While reading literature on wavelets, I have encountered keywords filters and dictionary atoms and their exact meaning is confusing for me. Basically, one can use them interchangeably and I won't see a problem. the following text is an example

Our objective in this section is to develop a method to generate a system of filters such that (i) the filters are translated versions of each other in the graph spectral domain, and (ii) the $M \cdot N$ dictionary atoms constructed by applying each generalized translation operator $T_{i}$ to each filter form a tight frame. More precisely, given an upper bound, $\lambda_{\text { max }},$ on the spectrum and a desired number of filters, $M,$ we want to find a kernel $\widehat{g}^{U}(\cdot)$ and constants $a$ and $A$ such that . . .

Would someone be able to comment on the how kernel, filter, and dictionary atoms differ with each other in this context? Thanks!

The text is from Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames (DOI:10.1109/TSP.2015.2424203).

  • $\begingroup$ a bit of context would be very helpful; as far as I can see, atom is a term from measure theory, or maybe set theory. The analytic tone of the rest of this suggests we're dealing with someone thinking in measure theory, but then again, the combination with the very discrete-math word dictionary implies we're more in the realm of set theory... $\endgroup$ – Marcus Müller Aug 4 '19 at 9:41
  • $\begingroup$ Kernel is even worse: in the context of image filtering, you can basically exchange filter impulse response by kernel (that's probably taken from the integral transform meaning of kernel), but in algebraic contextes, it means sets that map to zero. In other words, we'd probably have to read the chapters leading up to your paragraph to be sure how to answer your question. Without knowing at least the title and field of the text you're reading, I'm afraid it'll be impossible to help you with certainty, and "likely" is the best answer you could get. $\endgroup$ – Marcus Müller Aug 4 '19 at 9:44
  • $\begingroup$ A filter is a procedure mathematician applied to a signal to remove some "components" of it. Generally, filters are designed as linear element, so that, if the input signal is $S$, the output signal is $S*K$, where $K$ is known as the kernel. Kernel is a vector in a dimension, matrix in two dimension and tensor in three dimensions. Now, if you have one signal 1D or 2D and then applied one or various kernels, the outputs can be organized in an array known as "dictionary atoms", this dictionary has the representation of your signal and can be used in test phase to classifier a test signal. $\endgroup$ – Roger Figueroa Quintero Aug 4 '19 at 10:33
  • $\begingroup$ $*$ means convolution operation $\endgroup$ – Roger Figueroa Quintero Aug 4 '19 at 10:35
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    $\begingroup$ When the paper is published, I suggest to use its full title and DOI for long term access: Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames (DOI:10.1109/TSP.2015.2424203) $\endgroup$ – Laurent Duval Aug 4 '19 at 17:50

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