A Novel and Fast Adaptive Compressive Sampling Matching

Pursuit Algorithm

Jian Zhao

, Tingting Lu

, Jian Jia

, Chao Zhang

, Weiwen Su

, Rui Wang

, Shunli Zhang

School of Information Science and Technology, Northwest University, Xian, PR China,710127

Deparment of Mathematics, Northwest University, Xi’an, PR China,710127

zjctec@nwu.edu.cn

Keywords: Compressive Sensing; Adaptive; Compressive Sampling Matching Pursuit.

Abstract: A weakness of compressive sampling is that it needs the information of sparsity to approximate the

compressible signal. In this paper, an fast iterative reconstruction algorithm called Adaptive Compressive

Sampling Matching Pursuit is presented to solve the problem mentioned above, which delivers the same

guarantees as the best optimization-based approaches and get rid of the dependence on the information of

sparsity. Experimental results also demonstrate that the image reconstructed performance of the proposed

algorithm is improved in terms of PSNR, SNR and reconstructed time, compared to Orthogonal Matching

Pursuit (OMP) and Compressive Sampling Matching Pursuit (CoSaMP).

1 INTRODUCTION

Compressive sensing (CS) (Donoho, 2006) (Candes

and Wakin, 2008) (Baraniuk, 2007) is a relatively

novel theory in signal sampling, which is based on

sparse or compressible signal. Reconstruction

algorithm (Candes et al, 2006) is one of the most

active and challenging part of compressive sensing,

which is of great significance to accurately

reconstruct the signal and verify the sampling

accuracy.

However, current reconstruction algorithms

based on compressive sensing also have drawbacks.

Matching Pursuit (MP) algorithm (Mallet and Zhang,

1993) needs to go through multiple iterations to

obtain convergence, since the results of each iteration

may be sub-optimal due to non-orthogonal projection

of the signal on the selected atom sets (measurement

matrix column vector). To overcome the drawback of

MP, TroPP J et al. proposed Orthogonal Matching

Pursuit (OMP) (TroPP and Gilbert, 2007). But

OMP’s theoretical guarantee which ensures accurate

reconstruction is weaker than the minimum

l -norm

approach, so not all signals can be reconstructed

accurately. On the basis of OMP, Needell et al.

(Needell and Vershynin, 2009) and Donoho et

al.( Donoho et al, 2009) proposed Regularized

Orthogonal Matching Pursuit (ROMP) and Stagewise

Orthogonal Matching Pursuit (StOMP), respectively.

Their computing speeds are faster and their

reconstruction complexities are lower, compared to

OMP; yet their properties are poor. As a result, M

must be large enough to obtain better reconstructed

performance of the signal. Needen et al. proposed

Compressive Sampling Matching Pursuit (CoSaMP)

(Needell and TroPPJ, 2009). The algorithm offers

rigorous bounds on computational cost and storage. It

is likely to be extremely efficient algorithm for

practical problems because it requires only matrix

vector-multiplies with the sampling matrix. These

algorithms are built on the basis of known sparsity

, yet the sparsity

is often unknown in a practical

application.

In this paper, we propose a improved

reconstruction algorithm named Adaptive-CoSaMP

based on CoSaMP, according to the prior information

that reconstruction algorithm needs sparsity of

sampling signal (Davenport,M.A. and Wakin,M.B.,

2010). Property measures, such as PSNR and

reconstruct time, are used to evaluate the performance

of the proposed algorithm. Simulation results of

Adaptive-CoSaMP are compared with that of

CoSaMP and OMP.

Be advised that papers in a technically

unsuitable form will be returned for retyping. After

returned the manuscript must be appropriately

modified.

312

Jia J., Zhao J., Zhang C., Lu T., Su W., Wang R. and Zhang S.

A Novel and Fast Adaptive Compressive Sampling Matching Pursuit Algorithm.

