Cascading the two channel filter banks produces arbitrary filter banks. The transfer function of the cascade of filters and downsamplers is given by,
and is followed with downsampling by 
, where 
= depth of the cascade of path r and 
is an indicator function that selects the filter, 
or 
, at stage k in the filter path.
Since the two-channel filter bank implements an orthonormal signal expansion, the cascaded filter bank also implements an orthonormal expansion. In other words, the impulse response of filters 
and their appropriate shifts also form an orthonormal basis for 
[VK95]. The filter bank may be used to construct an expansion adaptively to the signal characteristics. However, in practice it is difficult to determine the best filter bank structure without first expanding the signal. One solution is to over-expand the signal using a full cascade, and then choose the paths that gives the best complete expansion. However, finding the best spatial-frequency decomposition for an image still does not compensate for non-stationarity. Images are inherently non-stationary signals. The wavelet packet algorithm adapts to the whole image and not to different regions as needed. Therefore, we extend the wavelet packet algorithm by including the next building block of the joint image expansion - segmentation.