Improved TV Algorithm Based on Adaptive Multiplier for Interference Hyperspectral Image Decomposition

Interference Hyperspectral Images (IHI) data acquired by Interference Hyperspectral Imaging Spectrometer exhibit many vertical interference stripes. The above characteristics will affect the application of dictionary learning and compressed sensing theory used on IHI data. According to the special characteristics of IHI data, many algorithms are proposed to separate the interference stripes layers and the background layers of IHI data in 2015, but the interference stripes layers are still not clean enough and the ideal background layers without interference stripes are also difficult to be obtained. In this paper, an improved total variation (TV) algorithm based on adaptive multiplier is proposed for IHI data decomposition. The value of the Lagrange multiplier is adaptive according to the unidirectional characteristics of IHI data. The proposed algorithm is used on Large Spatially Modulated Interference Spectral (LSMIS) images and is proved to provide better experimental results than the current algorithms both visually and quantitatively.


Introduction
The technology of interference hyperspectral imaging [1][2][3] is very powerful in the field of remote sensing, which can get the spatial and spectral information of the interested targets, and has been widely used in many fields, such as meteorology, geology, environmental monitoring, and military.The interference hyperspectral spectrometer has been successfully equipped in the "Chang'E" lunar exploration satellite.Interference hyperspectral imaging has become the research focus in the recent years.IHI is a kind of threedimensional massive data, which has high resolution and will also lead to the difficulty on the storage and transmission on remote sensing.According to the special characteristics of IHI data, it is necessary to design efficient compression methods.The current compression methods for IHI include predictive algorithms [4][5][6], transform algorithms [7,8], vector quantization algorithms [9], and data coding algorithms [10,11].
Due to the special image principle, there are many position-fixed interference stripes exist in each frame of IHI data, which will seriously affect the results of IHI compression and reconstruction [8].The inherent characteristics of IHI data will seriously impact the direct application of many traditional methods, such as the predictive coding and wavelet transform, which also cannot meet the precondition of the compressed sensing theory.
In 2015, morphological component analysis (MCA) and total variation (TV) are adopted to separate IHI data, respectively [12,13].A frame of IHI data is decomposed into an interference stripes layer and a background layer.But the results are not satisfied as there are still obvious vertical interference stripes in background layers in [12] and the interference stripes layers are still not clean enough in [13].
However, an improved TV algorithm based on adaptive multiplier is proposed in this paper.The expected decomposition result is to obtain background layers without vertical interference stripes, and the interference stripes layers are clean enough without any residual background pixels, which is the basic idea of the proposed method.The value of the Lagrange multiplier is adaptive according to the unidirectional characteristics of IHI data; that is, the horizontal variations of background layers and the vertical variations of interference stripes layers should be small enough, and the optimal value of the Lagrange multiplier will be helpful for obtaining the optimal solution.The imaging principle and characteristics of IHI will be introduced in the next section.The TV algorithm will be introduced in Section 3.An improved TV algorithm based on adaptive multiplier will be proposed in Section 4. Experiments and analysis will be given in Section 5 and the conclusion will be given in Section 6.

Imaging Principle and Characteristics of IHI
Figure 1 shows the equivalent optical path in lateral shearing interferometer. is the distance between  1 and  2 , which are the two separated rays by the light from a ground point.In the interferometer, Fourier transform lens (FTL) is the main imaging equipment.The Optical Path Difference (OPD) of point  in Charge Coupled Device (CCD) is zero.
The OPD of point  on the CCD detector is where  FTL is the focus of Fourier lens.
According to the theory of Fourier transform [14], the interference curve can be expressed as in which () is the spectral distribution of source,  max and  min are the maximum and minimum of wavenumber, respectively, and  represents OPD of this interference curve.Because () is a real and even function, its Fourier transform  must be a real and even function too [14].So formula (2) equals The spectrum curve can be obtained by inverse Fourier transform of the interference curve according to the basic Fourier transform relationship.The inverse Fourier transform of formula (2) is is the maximum OPD.The corresponding cosine transform is Figure 2 shows the sketch map of three-dimensional IHI data produced by LASIS [1][2][3].
The main characteristics of IHI data are as follows.First, IHI is not directly imaging of light, so there are obvious vertical interference stripes in each frame of IHI.
Second, IHI is three-dimensional data formed through push-broom technique, so the background of IHI has horizontal shift between frames.
To get better result when the dictionary learning and compressed sensing algorithms are applied on IHI data, TV algorithm [15][16][17][18] is adopted to separate the interference stripes from the background in [13].

TV Algorithm for IHI Data Decomposition
The effect of interference stripes in one frame of IHI data  is assumed to be an additive layer   , while the background pixels of IHI data are assumed to be in another layer   .So the decomposition process can be formulated as follows: TV algorithm is proposed for IHI data decomposition in 2015 [13].As the interference stripes in   are vertical, so their variations are mainly concentrated along the -axis.In mathematical words, most pixels of the stripes have the property as follows: where TV  and TV  are horizontal and vertical total variations, respectively.The formula (7) can also be written as [15][16][17][18] ∫ ,∈ So the following function is proposed in [19] for IHI data decomposition: where  is the Lagrange multiplier which quantifies the degree of smoothness across the -axis.The formula (9) can also be written as The traditional gradient descent algorithm is adopted to get the optimal result of   in the formula (10) in [19], and the optimal result of   will also be obtained through the difference between  and   .

