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Forward-backward pursuit (FBP) algorithm is a novel two-stage greedy approach. However once its forward and backward steps were determined during iteration, it would make computing time increased and affected the reconstruction efficiency. This paper presents a algorithm called forward-backward pursuit algorithm based on weak selection (SWFBP) by introducing threshold strategy into FBP algorithm, and in view of that in the first few iterations, most of the atoms which are selected are right, so this part of atoms are directly incorporated into support set instead of using backward strategy to reduce them. Flexible forward and backward steps accelerate the speed of atom selecting and improve the reconstruction accuracy. We compared SWFBP and FBP algorithm via one-dimensional signal and two-dimensional image reconstruction experiments. The simulation results demonstrate that compared with FBP, SWFBP algorithm has superior performance, including higher PSNR, faster computing speed and lower recovery time.

Compressed Sensing (CS) [

The main research of CS theory includes signal sparse transformation, design of measurement matrix and signal reconstruction algorithm. At present, the common algorithms used in CS theory are mainly divided into two categories: one is linear programming algorithm based on optimization norm l 1 , which approximates the original signal by converting the non-convex problem into convex problem. It mainly includes Basis Pursuit (BP) [

On the basis of TST algorithm, N. B. Karahanoglu and H. N. Erdogan proposed Forward-Backward Pursuit (FBP) in reference [

To address the above issues of FBP, forward-backward pursuit algorithm based on weak selection (SWFBP) was proposed by introducing threshold strategy into FBP algorithm to calculate the forward step, by which can make the forward step flexible. And for the first few iterations, most of the atoms which are selected are right, and they don’t need backward strategy, so this part of atoms would be directly incorporated into support set. Through above two methods, SWFBP has flexible forward and backward steps which would bring down the computing complexity and improve the reconstruction performance. The simulation results show that SWFBP is a more efficient greedy pursuit algorithm without prior information of the sparsity compared with FBP.

The core of CS theory is choosing an appropriate measurement matrix, by which can project a high-dimensional sparse signal to a low dimensional space, then use reconstruction algorithm to reconstruct the original high-dimensional signal from low dimension projection value. Specifically, suppose that x is a K-sparse, N-dimensional digital signal, that is, only K of x are valid and nonzero. We are interested in the case when K ≪ N . One can reconstruct signal x with the following equation:

y = Φ x (1)

where y is an M-dimensional vector indicating the measurement value. Φ represents an M ∗ N matrix (M < N), which called measurement matrix as well. Reference [

min ‖ x ‖ 0 subject to y = Φ x (2)

However, l 0 minimization problem is NP-hard, while greedy pursuit algorithms provide a favorable tool to solve this problem for approximate solutions. Reference [

min ‖ x ‖ 1 subject to y = Φ x (3)

FBP algorithm is a greedy pursuit algorithm based on l 0 norm minimization problem, which extends the signal support set by forward and backward step. It receives wide attention due to its high reconstruction accuracy and the feature without prior information of the sparsity. The first stage of FBP algorithm is the forward step which expands the support set by indices of α largest magnitude elements in Φ T r l − 1 , where we call α the forward step size. The second step is removing β smallest magnitude projection coefficients in ‖ y − Φ T ˜ l w ‖ 2 . Similar to α , β is referred to as the backward step size. The forward step size is chosen larger than the backward step size, hence the support estimate is enlarged by α − β atoms at each iteration.

When the sparse degree of signal is relatively small, the residual will constantly decrease by the process of iteration until it meet the termination conditions ‖ r l ‖ 2 ≤ ε ‖ y ‖ 2 ( ε is the termination threshold of iteration). As well, when the sparse degree is a large one, then it will exit the iteration by the termination conditions | T l | ≥ l max . A schematic diagram of the FBP algorithm is depicted in

Algorithm 1: FBP Algorithm

Input: Measurement matrix Φ , measurement vector y

Define: Forward step α , backward step β , largest number of iterations l max , termination threshold of iteration ε

Initialize: T 0 = ∅ , r 0 = y , l = 0

while true do

l = l + 1

Forward update:

T f = arg max J : | J | = α ‖ Φ J T r l − 1 ‖ 1

T ˜ l = T l − 1 ∪ T f

w = arg min w ‖ y − Φ T ˜ l w ‖ 2

Backward update:

T b = arg min J : | J | = β ‖ w J ‖ 1

T l = T ˜ l − T b

Projection:

w = arg min w ‖ y − Φ T l w ‖ 2

r l = y − Φ T l w

if ‖ r l ‖ 2 ≤ ε ‖ y ‖ 2 or | T l | ≥ l max then

break

end if

end while

x ˜ = 0

x ˜ T l = w

Output: x ˜

The basic framework of SWFBP algorithm is similar to FBP algorithm. In the first few iterations, we select atoms by weak selection strategy. As most of these atoms are right, so join these atoms into support set directly. For the following iterations, first of all, through threshold strategy to expand the forward support set, and the forward step α is equal to the length of this set. Then calculate the orthogonal projection of the measurement vector y on this support set. Next, use α to calculate the backward step β and delete the indexes of β smallest orthogonal projection coefficients. Last, calculate the new projection coefficients and update the residuals. Terminal condition of iteration is the same as FBP algorithm. A schematic diagram of the SWFBP algorithm is depicted in

