HYPERSPECTRAL IMAGE DENOISING WITH CUBIC TOTAL VARIATION MODEL

Image noise is generated unavoidably in the hyperspectral image acquision process and has a negative effect on subsequent image analysis. Therefore, it is necessary to perform image denoising for hyperspectral images. This paper proposes a cubic total variation (CTV) model by combining the 2-D total variation model for spatial domain with the 1-D total variation model for spectral domain, and then applies the termed CTV model to hyperspectral image denoising. The augmented Lagrangian method is utilized to improve the speed of solution of the desired hyperspectral image. The experimental results suggest that the proposed method can achieve competitive image quality.


INTRODUCTION
Hyperspectral remote sensing image acquision is a complicated process, in which image noise is generated unavoidably.Hyperspectral image noise has a negative effect on subsequent image processing and analysis, such as classification, target detection, unmixing and etc.Therefore, hyperspectral image denoising aims at removing the noise included in hyperspectral images and supporting improved image analysis capabilities.
Othman and Qian [1] proposed a hybrid spatial-spectral derivative-domain wavelet shrinkage noise reduction approach.
Chen and Qian [2,3] where M is the total number of pixels in band b of the hyperspectral image.
Secondly, from the perspective of the 1-D spectral domain pixel by pixel, each pixel of the hyperspectral image is a 1-D spectral signal.The 1-D TV model [8] for the m th pixel of the hyperspectral image can be written as: where :,m u is the 1-D spectral signal of the m th pixel of the hyperspectral image, and where  represents the weight of spectral dimension relative spatial domain.

CTV based Hyperspectral Image Denoising
In recent years, the maximum a posteriori (MAP) [9] estimation theory, which inherently includes prior constraints in the form of prior probability density functions, has been attracting attention and enjoying increasing popularity.Based on the MAP estimation theory, the denoising model for a hyperspectral image can be represented as the following constrained least squares problem: where g represents the observed hyperspectral image, the The desired hyperspectral image can be solved by optimizing the cost function shown in (6).Because of the high dimension property of hyperspectral images and the non-difference property of the proposed CTV model, how to efficiently resolve the hyperspectral image denoising model is very important.In this paper, the augmented Lagrangian method [10] is utilized to optimize the CTV denoising model.

Simulation Results
The hyperspectral image used in this experiment is a remote sensing image of size 307 280 

Real Results
The image data used in this experiment is the AVIRIS suggest that the proposed method can achieve competitive image quality.
where x  and y  represent the gradient operators of the horizontal and vertical directions, and M is the total number of pixels in gray-level image v .Now, we consider the total variation model for hyperspectral image.Firstly, from the perspective of the 2-D spatial domain band by band, each band of the hyperspectral image is a 2-D gray-level image signal.Therefore, the most direct TV model of hyperspectral image is to add the standard TV model of each band and we can get the following 2-D TV model: where ,: b u represents the b th band of hyperspectral image u , B is the number of bands of hyperspectral image, and   ,: b R u represents the standard TV model for the b th band of hyperspectral image:

z
represent the gradient operator of the spectral domain.It should be noted that the hyperspectral image cube exhibits a 3-D concept.Therefore, it is natural for us to combine the 2-D total variation model for spatial domain with the 1-D total variation model for spectral domain and propose the termed cubic total variation model for the hyperspectral image, which can be written as:

term 2 2  2 
g u represents the data fidelity between the observed noisy image and the original clean image, and R(u) is the regularization item, which gives a prior model of the original clear hyperspectral image. is the regularization parameter which controls the relative contribution of data fidelity term 2 g u and regularization term   R u .We apply the proposed CTV model in(5) to the hyperspectral image denoising framework and obtain the following cost function:

.
collected with HYDICE from Washington DC Mall.In the simulated process, zero-mean Gaussian-noise and salt-and-pepper noise was added to the hyperspectral image.The denoising result with traditional band-by-band TV model is compared as benchmark.The parameters of the proposed method are set as 12 Bands 1 and 21 of the original image, noisy image, denoising results with traditional TV model and with CTV model are shown in Fig. 1.The objective evaluations results are shown in Table 1.It is clearly observed that the proposed hyperspectral image denoising method outperforms the the hyperspectral image denoising method with the traditional band-by-band TV model in terms of both both the quantitative measurements and visual evaluation.

Fig. 1
Fig. 1 The denoising result of Washington DC Mall image.

CUBIC TOTAL VARIATION MODEL BASED HYPERSPECTRAL IMAGE DEBOISING 2.1 Cubic Total Variation (CTV) Model
[6]sotropic diffusion was proposed in[5].Recently, Chen et al.[6]proposed a new hyperspectral image denoising algorithm by adding a PCA transform before using wavelet shrinkage; first, a PCA transform was implemented on the original hyperspectral image, and then the low-energy PCA output channel was denoised with wavelet shrinkage denoising processes.edge-preservingproperty.The standard TV model for gray-level image looks like

Table 1
Objective evaluations of denoising results