Reweighted Laplace Prior Based Hyperspectral Compressive Sensing for Unknown Sparsity: Supplementary Material

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1 Reweighted Laplace Prior Based Hyperspectral Compressive Sensing for Unknown Sparsity: Supplementary Material Lei Zhang, Wei Wei, Yanning Zhang, Chunna Tian, Fei Li School of Computer Science, Northwestern Polytechnical University, Xi an, 77, China School of Electronic Engineering, Xidian University, Xi an, 771, China {weiweinwpu, This document is the Supplementary Material for the CVPR 1 paper Reweighted Laplace Prior Based Hyperspectral Compressive Sensing for Unknown Sparsity, which includes three contents: Section1: Derivation of the reweighted Laplace prior. Section: Derivation of the optimization procedure. Section3: More experimental results. 1. Derivation of the Reweighted Laplace Prior In this section, we present the detailed derivation of the intuitive reweighted Laplace prior from the hierarchical structure. For each columny i iny, we have py i py i γpγ κdγ { } exp 1 yt i Σ n y y b i n b j1 n b j1 n b j1 π n b/ Σ y 1/ exp κ j exp j1 { } y ji κj γ j exp κ jγ j κ jγ j dγ dγ π nb/ Σ y 1/ κ j / j exp y ji +κ jγj γ j dγ j π 1/γ/ κ 1/ j exp κ j y ji exp Ky i 1 n b K, 1/ κj y ji π exp κ κ j yji j yji }{{} GIGγ j κ j,y ji,1/ γ / j exp y ji +κ jγ j γ j where Σ y diag [γ 1,...,γ nb ] T1 is the covariance matrix of Y, K diag [ κ 1,..., κ nb ] T is the weight matrix and GIGγ j κ j,y ji,1/ represents the pdf of generalized inverse Gaussian distribution. It can be seen that the proposed hierarchical prior results in a reweighted Laplace prior on each column ofy with weight matrix K. indicates equal contributions. 1 For a vector x, diagx denotes a diagonal matrix with elements from x. For a matrix X, diagx denotes extracting the diagonal elements from X to form a vector. dγ j /1/$31. 1 IEEE

2 . Derivation of the Optimization Procedure In this section, we present the detailed derivation of sparsity learning over γ and noise estimation over λ in the optimization procedure of the main paper..1. Sparse Signal Recovery: Optimizing for γ GivenY,λand κ, the sub-problem overγ is derived as γ Y Σ y +log Σ by + whereσ by Σ n +ADΣ y D T A T. According to matrix algebra, we rewritelog Σ by term as i1 κ i γ i logκ i n p log Σ by log Σ n +ADΣ y D T A T log I mb +Σ n ADΣ y D T A T +log Σ n log Σ y +D T A T Σ n AD +log Σ y +log Σ n, 3 where I mb R m b m b is an identity matrix. Substituting Eq. 3 into Eq. and removing those irreverent terms to γ, we have a new sub-problem overγ as γ Y Σ y +log Σ y +D T A T Σ n AD +log Σ y + 1 n p i1 κ i γ i 4 Since the concave function f γ log Σ y +D T A T Σ n AD over γ [γ1,...,γ n b ] T results in Eq. 4 to be a nonconvex optimization, we utilize a conjugate function in convex optimization [3] to transform this noncovex optimization into a convex one by finding a strict upper bound off γ f γ log Σ y +D T A T Σ n AD z T γ f z, z where f z is the concave conjugate function of f γ, z [z 1,...,z nb ] T. It can be proved that the equation in Eq. only succeeds when z i γ log Σ y +D T A T Σ n AD i [ Σ tr y +D T A T Σ n AD γ Σ y +D T A T Σ n AD ] i tr Σ y +D T A T Σ n AD e γ j e T j +D T A T Σ n AD i γ j1 i [ Σ tr y +D T A T Σ n AD ] ei e T i tr [ e T i Σ y +D T A T Σ n AD ei ], where e i is the ith standard unit vector in R n b. To decrease the inverse operations in Eq. 6, we use matrix algebra to simplify the formula as [ z i tr e T i Σ y +D T A T Σ n AD ] ei { tr e T i [Σ y Σ y D T A T Σ n +ADΣ y D T A T } ADΣy ]e i 7 [ ] tr e T i Σ y Σ y D T A T Σ by ADΣ y e i. In summary, the formula ofz can be written as z diag Σ y Σ y D T A T Σ by ADΣ y, 8 6

