Sparse Unmixing using an approximate L 0 Regularization Yang Guo 1,a, Tai Gao 1,b, Chengzhi Deng 2,c, Shengqian Wang 2,d and JianPing Xiao 1,e

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1 International Conference on Information Sciences, Machinery, Materials and Energy (ICISMME 5) Sparse Unmiing using an approimate L Regularization Yang Guo,a, ai Gao,b, Chengzhi Deng,c, Shengqian Wang,d and JianPing Xiao,e School of Jiangi Science & echnology Normal University, Nanchang Jiangi Province, china School of Nanchang Institute of echnology, Nanchang Jiangi Province, china a accuracygy@gmail.com, b @qq.com, c dengchengzhi@6.com, d sqwang3@63.net, e iaojianping89@63.com Keywords: Sparse unmiing, approimate sparsity, the linear miture model, and approimate sparsity regularizer Abstract. Recently, sparse unmiing focuses on finding an optimal subset of spectral signatures in a large spectral spetral library. In most previous work concerned with the sparse unmiing, the linear miture model has been widely used to determine and quantify the abundance of materials in mied pieels[]. In this paper, we propose a new sparse unming method based on an approimate sparsity regularization model[]. he approimate sparsity regularizer is much easier to solve than the L regularizer and has stronger sparsity than the L regularizer. What s more, a variable splitting and augmented Lagrangian methods introduced in to solve the proposed problem. Our numerical results on sparse unmiing illustration the efficiency of approimate sparsity a under the SUnSAL algorithm framework, compared to the L. Introduction Spectral unmiing is an effectively way to hyperspectral date analysis. o deal this problem, the spectral unmiing technique was proposed, which estimates the fractional signatures of pure spectral signatures in each mied piel. Based on the relationship of photons interact with material, mied piel model can be divided into two basic models: the linear miture model and the nonlinear miture model[3]. As the linear miture model is ease of implementation and fleibility, It s has been widely used into many different applications. In this paper, we proposed a novel compound regularization based hyperspectral unmiing method, which eploits the approimate sparsity. he approimate sparsity L regularized, which provides much easier to solve than L regularized, and better sparsity than L regularized. Eperimental results also show that the proposed method can effectively improve the SRE of hyperspectral unmiing. Sparse unmiing Linear miture model he linear miture model assumes minimal secondary reflections and multiple scattering effects in the data collection procedure, and hence the measured spectra can be epressed as a linear combination of the spectral signatures of materials present in the mied piel. he LMM can be formulated as follows: y = Mα + n, () where y is a L column vector of observed hyperspectral piel, L is the number of the spectral bands, M is an L q matri standing for the q endmembers, α is a q abundance vector, n represents a L vector of error and noise. here are two constraints are widely used in the linear miture model: the abundance non-negativity and abundance sum-to-one, as follows: α, () 5. he authors - Published by Atlantis Press 9

2 q αi =, (3) i= Sparse Unmiing he idea of Sparse unmiing is to find a linear combination of endmembers for each observed piel from a large spectral library. Given a known large spectral library A, the sparse unmiing can be written as[4] : y = A + n (4) L p Where A acts as the available spectral library, which is a large matri, A R containing p endmembers. L denotes the number of bands, and p is the number of endmembers in A. Bearing the LMM and sparse unmiing theory in mind, we consider a general minimization problem as: min s. t. y A δ,, X = (5) Where denotes the L of the vector, δ is the tolerated error and modeling error. However, in terms of computational compleity, the L optimization problem is a typical NP-hard problem, and it was difficult to solve until Candes and ao[5,6] proved that the L can be replaced by the L under a certain condition of the restricted isometric property(rip). herefore, the optimization problem is relaed to alternative conve optimization problem can be written as: min s. t. y A δ,, X = (6) Where denotes the L. he constrained optimization problem can be converted into an unconstrained Lagrangian version,as follows: min A y + λ + q () + {}( ) (7) R+ Where λ is a non-negative regularization parameter which controls the relative weights of two objective function. he () and {}( ) represent the ANC and ASC, respectively. q R + Proposed model and algorithm Unmiing Model Based Approimation L While L regularization provides the best conve approimation to L regularization and it is computationally efficient. However, L regularizer can not obtain a satisfactory solution. In this paper, we consider using a continuous function to approimate L sparse unmiing method, which provides smooth measure of L and better sparsity than L regularizer. he smoothed L can be written as[7]: ep( ) ep( ) f () = (8) ep( ) + ep( ) he parameter is a positive constant and. As approaches to zero, we have: = lim f () = (9) or approimately can be denotes: f ( ) () 9

