Image Analysis, Classification and Change Detection in Remote Sensing, 4th Edition

3.2.3.4    The Daubechies D4 scaling function

A family of MRAs which is very useful in digital image analysis is generated by the Daubechies scaling functions and their associated wavelets (Daubechies, 1988). The Daubechies D4 scaling function, for example, can be derived by placing the following two additional requirements on an MRA (Aboufadel and Schlicker, 1999):

1.  Compact support: The scaling function ϕ(x) is required to be zero outside the interval 0 < x < 3. This means that the refinement coefficients c_k vanish for k < 0 and for k > 3. To see this, note that

$c_{-3}=2\langle \varphi \left(x\right),\varphi \left(2x+3\right)\rangle ={\displaystyle {\int }_0^3\varphi \left(x\right)\varphi \left(2x+3\right)dx=0}$

and similarly for k ≤ −4 and for k ≥ 6. Therefore, from the dilation equation,

$\varphi \left(-1/2\right)=0=c_{-2}\varphi \left(-1+2\right)+c_{-1}\varphi \left(-1+1\right)+\dots \text{implying}{c}_{-2}=0$

and similarly ...