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In this paper, a new method of combination single layer wavelet transform and compressive sensing is proposed for image fusion. In which only measured the high-pass wavelet coefficients of the image but preserved the low-pass wavelet coefficient. Then, fuse the low-pass wavelet coefficients and the measurements of high-pass wavelet coefficient with different schemes. For the reconstruction, by using the minimization of total variation algorithm (TV), high-pass wavelet coefficients could be recovered by the fused measure ments. Finally, the fused image could be reconstructed by the inverse wavelet transform. The experiments show the proposed method provides promising fusion performance with a low computational complexity.

Image fusion is the technique that integrates complementary and redundant information of multiple images to obtain a composite one, which contains more comprehensive description than any of the individual image. As a result of the processing, the fused image is more useful for human and machine perception or further image processing tasks such as object detection and recognition. By now, many well-known fusion algorithms have been proposed [

In recent years, a new technique for simultaneous data sampling and compression known as compressive sensing (CS) [

For image fusion in CS, one natural way is to perform traditional fusion among the reconstructed multiple images after separate recover of each image from the measurements. However, in order to save storage space and reduce the computational complexity, a better method is to directly perform fusion on the measurements, and then to reconstruct the fused image from the fused measurements. Recently, several different fusion strategy based on CS which have been proposed, e.g., a simple maximum selection rule [

In this paper, we proposed an image fusion scheme in a general CS framework. Specifically, we just take single layer wavelet decomposition for image and only measured the high-pass wavelet coefficients of the image with Fast Walsh Hadamard Transform [

This paper is organized as follows. In Section 1, we provide a brief review of CS. The proposed fusion scheme based on CS is described in Section 3. Some experimental results and a discussion are given in Section 4. Finally, Section 5 ends this paper with a conclusion.

Consider a real-valued, finite-length, one-dimensional, discrete-time signal

where

In CS, we can not measure or encode

where

To solve the inverse transform from Equation (2), some non-linear recovery algorithms for such ill-posed problems have been developed [

where x is an

In order to obtain the measurements of the original images, a type of structured measurement matrices called Fast Walsh Hadamard Transform matrices will be used [

And in general

This is also known as the Hadamard-ordered Walsh Hadamard matrix. Other orders, such as sequency order, dyadic order, and so on can be obtained by reordering the rows of the Hadamard matrix. Walsh Hadamard matrices 'in various orders have recently received growing concern due to the extensive application in engineering field.

To achieve the Fast Walsh Hadamard transform matrix, it is necessary to understand the “so-called Kronecker product”.

For any two matrices

In order to better understand the meaning of the Kronecker product, we define two new operators vec and mtx. The vec operator is to make all the columns of a matrix into a vector, and mtx is the inverse operator of vec that separates the vector into several equal-length vectors and forms a matrix.

According to “Kronecker product” theorem, Matrix

where

Detailed derivation process in [

For any two vector x with the length of

Due to the Basic KP theorem, it follows

A implementation of the fast

Let

In each wavelet sub-band, we set the same sampling rate, so the number of measurements for each sub-band are the same.

Multi-resolution wavelet decomposition has a noticeable advantage in the representation of a signal. In our fusion scheme, we take a single-level CDF-97 wavelet transform to decompose the original image into approximation coefficients and the detail coefficients. High frequency sub-band include the main energy of the original image and can considered to be sparse, but the scale coefficients low frequency sub-band is not considered to be sparse. So the low-frequency and high frequency coefficients multiplied with measurement matrix A together will destroyed the correlation between low-fre- quency approximate weight coefficient and the detail coefficients and lead to a poor reconstruction results. Hence, we preserving the low-pass wavelet coefficients and only measured the high-pass wavelet coefficients of the image. Then we fuse them with different fusion rules.

A) Fusion rule I

Let

where

B) Fusion rule I

Image structural similarity (SSIM) [

The structural similarity (SSIM) is designed by modeling any image distortion as the combination of structure comparison function

where

For image fusion, it is useful to apply the SSIM locally rather than globally. In our experiments

At last, the mean SSIM (MSSIM) index to measure the similarity of two images defined as follows:

where

For the detail coefficients fusion, we let

where

Consequently the fused high-frequency coefficient can be obtained through the TV minimization scheme based on augmented Lagrangian and alternating direction algorithms (TVAL3), see [

Algorithm 1. Image fusion in compressed sensing.

C) Algorithm

We recover the fused high-frequency coefficient from the fused high-frequency measurement using the TVAL3 algorithm [

In this section, three groups of test images are employed for the performance evaluation to illustrate the effectiveness of the proposed approach. Three pairs of source images including computed tomography (CT) and magnetic resonance imaging (MRI) images shown in

In our experiment, We compare the proposed method with method in [

rule (CS_MS) and [

For further comparison, several objective criteria are used to evaluate the fusion results. The first criterion is the average gradient (AG), which is commonly used to evaluate the clarity of image, the greater AG is, the sharper is the image. The next two criterion is the mutual information (MI) and edges keep degrees (

The performance evaluation of fusion results for three methods is shown in

Image | Method | AG | MI | Q | T (s) |
---|---|---|---|---|---|

Med 256 * 256 | CS-MAV | 8.67 | 1.56 | 0.51 | 583 |

CS-SD | 8.06 | 3.74 | 0.50 | 547 | |

Ours | 8.02 | 3.93 | 0.77 | 3 | |

Military 256 * 256 | CS-MAV | 3.65 | 2.03 | 0.39 | 468 |

CS-SD | 1.21 | 3.34 | 0.26 | 364 | |

Ours | 4.01 | 3.42 | 0.49 | 3 | |

Clock 512 * 512 | CS-MAV | 3.57 | 5.22 | 0.46 | 7280 |

CS-SD | 1.97 | 6.75 | 0.52 | 7269 | |

Ours | 4.35 | 6.79 | 0.57 | 19 |

In the paper, we put forward a kind of effective image fusion scheme based on compressive measurements. Compare with traditional method, we firstly take single layer wavelet transform for original image. Then, according to the characteristics of high frequency coefficients sparse, we only measured the high-pass wavelet coefficients with a low sampling rate. At last, we fuse low-frequency wavelet coefficient and high- frequency measurement with different fusion rules and recovery them with a new efficient TV minimization algorithm. Simulation results indicate that proposed scheme provides efficiency and promising fusion results. In addition, owing to the proposed scheme only needs incomplete measurements rather than acquiring all the samples of the whole image, the computational complexity significantly reduces. Moreover, because of using Fast Walsh Hadamard Transform, the CPU running time of our method for fusion and CS reconstruction is far less than other two methods.

CS image fusion is still in the stage of exploration, many problems remain to solve (e.g., the relationship between the original image and measurements and how to design the best fusion rule). Therefore the study of new and more advanced fusion rules matched with CS principle is another research topic in our future work.

This work was supported by the Gansu Academy of Sciences Youth Foundation under Grant 2015-QN-06. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Bo Zheng. The authors are with the Institute of sensing technology, Gansu Academy of Sciences (e-mail: glxygh@163.com; zhengboem@163.com).

Yang, G.H., Xu, W.D., Zheng, B., Ma, F.L., Yang, X.H., Ma, H.W., Zhang, H.X. and Han, G.L. (2016) Compressed Sensing Based on the Single Layer Wavelet Transform for Image Fusion. Journal of Computer and Communications, 4, 107-116. http://dx.doi.org/10.4236/jcc.2016.415010