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A new denoising algorithm is proposed according to the characteristics of hyperspectral remote sensing image (HRSI) in the curvelet domain. Firstly, each band of HRSI is transformed into the curvelet domain, and the sets of subband images are obtained from different wavelength of HRSI. And then the detail subband images in the same scale and same direction from different wavelengths of HRSI are stacked to obtain new 3-D datacubes of the curvelet domain. Again, the characteristics analysis of these 3-D datacubes is performed. The analysis result shows that each new 3-D datacube has the strong spectral correlation. At last, due to the strong spectral correlation of new 3-D datacubes, the multiple linear regression is introduced to deal with these new 3-D datacubes in the curvelet domain. The simulated and the real data experiments are performed. The simulated data experimental results show that the proposed algorithm is superior to the compared algorithms in the references in terms of SNR. Furthermore, MSE and MSSIM in each band are utilized to show that the proposed algorithm is superior. The real data experimental results show that the proposed algorithm effectively removes the common spotty noise and the strip noise and simultaneously maintains more fine features during the denoising process.

Hyperspectral remote sensing image (HRSI) can be viewed as three-dimensional data consisting of one-dimensional spectral information and two-dimensional spatial information. With the fast development of hyperspectral remote sensing technology, HRSI can describe the characteristics of Earth objects more comprehensively and explicitly therefore, it is widely applied in many fields including agriculture, forestry, geological surveys, environmental monitoring, and military recon. Although over the last decades the development of imaging spectrometers is rapid, HRSI is still affected by many complex factors during the processing of acquisition and transmission, which will produce a mass of noises. The data that are contaminated with noise can cause a failure to extract valuable information and hamper further interpretation. In presence of noise in the image, extraction of all the useful information becomes difficult and noise can lead to artefacts and loss of spatial resolution [

Though the spatial correlation of HRSI is weaker than the nature image, the two-dimensional spatial information of HRSI is similar to the nature image [

In recent years, many denoising methods for HRSIs are constantly introduced with the development of hyperspectral remote sensing technology. Most of denoising methods combine correlation of spatial and spectral domain. Currently, wavelet denoising methods are widely used. The seminal work on signal denoising via wavelet thresholding or shrinkage proposed by Donoho and Johnstone [

Since wavelet has good time-frequency-localization property and multiresolution analysis property, it is widely and successfully applied in several fields [

Compared with normal three-dimensional data cube of fixed variance of additive noise, the noise level of HRSI may vary dramatically from band to band. The noise standard deviation in each band of HRSI is not constant; in particular, there exist some bands at which the atmosphere absorbs so much light that the signal received from the surface is unreliable [

In the rest of this paper, the mathematical tools curvelet transform and MLR model are introduced in Section

In the section, the mathematical tools curvelet transform and MLR model are introduced. The curvelet transform, pioneered by Candès and Donoho, is shown to be optimal in a certain sense for functions in the domain with curved singularities [

The curvelet transform is a new multiresolution analysis framework and widely applied in various image processing problems. The curvelet decomposition can be equivalently stated in the following four steps: (1) subband decomposition, (2) smooth partitioning, (3) renormalization, and (4) ridgelet transform. In short, the curvelet obtained by bandpass filtering of multiscale ridgelets with passband is rigidly linked to the scale of spatial localization. The discrete curvelet transform [

The digital image is used as an example to introduce the discrete curvelet transform.

Perform the binary wavelet transform with

Let the size of initial block (the subband in the finest scale) be

For

partition the subband

apply the digital ridgelet transform to each block;

If

Since each step of the previous decomposition process is invertible, the inverse curvelet transform is an invertible process.

The curvelet transform overcomes the major drawback that wavelets cannot really represent two-dimensional objects with edges sparsely and captures more directional information besides the horizontal, vertical, and diagonal directions. The system approximately obeys the scale relationship

It is known that the curvelet transform competes surprisingly well with the ideal adaptive rate. The approximation error is obtained as

Curvelet is optimal in the sense that no other representation can yield a smaller asymptotic error with the same number of terms. Because of its surprising properties for image processing, a fast and accurate discrete curvelet transform operating on digital data is necessary. Candès et al. [

Due to the strong spectral correlation of new 3-D datacubes in the curvelet domain, the MLR model is introduced to predict the representation of pure HRSI in the curvelet domain. It is assumed that the HRSI has

HRSI is a datacube, having two spatial dimensions and a third spectral dimension. Fixing the wavelength band yields a 2-D image of the scene at a particular wavelength. So it may also be visualized as a stack of 2-D band images, each corresponding to a certain wavelength [

Let

The process of obtaining new 3-D datacubes

Let

The spectral correlation factor of

The spectral correlation factor of

From Figure

In this section, we summarize our denoising algorithm. According to the correlation factor of HRSI in the curvelet domain in Section

The denoising process is as follows (Figure

Input the noisy datacube

A

Set aside the coarsest scale coefficients

MLR is performed on

For

The inverse curvelet transform (ICT) is performed to obtain the denoised image

Output the denoised datacube

Block diagram of the proposed method in this paper.