3 Substituting Eq. into Eq. 4, we obtain the revised subproblem γ Y Σ y +z T γ +log Σ y + 1 n p i1 n b ȳ κ i γ i i +z i whereȳ i denotes theith element of ȳ diagyy T [ȳ 1,...,ȳ nb ] T. Since γ, the solution of γ i is 4κ i n p ȳ γi new i +z i+n p n p. κ i.. Noise Estimation: Optimizing for λ With the recovered sparse signaly and prior parameter γ, we have the following sub-problem overλ i1 γ i +logγ i + κ iγ i n p, 9 ADY G +log Σ by 11 λ np Σ n whereσ by Σ n +ADΣ y D T A T. Similar to the optimization ofγ, we a conjugate function to find the strict upper bound of the concave functionφλ log Σ by log Σ n +ADΣ y D T A T as follows φλ log Σ n +ADΣ y D T A T α T λ φ α, α 1 whereφ α is the concave conjugate function ofφλ,σ by Σ n +ADΣ y D T A T,Σ n diagλ andα [α 1,...,α mb ] T. It can be proved that the equation of Eq. 1 only succeeds when α i λi log Σ n +ADΣ y D T A T [ tr ξ T i Σn +ADΣ y D T A T ] ξi tr ξ T i Σ by ξ i 13 whereξ i is theith standard unit vector inr m b. Further, the formula of α can be written as α diag Σ by Substituting Eq. 1 into Eq. 11, we can obtain the revised sub-problem overλas λ ADY G m b +α T λ np Σ n i1 q i +α i λ i λ i 14 1 where q i denotes theith element of q diagqq T [ q 1,..., q mb ] T,Q ADY G/ n p. Sinceλ, the solution over λ is λ new q i i. 16 α i 3. More Experimental Results In this section, we provide more experimental results on the orthogonal transformation based HCS and dictionary based HCS Performance Evaluation on Orthogonal Transformation Based HCS Given Haar wavelet as linear basis matrix D, we conduct [9], St [], [6], FL [1], RCS [4] and the proposed to reconstruct the HSI from the linear measurementsg, which is corrupted with additive Gaussian white noise. The HSI comes from three real hyperspectral datasets, namely INDIANA, URBAN and PAVIAU. Figure 1 and Figure give the visual reconstruction results of all competing methods with sampling rate ρ. when SNR of measurements is db anddb. The results in Figure 1 and Figure indicate that the proposed has the most approximate visual reconstruction results among all methods. This conclusion is consistent with those in the main paper.

4 a b c d e f g Figure 1. Visual reconstruction results of the th band from INDIANA, the 6th band from URBAN and the 9th band from PAVIAU with sampling rate ρ. when SNR of measurements is db. All the comparison methods recover the HSI in a wavelet transform based HCS framework. a, b St, c Lasso, d FL, e RCS, f, g Original bands. a b c d e f g Figure. Reconstruction results of the th band from INDIANA, the 6th band from URBAN and the 9th band from PAVIAU with sampling rate ρ. when SNR of measurements is db. All the comparison methods recover the HSI in a wavelet transform based HCS framework. a, b St, c Lasso, d FL, e RCS, f, g Original bands. 3.. Performance Evaluation on Dictionary Based HCS In hyperspectral domain, the HSI can be linearly represented by a spectrum dictionary where each spectrum in the dictionary is termed as an endmember. Given endmembers [] extracted from each HSI by NFINDR [] as the dictionary D to transform the HSI into a sparse matrix Y, [9], St [], [6], [7] and [8] are adopted as the comparison methods to reconstruct HSI with sampling rate ρ ranging from.1 to.9. To simulate the random noise in compressive sensing procedure, we add Gaussian white noise to the measurements, which results in the SNR of measurements to bedb anddb. Similar to the main paper, INDIANA, URBAN and PAVIAU are employed as the test data. Three

