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The Scaling Equation

Module by: Phil Schniter. E-mail the author

Summary: This module covers scaling equations and coefficients for the discrete wavelet transform.

Consider the level-1 subspace and its orthonormal basis:

V 1 =span⁢ φ 1 , n ⁢t n∈Z V 1 span φ 1 , n t n
(1)
φ 1 , n ⁢t=12⁢φ⁢12⁢t−n φ 1 , n t 1 2 φ 1 2 t n
(2)
Since V1⊂V0 V1 V0 (i.e., V 0 V 0 is more detailed than V 1 V 1 ) and since φ 1 , 0 ⁢t∈ V 0 φ 1 , 0 t V 0 , there must exist coefficients h⁢n n∈Z h n n such that
φ 1 , 0 ⁢t=∑n=−∞∞h⁢n⁢ φ 0 , n ⁢t φ 1 , 0 t n h n φ 0 , n t
(3)
⇔ 12⁢φ⁢12⁢t=∑n=−∞∞h⁢n⁢φ⁢t−n ⇔ 1 2 φ 1 2 t n h n φ t n
(4)

Scaling Equation

φ⁢t=2⁢∑n=−∞∞h⁢n⁢φ⁢2⁢t−n φ t 2 n h n φ 2 t n
(5)
To be a valid scaling function, φ⁢t φ t must obey the scaling equation for some coefficient set h⁢n h n .

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