Abstract
In this paper, we aim to prove the possibility to reconstruct a bicomplex sparse signal, with high probability, from a reduced number of bicomplex random samples. Due to the idempotent representation of the bicomplex algebra, this case is similar to the case of the standard Fourier basis, thus allowing us to adapt in a rather easy way the arguments from the recent works of Rauhut and Candés et al.
Highlights
An important problem in signal processing is the possibility of reconstruction of a given signal from a few of its samples
The main idea behind Compressed Sensing (CS) is that under certain conditions on the sampling matrix one can obtain a sparse reconstruction of the signal, i.e., if we have an a-priori knowledge that the representation of the signal in a given dictionary has only a few non-zero coefficients we can reconstruct it from those few samples by means of a simple basis pursuit procedure
We propose to follow the idea of the reconstruction of the signal by basis pursuit via a 1-minimization procedure [2, 4, 5]
Summary
An important problem in signal processing is the possibility of reconstruction of a given signal from a few of its samples. The condition on the sampling matrix, or RIP condition, states that the matrix behaves almost like an isometry This represents a problem since in practice this is almost never the case In [12] the authors point out that the so-called IHS-color spaces representation (i.e., Intensity-Hue-Saturation) which has broad applications, in human vision, can be mathematically represented by having values in bicomplex numbers. This means that a color image based on this coding scheme can be represented by a function depending on two variables and taking. Section 4. is dedicated to the detailed proof of the main theorems
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