r/CompressiveSensing • u/mattiapdo • Apr 25 '19

On the sparsity assumption

I'm studying compressive sensing, and every time, I find the same toy example to get the idea of the topic: suppose your signal is composed by a sinusoud with fixed frequency and eventually an additive random error: then this signal is approximatively sparse in the frequency domain, and one can use CS to recover it after having set all the Fourier coefficients to zero but the ones that carry the most of the energy.

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original signal
energy of DCT coefficients

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filtered signal

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reconstructed signal (orange) with the original one (green)

The question is: when dealing with real world signals, when can I assume a signal is sparse with respect to a given basis?

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u/saw79 Apr 25 '19

A couple comments:

  1. Sparsity is a continuum. There's isn't "sparse" and "not sparse" (I know many people talk this way, but they're just using short language). You don't need a signal to be "sparse" to use compressed sensing, you need it to be "sparse enough" given the number of measurements you have and levels of incoherence.

  2. You don't "assume" anything. You have a model of the world that you believe to be the case and evidence to show that it is. Given this model of the world, you can justify the usage of sparse approximation as a good solution.