Image compression using an overcomplete discrete wavelet transform

Distributed Peer-to-Peer Target Tracking in Wireless Sensor Networks

“…With the discrete translation of function f ( x ) on a dyadic scale δ a =2 j and discrete translation δ b = 2 j k , the discrete wavelet transform (DWT) can be presented as follow [15]: $\mathit{\text{DSWT}}\{f(x);2^j,2^{j}k\}=C_{j,k}=WT\{f(x);{\delta }_a=2^j,{\delta }_b=2^{j}k\}$where C j,k is the wavelet coefficients of the function f ( x ), and the wavelet function forms a orthogonal and complete dyadic family: ${\psi }_{j,k}(x)=2^{-\frac{1}{2}j}\psi (2^{-j}x-k)$…”

“…With DWT, the f ( n ) can be written as a multiresolution decomposition on J levels, j =1,…, J , given by [15]: $f(n)=\sum _{j=1}^J\sum _{k\in Z}{c}_{j,k}{\tilde{g}}_j(n-2^{j}k)+\sum _{k\in Z}{e}_{j,k}{\tilde{h}}_{J}false(n-2^{J}kfalse)$where g̰ jj ( n − 2 j k ) is the synthesis wavelets and discretely equal to ψ j , k , and the scaling coefficients e j , k is defined as: $e_{j,k}=\sum _{n}ffalse(nfalse)h_J^{*}false(n-2^{J}kfalse)$where $h_J^{*}(n-2^{J}k)$ is the scaling sequences.…”

“…For discrete signals f ( n ), n∈Z , the DWT is defined as: where is the discrete equivalent of the . With DWT, the f ( n ) can be written as a multiresolution decomposition on J levels, j =1,…, J , given by [ 15 ]: where g̰ jj ( n − 2 j k ) is the synthesis wavelets and discretely equal to ψ j , k , and the scaling coefficients e j , k is defined as: where is the scaling sequences.…”