Discretized CWT vs. DWT vs. FWT vs. MRA By reading some books about wavelets, it seems that Discretized CWT is not the same thing as DWT.

Can we give a classification of Discretized CWT vs DWT vs FWT etc. ?

A few remarks :


All I read about DWT or MRA (multi-resolution analysis) involves a dyadic factor (2)  for $a$ in $\frac{1}{\sqrt{a}} \Psi(\frac{t-b}{a})$, ie $a=2^m$. Is it always true ?
Remark : This is good in order to have a non-redondant transform (see B Z's answer here Scalogram (and related nomenclatures) for DWT?), but the graphical representation of such a transform isn't very satisfying for visual analysis of a signal.
The fact that there is $ a=2^m$ involved implies that the frequency bands studied [in the case where the mother-wavelet has a narrow-band fourier transform, e.g. Morlet wavelet http://ieeexplore.ieee.org/ieee_pilot/articles/06/ttg2009061375/assets/img/article_1/fig_3/large.gif] are things like [20hz,40hz], [40hz,80hz], [80hz, 160hz], etc.
Thus we cannot hope a good graphical representation with precise frequency resolution. It seems that this kind of transform is more suited for "approximation + details" than for visual frequency analysis (scalogram, etc.)
On the other hand, a discretization of Continuous wavelet transform allows to take $a=1.0004^m$ for example, and thus have a more precise frequency resolution, for one who wants to do a graphical representation (scalogram).
For the CWT the values in your definition can be any $a\in \mathbb R^+$ and $b\in \mathbb R$ if applied to functions. The CWT can also be applied to discrete signals (discretized CWT) with $b$ restricted to appropriate discrete values (and any $a$, but typically $2^{j/v}$ where $v$ is the "voices/octave" of the transformation). Now when we restrict $a$ to some discrete values $2^j$ we get the DWT, which is an orthogonal transformation.

(Small edit to correct the scale : $2^{j/v}$ instead of $a^{j/v}$)