This paper suggests an upside-down tree-based orthogonal matching pursuit (UDT-OMP) compressive sampling signal reconstruction method in wavelet domain. An upside-down tree for the wavelet coefficients of signal is constructed, and an improved version of orthogonal matching pursuit is presented. The proposed algorithm reconstructs compressive sampling signal by exploiting the upside-down tree structure of the wavelet coefficients of signal besides its sparsity in wavelet basis. Compared with conventional greedy pursuit algorithms: orthogonal matching pursuit (OMP) and tree-based orthogonal matching pursuit (TOMP), signal-to-noise ratio (SNR) using UDT-OMP is significantly improved.

Compressive sampling (CS) [

MP algorithm is computationally efficient and often features good performance; however, when the basis

In this paper, we present an improved OMP signal recovery algorithm by employing an upside-down tree structure of signal in wavelet domain (we refer to this tree-based algorithm as UDT-OMP). Proposed algorithm is evaluated by signal-to-noise ratio (SNR) as a measure of quality of reconstructed signal. We have compared the performance of UDT-OMP with OMP and tree-based orthogonal matching pursuit (TOMP) with SNR as a function of number of measurements.

CS is a novel sampling paradigm that goes against the common wisdom in data acquisition. CS states that a sparse or compressible signal can be recovered from a small salient set of random projections. To make it possible, there are two fundamental premises [

CS also extends to that so-called compressible signals that are not exactly sparse but can be closely approximated as such (i.e., wavelet coefficients of signal and image). Sparse signals have coefficients

Since

MP approximation is improved by orthogonalizing the directions of projection with a Gram-Schmidt procedure; it is known as orthogonal matching pursuit (OMP). This orthogonalization was introduced in [

In CS, both BP and OMP reconstruct signal based on sparsity without considering any other particular structure that may exist in the signal. The wavelet transform of a piecewise smooth signal (many punctured real-world phenomena give rise to such signal), an important subclass of sparse signals, yields a sparse, structured representation of signals in this class: the significant coefficients tend to form a connected subtree of the wavelet coefficient tree. Figure

Example of sparse representations in the wavelet domain: (a) piecewise smooth signal; (b) wavelet transform of piecewise smooth signal, only few coefficients are significant and they are compressible or near sparse; (c) the significant coefficients are well organized in tree structure across the scales; (d) binary tree for piecewise smooth signal, the significant wavelet coefficients arise from the discontinuities in the signal (the black circles denote the large wavelet coefficients).

In this work, we only focus on 1D signals, and similar arguments apply for 2D and multidimensional signals. Consider a signal

According to the statistical analysis in wavelet domain, wavelet coefficients have the following two properties [

This tree structure was exploited by previous CS reconstruction algorithms known as iterative reweighted

In previous tree-based CS reconstruction algorithms, trees are assumed to be upright, which means that those significant coefficients propagate from coarser scale to finer scale. Examining the tree structure in Figure

The input of UDT-OMP is a dictionary

UDT-OMP evaluates the projection of each single column vector and searches for the maximum projection. According to the columns associated with the maximum projections, UDT-OMP constructs an upside-down subtree upward coarser scale with depth of

In (

UDT-OMP algorithm consists of two steps. We first limit our search space in the columns associated with the scaling coefficients and then in the columns associated with the wavelet coefficients. Let _{0} to be empty. In step 1, we select all columns of

All vectors in _{0}. At end of step 1, the residual is updated by

In the first iteration, we initialize the counter

According to the maximum projection position

The pseudocodes of UDT-OMP are described as in Algorithm

Inputs: CS matrix

iterative number

Outputs: approximated signal

Initialize: ^{J}

(1)

(2)

(3)

(4) construct upside-down tree

(5)

(6)

(7)

return

For simplicity, sparsity level

To demonstrate the advantage of upside-down tree structure, we evaluated UDT-OMP algorithm by comparing the performance of OMP, TOMP, and UDT-OMP. We used piecewise smooth signal of length

In the first experiment, we reconstructed the signal from 300 random measurements by OMP, TOMP, and UDT-OMP, respectively. Figure

An example piecewise smooth signal of length 1024 and its reconstructions from 300 random measurements using OMP, TOMP, UDT-OMP. (a) Original signal. (b) OMP reconstructed signal, SNR = 27.63 dB. (c) TOMP reconstructed signal, SNR = 34.57 dB. (d) UDT-OMP reconstructed signal, SNR = 39.62 dB.

In the second experiment, we reconstructed the test signal from different numbers of measurements using OMP, TOMP, and UDT-OMP. Numbers of measurements over the range of 100 to 400 in increment of 30 are tested. 100 trials are repeated for each specific number, and the averaged reconstruction SNR is plotted in Figure

Comparison between the performance of OMP, TOMP, and UDT-OMP.

This paper introduced an upside-down tree structure weighting scheme for OMP algorithm in wavelet domain CS signal reconstruction. The UDT-OMP weights the nodes that connect in the subtree with significant values. Different from tree structure presented in the previous CS recovery algorithms, UDT-OMP constructs the tree using upside-down structure rather than upright structure. UDT-OMP weights the projections that should have larger coefficients. The experimental results show that our method outperforms OMP and TOMP, and it can achieve more accurate approximation in piecewise smooth signal reconstruction. In this paper, we only considered the constant weight value. We can also adopt different weight values at different scales.

The work is partially supported by National Natural Science Foundation of China (no. 60827001) and China Scholarship Council (no. 2009607046).