We investigate an $M$-channel critically sampled filter bank for graphsignals where each of the $M$ filters is supported on a different subband ofthe graph Laplacian spectrum. We partition the graph vertices such that the$m^{th}$ set comprises a uniqueness set for signals supported on the $m^{th}$subband. For analysis, the graph signal is filtered on each subband anddownsampled on the corresponding set of vertices. However, the classicalsynthesis filters are replaced with interpolation operators, circumventing theissue of how to design a downsampling pattern and graph spectral filters toensure perfect reconstruction for signals that do not reside on bipartitegraphs. The resulting transform is critically sampled and graph signals areperfectly reconstructable from their analysis coefficients. We empiricallyexplore the joint vertex-frequency localization of the dictionary atoms andsparsity of the analysis coefficients, as well as the ability of the proposedtransform to compress piecewise-smooth graph signals.
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机译：我们研究了一个$ M $通道的关键采样滤波器组，以获取图信号，其中每个$ M $滤波器都在图拉普拉斯谱的不同子带上得到支持。我们对图的顶点进行划分，以使$ m ^ {th} $集包含$ m ^ {th} $子带上支持的信号的唯一集。为了进行分析，图形信号在每个子带上进行过滤，并在相应的一组顶点上进行降采样。然而，经典合成滤波器被插值运算符所取代，从而避免了如何设计下采样模式和图形频谱滤波器的问题，从而确保了对不存在于二部图上的信号的完美重建。严格地对所得的变换进行采样，并且根据其分析系数可以完美地重建图形信号。我们经验地探讨了字典原子的联合顶点频率局部化和分析系数的稀疏性，以及所提出的变换压缩分段平滑图形信号的能力。
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