We discussed the method of Continuous Basis Pursuit introduced in recent papers Ekanadham et al. for decomposing a signal into a linear combination continuously translated copies of a small set of elementary features. A standard method for recovering the time-shifts and amplitudes for these features is to introduce a dictionary consisting of many shifted copies of the features, and then use basis pursuit or a related method to recover a representation of the signal as a sparse linear combination of these shifted copies of the elementary waveforms. Accurately representing the signal requires relatively close spacing of the dictionary elements; however, such close spacing yields highly correlated dictionary elements, which decreases the performance of basis pursuit and related recovery algorithms.
With Continuous Basis Pursuit, Ekanadham et al. circumvent this problem by first augmenting the dictionary to allow for continuous interpolation, either using a first- or second- order Taylor interpolation or using an interpolation based on trigonometric splines. These augmentations increase the ability to accurately represent the signal as a sparse sum of features, without introducing too much additional correlation in the set of dictionary vectors, and the smooth interpolation allows the recovery of continuous, rather than discrete, time shifts. This kind of decomposition of a signal into (continuously) shifted copies of a few basic features is useful in the spike-sorting problem, for example.
- Ekanadham, Chaitanya, Daniel Tranchina, and Eero P. Simoncelli. “Recovery of sparse translation-invariant signals with continuous basis pursuit.” Signal Processing, IEEE Transactions on 59.10 (2011): 4735-4744.
- Ekanadham, Chaitanya, Daniel Tranchina, and Eero Simoncelli. “A blind deconvolution method for neural spike identification.”