Improved MCA–TV algorithm for interference hyperspectral image decomposition

Optics and Lasers in Engineering 75 (2015) 81–87

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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

Improved MCA–TV algorithm for interference hyperspectral image decomposition$ Jia Wen a,b,n, Junsuo Zhao b, Wang Cailing c a

School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China Science and Technology on Integrated Information System Laboratory, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China c College of Computer Science, Xi’an Shiyou University, Xi’an 710065, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 1 November 2014 Received in revised form 7 May 2015 Accepted 4 July 2015 Available online 25 July 2015

The technology of interference hyperspectral imaging, which can get the spectral and spatial information of the observed targets, is a very powerful technology in the field of remote sensing. Due to the special imaging principle, there are many position-fixed interference fringes in each frame of the interference hyperspectral image (IHI) data. This characteristic will affect the result of compressed sensing theory and traditional compression algorithms used on IHI data. According to this characteristic of the IHI data, morphological component analysis (MCA) is adopted to separate the interference fringes layers and the background layers of the LSMIS (Large Spatially Modulated Interference Spectral Image) data, and an improved MCA and Total Variation (TV) combined algorithm is proposed in this paper. An update mode of the threshold in traditional MCA is proposed, and the traditional TV algorithm is also improved according to the unidirectional characteristic of the interference fringes in IHI data. The experimental results prove that the proposed improved MCA–TV (IMT) algorithm can get better results than the traditional MCA, and also can meet the convergence conditions much faster than the traditional MCA. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Interference hyperspectral image (IHI) Morphological component analysis (MCA) Total Variation (TV) Sparse representation Compressed sensing

1. Introduction Interference hyperspectral imaging [1–3] is a very powerful technology in the field of remote sensing. This technology can get the spatial and spectral information of the interested targets, which has been widely used in many fields, such as meteorology, geology, military and environmental monitoring and so on. 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 three-dimensional 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, it is necessary to design efficient compression methods, such as the compression methods for IHI include predictive algorithms [4–6], transform algorithms [7,8], vector quantization algorithms [9] and data coding algorithms [10,11]. Due to the special image principle, there are many positionfixed interference fringes exist in the frames of the IHI data, which ☆ This work was supported by the National Natural Science Foundation of China (Grant nos. 61401439, 41301382). n Corresponding author at: School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China. E-mail address: [email protected] (J. Wen).

http://dx.doi.org/10.1016/j.optlaseng.2015.07.001 0143-8166/& 2015 Elsevier Ltd. All rights reserved.

will seriously affect the results of IHI compression and reconstruction [8]. These inherent characteristics of IHI data will seriously impact the directly application of many traditional methods, such as the predictive coding, adaptive lifting wavelet transform and so on. The characteristics also cannot meet the precondition of the popular theory called compressed sensing. In [6], the structure of IHI data is modified with the corresponding column extracted mode, but the interference fringes still cannot be completely removed. In [7], tensor theory is adopted in the IHI compression, but the influence of the interference fringes is not considered. In [8], interference fringes have been removed in the high frequency domain with changing order of wavelet transform, but the interference fringes still cannot be removed in the low frequency domain. However, MCA algorithm is adopted in this paper, which can be used to separate the interference fringes from the background, and an improved MCA and TV combined algorithm (IMT) is also proposed in this paper to improve both the decomposition result and the computational efficiency of the traditional MCA. The working procedure of interferential spectrometer will be introduced in the next section. The imaging principle and characteristics of IHI will be introduced in Section 3. The morphological component analysis (MCA) algorithm will be introduced in Section 4. An improved MCA algorithm, which can improve the computational efficiency of the traditional MCA, will be proposed in Section 5. In order to further eliminate the interference fringes, an improved

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MCA–TV algorithm will be proposed in Section 6. Experiments and analysis will be given in Section 7 and the conclusion will be given in Section 8.