Proposed TV Algorithm Based on Adaptive Multiplier
Although many methods [12,13] have been proposed in order to separate the interference stripes layers from the background layers, the decomposition results are still not completely satisfied.Vertical interference stripes still exist in background layers obviously in [12] and the interference stripes layers still contain the background pixels in [13].
In this paper, an improved TV algorithm based on adaptive multiplier is proposed for IHI data decomposition.The Lagrange multiplier  plays a key role in the process of decomposition.If  is too small, there will be residual stripes in the background layers   , while a large value of  will lead to oversmoothing.So the choice of an optimal Lagrange multiplier is very important in TV algorithm.The basic idea of the proposed algorithm is based on the special characteristics of IHI data.For the ideal result of decomposition,   should only contain the background pixels without any vertical stripes, as the special imaging principle of IHI data; the background layers should have unidirectional property because the values of most pixels are almost the same in the horizontal direction, while   should only contain the vertical interference stripes without any background pixels, which should have unidirectional property in vertical direction.As   and   should have unidirectional property in horizontal and vertical directions, respectively, the value of Lagrange multiplier  should be variant depending on the horizontal and vertical total variations of the layers.The improved TV algorithm based on adaptive multiplier is proposed as in Algorithm 1.
Genetic Algorithm (GA) [20] is adopted in the proposed algorithm for calculating the optimal Lagrange multiplier  and making it adaptive for different frames of IHI data.In ideal state, no interference stripes will exist in   , and only interference stripes will exist in   , so formula ( * ) in Algorithm 1 is used for the fitness function of GA, which can keep the final outputs   and   having the minimum of horizontal and vertical total variations, respectively.

Experiments and Analysis
Three groups of LSMIS IHI data will be chosen for experiment.The LSMIS data is 12 bytes' image of size 256 × 256.
Compared with the novel methods [12,13], the proposed method will be used for LSMIS data decomposition.The start value, end value, , and ini num are chosen to be 0, 3, 0.2, and 10 in the experiment, respectively.For LSMIS data 1-3, values of the optimal multiplier calculated by GA are 0.6273, 0.3963, and 1.4851, respectively.Experimental results for visual comparisons are shown in Figures 3-5, and the quantitative results for comparisons are shown in Tables 1 and  2.
In   residual stripes still exist at the top and bottom of interference stripes parts in (b), while in (d) and (f) the stripes are almost removed.And as there are few residual background pixels in the interference stripes layers produced by the proposed method, the layers in (g) are relatively much cleaner compared with (c) and (e).And as the results shown in Tables 1 and 2, the proposed method can obtain   and   with smaller total variations in horizontal and vertical directions, respectively.The reason of the experimental results is that in the regularization process of the proposed method, the unidirectional characteristics of ideal background layers and interference layers are both taken into account.

Conclusion and Prospect
IHI data has different characteristics from natural images due to the special imaging principle, which will affect the direct application of dictionary learning and compressed sensing algorithms.An improved TV algorithm based on adaptive multiplier is proposed for IHI data decomposition, which will separate IHI data into two different layers and make them sparsely represented and meet the precondition of compressed sensing.Compared with other decomposition methods for IHI data, the proposed method can obtain background layers and interference stripes layers with smaller total variations in horizontal and vertical directions, respectively.

Figure 1 :
Figure 1: The sketch map of the interference hyperspectral spectrometer.

Figure 3 :
Figure 3: Experimental results of LSMIS data 1.(a) Original LSMIS data 1.(b) Background layer produced by MCA.(c) Interference stripes layer produced by MCA.(d) Background layer produced by IMT.(e) Interference stripes layer produced by IMT.(f) Background layer produced by the proposed method.(g) Interference stripes layer produced by the proposed method.

Figure 4 :
Figure 4: Experimental results of LSMIS data 2. (a) Original LSMIS data 1.(b) Background layer produced by MCA.(c) Interference stripes layer produced by MCA.(d) Background layer produced by IMT.(e) Interference stripes layer produced by IMT.(f) Background layer produced by the proposed method.(g) Interference stripes layer produced by the proposed method.

Figure 5 :
Figure 5: Experimental results of LSMIS data 3. (a) Original LSMIS data 1.(b) Background layer produced by MCA.(c) Interference stripes layer produced by MCA.(d) Background layer produced by IMT.(e) Interference stripes layer produced by IMT.(f) Background layer produced by the proposed method.(g) Interference stripes layer produced by the proposed method.

Table 1 :
Horizontal total variations of interference stripes part in   .
by IMCA, IMT, and the proposed method are shown in (c), (e), and (g), respectively.We can clearly see that the

Table 2 :
Vertical total variations in   .