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Denoising star map data via sparse representation and dictionary learning Zhou Mingyuan a,∗ , Shi Ying b , Yang Jigang c a b c

School of Astronautics, Beihang University, Beijing, China Department of Instrument Science and Opto-electronics Engineering, Beihang University, Beijing, China Beijing Institute of Automation Control Equipment, Beijing, China

a r t i c l e

i n f o

Article history: Received 16 February 2014 Accepted 27 February 2015 Keywords: Sparse representation Redundant dictionary Star pattern recognition Noise suppression

a b s t r a c t As the highest precision devices of celestial navigation system [1], star sensors have been getting more and more attention in recent years. In which the star image positioning and recognition is the key technology of CNS, while the extraction of stars from star maps is the ﬁrst step. By the background noise, there are some error extractions when traditional methods are used, which can even lead to the failure of star map matching. To solve this problem, a denoising method based on overcomplete sparse representation is presented in this paper. This method uses the adaptive sparse decomposition of star map in the redundant dictionary to process the threshold, as a result, the reliability of star extraction is improved. The experimental results show that the correct rate of this method that extracting star after reducing background noise of star map is close to 100%. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction For star pattern recognition and attitude determination using star sensor is the method of celestial navigation with the highest precision. Celestial navigation with star sensor is the most precise navigation system. Star sensors use cameras’ detect unit shooting the star directly at a certain moment, and process the obtained star map by centroid extraction, star pattern recognition and attitude solution. And the instantaneous pointing information of star sensor is achieved. After the corresponding coordinate transformation according to the installation position of the star sensor in the aircraft, the attitude information of aircraft is ultimately obtained. The correct rate of star pattern recognition is affected by the accuracy and reliability of star image preprocessing directly, so the star image preprocessing is one of the key technologies of the star pattern recognition. The traditional methods used in star image preprocessing include method of gray weighed, centroid method with threshold, curved surface approximation and star extraction of centroid compensation, etc. [2,3]. Considering the inﬂuence of background noise, improved methods such as wavelet image denoising [4,5] and partial differential equation model for optics image denoising [6] are proposed by some scholars. By choosing an appropriate wavelet

∗ Corresponding author. Tel.: +86 18210068325. E-mail address: [email protected] (M. Zhou). http://dx.doi.org/10.1016/j.ijleo.2015.02.091 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

transform ﬁlter, the correlation between different characteristics extracted from the image can be greatly reduced, so it can express the one-dimensional signal with singular point perfectly. However, one-dimensional singular feature such as edge and texture included in two-dimensional image signal is not presented as well as the former circumstances. The mount of coefﬁcient is no doubt getting larger with the scale increasing, when presenting the image with wavelet by the linear singularity that approximated by point singularity, the ability of wavelet sparse representation decreased rapidly. Hence, a method for depressing noise of star map based on sparse representation [7] is put forward in this paper, so that the correct rate of star extraction is improved. Sparse representation has been widely researched in recent years, and the problems solved by which is to search for the most concise representation of a signal in terms of linear combination of atoms in an over-complete dictionary. Sparse representation tends to offer better effect in image denoising, therefore it is an extremely powerful tool for engineering [8,9]. But account for the scale of redundant dictionaries which consists of kinds of cascaded basic functions [10] is large, hindering the application in engineering practice. An image denoising method based on KSVD over-complete dictionaries learning is proposed in literature [11,12], and obtained very good results. All of the previous searches of sparse representation algorithm on restraining noise are simulated and analyzed based on adding Gauss white noise artiﬁcially, and we can generate a picture with noise in the same way by matlab shown in Fig. 1. However, there

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And it can be expressed as: y = Dx + b =

L

di xi + b

(2)

i=1

where the matrix D is the overcomplete dictionary, and di is one of the atoms of the D. x = [x0 , x1 ,. . .,xL ]T is the coefﬁcient matrix, and b is the residual component. The problem of sparse representation is to ﬁnd an L × 1 coefﬁcient vector x, such that y = Dx + b and x0 is minimized, i.e., x = min x0

s.t. y − d x i i ≤ ε i∈Dn×m

(3)

2

Fig. 1. The black background added the Gauss white noise with variance of 0.0005 simulated by matlab.

where x0 is the 0 norm and is equivalent to the number of nonzero components in the vector x. However, rarely vector is with the most coefﬁcients equal to zero strictly in the actual signal representation, so it is not effective enough to measure the sparseness by l0 norm. This is more so especially when there is noise in signal. Hence it turns to be an NP problem how to get the optimal solution. An approximate solution is put forward by replacing the 0 norm in formula (3) with the 1 norm, as follows: x = min x1

s.t. y − d x i i ≤ ε i∈Dn×m

(4)

2

In fact, this way to solve the NP problem is convex relaxation method and greedy tracking method, all of these algorithms and the improvements such as MP [14], OMP [15] and BP [16] solve the NP problem availably. 3. Algorithm of noise depression in image based on learning

Fig. 2. Mixed noise of real star map (ampliﬁed).

are various of irregular noises in original star map such as the noise from CCD image sensor, electronic circuit [13], background light, A/D conversion, atmospheric disturbance and from other unpredictable factor shot by star sensor in practical engineering, which is shown in Fig. 2. Although it is difﬁcult to distinguish by the naked eye, simulated noise is not as persuasive as the noise shot in real star map. Therefore, it aims to explore how well it works of sparse representation algorithm in real image denoising and improve the method of overcomplete dictionaries construction.

