Compressive sensing has attracted significant interest of researchers providing an alternative way to sample and reconstruct the signals. This approach allows us to recover the entire signal from just a small set of random samples, whenever the signal is sparse in certain transform domain. Therefore, exploring the possibilities of using different transform basis is an important task, needed to extend the field of compressive sensing applications. In this paper, a compressive sensing approach based on the Hermite transform is proposed. The Hermite transform by itself provides compressed signal representation based on a smaller number of Hermite coefficients compared to the signal length. Here, it is shown that, for a wide class of signals characterized by sparsity in the Hermite domain, accurate signal reconstruction can be achieved even if incomplete set of measurements is used. Advantages of the proposed method are demonstrated on numerical examples. The presented concept is generalized for the short-time Hermite transform and combined transform.

The Hermite polynomials and Hermite functions have attracted the attention of researchers in various fields of engineering and signal processing [

The paper is organized as follows. The theory behind the Hermite transform and the fast method for Hermite coefficients calculation is given in Section

The Hermite functions provide good localization and the compact support in both time and frequency domain [

Generally, the compressive sensing scenarios are focused on the new sampling strategy, which results in a large number of randomly missing samples compared to the standard sampling methods [

In the CS context, we are dealing with a small set of randomly chosen samples of

The reconstructed signal

In order to solve the previous minimization problem, first we need to calculate the initial Hermite transform using the available set of

Sum of squared values of Hermite functions for different

Let us assume that

In the context of compressive sensing, the STHT allows us to define two types of CS problem. A common CS problem is defined by assuming that the missing samples appear in the time domain, while the sparsity is exploited in the transform domain. However, the missing samples may also appear in the STHT domain, after applying certain filter forms such as the

(1) Assume that CS procedure is done in the time domain, such that the measurements are taken from each windowed signal part:

(2) Let us observe the missing values in the STHT domain, where the measurements vector is denoted by

On the basis of the STHT, we can also define the short-time combined transform as follows:

We can again observe the set of measurements corresponding to different signal parts:

In order to illustrate efficiency of the proposed method, let us observe a signal that is sparse in the Hermite transform domain. The time domain signal is shown in Figure

After calculating the initial Hermite transform using available signal samples, the threshold is applied to select signal support in the Hermite domain. The threshold is set empirically to

The selected components in the Hermite transform domain are shown in Figure

The original signal: (a) time domain representation, (b) Hermite transform of signal calculated using

Hermite coefficients of signal with missing samples -○, and the reconstructed Hermite coefficients -×.

The reconstructed signal: (a) time domain representation, (b) the absolute error between original and reconstructed signal, (c) for comparison: reconstructed signal based on the Fourier transform reconstruction approach (-°) and the corresponding error (solid line).

In this example we have observed different sparse signals (having 10 components) in the Hermite domain through the 1000 repetitions of the previously described procedure. The aim is to test how the accuracy of the proposed method changes for different number of missing samples. Namely, the number of missing samples was increased in steps of 10% starting from zero up to 80% of missing samples. For each instance on the

Additionally, the proposed approach is tested in the presence of external additive Gaussian noise. Namely, the MSE is calculated for different values of SNR (for each SNR, we assume 1000 realizations of random noise). The results are presented in Figure

(a) MSE calculated for 1000 realizations and for different number of missing samples, (b) MSE calculated for different values of SNR in 1000 realizations of external noise.

This example illustrates the concept of combined Hermite-Fourier transform and CS reconstruction. Observe the signal with the total length of

Missing samples and available measurements in time domain.

(a) Original short-time combined transform of full length signal (FT: Fourier transform, HT: Hermite transform), (b) short-time combined transform of available measurements.

The threshold based components selection and signal reconstruction (in analogy to (

Selecting the components of interest by applying threshold to the following: (a) FT components and (b) HT components.

Reconstructed signal components.

The possibility of using the Hermite transform in compressive sensing applications was explored in this work. The compressive sensing setup and signal reconstruction approach in the Hermite transform domain were defined. A simple and fast procedure for the total reconstruction of signals using selected Hermite transform coefficients provides the results very close to the original signal even when we deal with significant missing information. It is important to emphasize that the proposed concept in the Hermite transform domain can be also combined with other known compressive sensing solvers. The entire concept is generalized and extended by defining the short-time Hermite transform as well as the short-time combined transform. These two transforms open more possibilities to apply the compressive sensing approach in different scenarios.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Montenegrin Ministry of Science, Project Grant “New ICT Compressive Sensing Based Trends Applied to: Multimedia, Biomedicine and Communications” (ACRONYM: CS-ICT).