DOI: 10.5220/0006449603120317

In ISME 2016 - Information Science and Management Engineering IV (ISME 2016), pages 312-317

ISBN: 978-989-758-208-0

2 COMPRESSIVE SAMPLING

To enhance intuition, we focus on sparse and

compressible signals. For vectors x in





, the

“quasi-norm” (Candes and Romberg, 2007) is defined

sup ( ) { : 0}

xpxjx==≠ (2.1)

A signal x is called s-sparse if

sx ≤

. Compressible

signals are well approximated by sparse signals. In

compressive sampling theory, a sample is a linear

functional applied to a signal. The process of

collecting multiple samples is best viewed as the

action of a sampling matrix

Φ on the target signal. If

we take

samples or measurements of a signal in





, then dimension of the sampling matrix Φ is

mN×

The minimum number of measurements satisfies

2ms≥

on account of the following simple

argument. The sampling matrix must not map two

different s-sparse signals to the same set of samples.

Therefore, each collection of

columns from the

sampling matrix must be nonsingular. As a result,

some sparse signals are mapped to very similar sets

of samples, and it is unstable to invert the sampling

process numerically. Instead, Candes and Tao

proposed the stronger condition that the geometry of

sparse signals should be preserved under the action of

the measurement matrix (Needell and TroPPJ, 2009).

To quantify this idea, they defined the

r th restricted

isometry constant of a matrix

as the least number

for which

22 2

(1 ) +

δδ

−≤Φ≤（ 1 ）

whenever

r≤

(2.2)

We have written

•

for the

l vector norm.

When

< , these inequalities imply that each

collection of

r columns from Φ is nonsingular,

which is the minimum requirement for acquiring

(r/2)-sparse signals. When





≪1 , the sampling

operator very nearly maintains the

l distance

between each pair of (r/2)-sparse signals. In

consequence, it is possible to invert the sampling

process stably.

3 AN ADAPTIVE COMPRESSIVE

SAMPLING MATCHING

PURSUIT ALGORITHM

Compressive Sampling Matching Pursuit (CoSaMP)

is proposed by Candes and Donoho (Needell and

Vershynin, 2009). The CoSaMP algorithm selects the

reserved atom based on the known sparsity of the

approximation to be produced and removes a fixed

number of atoms combining backward thought, so

computing speed of each iteration of the CoSaMP

algorithm is slow to some extent. In this section, an

Adaptive-Compressive Sampling Matching Pursuit is

proposed based on CoSaMP. The algorithm gets rid

of the dependence on sparsity, reconstructs the

original signal through adaptively adjusting the step

size in iteration, and has less reconstruct time.

The algorithm is initialized with a trivial signal

approximation, which means that the initial residual

equals the unknown target signal. During each

iteration, Adaptive-CoSaMP performs five major

steps:

(1) Identification. The algorithm forms a proxy of the

residual from the current samples and locates the

largest components of the proxy.

(2) Support Merger. The set of newly identified

components is united with the set of

largest

components that appear in the current

approximation.

(3) Estimation. The algorithm solves a least-squares

problem to approximate the target signal on the

merged set of components.

(4) Pruning. The algorithm produces a new

approximation by retaining only the

t largest

entries in this least-squares signal approximation.

(5) Sample Update. Finally, the samples are updated

so that they reflect the residual, the part of the

signal that has not been approximated.

The main source code of the proposed algorithm

is summarized in Table 1.

4 EXPERIMENTAL RESULTS

AND DISCUSSION

In this section, the peak signal-to-noise ratio (PSNR)

and SNR (signal-to-noise ratio) are used to evaluate

the visual quality of the reconstructed image

PSNR is defined as

A Novel and Fast Adaptive Compressive Sampling Matching Pursuit Algorithm

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MSE

PSNR )

255

(log20

= , (4.1)

and SNR is defined as

MSE

jiF

SNR )

),(

(log20

= , (4.2)

where MSE is the mean square error between the

original image F and the reconstructed image

∧

. It

is given by

∑∑

−

)],(

),([

jiFjiF

MSE

. (4.3)

To further evaluate the effectiveness of the

proposed algorithm, the proposed algorithm is

compared with existing algorithm proposed in

reference (TroPP and Gilbert, 2007) and in reference

(Needell et al, 2009). Figure 2, Figure 3, Figure 3,

Figure 4 and Table 2 show the comparative results of

two test images：

Lena (

256256 × ) and Barbara( 256256 × ).

