Mathematical Conventions :: Advanced Concepts (Wavelet Toolbox™)

Mathematical Conventions
This chapter and the reference pages use certain mathematical conventions.

General Notation
Interpretation

Dyadic scale. is the level, 1/ or is the resolution.

Dyadic translation

Continuous time

or
Discrete time

Pixel

Signal or image. The signal is a function defined on or ; the image is defined on or .

Fourier transform of the function f or the sequence f

Continuous Time

Set of signals of finite energy

Energy of the signal

Scalar product of signals and

Set of images of finite energy

Energy of the image

Scalar product of images and

Discrete Time

Set of signals of finite energy

Energy of the signal

Scalar product of signals and

Set of images of finite energy

Energy of the image

Scalar product of images and

General Notation	Interpretation
	Dyadic scale. is the level, 1/ or is the resolution.
	Dyadic translation
	Continuous time
or	Discrete time
	Pixel
	Signal or image. The signal is a function defined on or ; the image is defined on or .
	Fourier transform of the function f or the sequence f
Continuous Time
	Set of signals of finite energy
	Energy of the signal
	Scalar product of signals and
	Set of images of finite energy
	Energy of the image
	Scalar product of images and
Discrete Time
	Set of signals of finite energy
	Energy of the signal
	Scalar product of signals and
	Set of images of finite energy
	Energy of the image
	Scalar product of images and

Wavelet Notation
Interpretation

A_j
j-level approximation or approximation at level j

D_j
j-level detail or detail at level j

phi
Scaling function

psi
Wavelet

Family associated with the one-dimensional wavelet, indexed by and

Family associated with the two-dimensional wavelet, indexed by

Family associated with the one-dimensional scaling function for dyadic scales ; it should be noted that .

Family associated with the one-dimensional for dyadic scales ; it should be noted that .

Scaling filter associated with a wavelet

Wavelet filter associated with a wavelet

Wavelet Notation	Interpretation
A_j	j-level approximation or approximation at level j
D_j	j-level detail or detail at level j
	Scaling function
	Wavelet
	Family associated with the one-dimensional wavelet, indexed by and
	Family associated with the two-dimensional wavelet, indexed by
	Family associated with the one-dimensional scaling function for dyadic scales ; it should be noted that .
	Family associated with the one-dimensional for dyadic scales ; it should be noted that .
	Scaling filter associated with a wavelet
	Wavelet filter associated with a wavelet