Block transform

Wavelet packet bases are designed by dividing the frequency axis in intervals of varying sizes. These bases are particularly well adapted to decomposing signals that have different behavior in different frequency intervals. If $f$ has properties that vary in time, it is then more appropriate to decompose $f$ in a block basis that segments the time axis in intervals with sizes that are adapted to the signal structures.

Block Bases

Block orthonormal bases are obtained by dividing the time axis in consecutive intervals $[a_{p},a_{p+1}]$ with

$\lim _{p\to -\infty }a_{p}=-\infty$ and $\lim _{p\to \infty }a_{p}=\infty$.

The size $l_{p}=a_{p+1}-a_{p}$ of each interval is arbitrary. Let $g=1_{[0,1]}$. An interval is covered by the dilated rectangular window

$g_{p}(t)=1_{[a_{p},a_{p+1}]}(t)=g({t-a_{p} \over l_{p}}).$

Theorem 1. constructs a block orthogonal basis of $L^{2}(\mathbb {R} )$ from a single orthonormal basis of $L^{2}[0,1]$.

Theorem 1.

if $\{e_{k}\}_{k\in \mathbb {Z} }$ is an orthonormal basis of $L^{2}[0,1]$, then

$\{g_{p,k}(t)=g_{p}(t){\frac {1}{\sqrt {l_{p}}}}e_{k}({\frac {t-a_{p}}{l_{p}}})\}_{(p,k)\in \mathbb {Z} }$

is a block orthonormal basis of $L^{2}(\mathbb {R} )$

Proof

One can verify that the dilated and translated family

$\{{\frac {1}{\sqrt {l_{p}}}}e_{k}({\frac {t-a_{p}}{l_{p}}})\}_{(p,k)\in \mathbb {Z} }$

is an orthonormal basis of $L^{2}[a_{p},a_{p+1}]$. If $p\neq q$, then $\langle g_{p,k},g_{q,k}\rangle =0$ since their supports do not overlap. Thus, the family $\{g_{p,k}(t)=g_{p}(t){\frac {1}{\sqrt {l_{p}}}}e_{k}({\frac {t-a_{p}}{l_{p}}})\}_{(p,k)\in \mathbb {Z} }$ is orthonormal. To expand a signal $f$ in this family, it is decomposed as a sum of separate blocks

$f(t)=\sum _{p=-\infty }^{+\infty }f(t)g_{p}(t),$

and each block $f(t)g_{p}(t)$ is decomposed in the basis $\{{\frac {1}{\sqrt {l_{p}}}}e_{k}({\frac {t-a_{p}}{l_{p}}})\}_{(p,k)\in \mathbb {Z} }$

Block Fourier Basis

A block basis is constructed with the Fourier basis of $L^{2}[0,1]$:

$\{e_{k}(t)=exp(i2k\pi t)\}_{k\in \mathbb {Z} }$

The time support of each block Fourier vector $g_{p,k}$ is $[a_{p},a_{p+1}]$ of size $l_{p}$. The Fourier transform of $g=1_{[0,1]}$ is

${\hat {g}}(w)={\frac {\sin(w/2)}{w/2}}exp({\frac {iw}{2}})$

and

${\hat {g}}_{p,k}(w)={\sqrt {l_{p}}}{\hat {g}}(l_{p}w-2k\pi )exp({\frac {-i2\pi ka_{p}}{l_{P}}}).$

It is centered at $2k\pi l_{p}^{-1}$ and has a slow asymptotic decay proportional to $l_{p}^{-}1\left\vert w\right\vert ^{-1}.$ Because of this poor frequency localization, even though a signal $f$ is smooth, its decomposition in a block Fourier basis may include large high-frequency coefficients. This can also be interpreted as an effect of periodization.

Discrete Block Bases

For all $p\in \mathbb {Z}$, suppose that $a_{p}\in \mathbb {Z}$. Discrete block bases are built with discrete rectangular windows having supports on intervals $[a_{p},a_{p-1}]$:

$g_{p}[n]=1_{[a_{p},a_{p+1}-1]}(n)$.

Since dilations are not defined in a discrete framework, bases of intervals of varying sizes from a single basis cannot generally be derived. Thus, Theorem 2 supposes an orthonormal basis of $\mathbb {C} ^{l}$ for any $l>0$ can be constructed. The proof is:

Theorem 2.

Suppose that $\{e_{k,l}\}_{0\leqslant k<l}$ is an orthogonal basis of $\mathbb {C} ^{l}$ for any $l>0$. The family

$\{g_{p,k}[n]=g_{p}[n]e_{k,l_{p}}[n-a_{p}]\}_{0\leqslant k<l_{p},p\in \mathbb {Z} }$

is a block orthonormal basis of $l^{2}(\mathbb {Z} )$.

A discrete block basis is constructed with discrete Fourier bases

$\{e_{k,l[n]}={\frac {1}{\sqrt {l}}}exp({\frac {i2\pi kn}{l}})\}_{0\leqslant k<l}$

The resulting block Fourier vectors $g_{p,k}$ have sharp transitions at the window border, and thus are not well localized in frequency. As in the continuous case, the decomposition of smooth signals $f$ may produce large-amplitude, high-frequency coefficients because of border effects.

Block Bases of Images

General block bases of images are constructed by partitioning the plane $\mathbb {R} ^{2}$ into rectangles $\{[a_{p},b_{p}]\times [c_{p},d_{p}]\}_{p\in \mathbb {Z} }$ of arbitrary length $l_{p}=b_{p}-a_{p}$ and width $w_{p}=d_{p}-c_{p}$. Let $\{e_{k}\}_{k\in \mathbb {Z} }$ be an orthonormal basis of $L^{2}[0,1]$ and $g=1_{[0,1]}$. The following can be denoted:

$g_{p,k,j}(x,y)=g({\frac {x-a_{p}}{l_{p}}})g({\frac {y-c_{p}}{w_{p}}}){\frac {1}{\sqrt {l_{p}w_{p}}}}e_{k}({\frac {x-a_{p}}{l_{p}}})e_{j}({\frac {y-c_{p}}{w_{p}}})$.

The family $\{g_{p,k,j}\}_{(p,k,j)\in \mathbb {Z} ^{3}}$ is an orthonormal basis of $L^{2}(\mathbb {R} ^{2})$.

For discrete images, discrete windows that cover each rectangle can be defined

$g_{p}=1_{[a_{p},b_{p}-1]\times [c_{p},d_{p}-1]}$.

If $\{e_{k,l}\}_{0\leqslant k<l}$ is an orthogonal basis of $\mathbb {C} ^{l}$ for any $l>0$, then

$\{g_{p,k,j}[n_{1},n_{2}]=g_{p}[n_{1},n_{2}]e_{k,l_{p}}[n_{1}-a_{p}]e_{j,w_{p}}[n_{2}-c_{p}]\}_{(k,j,p)\in \mathbb {(} Z)^{3}}$

is a block basis of $l^{2}(\mathbb {R} ^{2})$

References

1. St´ephane Mallat, A Wavelet Tour of Signal Processing, 3rd