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# Scaling Functions

Consider a set of functions

where the specific function is to be determined later. These functions have two integer subscripts or parameters and :

• The first subscript specifies the magnitude as well as the scale of the function. A greater corresponds to a larger magnitude but a more compressed function (higher frequency'' representing more details). This corresponds to the parameter representing frequency in the basis functions for Fourier expansion. When , becomes infinitely wide (zero frequency) with infinitely small magnitude, while , approaches a delta function with infinitely narrow width but infinitely high magnitude.
• The second subscript specifies the position (integer location, translation or shift) of the function. This does not have a counterpart in the basis functions for Fourier transform, where the position information is totally missing.
Replacing in the equation above by , we get:

indicating how a function at the level is related to the corresponding function at the level.

Corresponding to a specific index , a subset of functions spans a function space . In general, there are many such functions spaces: .

If a family of functions satisfy the follow requirements, they become a set of basis functions that span a function space, so that a given function can be represented as a linear combination of these bases:

• The functions have to be orthogonal to its integer translates:

• The function space spanned by a set of functions is a subspace of the function space spanned by the functions of a higher scale (with double-resolution and capable of representing more details):

• The only function contained in all subspaces is (constant of lowest resolution or frequency carrying no information), which is the only function contained in , i.e., .

• A function can be represented with arbitrary precision by including terms of progressively higher scales (greater ) for representing more details (high frequency components) of the function. Any square-integrable functions (with arbitrary details) can be represented in , i.e.,

where represents all square-integrable functions ( ).

Functions satisfying these requirements are called scaling functions. As the basis functions, a subset of scaling functions with a certain scale can only represent a function up to a certain level of details corresponding to the scale of the bases. All details in finer than the limit of the scales of are lost. However, such details can be better represented by the basis functions of the next higher scale . In general, space contains all functions representable in , as well as those functions with more details not representable in , i.e., , and a function can always be more precisely represented by increasing the scales (the parameter ) of the bases.

Note: The nested relations between the sequence of subspaces shown above is closely related to the Gaussian-Laplacian pyramid discussed earlier. Essentially they both state that a signal can be decomposed into a set of components each representing details of different levels contained in the signal. The images in the Laplacian pyramid correspond to the subspaces as they both represent the difference between two consecutive levels of details.

The basis functions are themselves members of the space they span as well as all the super-spaces:

And they can be expanded in the space of next higher scale with doubled resolution:

where are called scaling functions coefficients. Usually we let and drop the subscripts and to indicate that any scaling function can be expressed as a linear combination of the basis scaling functions of the space with the next higher resolution:

This is called refinement, dilation or MRA (multiresolution analysis) equation.

The first 4 panels show the scaling functions: , , , and , The 5th panel shows a function in space spanned by . The 6th panel shows as a special function in space .

Example: The Haar scaling function at level is defined as a square pulse of unit width and unit height:

Scaling function together with its shifted versions form a set of basis functions that span the space . The scaling functions of higher scales () for higher resolutions (for signal details) can be obtained by

For example, when , we have

A given function can be expanded as a linear combination of these scaling (basis) functions:

This is shown in panel 5 of the figure above.

Moreover, the Haar scaling functions in space are also functions in space and they can be expressed as a linear combination of the basis functions :

This is shown in panel 6 of the figure above. In particular,

But as

the above can be written as

where the two coefficients are the first row of the Haar matrix .

Next: Wavelet Functions Up: wavelets Previous: Haar Wavelets
Ruye Wang 2008-12-16