Using continuous verses discrete wavelet transform in digital applications I am familiar with much of the mathematical background behind wavelets. However when implementing algorithms on a computer with wavelets I am less certain about whether I should be using continuous or discrete wavelets. In all reality everything on a computer is discrete of course, so it seems obvious that discrete wavelets are the right choice for digital signal processing. However according to wikipedia it is the continuous wavelet transform that is primarily used in (digital) image compression as well as a large number of other digital data processing activities. What are the pros and cons to consider when deciding whether to use an (approximate) continuous wavelet transform instead of an (exact) discrete wavelet transform for digital image or signal processing?

P.S. (Checking an assumption here) I am assuming continuous wavelet transforms are used in digital processing by simply taking the value of the continuous wavelet at evenly spaced points and using the resulting sequence for wavelet computations. Is this correct?

P.P.S. Usually wikipedia is pretty precise about mathematics, so I am assuming that the applications in the article on Continuous Wavelet Transforms are in fact applications of the Continuous Wavelet Transform. Certainly it mentions some that are specifically CWT so there is clearly some use of CWT in digital applications.
As Mohammad stated already the terms Continuous Wavelet Transforms (CWT) and Discrete Wavelet Transforms (DWT) are a little bit misleading. They relate approximately as (Continuous) Fourier Transform (the math. integral transform) to DFT (Discrete Fourier Transform).

In order to understand the details it is good to see the historical context. The wavelet transform was originally introduced in geophysics by Morlet, and was basically a Gabor transform with a Window that grows and shrinks together with the selected scale/frequency. Later Daubchies (a physicist-ett from Belgium) realized that by choosing special orthogonal wavelet bases the infinitely redundant CWT can be critically sampled on a dyadic grid. From the resulting DWT the corresponding full CWT can be obtained by convolving the DWT with the reproducing kernel of the respective wavelet. The reproducing kernel is the CWT of the wavelet itself.

Daubchies findings gave a big boost to the wavelet theory in the early 80ies. The next big result was that the DWT can be computed very efficiently (this is sometimes called FWT [fast WT] as well) by using techniques from the theory of filterbanks, namely quadrature mirror filters (QMF) together with downsampling filterbanks. By constructing special QMFs the corresponding DWT can be computed via filtering and downsampling, which is the state-of-the-art algorithm to compute DWTs today. You do not need the scaling function to compute the DWT, it is just an implementation detail that FWT process.

Concerning the application side the CWT is the more ideal candidate for signal or time series analysis due to its more fine grained resolution and is usually chosen in most tasks (e.g. singularity detection). The DWT is more of interest in the context of fast non redundant transforms. The DWT has a very good energy compactification and is thus a good candidate for lossy compressions and signal transmissions.



Hope that clarified things.