Signal-to-noise ratio (SNR) is a key parameter on measuring the HRSI quality. So in this paper, we utilize SNR to evaluate the proposed algorithm. Here the SNR is defined as

The simulated experiment of the noise reduction is carried out on AVIRIS images, Cuprite, Jasper Ridge, Low Altitude, Lunar Lake, and Moffett Field provided by JPL, NASA. The size of datacube we extracted from the Cuprite, Jasper Ridge, Low Altitude, Lunar Lake, and Moffett Field for testing is 256 × 256 × 224 (width × height × band). Figure

Band no.80 of AVIRIS images.

Cuprite

Jasper Ridge

Low Altitude

Lunar Lake

Moffett Field

In order to indicate that the nature image denoising methods cannot be immediately used for HRSI noise reduction, the 2-D complex wavelet with bivariate shrinkage (CWBS) [

SNR of AVIRIS data Cuprite, Jasper Ridge, Low Altitude, Lunar Lake, and Moffett Field.

SNR | CWBS | CD | HSSNR | PCABS | The proposed | |
---|---|---|---|---|---|---|

Cuprite | 600 | 1368.3 | 907.6 | 2712.6 | 8192.8 | 10952.0 |

Jasper Ridge | 720.6 | 387.8 | 2365.9 | 6135.2 | 7503.0 | |

Lunar Lake | 2869.5 | 1844.2 | 2791.4 | 9787.5 | 12516.0 | |

Low Altitude | 530.7 | 220.8 | 2000.4 | 4351.5 | 5140.3 | |

Moffett Field | 530.1 | 253.7 | 2267.8 | 5757.7 | 6427.1 |

In order to deeply analyze the proposed algorithm, the mean square of errors (MSE) in each band and the mean structural similarity (MSSIM) [

From Table

The MSE and the MSSIM in each band obtained by PCABS and the proposed method.

Cuprite

Jasper Ridge

Lunar Lake

Low Altitude

Moffett Field

The computational complexity, that is, the number of floating-point operations (flops), of the proposed method can be analyzed as follows. The complexity of the curvelet transform is in order of

The computational time of AVIRIS data Cuprite, Jasper Ridge, Low Altitude, Lunar Lake, and Moffett Field (units: s).

HSSNR | PCABS | The proposed | |
---|---|---|---|

Cuprite | 145.12 | 185.25 | 810.09 |

Jasper Ridge | 145.12 | 188.29 | 847.60 |

Lunar Lake | 148.58 | 184.10 | 809.97 |

Low Altitude | 146.98 | 201.05 | 860.91 |

Moffett Field | 146.65 | 185.30 | 858.04 |

In this paper the OMIS (operational modular imaging spectrometer) data that is developed by the Shanghai Institute of Technical Physics of the Chinese Academy of Sciences is used for real data experiment to verify the correctness and performance of algorithm. It has 128 spectral bands ranging from visible to thermal infrared wavelength. The size of datacube we extracted from OMIS data for testing is 256 × 256 × 128 (width × height × band). We perform denoising for the original OMIS data. Figure

Band nos. 20, 40, 60, 80, and 100 of (a) the original image and (b) the denoised image obtained by the PCABS, (c) the proposed algorithm, and (d) the difference between the original image and the denoised image obtained by PCABS, as well as (e) the difference between the original image and the denoised image obtained by the proposed algorithm.

In this paper, the spectral correlation of HRSI in the curvelet domain is discussed. By the analysis, in the curvelet domain, the strong spectral correlation of the hyperspectral remote sensing image is kept; even in some directions and scales it becomes stronger. So a new denoising algorithm is proposed; the MLR is performed in the curvelet domain to denoise the HRSI.

Simulated experimental result shows that the proposed method improves the quality of HRSI significantly in terms of SNR, MSE of each band, and MSSIM of each band. It is also seen that the denoised results obtained by the two algorithms are not content in the bands 1–5. This is a problem that will be studied in the future. For the real OMIS data, the results show that the proposed method is valid. The proposed method obtains better results in terms of detail preservation and noise removal during the denoising process.

This work was supported by the National Natural Science Foundation of China under Project 61101183 and Project 41201363.

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