2. Working procedure of interferential spectrometer (IS) A general scheme of IS working on satellite is shown in Fig. 1, which is represented by a lateral shearing interferometer and a CCD array detector carried on the satellite. Fig. 1 also shows the main working procedure of IS [12,13].

Fig. 2. The sketch map of the interference hyperspectral spectrometer.

(a) On the satellite, light from infinite distance is separated into two rays in the interferometer, and will form two slim targets on the fore-focal planes producing coherent light source; (b) According to optical coherent theory, interferogram is collected from a CCD array detector by the optical collection system; (c) Spectrum will be recovered by Fourier transform on the ground.

3. Characteristics of IHI data Fig. 2 shows the equivalent optical path in lateral shearing interferometer in Fig. 1 [12,13]. d is the distance between S1 and S 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 OPD of point O in CCD is zero. The OPD of point P on the CCD detector is: d x ¼ d sin θ ¼ y f FTL

ð1Þ

where f FTL is the focus of Fourier lens. According to the theory of Fourier transform [12], the interference curve can be expressed as: Z kmax Z kmax j2π k yd IðxÞ ¼ BðkÞej2π kx dk ¼ BðkÞe f FTL dk ð2Þ kmin

kmin

In which BðkÞ is the spectral distribution of source, kmax and kmin are the extremums of wavenumber, x represents OPD of this interference curve. Because BðkÞ is a real and even function, its Fourier transform must be a real and even function too [12]. So formula (2) equals:   Z kmax Z kmax d IðxÞ ¼ BðkÞ cos ð2π kxÞdk ¼ BðkÞ cos 2π ky dk ð3Þ f FTL kmin kmin 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: Z δm Z δm  j2π kf yd FTL dx BðkÞ ¼ IðxÞe  j2π kx dx ¼ IðxÞe ð4Þ 0

0

Fig. 3. Three dimensional LASIS IHI data.

δm is the maximum OPD. The corresponding cosine transform is: BðkÞ ¼

Z δm 0

Z δm 0

IðxÞ cos ð2π ky

d Þdx f FTL

ð5Þ

Fig. 3 shows the sketch map of three dimensional IHI data produced by LASIS [1–3]. The main characteristics of IHI data are as follows: First, there are vertical interference fringes in each frame of IHI. Second, the background of IHI has horizontal shift between frames. To get better result when the traditional compression algorithm is applied on IHI data, MCA algorithm [14] is used to separate the interference fringes from the background in this paper.

4. Morphological component analysis, MCA Suppose the original image signal X contains M different signal layers X i , i ¼ 1; 2; ⋯; M. X is the sum of the M signal layers, i.e., X ¼ X 1 þX 2 þ ⋯ þ X M . The basic thought of MCA [14] is that there is a group of dictionary can only sparsely represent the ith layer X i but cannot sparsely represent other layers. MCA is adopted to separate the interference fringes from the background in this paper. X is one frame of IHI data containing N pixels, which is composed of two parts, i.e., interference fringes layer X I and background layerX B : X ¼ XI þ XB

Fig. 1. Working procedure of interferential spectrometer.

IðxÞ cos ð2π kxÞdx ¼

ð6Þ

In MCA algorithm, each layer signal can be sparsely represented in a corresponding dictionary. Let DI ; DB A ℝNL be an overcomplete dictionary of L prototype signal-atoms, and suppose layer signals X I ; X B can be represented as a sparse linear combination of these atoms. That is, the interference fringes layer X I can be written as X I ¼ DI αI , while the background layer X B can be written as X B ¼ DB αB . αI and αB , which are vectors with very few (⪡L) nonzero entries, are the corresponding sparse coefficients of X I ; X B respectively.

