Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 286590, 9 pages

http://dx.doi.org/10.1155/2015/286590

## Compressive Sensing Approach in the Hermite Transform Domain

University of Montenegro, Faculty of Electrical Engineering, Dzordza Vasingtona bb, 81000 Podgorica, Montenegro

Received 20 August 2015; Accepted 16 November 2015

Academic Editor: Yuqiang Wu

Copyright © 2015 Srdjan Stanković et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

The Hermite polynomials and Hermite functions have attracted the attention of researchers in various fields of engineering and signal processing [1–7], such as in quantum mechanics (harmonic oscillators), ultra-high band telecommunication channels, and ECG data compression using Hermite functions representation of the QRS complexes. A set of Hermite functions forming an orthonormal basis is suitable for approximation, classification, and data compression tasks [3]. Since the Hermite functions are eigenfunctions of the Fourier transform, time and frequency spectra are simultaneously approximated. Here, we are especially interested in a class of signals that are sparse in the Hermite transform domain. Note that, generally, such signals are not sparse in the Fourier transform domain. In the light of compressive sensing (CS) theory [8–12], we propose the method for efficient reconstruction of signals from its incomplete set of samples using the Hermite transform. The number of Hermite functions used in this approach is much smaller compared to the original length of the signal. The proposed approach is useful in the applications where the significant information is missing and the total signal reconstruction is required. Note that the large amount of missing signal samples may occur as a consequence of the compressed sampling strategy but also as a consequence of discarding damaged signal parts [13–15]. The theory is illustrated through examples showing that the Hermite transform based CS for certain types of signals can outperform the Fourier transform related reconstructions. Furthermore, in analogy with the time-frequency analysis based on the Fourier transform, the short-time Hermite transform is defined as a linear representation that reveals the local behavior of windowed signal parts. If the signal components are of short duration, then the short-time Hermite transform is more suitable for compressive sensing than the standard Hermite transform. Finally, the possibility of combining different transforms depending on the signal characteristics is explored.

The paper is organized as follows. The theory behind the Hermite transform and the fast method for Hermite coefficients calculation is given in Section 2. The formulation of compressive sensing approach in the Hermite transform domain is given in Section 3. The possibilities to exploit other sparsity domains based on the short-time Hermite transform and combined transform are presented in Section 4. The experimental evaluation of the proposed approach is given in Section 4, while the concluding remarks are given in Section 5.

#### 2. Hermite Transform

The Hermite functions provide good localization and the compact support in both time and frequency domain [4]. The th order Hermite function is defined as follows: The Hermite functions provide an orthonormal basis set for an optimal representation of different signals using the fewest number of basis functions. Signal expansion into Hermite functions, known as the Hermite transform, has been used for both 1D and 2D signals in various applications. The Hermite expansion for a signal can be defined as follows:where are the Hermite functions and is the number of functions used for the approximation. The number of Hermite functions could be usually much smaller than the number of signal samples (). The Hermite coefficients can be defined by using the Hermite polynomials as follows:where represents the Hermite polynomial. An efficient procedure for calculation of Hermite coefficients can be done by applying the Gauss-Hermite quadrature [5]: where are zeros of Hermite polynomials. By using the Hermite functions instead of polynomials, a simplified expression is obtained: The constants are calculated as follows:

#### 3. Compressive Sensing Formulation in the Hermite Transform 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 [8]. Hence, based on a small set of acquired measurements, the entire signal needs to be reconstructed. The missing samples in compressive sensing generally cannot be recovered using standard interpolation methods due to the complexity of nonstationary signals in real applications. Namely, the interpolation methods such as polynomial fit, cubic spline interpolation, or similar usually assume certain model function, which is mostly inappropriate for time domain signal modeling. Therefore, the compressive sensing reconstruction is formulated in the literature as an optimization problem (rather than interpolation) which reconstructs the signal by finding the sparsest transform domain solution corresponding to the available small set of samples.

In the CS context, we are dealing with a small set of randomly chosen samples of . Let us assume that we have only out of available samples ( is the total number of signal samples and ). The vector of available measurements is denoted as . Now we may write where is original full signal (of length ) written in the vector form, while () is the measurement matrix. The original signal can be expressed using the Hermite transform as follows: where is the vector of Hermite transform coefficients, while is the inverse Hermite transform matrix of size (): The direct Hermite transform matrix is given by . In the extended form, (9) can be written as follows:The Hermite basis functions are calculated using the fast recursive realization defined as follows:According to (8) and (9), we have

The reconstructed signal can be obtained as a solution of linear equations with unknowns. The system is undetermined and can have infinitely many solutions. Now we assume that the signal is sparse in the Hermite transform domain. It means that the observed signal can be efficiently represented by a very small number of Hermite expansion coefficients, such that . Therefore, the optimization based mathematical algorithms should be employed to search for the sparsest solution. A near optimal solution is achieved by using the norm based minimization as follows: where is the Hermite transform vector of signal .

In order to solve the previous minimization problem, first we need to calculate the initial Hermite transform using the available set of samples with the time support . Therefore, we observe the signal in the following form: The initial vector of Hermite transform coefficients can be then calculated using Hermite functions () as follows: where contains only the columns of that correspond to instants . Alternatively, we can write In order to determine the signal support in the Hermite transform domain, the initial vector of the Hermite transform coefficients is compared by the threshold : The exact values of Hermite coefficients at positions selected in vector are obtained as a solution of the CS problem:The CS matrix is obtained from the inverse Hermite transform matrix , using columns that correspond to frequencies and rows corresponding to measurements with support . The system is solved in the least square sense as follows:where () denotes the conjugate transpose operation.

*Analysis of Components Reconstruction*. Let us consider the isometry property of Hermite transform matrix : The previous equation can be written as follows:Now, observe the first term on the right side given by , particularly the sum of squared values of different Hermite functions. Figure 1 illustrates the sum of squared values of different Hermite functions calculated in the zeros of Hermite polynomials for = (128, 100, 70 and 50). Unlike in the case of the Fourier transform basis, where the sum of absolute values of complex exponential basis would be constant for any frequency , from Figure 1 we can note an approximate low-pass characteristic of the curves, meaning that the lower-order coefficients are favored compared to the higher order coefficients. Consequently, when applying the threshold for components detection, it would be easier to detect a set of low-order coefficients than the high-order ones.