Một Phương Pháp Subspace MM Cục Bộ Để Giải Quyết Các Vấn Đề Biến Thiên Bị Ràng Buộc Trong Khôi Phục Hình Ảnh

Journal of Mathematical Imaging and Vision - Tập 65 - Trang 253-276 - 2022
Emilie Chouzenoux1, Ségolène Martin1, Jean-Christophe Pesquet1
1Université Paris-Saclay, Inria, CentraleSupélec, Centre de Vision Numerique, Gif-sur-Yvette, France

Tóm tắt

Bài báo này giới thiệu một thuật toán subspace mới được điều chỉnh dựa trên yếu tố chính là tối thiểu hóa (P-MMS) để giải quyết các vấn đề tối ưu có ràng buộc và mượt mà. Tóm lại, cách tiếp cận của chúng tôi bao gồm việc nhúng một thuật toán subspace trong một quy trình hình phạt ngoại thất không chính xác. Chiến lược subspace, kết hợp với bước tìm kiếm kích thước bước tối thiểu hóa - majoration, tận dụng lợi thế lớn từ độ mượt mà của hàm chi phí đã bị phạt, trong khi phương pháp hình phạt cho phép xử lý một loạt các ràng buộc. Nhược điểm chính của các phương pháp hình phạt ngoại thất, cụ thể là tình trạng không ổn định khi giá trị của tham số hình phạt lớn, được khắc phục bằng cách sử dụng một kỹ thuật giống như vùng tin cậy. Độ hội tụ của thuật toán kết quả được phân tích. Các thí nghiệm số được thực hiện trên hai ứng dụng phục hồi hình ảnh quy mô lớn cho thấy rằng, so với các thuật toán tiên tiến nhất, phương pháp đề xuất hoạt động tốt về thời gian tính toán.

Từ khóa

#Tối ưu hóa có ràng buộc #Phương pháp subspace #Hình phạt ngoại thất #Khôi phục hình ảnh #Tối thiểu hóa #Thuật toán MM