J. Wen et al. / Optics and Lasers in Engineering 75 (2015) 81–87

The sparsity of signal can be quantified pby the ℓ0 norm P ‖α‖0 ¼ fi : αðiÞ a 0g or ℓp norm ‖α‖p ¼ ð αðiÞ Þ1=p , p a 0. In this paper ℓ1 norm is chosen. The sparse decomposition can be transformed into an optimization problem as follows: n o opt ¼ arg min ‖αI ‖1 þ‖αB ‖1 þ ω‖X DI αI  DB αB ‖22 αopt ð7Þ I ; αB fαI ;αB g

where ω is regularization coefficients. In MCA, the orthogonal basis of curvelet transform [15] is used to sparsely represent the background layer, and the orthogonal basis of DCT is used to sparsely represent the interference fringes layer. DB ,DI represent the orthogonal basis of curvelet transform and DCT respectively. The process of MCA decomposition is shown as follows: Step 1, initialize the scaling parameters λ, the paraΛ threshold Λ meter Lmax , the maximum iterations n, X B ¼ X, X I ¼ 0, and δ ¼ λLmax . Step 2, decomposition. Λ 1) Update the background layer X B , and calculate the residual R: Λ

Λ

R ¼ X X B X I

ð8Þ Λ

Calculate the curvelet coefficients of X B þ R: Λ

αB ¼ DBþ ðX B þ RÞ

ð9Þ Λ

Get αB by processing αB with soft threshold δ, update X B : Λ

X B ¼ DB α B

ð10Þ Λ

2) Update the interference fringes layer X I , and calculate the residual R: Λ

Λ

R ¼ X X I  X B

ð11Þ Λ

Calculate the DCT coefficients of X I þ R: Λ

αI ¼ DIþ ðX I þ RÞ

ð12Þ Λ

Get αI by processing αI with soft thresholdδ, update X I : Λ

X I ¼ DI αI

ð13Þ

Step 3, update threshold δ:

δ ¼ δ  λ=n

83

coefficients αB are gotten by processing αB with soft threshold δ. In order to ensure at least one coefficient can be selected to be αB , δ should be smaller than the maximum of the sparse coefficients αkB , i.e.:

δ o maxðabsðαkB ÞÞ

ð15Þ

As formulas (10) and (13) show, sparse coefficients αkB are the projection of the background signal in the orthogonal basis of curvelet transform, while sparse coefficients αkI are the projection of the interference fringes signal in the orthogonal basis of DCT. Here we let αk0 I represent the projected coefficients of the background layer in DCT dictionary, and let αk0 B represent the projected coefficients of the interference fringes layer in curvelet dictionary: Λ

αk0I ¼ DIþ ðX  X kI Þ

ð16Þ

Λ

αk0B ¼ DBþ ðX  X kB Þ

ð17Þ

The relationship of the projected coefficients and the dictionaries is shown as Fig. 4, and the process of the proposed IMT algorithm is shown as Fig. 5: The basic thought of MCA is that the current orthogonal basis can only sparsely represent the corresponding layer but cannot sparsely represent other layers. The goal of thresholding the background layer's sparse coefficients αkB is to remove the sparse coefficients of the interference fringes layer, so the threshold value δ should be greater than the maximum value of αk0I , which are the projected coefficients of the background layer in DCT basis:

δ 4 maxðabsðαk0I ÞÞ

ð18Þ

Due to the above analysis, δ should meet the condition: k maxðabsðαk0 I ÞÞ o δ o maxðabsðαB ÞÞ

ð19Þ

In this paper, the adaptive threshold is chosen to be the mean value, i.e.,

maxðabsðαk0 ÞÞ þ maxðabsðαkB ÞÞ I , 2

and the final threshold to process Λ

the coefficients αkB for background layer X kB will be gotten as the minimum of the adaptive threshold and the linear threshold: ! λ maxðabsðαk0I ÞÞ þ maxðabsðαkB ÞÞ δ ¼ min δ  ; ð20Þ n 2 Similarly, the final threshold to process the coefficients αkI for Λ k interference fringes layer X I will be gotten as follows: ! λ maxðabsðαk0B ÞÞ þ maxðabsðαkI ÞÞ δ ¼ min δ  ; ð21Þ 2 n

ð14Þ 6. Improved Total Variation algorithm

Step 4, to do the decomposition; If δ 4 λ, go to Step 2 and continue Λ Else, stop the iteration. X B is the final result of background Λ layer, X I is the final result of interference fringes layer.