5 St St St PSNRdb St PSNRdb St PSNRdb St a b c d e f Figure 3. The PSNR curves and SAM bar charts of different methods on three datasets under SNRdb. abc PNSR curves. def SAM bar charts. 4 4 SAMdegree St SAMdegree St SAMdegree St PSNRdb PSNRdb PSNRdb SAMdegree 4 3 SAMdegree 1 SAMdegree St St St a b c d e f Figure 4. The PSNR curves and SAM bar charts of different methods on three datasets under SNRdb. abc PNSR curves. def SAM bar charts Table 1. The SSIM scores of different methods on three datasets under SNR db. Results on INDIANA dataset ρ.1 ρ. ρ.3 ρ.4 ρ. ρ.6 ρ.7 ρ.8 ρ.9 [9] St [] [6] [7] [8] Results on URBAN dataset ρ.1 ρ. ρ.3 ρ.4 ρ. ρ.6 ρ.7 ρ.8 ρ.9 [9] St [] [6] [7] [8] Results on PAVIAU dataset ρ.1 ρ. ρ.3 ρ.4 ρ. ρ.6 ρ.7 ρ.8 ρ.9 [9] St [] [6] [7] [8] evaluation measures including PSNR, SAM and SSIM, are utilized to evaluate the reconstruction performance, which record the average performance of each method after Monte-Carlo runs, including PSNR, SAM and SSIM. Under two levels of noise, the results of PSNR and SAM on three datasets are shown in Figure 3 and Figure 4, respectively. It is clear that the proposed gives the highest PSNR among all comparison methods. The SAM values of the proposed are lower than those of other methods in most cases. Additionally, the SSIM scores on three datasets are given in Table 1 and Table. Under two different levels of noise, it can be seen that the proposed gives the highest SSIM scores in most cases. The results evaluated by three measures indicate that the proposed outperforms other methods on the reconstruction accuracy of HSI with the dictionary based sparsification strategy. References [1] S. D. Babacan, R. Molina, and A. K. Katsaggelos. Bayesian compressive sensing using laplace priors. Image Processing, IEEE Transactions on, 191:3 63,.

6 Table. The SSIM scores of different methods on three datasets under SNR db. Results on INDIANA dataset ρ.1 ρ. ρ.3 ρ.4 ρ. ρ.6 ρ.7 ρ.8 ρ.9 [9] St [] [6] [7] [8] Results on URBAN dataset ρ.1 ρ. ρ.3 ρ.4 ρ. ρ.6 ρ.7 ρ.8 ρ.9 [9] St [] [6] [7] [8] Results on PAVIAU dataset ρ.1 ρ. ρ.3 ρ.4 ρ. ρ.6 ρ.7 ρ.8 ρ.9 [9] St [] [6] [7] [8] a b c d e f g Figure. Reconstruction results of the th band from INDIANA, the 6th band from URBAN and the 9th band from PAVIAU with sampling rate ρ. when SNR of measurements is db. All the comparison methods recover the HSI in a dictionary based HCS framework. a, b St, c Lasso, d FL, e RCS, f, g Original bands. [] J. Bioucas-Dias and A. Plaza. Hyperspectral unmixing: Geometrical, statistical, and sparse regression-based approaches. In Proc. SPIE, volume 78, page 78A,. [3] S. Boyd and L. Vandenberghe. Convex optimization. Cambridge university press, 4. [4] E. J. Candes, M. B. Wakin, and S. P. Boyd. Enhancing sparsity by reweighted l1 imization. Journal of Fourier analysis and applications, 14-6:877 9, 8. [] D. L. Donoho, Y. Tsaig, I. Drori, and J.-L. Starck. Sparse solution of underdetered systems of linear equations by stagewise orthogonal matching pursuit. Information Theory, IEEE Transactions on, 8:94 111, 1. [6] B. Efron, T. Hastie, I. Johnstone, R. Tibshirani, et al. Least angle regression. The Annals of statistics, 3:47 499, 4. [7] C. Li, T. Sun, K. F. Kelly, and Y. Zhang. A compressive sensing and unmixing scheme for hyperspectral data processing. Image Processing, IEEE Transactions on, 13: 1, 1.

7 a b c d e f g Figure 6. Reconstruction results of the th band from INDIANA, the 6th band from URBAN and the 9th band from PAVIAU with sampling rate ρ. when the SNR of measurements is db. All the comparison methods recover the HSI in a dictionary based HCS framework. a, b St, c Lasso, d FL, e RCS, f, g Original bands. [8] G. Martin, J. M. Bioucas-Dias, and A. Plaza. Hyperspectral coded aperture hyca: A new technique for hyperspectral compressive sensing. In Signal Processing Conference EUSIPCO, 13 Proceedings of the 1st European, pages 1. IEEE, 13. [9] J. A. Tropp and A. C. Gilbert. Signal recovery from random measurements via orthogonal matching pursuit. Information Theory, IEEE Transactions on, 31: , 7. [] M. E. Winter. N-findr: an algorithm for fast autonomous spectral end-member deteration in hyperspectral data. In SPIE s International Symposium on Optical Science, Engineering, and Instrumentation, pages International Society for Optics and Photonics, 1999.