Tài liệu tham khảo

Abboud, F., Chouzenoux, E., Pesquet, J.C., Chenot, J.H., Laborelli, L.: Dual block-coordinate forward-backward algorithm with application to deconvolution and deinterlacing of video sequences. J. Math. Imaging Vis. 59(3), 415–431 (2017) Afonso, M.V., Bioucas-Dias, J.M., Figueiredo, M.A.T.: An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans. Image Process. 20(3), 681–695 (2010) Afonso, M.V., Bioucas-Dias, J.M., Figueiredo, M.A.T.: A fast algorithm for the constrained formulation of compressive image reconstruction and other linear inverse problems. In: 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 4034–4037. IEEE (2010) Allain, M., Idier, J., Goussard, Y.: On global and local convergence of half-quadratic algorithms. IEEE Trans. Image Process. 15(5), 1130–1142 (2006) Anscombe, F.J.: The transformation of Poisson, binomial and negative-binomial data. Biometrika 35(3/4), 246–254 (1948) Bahmani, S., Raj, B., Boufounos, P.T.: Greedy sparsity-constrained optimization. J. Mach. Learn. Res. 14(3), 807–841 (2013) Bauschke, H.H., Combettes, P.L., et al.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, vol. 408. Springer, Berlin (2011) Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009) Belzunce, M.A.: High-resolution teterogeneous digital PET [18F]FDG brain phantom based on the BigBrain Atlas (2018). https://doi.org/10.5281/zenodo.1190598 Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, London (2014) Bolte, J., Daniilidis, A., Lewis, A.: A nonsmooth Morse-Sard theorem for subanalytic functions. J. Math. Anal. Appl. 321(2), 729–740 (2006) Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1–2), 459–494 (2014) Bonettini, S., Zanella, R., Zanni, L.: A scaled gradient projection method for constrained image deblurring. Inverse Probl. 25(1), 015002 (2008) Bonnans, J.F., Gilbert, J.C., Lemaréchal, C., Sagastizábal, C.A.: Numerical Optimization: Theoretical and Practical Aspects. Springer, Berlin (2006) Boyd, S., Parikh, N., Chu, E.: Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Now Publishers, Delft (2011) Branch, M.A., Coleman, T.F., Li, Y.: A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems. SIAM J. Sci. Comput. 21(1), 1–23 (1999) Byrd, R.H., Gilbert, J.C., Nocedal, J.: A trust region method based on interior point techniques for nonlinear programming. Math. Program. 89(1), 149–185 (2000) Byrd, R.H., Hribar, M.E., Nocedal, J.: An interior point algorithm for large-scale nonlinear programming. SIAM J. Optim. 9(4), 877–900 (1999) Byrd, R.H., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16(5), 1190–1208 (1995) Carrillo, R.E., McEwen, J.D., Wiaux, Y.: Sparsity Averaging Reweighted Analysis (SARA): a novel algorithm for radio-interferometric imaging. Mon. Not. R. Astron. Soc. 426(2), 1223–1234 (2012) Castillo, E.: Quadratic penalty method for intensity-based deformable image registration and 4DCT lung motion recovery. Med. Phys. 46(5), 2194–2203 (2019) Celis, M.R., Dennis, J.E., Tapia, R.A.: A trust region strategy for equality constrained optimization, Tech. rep. Rice University Houston TX Department of Mathematical Sciences (1984) Chambolle, A., Dossal, C.: On the convergence of the iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm’’. J. Optim. Theory Appl. 166(3), 968–982 (2015) Chan, T.F., Mulet, P.: On the convergence of the lagged diffusivity fixed point method in total variation image restoration. SIAM J. Numer. Anal. 36(2), 354–367 (1999) Cherni, A., Chouzenoux, E., Duval, L., Pesquet, J.C.: SPOQ \( \ell _p \)-over-\(\ell _q \) regularization for sparse signal recovery applied to mass spectrometry. IEEE Trans. Signal Process. 68, 6070–6084 (2020) Chouzenoux, E., Idier, J., Moussaoui, S.: A Majorize-Minimize strategy for subspace optimization applied to image restoration. IEEE Trans. Image Process. 20(6), 1517–1528 (2010) Chouzenoux, E., Jezierska, A., Pesquet, J.C., Talbot, H.: A Majorize-Minimize subspace approach for \(\ell _2-\ell _0\) image regularization. SIAM J. Imaging Sci. 6, 563–591 (2013) Chouzenoux, E., Pesquet, J.C.: Convergence rate analysis of the Majorize-Minimize subspace algorithm. IEEE Signal Process. Lett. 23(9), 1284–1288 (2016) Coleman, T.F., Li, Y.: An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6(2), 418–445 (1996) Combettes, P.L., Pesquet, J.C.: Image restoration subject to a total variation constraint. IEEE Trans. Image Process. 