# Scalogram (and related nomenclatures) for DWT?

My understanding of the scalogram is that, for a particular row, the scores of the projection of the input signal with the wavelet at a particular displacement is shown. Across rows, the same thing applies, but for dilated version of the wavelet. I thought that scalograms can be defined for all types of wavelet transforms, that is, for the:

1. Continuous wavelet transform
2. Discrete wavelet transform
3. Redundant wavelet transform

However upon further investigation is seems that the scalogram is only definable for the CWT. Based on this I have multiple inter-related questions that google has not sufficed for ATM.

Questions:

1. Is it true that the scalogram is not defined for the DWT or RWT? If so, why not?
2. Let us say an $N$ length signal has a 10-level decomposition by using DWT. If all levels are plotted as an image, (that is, a $10xN$ image), what is this image called?

As an example of a DWT 'scalogram', here is one for AWGN: 1. Concerning the same signal, suppose we instead plot the approximation MRA of the signal at all levels. (So again, a $10xN$) image. What is this image called in proper terminology? For example, here I have shown approximation MRAs and detail MRAs for AWGN. (Clearly they are not the same as 'scalogram' of DWT).  Thanks!

• It looks like MatLab's implementation of the DWT doesn't impose dyadic scaling to avoid redundency. The MRA must impose this. Notice how the blocks of information get wider as you progress down the MRA plot. The scale changes by a factor of 2 with each subsequent analysis. – user2718 Jan 24 '13 at 23:05
• Beware, your first scalogram is incorrectly drawn. – Alexei Averchenko Aug 26 '13 at 11:33

1. Continuous wavelet transform is suitable for a scalogram because the analysis window can be sized and placed at any position. This flexibility allows for the generation of a smooth image in both the time in scale (analogous to frequency) directions. The continuous wavelet transform is a redundant transform because the analysis window can overlap. In fact the CWT is considered infinitely redundant.

2. Discrete wavelet transform is a non-redundant transform. It was developed so there would be a one to one correspondence between the information in the signal domain and the transform domain. This tight correspondence makes the DWT more suitable for use in signal reconstruction. The analysis windows are fixed in both the time and scale directions, so if you plot the resulting DWT coefficients you will end up with a grid of boxes that start out large at one end of the scale axis and end up small at the other end. This representation isn't very satisfying for visual analysis of a signal. It certainly can be done, but I haven't seen anyone bother to do it. The plot is also referred to as a scalogram.

3. Redundant Wavelet Transform: I had no previous experience with this, but thanks to comments from the OP, I found that the RWT or Stationary Wavelet Transform (SWT) is a discrete wavelet transform that has redundancy introduced to make the transform translation invariant. Furthermore, I found a reference that does a nice comparison of transform types as they apply to speech analysis. In this article, the transform results are all plotted and for any case of wavelet transform, the plots are all referred to as scalograms (this includes the DWT and a version of RWT). You can see how the various transform types present themselfs visually in the article. For reference, here is a link to the article: http://www.math.purdue.edu/~lipeijun/paper/2005/End_Gen_Li_Fra_Sch_JASA_2005.pdf

MRA - My encounter with this term is in association with multiresolution analysis. This applies to all wavelet transform types, but usually is discussed in the context of the DWT and its realization as a set of filter banks. In this context the result of a MRA is the same as the result of a DWT and the plot of such results (a plot of a set of numbers) would still be a scalogram. Here is another paper that discusses MRA: http://alexandria.tue.nl/repository/books/612762.pdf

The following is an example of CWT and DFT Scalograms: • Thanks Bruce. The RWT also goes by the Stationary Wavelet Transform. I do not think it is the same as the CWT, but I could be wrong, as I am weak on this point. Regarding Q2) What does one call an image of all DWT co-efficients plotted across scale, and regarding Q3), what does one call an image plot of the approximation MRAs of a DWT? Thanks! – Spacey Jan 24 '13 at 19:04
• I updated my answer based on your comments. I wasn't familiar with the RWT, so thanks for the reference. Live and learn :-) Hope this is helpful. – user2718 Jan 24 '13 at 20:47
• Bruce, thanks once again. I do not think however that the MRA in the DWT context shows the same thing as the scalogram. (See my edited post for images with a signal being AWGN). If can accept that a the first image is a DWT scalogram, but what would the other images be called in the field? Just MRAs? I am BTW still suspicious of scalogram existing for anything other than CWT, as my wavelet book only computes it for CWTs, and MATLAB's own library claims that a scalogram is only supported for CWTs. This adds to the confusion. – Spacey Jan 24 '13 at 21:10
• MRA and DWT certainly look different, I'll have to agree here, but not sure why. I understand the confusion with math programs. I'm using Mathematica and it has similar segregation of ideas. Also they don't expose their implementation, so you generally have to guess and do trial and error work to figure out what you are getting. – user2718 Jan 24 '13 at 22:32
• Regarding the term scalogram, I too have not seen this in common use with anything other than the CWT, but the first paper I referenced employs the term for DWT based plots as well. I think this is just a matter of convention. – user2718 Jan 24 '13 at 22:39