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 ${\displaystyle f}${\displaystyle f} has properties that vary in time, it is then more appropriate to decompose ${\displaystyle f}${\displaystyle f} in a block basis that segments the time axis in intervals with sizes that are adapted to the signal structures.

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

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

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

${\displaystyle g_{p}(t)=1_{[a_{p},a_{p+1}]}(t)=g({t-a_{p} \over l_{p}}).}${\displaystyle 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 ${\displaystyle L^{2}(\mathbb {R} )}${\displaystyle L^{2}(\mathbb {R} )} from a single orthonormal basis of ${\displaystyle L^{2}[0,1]}${\displaystyle L^{2}[0,1]}.

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

${\displaystyle \{g_{p,k}(t)=g_{p}(t){\frac {1}{\sqrt {l_{p}}}}e_{k}({\frac {t-a_{p}}{l_{p}}})\}_{(p,k)\in \mathbb {Z} }}${\displaystyle \{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 ${\displaystyle L^{2}(\mathbb {R} )}${\displaystyle L^{2}(\mathbb {R} )}

One can verify that the dilated and translated family

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

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

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

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

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

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

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

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

and

${\displaystyle {\hat {g}}_{p,k}(w)={\sqrt {l_{p}}}{\hat {g}}(l_{p}w-2k\pi )exp({\frac {-i2\pi ka_{p}}{l_{P}}}).}${\displaystyle {\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 ${\displaystyle 2k\pi l_{p}^{-1}}${\displaystyle 2k\pi l_{p}^{-1}} and has a slow asymptotic decay proportional to ${\displaystyle l_{p}^{-}1\left\vert w\right\vert ^{-1}.}${\displaystyle l_{p}^{-}1\left\vert w\right\vert ^{-1}.} Because of this poor frequency localization, even though a signal ${\displaystyle f}${\displaystyle 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.

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

${\displaystyle g_{p}[n]=1_{[a_{p},a_{p+1}-1]}(n)}${\displaystyle 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 ${\displaystyle \mathbb {C} ^{l}}${\displaystyle \mathbb {C} ^{l}} for any ${\displaystyle l>0}${\displaystyle l>0} can be constructed. The proof is:

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

${\displaystyle \{g_{p,k}[n]=g_{p}[n]e_{k,l_{p}}[n-a_{p}]\}_{0\leqslant k<l_{p},p\in \mathbb {Z} }}${\displaystyle \{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 ${\displaystyle l^{2}(\mathbb {Z} )}${\displaystyle l^{2}(\mathbb {Z} )}.

A discrete block basis is constructed with discrete Fourier bases

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

The resulting block Fourier vectors ${\displaystyle g_{p,k}}${\displaystyle 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 ${\displaystyle f}${\displaystyle f} may produce large-amplitude, high-frequency coefficients because of border effects.

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

${\displaystyle g_{p,k,j}(x,y)=g({\frac {x-a_{p}}{l_{p}}})g({\frac {y-c_{p}}{w_{p}}}){\frac {1}{\sqrt {l_{p}w_{p}}}}e_{k}({\frac {x-a_{p}}{l_{p}}})e_{j}({\frac {y-c_{p}}{w_{p}}})}${\displaystyle 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 ${\displaystyle \{g_{p,k,j}\}_{(p,k,j)\in \mathbb {Z} ^{3}}}${\displaystyle \{g_{p,k,j}\}_{(p,k,j)\in \mathbb {Z} ^{3}}} is an orthonormal basis of ${\displaystyle L^{2}(\mathbb {R} ^{2})}${\displaystyle L^{2}(\mathbb {R} ^{2})}.

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

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

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

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

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