It is assumed that D is the dictionary and adaptive updates are performed by using K-SVD method in this paper in order to depress noise in image effectively with the lossless image as possible at the same time. So a thought is put forward based on the former researchers [17] that dividing the image into blocks and suppressing noise by iterative residuals. KSVD is another method of atomic base training which was put forward by Aharon et al. [12]. The main contribution lies in the update of atomic base, and it is not necessary to inverse matrix, but process the atoms in base one by one. Meanwhile, the atom base of KSVD and the corresponding representation coefﬁcient are updated at the same time, which save the training time. KSVD method names from the core step used singular value decomposition, and repeats K times to ﬁnish. SVD is deﬁned as follows: it is assumed A is real matrix with the size m × n, and rank(A) = r, then there must exist m order orthogonal matrix U and n order matrix V subject to,

2. Method of sparse representation of signal

T

A = UV = U

The general thought of sparse representation is to select the information structure that contains being expressed signals as far as possible. Sparse decomposition of signals is to express the signal by choosing some atoms which are best in linear combination from overcomplete dictionaries. In fact it is a kind of process of approaching. It is assumed that y ∈ Rn is the original signal, D is assemblage of the n-dimensional unit length vector dr with the amount of L, namely, D=

di ∈ Rn di = 1,

1≤i≤L

(1)

˙

0

0

0

VT,

UU T = I, VV T = I

(5)

In which ˙ = diag(1 , 2 , · · · , r ), 1 ≥ 2 ≥ · · · ≥ r > 0 are all nonzero singular values of A with the amount of r, and columns of U and V are respectively as the feature vector AAT . According to formula (4), when meeting the sense of minimum mean square error, we can update the dictionary by using the method of atomic update one by one by iteration. Dictionary learning can be integrated into the Bayesian maximum a posteriori estimation (MAP) [18], in other words, if dictionary D is unknown as well, for a given image, the atom dˆ ij of sparse representation can be achieved by solving NP problem [19], then using the K-SVD

M. Zhou et al. / Optik 126 (2015) 1133–1137

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algorithm to update the dictionary D, and improve the precision of dictionary D by residual iteration afterwards. When both dˆ ij and dictionary D meet the requirements, the clean image is acquired by formula (7) in literature [17],

⎛ Xˆ = ⎝I +

⎞−1 ⎛ RijT Rij ⎠

⎝Y +

ij

⎞ RijT Ddˆ ij ⎠

(6)

ij

In which X is the original image to be estimated and the Y is the observed available noisy version of it. Concrete steps of the algorithm are as follows: √ 1. Initializing, an image Y is given with noise, whose size is N × √ N. Construct the initial dictionary D0 , which is discrete cosine transform dictionary (DCT). √ 2. Divide the image Y into some blocks with the size n × √ n, solve the NP problem by OMP algorithm, namely ∀ d = s.t. Rij X − Ddij ≤ ε, and ﬁnd out the approximin d 1 2 mate solution dˆ ij . 3. Update the dictionary for each column in turn, ∀dij = / 0, ﬁnd out the assemblage ωl of small image blocks Rij X, in which 1 ≤ l ≤ k. 4. For each image block Rij X ∈ ωl , compute the residuals eijl = Rij Xij −

dm dˆ ij (m), then the residual matrix is achieved.

m= / l

5. Have singular value decomposition to the residual matrix (SVD), which is subject to El = UVT , update the dictionary atoms d˜ l and dij (l), then return to step 2, J times for iteration. 6. After iteration for J times, both updated dictionary D and sparse representation matrix dˆ are achieved. Because of the image block used in the algorithm are overlapping, so we can take the average processing for the overlapped blocks in denoised image. Finally, the denoised image is achieved according to formula

⎛

(6), Xˆ = ⎝I +

⎞−1 ⎛

RijT Rij ⎠

ij

⎝Y +

Fig. 4. The dictionary trained on patches from the original image.

64 ms. In order to avoid the effect of atmospheric refraction, the direction of camera lens was toward the zenith. The original star image is shown in Fig. 3: Depressing the noise of original star map, the speciﬁc steps are as follows: 1) Take each pixel as the center for the noisy image itself, and expand into block with the size 8 × 8, then a series of blocks as training samples can be produced. 2) Initial dictionary is made up of overcomplete dictionary DCT radical.

⎞

RijT Ddˆ ij ⎠, In which is

ij

a Lagrange multiplier and I is identity matrix.

4. Simulations and results In this work, the real star image is given with the size 1024 × 1024, and it was shot in 22h52m19s, August 18th, 2013 by Flea2 industrial camera, with the gain 24 and the integration time

Fig. 3. The real star map.