Table 1 The main source code of the Adaptive-CoSaMP algorithm

Adaptive-CoSaMP algorithm

Input: Measurement matrix Φ , Sampling vector

, halting criterion

, iteration

Output: An sparse approximation

of the target signal

Residual

ry=

Initial step

1t =

Index set of values A=

∅

S =∅

stage=0

0k =

repeat

1kk←+

() * ( 1)kk

−

←Φ {Form signal proxy}

()

sup ( )

Spg← {Identify large components}

AS←∪

{Merge supports}

*1*

|(( ) )

−

←ΦΦ Φ

{Signal estimation by least-squares}

b ←

b← {Prune to obtain next approximation}

(1)

rr x

−

=−Φ {Update current samples}

tage stage=+

*t stage t=

until halting criterion true

Table 2 SNR, PSNR and reconstructed time comparison between our proposed method and method in (TroPP and Gilbert,

2007)and(Needell et al, 2009) for host images in the case of compression ratio respectively 0.4 and 0.5

Compression ratio (0.5) Compression ratio (0.4)

Reference

(TroPP and

Gilbert ,

2007)

Reference

(Needell

et al,

2009)

Proposed

algorithm

Referenc

e (TroPP

and

Gilbert ,

2007)

Referenc

e(Needell

et al,

2009)

Proposed

algorithm

Lena

PSNR 31.59 30.85 33.84 25.18 26.45 29.09

SNR 16.00 17.39 18.99 8.31 9.22 9.5

Time 69.35 70.72 55.19 18.64 8.67 7.08

Barbara

PSNR 29.79 31.06 31.66 21.24 20.48 21.22

SNR 14.46 16.56 18.11 6.54 6.54 8.79

Time 79.24 75.12 57.64 21.94 9.15 7.64

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(a) (b)

Figure 1.Original images:(a) Lena image; (b) Barbara image

(a) OMP (b) CoSaMP (c)Adaptive-CoSaMP

Figure 2.Reconstructed images of Lena image in the case of compression ratio 0.5

(a) OMP (b) CoSaMP (c)Adaptive-CoSaMP

Figure 3.Reconstructed images of Barbara image in the case of compression ratio 0.5

(a) OMP (b) CoSaMP (c)Adaptive-CoSaMP

Figure 4.Reconstructed images of Lena image in the case of compression ratio 0.4

From the Table 2, we can observe that our

algorithm has higher SNR values and PSNR values

for all the compression ratio of reconstruction,

compared to algorithms in (TroPP and

Gilbert , 2007)

and. (Davenport,M.A. and Wakin,M.B., 2010) And

the reconstructive time of the scheme is less than that

of reference (TroPP and Gilbert , 2007) and

reference(Needell et al, 2009). From Figure 2, Figure

3, Figure 4 and Figure 5, it is not hard to observe that

A Novel and Fast Adaptive Compressive Sampling Matching Pursuit Algorithm

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(a) OMP (b) CoSaMP (c)Adaptive-CoSaMP

Figure 5.Reconstructed images of Lena image in the case of compression ratio 0.4

proposed algorithm has better visual qualities of the

reconstructed images, compared to algorithms in

reference (TroPP and Gilbert, 2007) and

reference(Needell et al, 2009). Hence we can

conclude that the proposed algorithm-Aaptive-

CoSaOMP algorithm is superior to other algorithms

for reconstructing signal .

5 CONCLUSION

In this paper, we discuss compressive sampling theory

which is one of the most active and challenging

subject in signal processing in recent years. Taking

advantage of the greedy iterative algorithm often-

used in compressive sensing, an improved matching

pursuit algorithm—Adaptive-CoSaMP algorithm,

which is based on compressive sampling matching

pursuit algorithm, is proposed. The proposed

algorithm not only allows a accurate reconstruction of

signal in the case of unknown sparsity K, but also can

gradually update to approximate the original signal by

setting the step value. Simulation results also shows

that the improved algorithm has significantly

improvement in reconstruction effect of the image

whether from the visual effects of the reconstructed

image or from the PSNR value of the reconstructed

image.

ACKNOWLEDGEMENTS

This work was supported by National Natural Science

Foundation of China (No. 61379010，61572400) and

Natural Science Basic Research Plan in Shaanxi

Province of China (No.2015JM6293).

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