Cascade algorithm
Encyclopedia
In the mathematical
topic of wavelet
theory, the cascade algorithm is a numerical method for calculating function values of the basic scaling and wavelet functions of a discrete wavelet transform
using an iterative algorithm. It starts from values on a coarse sequence of sampling points and produces values for successively more densely spaced sequences of sampling points. Because it applies the same operation over and over to the output of the previous application, it is known as the cascade algorithm.
The iterations are defined by
For the kth iteration, where an initial φ(0)(t) must be given.
The frequency domain estimates of the basic scaling function is given by
and the limit can be viewed as an infinite product in the form
If such a limit exists, the spectrum of the scaling function is
The limit does not depends on the initial shape assume for φ(0)(t). This algorithm converges reliably to φ(t), even if it is discontinuous.
From this scaling function, the wavelet can be generated from
Plots of the function at each iteration is shown in Figure 1.
Successive approximation can also be derived in the frequency domain.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
topic of wavelet
Wavelet
A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...
theory, the cascade algorithm is a numerical method for calculating function values of the basic scaling and wavelet functions of a discrete wavelet transform
Discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled...
using an iterative algorithm. It starts from values on a coarse sequence of sampling points and produces values for successively more densely spaced sequences of sampling points. Because it applies the same operation over and over to the output of the previous application, it is known as the cascade algorithm.
Successive approximation
The iterative algorithm generate successive approximations to ψ(t) or φ(t) from {h} and {g} filter coefficients. If the algorithm converges to a fixed point, then that fixed point is the basic scaling function or wavelet.The iterations are defined by
For the kth iteration, where an initial φ(0)(t) must be given.
The frequency domain estimates of the basic scaling function is given by
and the limit can be viewed as an infinite product in the form
If such a limit exists, the spectrum of the scaling function is
The limit does not depends on the initial shape assume for φ(0)(t). This algorithm converges reliably to φ(t), even if it is discontinuous.
From this scaling function, the wavelet can be generated from
Plots of the function at each iteration is shown in Figure 1.
Successive approximation can also be derived in the frequency domain.