Fig. 5. Part of the original star map and the noise depressed star map after sparse representation.

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Fig. 6. The extracted star from the denoised image after sparse representation with the threshold 4.

3) Updating the initial dictionary by using KSVD algorithm, and get the adaptive overcomplete dictionary shown in Fig. 4. 4) Complete the image of noise reduction according to section third of the denoising method. The contrast of star map before and after noise is as shown in Fig. 5. The connected domain method is used in this paper to extract star image point. The least pixel threshold of star is set to 4 when extracting star and the mount of extracted stars from both the two algorithms is the same. The 16 extracted stars from noise depressed image are shown in Fig. 6. When the threshold is set to 3, the traditional method can extract to 44 stars, however there begins to be error extraction this moment and the number of error stars has reached to 7, which is shown in Fig. 7 with red dot. In contrast, there are 28 stars extracted in the same star map after the process of sparse representation, which is shown in Fig. 8. Those stars are all found in the Stellarium, a software of professional virtual planetarium, which is the most important. The extraction accuracy is 100%. In order to verify the reliability of the algorithm, the comparison of star extracting is performed 120 times again. There are always some degrees of error extraction in traditional method, with the correct rate ranging from 85% to 90%. While in this method, the accuracy of star extraction is 100%.

Fig. 8. The star extraction after processing by sparse representation.

Table 1 The series number of star that recognized from this method corresponds to the asterisk of real star. Serial number

Real asterisk

Serial number

Real asterisk

1 2 3 4 5 6 7 8 9 10 11 12 13 14

HIP99889 HIP100437 HIP102724 HIP102098 HIP99685 HIP102843 HIP98194 HIP102155 HIP100069 HIP99968 HIP100453 HIP101949 HIP98921 HIP100907

15 16 17 18 19 20 21 22 23 24 25 26 27 28

HIP100044 HIP101756 HIP100574 HIP99303 HIP100108 HIP99770 HIP99031 NGC6871 HIP102589 HIP102062 HIP101214 HIP101067 HIP100515 HIP100501

The series number of star that recognized from the presented method with the corresponding asterisk of real star is shown in Table 1: The error extraction of star can increase the difﬁculty and complexity of the star pattern recognition, reduce the accuracy of star pattern recognition and even lead to false matches, which would obtain error attitude result and serious consequence. Therefore, it is very important to depress the noise of original star map to increase the accuracy of star extraction in star pattern recognition. 5. Conclusion and prospect

Fig. 7. The traditional method of star extraction with the threshold of 3.

In this paper a novel algorithm for denoising star image is put forward, which affectivity is demonstrated by the tests. The algorithm is simple, ﬂexible, high efﬁciency and excellent ability of resisting noise. It is basically on the basis of sparse representation of an original image shot in wild, restrains the noise in order to reduce the error extraction of star in a star image so that to improve the correct rate of star pattern recognition. Compared with the traditional extraction of star in the image, the method in this paper is more stable. The extraction of star is the ﬁrst step in star pattern recognition, and it needs to subdivide the centroid of the star [20] in order to improve the precision of the center in the next step. Super resolution is undoubtedly one of the best methods. Future theoretical work on the general behavior of this algorithm is on our further research agenda.

M. Zhou et al. / Optik 126 (2015) 1133–1137

Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant No. 61074184 and Grant No. 61233005. References [1] T. Sun, F. Xing, Z. You, Accuracy measurement of star trackers based on astronomy, J. Tsinghua Univ. (Sci. Technol.) 52 (4) (2012) 430–435. [2] X. Wei, G. Zhang, J. Jiang, Subdivided locating method of star image for star sensor, J. Beijing Univ. Aeronaut. Astronaut. 29 (9) (2003) 812–815. [3] H. Jia, Star Centroid Estimation and Star Identiﬁcation of High Accuracy Star Tracker, National University of Defense Technology, 2010. [4] L. Ma, Y. Sun, Application of wavelet domain Wiener ﬁltering analysis to noise elimination in star-image denoising, Infrared Laser Eng. 33 (1) (2004) 55–58. [5] L. Ma, Y. Sun, The method of star-image denoising based M-channel wavelet transforms and space-shield ﬁltering, Infrared Technol. 26 (4) (2004) 45–48. [6] N. Qiao, B. Zou, Nonlocal orientation diffusion partial differential equation model for optics image denoising, OPTIK 124 (14) (2013) 1889–1891. [7] H. Huang, F. Da, Sparse representation-based classiﬁcation algorithm for optical Tibetan character recognition, OPTIK 125 (3) (2014) 1034–1037. [8] N. Zhu, T. Tang, S. Tang, D. Tang, F. Yu, A sparse representation method based on kernel and virtual samples for face recognition, OPTIK 124 (23) (2013) 6236–6241. [9] J.-X. Mi, D. Lei, J. Gui, A novel method for recognizing face with partial occlusion via sparse representation, OPTIK 124 (24) (2013) 6786–6789.

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