13(9), 1213–1222 (2004) Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Fixed-point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer (2011) Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158(2), 460–479 (2012) Condat, L.: Fast projection onto the simplex and the \(\ell _1\)-ball. Math. Program. 158(1–2), 575–585 (2015) Conn, A.R., Gould, N.I.M., Toint, P.: A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28(2), 545–572 (1991) Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust Region Methods. Society for Industrial and Applied Mathematics, Philadelphia (2000) Cragg, E.E., Levy, A.V.: Study on a supermemory gradient method for the minimization of functions. J. Optim. Theory Appl. 4(3), 191–205 (1969) Curtis, F.E., Gould, N.I.M., Robinson, D.P., Toint, P.L.: An interior-point trust-funnel algorithm for nonlinear optimization. Math. Program. 161(1–2), 73–134 (2017) Di Pillo, G., Grippo, L.: Exact penalty functions in constrained optimization. SIAM J. Control. Optim. 27(6), 1333–1360 (1989) Dupe, F.X., Fadili, J.M., Starck, J.L.: A proximal iteration for deconvolving Poisson noisy images using sparse representations. IEEE Trans. Image Process. 18(2), 310–321 (2009) Durand, S., Fadili, J., Nikolova, M.: Multiplicative noise removal using \(\ell _1\) fidelity on frame coefficients. J. Math. Imaging Vis. 36(3), 201–226 (2009) Dussault, J.P.: Numerical stability and efficiency of penalty algorithms. SIAM J. Numer. Anal. 32(1), 296–317 (1995) Elad, M., Matalon, B., Zibulevsky, M.: Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization. Appl. Comput. Harmon. Anal. 23(3), 346–367 (2007) Erdogan, H., Fessler, J.A.: Monotonic algorithms for transmission tomography. In: 2002 5th IEEE EMBS International Summer School on Biomedical Imaging, pp. 14–pp. IEEE (2002) Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. SIAM, Philadelphia (1990) Fletcher, R.: A penalty method for nonlinear constraints. In: Numerical Optimization 1984, pp. 26–40. SIAM Publications (1985) Fletcher, R., Reeves, C.M.: Function minimization by conjugate gradients. Comput. J. 7(2), 149–154 (1964) Florescu, A., Chouzenoux, E., Pesquet, J.C., Ciuciu, P., Ciochina, S.: A Majorize-Minimize memory gradient method for complex-valued inverse problems. Signal Process. 103, 285–295 (2014) Foi, A.: Clipped noisy images: heteroskedastic modeling and practical denoising. Signal Process. 89(12), 2609–2629 (2009) Gould, N.I.M.: On the convergence of a sequential penalty function method for constrained minimization. SIAM J. Numer. Anal. 26(1), 107–128 (1989) Gould, N.I.M., Toint, P.L.: Nonlinear programming without a penalty function or a filter. Math. Program. 122(1), 155–196 (2010) Grapiglia, G.N., Yuan, J., Yuan, Y.X.: A subspace version of the Powell-Yuan trust-region algorithm for equality constrained optimization. J. Oper. Res. Soc. China 1(4), 425–451 (2013) Harizanov, S., Pesquet, J.C., Steidl, G.: Epigraphical projection for solving least squares Anscombe transformed constrained optimization problems. In: International Conference on Scale Space and Variational Methods in Computer Vision, pp. 125–136. Springer Berlin Heidelberg (2013) Ivanov, V.K., Vasin, V.V., Tanana, V.P.: Theory of Linear Ill-Posed Problems and its Applications. De Gruyter, New York (2013) Konnov, I.V.: An approximate penalty method with descent for convex optimization problems. Russ. Math. 63(7), 41–55 (2019) Lee, J.H., Jung, Y.M., Yuan, Y., Yun, S.: A subspace SQP method for equality constrained optimization. Comput. Optim. Appl. Int. J. 74(1), 177–194 (2019) Liu, D.C., Nocedal, J.: On the limited memory BFGS method for large scale optimization. Math. Program. 45(1–3), 503–528 (1989) Luenberger, D.G.: Optimization by Vector Space Methods. Wiley, New York (1997) Mallat, S.: A Wavelet Tour of Signal Processing. Elsevier, Amsterdam (1999) Miele, A., Cantrell, J.W.: Study on a memory gradient method for the minimization of functions. J. Optim. Theory Appl. 3(6), 459–470 (1969) Murtagh, B.A., Saunders, M.A.: Large-scale linearly constrained optimization. Math. Program. 