In Section 5 we proposed an IMCA algorithm which only can improve the efficiency of the traditional MCA. The experimental

5. Improved morphological component analysis, IMCA In the traditional MCA, the threshold δ is decreased in a linear mode until the iteration is stopped. This linear mode will spend too much time and seriously affect the computational efficiency. In this paper an improved MCA algorithm is proposed as follows: In order to improve the computational efficiency of the MCA algorithm, an adaptive mode will be proposed to decrease the threshold instead of the linear mode. At the kth iteration, the sparse

Fig. 4. Relationship of the projected coefficients and the basis.

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energy of the image X which is calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X EðXÞ ¼ Xðx; yÞ2

ð28Þ

x;y A X

The process of the proposed IMT algorithm is shown as Fig. 5.

7. Experiments and analysis

Fig. 5. The process of the proposed IMT algorithm.

decomposition results of MCA are shown in Section 7. From the background layer shown as (b) in Figs. 6–8, it is clear to see that MCA cannot completely eliminate the interference fringes only using MCA algorithm. In order to further eliminate the fringes from the backΛ ground layer X B produced by MCA, the traditional TV algorithm [16–20] is adopted and improved according to the unidirectional characteristic of the interference fringes. As the interference fringes are vertical in IHI, so their variations are mainly concentrated along the x-axis. In mathematical words, most pixels of the fringes have the property as follows: TVy ðX I Þ⪡TVx ðX I Þ

ð22Þ

where TVx and TVy are horizontal and vertical Total Variations, respectively. The formula (22) can also be written as [16–19]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   Z   Z ∂X I ðx; yÞ 2 ∂X I ðx; yÞ 2 ⪡ ð23Þ ∂y ∂x x;y A X x;y A X So in order to further eliminate interference fringes, the following function is proposed in this paper: X B ¼ arg minfTVy ðX  X B Þ þ βTVx ðX B Þg

ð24Þ

XB

where β is a regularization parameter that quantifies the degree of smoothness across the x-axis. Formula (24) can also be written as: 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9     Z > X B0 ¼ X B > <  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ðXðx;yÞ  X ðx;yÞÞ2 R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂X ðx;yÞ2 R ð26Þ B B ∂ þβ > ∂y ∂x x;y A X X > > : XB n þ 1 ¼ XB n  η ∂X B n 4

where η is the step parameter and X B is the background layer produced by MCA. The iterative procedure is stopped when   4   EðX B n þ 1 Þ  EðX B Þ   ZThd ð27Þ 4     EðX B Þ where Thd is the threshold parameter, and EðXÞ represents the

Three groups of LSMIS IHI data will be chosen for experiment. The LSMIS data is 12 bytes image of size 256  256. The MCA and IMT will be used to separate the interference fringes from the background respectively. The maximum number of iteration n is 60, and the parameters β ,η and Thd are chosen to be 2.5, 0.2 and 0.025 in the experiment, respectively. In the experiment, to compare the ability of removing fringes, results of MCA and IMT1 algorithm (which is the traditional MCA combined with improved TV algorithm, in the following we call it IMT1) are compared, and to compare the computational efficiency, results of IMT1 and IMT2 (which is improved MCA with the mode of adaptive threshold, combined with improved TV algorithm, in the following we call it IMT2) are also compared. Experimental results for LSMIS data 1, 2 and 3, are shown in Figs. 6–8, respectively. The comparison of the experimental results for LSMIS data is shown in Tables 1–2. SNR in Table 1 is calculated as follows: N P

