The authors investigate the characteristics and performance of joint (single-step) and sequential (two-step) approaches to creating sparse and structured multiresolution representations of audio and music signals derived using sparse overcomplete methods. A joint approach, such as molecular matching pursuit, attempts to find structures in a signal as part of the decomposition process, while a sequential approach, such as agglomerative clustering, attempts to find structures in the completed decomposition of a signal. Each of these approaches have different benefits and drawbacks. For a joint approach, it is computationally convenient that the decomposition and structuring are done simultaneously, but usually only simple structural relations are possible. For a sequential approach, one is working in a parameter space of much smaller dimension than the original signal, but the computation is higher since the decomposition and the structure building are two separate processes. Results from these approaches using real audio and music signals will be compared and contrasted, and will contribute to our goal of creating an enhanced interface between the content of audio and music signals, e.g., onsets, notes, voices, and their multiresolution sparse atomic decompositions.

In this article we provide an overview of dictionary-based methods (DBMs) -also called sparse approximation -and review recent work in the application of such methods to working with signals, in particular audio and music signals. As Fourier analysis is to additive synthesis, DBMs can be seen as the analytical counterpart to granular synthesis since a signal is rebuilt by a linear combination of heterogeneous atoms selected from a user-defined dictionary. We demonstrate how DBMs provide novel means for analyzing, visualizing, and transforming audio signals by creating multiresolution and parametric descriptions of their contents.

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