Algorithm 2: SWFBP Algorithm

Input: Measurement matrix Φ , measurement vector y

Define: Threshold parameter θ , value of step difference δ , largest number of iterations l max , threshold of iteration numbers γ , termination threshold of iteration ε

Initialize: T 0 = ∅ , r 0 = y , l = 0

while true do

l = l + 1

Forward update:

u = a b s ( Φ T r l − 1 )

J = find ( u ≥ θ ∗ max (u))

the forward step α = length (J)

if l < = γ then

T l = T l − 1 ∪ J

else

Backward update:

T ˜ l = T l − 1 ∪ J

w = arg min w ‖ y − Φ T ˜ l w ‖ 2

the backward step β = α − δ

T b = arg min J : | J | = β ‖ w J ‖ 1

T l = T ˜ l − T b

end if

Projection:

w = arg min w ‖ y − Φ T l w ‖ 2

r l = y − Φ T l w

if ‖ r l ‖ 2 ≤ ε ‖ y ‖ 2 or | T l | ≥ l max then

break

end if

end while

x ˜ = 0

x ˜ T l = w

Output: x ˜

Simulation environment was MATLAB R2015b. We experiment with one- dimensional sparse signal (Gaussian sparse signal and 0 - 1 sparse signal) and two-dimensional international standard test image lena.bmp by using FBP and SWFBP algorithm. Measurement matrix Φ obeys Gaussian distribution.

Experimental parameters are setting as follows: length of the sparse signal N = 256, sampling numbers M = 102, sparsity level K = 20, largest number of iterations l max = 10 , termination threshold of iteration ε = 10 − 6 , threshold parameter θ = 0.95 , value of step difference δ = 2 . For FBP algorithm, the forward step α = 4 , 10, 20 separately, and corresponding to α , the backward step β = 2 , 8, 18. Simultaneously, for SWFBP algorithm, the relationship between α and β is β = α − 2 . For each combination of (K, M), do 500 times independent experiments to compare the frequency of exact reconstruction, average recovery time and average normalized mean-squared error (ANMSE). The condition of exact reconstruction is described as follow:

‖ x − x ^ ‖ 2 / ‖ x ‖ 2 ≤ 10 − 8 (4)

And ANMSE was calculated by Equation (5).

ANMSE = 1 500 ∑ i = 1 500 ‖ x i − x ^ i ‖ 2 2 / ‖ x i ‖ 2 2 (5)

Among (5), x ^ i represents the reconstruction signal for x i .

In SWFBP algorithm, parameters involved are α , β , θ and δ . Among them, α and β are mainly decided by θ . In this part, we experiment with different θ under the condition of β = α − δ and δ = 2 for SWFBP algorithm.

In order to further illustrate the reconstruction performance of SWFBP algorithm, in this part, we experiment with two-dimensional standard image 256 × 256 Lena. The parameters are setting the same as part 3.1. We choose two indicators to judge the reconstruction performance of algorithm: Recovery time, Peak Signal to Noise Ratio (PSNR), through the following two formulas to calculate PSNE.

PSNR = 20 ∗ lg ( 255 / M S E ) (6)

MSE = 1 M ∗ N ∑ m = 1 M ∑ n = 1 N ( x ^ ( m , n ) − x ( m , n ) ) 2 (7)

Indicators | Recovery time (s) | PSNR (dB) | ||||||
---|---|---|---|---|---|---|---|---|

Sampling rate (M/N) | 0.3 | 0.4 | 0.5 | 0.6 | 0.3 | 0.4 | 0.5 | 0.6 |

FBP | 3.075 | 5.523 | 6.584 | 7.407 | 18.221 | 24.246 | 26.695 | 28.768 |

FBP | 3.301 | 5.885 | 7.733 | 9.585 | 19.468 | 24.319 | 26.892 | 29.010 |

FBP | 4.103 | 5.967 | 8.479 | 10.727 | 21.011 | 24.455 | 26.931 | 29.065 |

SWFBP | 1.631 | 2.126 | 2.691 | 2.979 | 22.934 | 25.559 | 27.141 | 29.523 |

On the basis of FBP algorithm, in this paper, SWFBP algorithm was proposed. By adapting a simple threshold strategy to FBP algorithm, the proposed algorithm SWFBP can be flexibly the forward and backward steps, and thus to combine the high reconstruction accuracy and flexible atom selecting mechanism of weak selection together. The simulation results demonstrate that SWFBP algorithm provides a better performance and lower computation cost compared to FBP algorithm. In particular, while the sparse signal obeys Gaussian distribution, the exact reconstruction frequency and accuracy are better than FBP, and while the sparse signal obeys 0 - 1 distribution, the exact reconstruction frequency and precise are approximate to FBP’s. Reconstruction results of two-dimensional images show that the reconstruction performances of SWFBP have certainly improved while its recovery time is decreased obviously. Despite of this, the atom selecting method is still not match enough so far, and we hope to find a solution to perfect the atom selecting mechanism in the future.

This work was supported by the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130031110032 and No. 20130031110033).

Sun, G.L., Zhang, Y.Y. and Jia, J. (2017) Improvement of Forward-Backward Pursuit Algorithm Based on Weak Selection. Journal of Computer and Communications, 5, 9-19. http://dx.doi.org/10.4236/jcc.2017.51002