14(1), 41–72 (1978) Musse, O., Heitz, F., Armspach, J.P.: Topology preserving deformable image matching using constrained hierarchical parametric models. IEEE Trans. Image Process. 10(7), 1081–1093 (2001) Narkiss, G., Zibulevsky, M.: Sequential subspace optimization method for large-scale unconstrained optimization. Tech. rep, Technion, Israel Institute of Technology (2005) Nikolova, M.: Weakly constrained minimization: application to the estimation of images and signals involving constant regions. J. Math. Imaging Vis. 21(2), 155–175 (2004) Nikolova, M., Ng, M.K.: Analysis of half-quadratic minimization methods for signal and image recovery. SIAM J. Sci. Comput. 27(3), 937–966 (2005) Nocedal, J., Wright, S.: Numerical Optimization. Springer, Berlin (2006) Powell, M.J.D.: Variable metric methods for constrained optimization. In: Mathematical Programming the State of the Art, pp. 288–311. Springer (1983) Pustelnik, N.: Méthodes proximales pour la résolution de problèmes inverses: application à la tomographie par émission de positrons. Ph.D. thesis, Université Paris-Est (2010). https://pastel.archives-ouvertes.fr/tel-00559126 Pustelnik, N., Benazza-Benhayia, A., Zheng, Y., Pesquet, J.C.: Wavelet-based image deconvolution and reconstruction. In: Wiley Encyclopedia of Electrical and Electronics Engineering, pp. 1–34. American Cancer Society (2016) Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992) Setzer, S., Steidl, G., Teuber, T.: Deblurring Poissonian images by split Bregman techniques. J. Vis. Commun. Image Represent. 21(3), 193–199 (2010) Sghaier, M., Chouzenoux, E., Pesquet, J.C., Muller, S.: A novel task-based reconstruction approach for digital breast tomosynthesis. Med. Image Anal. 77, 102341 (2022) Shanno, D.F., Marsten, R.E.: Conjugate gradient methods for linearly constrained nonlinear programming. In: Mathematical Programming Studies, pp. 149–161. Springer Berlin Heidelberg (1982) Shultz, G.A., Schnabel, R.B., Byrd, R.H.: A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties. SIAM J. Numer. Anal. 22(1), 47–67 (1985) Sun, Q.Y., Wang, C.Y., Shi, Z.J.: Global convergence of a modified gradient projection method for convex constrained problems. Acta Math. Appl. Sin. Engl. Ser. 22(2), 227–242 (2006) Thai, T.H., Cogranne, R., Retraint, F.: Camera model identification based on the heteroscedastic noise model. IEEE Trans. Image Process. 23(1), 250–263 (2013) Toint, P.L.: Global convergence of a class of trust-region methods for nonconvex minimization in Hilbert space. IMA J. Numer. Anal. 8(2), 231–252 (1988) Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38(3), 667–681 (2013) Wald, A., Schuster, T.: Sequential subspace optimization for nonlinear inverse problems. J. Inverse Ill-Posed Probl. 25(1), 99–117 (2017) Wang, Y., Ma, S.: A fast subspace method for image deblurring. Appl. Math. Comput. 215(6), 2359–2377 (2009) Wang, Z.H., Wen, Z.W., Yuan, Y.X.: A subspace trust region method for large scale unconstrained optimization. In: Numerical Linear Algebra and Optimization, pp. 264–274. Science Press (2004) Wang, Z.H., Yuan, Y.X.: A subspace implementation of quasi-Newton trust region methods for unconstrained optimization. Numer. Math. 104(2), 241–269 (2006) Yu, C., Zhao, J., Wang, Y., Wang, C., Geng, W.: Separation and imaging diffractions by a sparsity-promoting model and subspace trust-region algorithm. Geophys. J. Int. 208(3), 1756–1763 (2017) Yuan, Y.X.: Subspace methods for large scale nonlinear equations and nonlinear least squares. Optim. Eng. 10(2), 207–218 (2008) Yuan, Y.X.: Recent advances in trust region algorithms. Math. Program. 151(1), 249–281 (2015) Zhang, B., Zhu, Z., Li, S.: A modified spectral conjugate gradient projection algorithm for total variation image restoration. Appl. Math. Lett. 27, 26–35 (2014) Zibulevsky, M.: SESOP-TN: Combining sequential subspace optimization with truncated Newton method. Tech. rep, Computer Science Department, Technion (2008) Zibulevsky, M., Elad, M.: \(\ell _1-\ell _2\) optimization in signal and image processing. IEEE Signal Process. Mag. 27(3), 76–88 (2010)