SNR ¼ 10  log 10

X 2 ðiÞ

i¼1 N P i¼1

ð29Þ

ðXðiÞ X I ðiÞ X B ðiÞÞ2

From (b) and (d) in Figs. 6–8, we can clearly see that there are still some interference fringes in (b) which is decomposed by MCA, but the fringes are almost removed in (d) which is decomposed by IMT1, and from (c) and (e) in Figs. 6–8, we can also see that the interference fringe layers decomposed by IMT1 are much smoother than MCA. From Table 1, we can see IMT2 can get almost the same variance of fringes as IMT1. Table 2 shows that although the SNR of IMT2 is a little lower than the traditional MCA and IMT1, the running time is reduced and the computational efficiency has been improved obviously. The reason of the experimental result is that in the proposed IMT algorithm, the unidirectional characteristic of interference fringes has been taken into consideration, and improved TV algorithm is also proposed according to this special characteristic. IMT1 will make the background layers get much more useful information, and make the corresponding fringes layer seem much smoother. IMT2 is only to improve the computational efficiency, and from Tables 1–2 we can see IMT2 can get almost the same result as IMT1, so the experimental result of IMT2 are not shown in this paper for concision.

8. Conclusion and prospect Compressed sensing is one of the research focuses in the recent years. Its precondition is the processed data can be sparsely represented. IHI data is formed with the special imaging principle, its special characteristics will seriously impact the directly applied result of the compressed sensing theory and many traditional compression algorithms. MCA algorithm is adopted for decomposing the IHI data, and according to the unidirectional characteristic of interference fringes in LSMIS, IMT (improved MCA and TV algorithm combined) is proposed in this paper. Experimental results have proved that the proposed IMT not only can get better the results of MCA decomposition, but also can obviously improved the computational efficiency of

J. Wen et al. / Optics and Lasers in Engineering 75 (2015) 81–87

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Fig. 6. Experimental results of LSMIS data 1 (a) original LSMIS data 1, (b) background produced by MCA, (c) interference fringes produced by MCA, (d) background layer produced by IMT1 and (e) interference fringes produced by IMT1.

Fig. 7. Experimental results of LSMIS data 2 (a) original LSMIS data 2, (b) background produced by MCA, (c) interference fringes produced by MCA, (d) background layer produced by IMT1, and (e) interference fringes produced by IMT1.

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J. Wen et al. / Optics and Lasers in Engineering 75 (2015) 81–87

Fig. 8. Experimental results of LSMIS data 3 (a) original LSMIS data 3, (b) background produced by MCA, (c) interference fringes produced by MCA, (d) background layer produced by IMT1 and (e) interference fringes produced by IMT1.

References

Table 1 Variance comparison of experimental results for LSMIS data. LSMIS data 1 Variance in fringe parts Original image 3.1193e þ 04 Background using 7.0414e þ 03 MCA Background using 6.6920e þ 03 IMT1 Background using 6.7255eþ 03 IMT2

LSMIS data 2 Variance in fringe parts

LSMIS data 3 Variance in fringe parts

3.1663e þ 04 1.9459e þ 04

2.5708e þ04 2.2704e þ04

1.8224eþ 04

2.1936e þ04

1.8431e þ 04

2.2263e þ04

Table 2 Time and SNR Comparison of experimental results for LSMIS data.

MCA IMT1 IMT2

LSMIS data 1

LSMIS data 2

LSMIS data 3

Time (s)

SNR (dB)

Time (s)

SNR (dB)

Time (s)

SNR (dB)

43.692562 43.814790 10.263919

36.9135 37.7720 37.5927

44.284072 44.415981 11.421851

40.6557 40.9595 40.9357

43.167380 43.292591 9.915717

38.6787 38.9340 38.8991

MCA, which will provide a very good solution for further application of the compressed sensing theory and other traditional compression methods for IHI data.

Acknowledgments The research work was supported by National Natural Science Foundation of China under Grant nos. 61401439 and 41301382.

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Jia Wen, Aug 1983, received the Bachelor and Master degree from Nanjing University, received the Ph.D. degree from Xi'an Jiaotong University. His research interests include sparse representation and compressive sensing, image and video processing.