3 Define the continuous multicariate approimate sparsity function as: m F () = f ( ) () As we know, the larger value of, the smoother F () and worse approimation to L ; the smaller value of, the closer behavior of F () to L. i= i Algorithms for approimative approach with smoothed L In this work, we use the approimative approach L to replace the L in(7), as follows: min A y + λg( ) + q () + {}( ) () R+ g ( ) = F () (3) In this work, we used the variable splitting and augmented Lagrangian algorithm to solve(), and we introduce intermediate variable u, then transform problem(4) into an equivalent problem, min A y + λg( u) + q ( u) + {}( ) (4) R+ he augmented Lagrangian of problem(4) is µ Lud (,, ) = A y + λgu ( ) + q ( u) + {}( ) + u d (5) R+ he algorithm is shown in Algorithm. Algorithm. Unmiing algorithm based on approimative approach L. Initialization and parameters setting: set k=,, u, d k k k.repeat: + arg min Lu (,, d ) k u + u k+ k k+ k+ d d u k k arg min L (, u, d) ( ) 3. until some stopping criterion is satisfied Eperiments Having presented our method in the previous section, we now turn our attention to demonstrate its utility for sparse unmiing. he proposed model are compared with the SUnSAL method. All the considered models take into account the abundance non-negativity constraint. Here, we employ synthetic data and real-world data in order to evaluate the performances of the algorithms. he signal-to-reconstruction(sre) is used to evaluate the accuracy of unmiing mothods, which is defined as follows: E( ) SRE ( db ) = *log E( ) Here, denotes the true abundances matri, represents the estimated one each column of which corresponds with the abundances of a piel, and E () denotes the epectation function. Generally speaking, the larger the SRE is, the more the estimation approimates the truth. he ma iteration number iterma and the iteration stopping criterion ε stop are set to and.. 9

4 Synthetic data In this eperiment, the spectral library is the United States Geological Survey(USGS) digital spectral library, which contains 4 materials with 4 spectral bands distributed uniformly in the interval.4-.5 m m. able shows SRE(dB), obtained in the simulated dataset, for all the SNR levels considered and for different values of the parameter, such as λ and. Real data he hyperspectral dataset used in the real data eperiments is the United States Geological Survey(USGS) digital spectral library. he size of the test area we chose was 5 9-piel subset. he USGS library containing 498 pure endmember signatures are measured for 4 spectral bands in the interval.4-.5 m m. with nominal spectral resolution of m m. Prior to the analysis, bands-, 5-5, 5-7, and3-4 were removed due to water absorption and low SNR in those bands, leaving a total of 88 spectral bands. In our eperiments, we use spectral obtained from this library as input to the unmiing methods, and make a qualitative analysis of the performances of different sparse unmiing methods. able. SRE(dB) comparison between different algorithms on the synthetic data Data cube SNR(db) SUnSAL ASLSU λ = λ =, =. DC (k=4) λ = 4.75 λ = λ = 4, = λ = 7, =. Fig..Estimated abundance fractions with the different methods for the Cuprite data As the smoothed Approimation L model behave much better than L model, we only display 93

5 the results obtained by these two models. As shown in Fig, the varying degrees of unmiing accuracy for the two typical minerals, Buddingtonite and Chalcedony. Compared with SUnSAL, the ASLSU algorithm is closer the classification maps produced by the SUGS etracorder algorithm. However, generally speaking, we can conclude that our algorithm outperforms the SUnSAL algorithm. Conclusions o improve the accuracy of spectral unmiing, we consider using the L to replace the L to measure the SNR, and propose a new smooth function to approimate the L. we also have used an effective method basde on variable splitting and augmented Lagrangian algorithm to solve the approimate L problem. Eperimental results on both the synthetic data and real data gives sparser and more accuate of our new models. Acknowledgements his work was financially supported by the National Natural Science Foundation of China under the Grants(666), and References [] N. Keshava. A survey of spectral unmiing [J]. Lincoln Lab. J., 3, 4(4): [] A. Plaza, Q. Du, J. Bioucas-Dias, X. Jia and F. Kruse. Foreword to the Special Issue on Spectral Unmiing of Remotely Sensed Data [J]. IEEE ransactions on Geoscience and Remote Sensing,, 49(): [3] J. Bioucas-Dias and A. Plaza. An Overview on Hyperspectra Unmiing: Geometrical, Statistical and Sparse Regression Based Approaches. IEEE International Geoscience and Remote Sensing Symposium (IGARSS'), Vancouver, Canada,. [4] M. D. Iordache, J. M. Dias, A. Plaza. Sparse unmiing of hyperspectral data [J]. IEEE rans. Geosci. Remote Sens.,, 49(6): [5] E. Candès and. ao, Decoding by linear programming, IEEE ransaction on Information heory 5 ()(5) [6] E. Candès and. ao, Near-optimal signal recovery from random projections: Universal encoding strategies, IEEE ransaction on Information heory 5 () (6) [7] G.H. Mohimani, M. B. Zadeh, A fast approach for overcomplete sparse decomposition based on smoothed L, IEEE ransaction on signal processing 57 () (9)