(2x)+(1+i)f_{n}(2x-1)\\f_{2n+1}(x)&=(1+i)f_{n}(2x)+(1-i)f_{n}(2x-1)\end{alignedat}}}
Property of fn
{ f j | j = 2 N , … , 2 N + 1 − 1 } {\displaystyle \{f_{j}|j=2^{N},\dots ,2^{N+1}-1\}} is an orthogonal basis for V N {\displaystyle V_{N}} , where V N {\displaystyle V_{N}} is the space of all possible approximations at the resolution 2 N {\displaystyle 2^{N}} of functions in L 2 [ 0 , 1 ) {\displaystyle L^{2}[0,1)} . For each n ≥ 1 {\displaystyle n\geq 1} , ∫ 0 1 f n ( x ) d x = 1 {\displaystyle \int _{0}^{1}f_{n}(x)dx=1}
For each n ≥ 1 {\displaystyle n\geq 1} , f n ( x ) = ∏ j = 0 ℓ ( n ) − 1 r ~ v j ( n ) ( 2 j x ) ∈ [ 0 , 1 ] {\displaystyle f_{n}(x)=\prod _{j=0}^{\ell (n)-1}{\tilde {r}}_{v_{j}(n)}(2^{j}x)\in [0,1]}
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