| 摘要: | 盲蔽訊號源分離（blind source separation, BSS）已在遙測、生物信息、機器學習等領域皆有成功應用。基於凸幾何之BSS不需仰賴不實際的訊號源統計獨立假設，因此更為實用，其中克雷格單形（Craig simplex, CS）已主導BSS過去二十年的發展。但計算CS需解一個非凸的優化問題，其效能在訊號源重度混合時嚴重降低。為解決此問題，我在最近的SIAM論文中採用泛函分析裡的約翰橢球（John ellipsoid, JE）設計一個從未有人思考過的BSS準則，不但證明其源辨識力和CS一樣強，更證明其只需解一個凸的錐形規劃。此振奮的突破被所有SIAM審稿人評為優雅、非常原創。我相信只要JE能被快速計算（本計畫重點議題之一），其將在十年內取代CS在BSS中的地位。此外，我最近驚訝地發現JE也有處裡BSS病態混合系統（困擾國際知名生醫團隊之複雜議題）之巨大潛力，相關新穎理論亦會在本計畫被建立。另一方面，因高解析度（high-resolution, HR）高光譜圖像（hyperspectral image, HSI）極為昂貴，電腦視覺領域裡較經濟的做法乃藉由耦合非負矩陣分解（coupled non-negative matrix factorization, CNMF）之數據融合法達到高光譜超解析（hyperspectral super-resolution, HySure）。但傳統優化理論已無法處理HSI巨大維度帶來的大規模CNMF，我在最近的IEEE論文中採用大數據優化理論（分佈式ADMM優化理論）來設計快速CNMF演算法，且突破了最尖端的HySure表現（在三種不同高光譜傳感器、及四個重要指標上(含PSNR、RMSE等)皆大幅超越現存六種最先進的HySure方法）。然而一個更根本而未解的問題乃是CNMF解析力之理論保證，其將於本計畫被探索，不像NMF解析力已在近年機器學習文獻被廣泛探討，CNMF解析力至今仍是個謎。此外CNMF所需之HR配對圖像（亦即覆蓋空間區域相同之HR全色圖、RGB圖像、或多光譜圖像）在某些應用中不可得（如太空/行星探索之衛星圖像），本計畫亦將探討無配對圖像之HySure前瞻理論與方法，此方向從未被先前的HSI文獻所嘗試過，但其對遙測之深遠影響力與經濟效益可被預見。 ;Blind source separation (BSS), aiming at unsupervisedly recovering the underlying sources from their mixtures, has found many successful scientific applications, including remote sensing, bioinformatics, machine learning and text mining. Unlike conventional BSS methods (e.g., independent component analysis (ICA)), convex geometry (CG) based BSS does not require unrealistic assumptions (e.g., source independence in ICA). A seminal CG criterion is based on the so-called Craig simplex, which has led to significant/substantial advances of BSS in the past two decades. Nevertheless, it requires solving a non-convex optimization problem, and such non-convexity seriously degrades its effectiveness when the sources are heavily mixed, as demonstrated in our recent SIAM paper. As a cutting-edge breakthrough, I employed the John ellipsoid in functional analysis to design a novel CG criterion, for which I proved that its source identifiability is as strong as that of Craig simplex, but only requiring solving a conic programing that is convex. This exciting discovery has been accepted with all the SIAM reviewers evaluating it as elegant and very original contribution. I believe that John ellipsoid will replace Craig simplex in the coming decade, if the large-scale optimization for computing it can be solved very fast (a critical issue for this 5-year project). Moreover, I recently surprisingly found that John ellipsoid also has the potential in attacking the tough BSS scenario wherein the mixing system is ill-conditioned (a big issue bothering some internationally well-known groups, e.g., CBIL at Virginia Tech). The geometry meaning of such ill-conditioning and the associated novel BSS theory will also be investigated in this project.On the other hand, hyperspectral super-resolution (HySure), aiming at improving the spatial resolution of a hyperspectral image (HSI), receives enormous attentions from recent computer vision area, as the hundreds of densely populated spectral bands in typical HSI make its high-resolution (HR) data acquisition very expensive. An economical approach, known as coupled non-negative matrix factorization (CNMF), achieves HySure by fusing a cheaper low-resolution (LR) HSI with its counterpart HR panchromatic or multispectral images that cover the same spatial scene. Unlike 3-band RGB image, conventional optimization theory (e.g., interior-point method) can no longer handle the large-scale CNMF induced by the huge dimensionality of hyperspectral data. I recently employed a big data optimization theory, i.e., the distributed optimization ADMM theory, to design a fast CNMF algorithm. Although it has achieved state-of-the-art HySure performance with provable guarantee of stationary convergence, as reported in my recent IEEE paper, the fundamental issue regarding CNMF identifiability remains a critical yet unresolved problem (unlike the extensively studied NMF identifiability in recent machine learning literature), which will also be investigated in this project. Moreover, CNMF requires a HR counterpart that is, however, often unavailable in real-world applications (e.g., satellite imagery for space/planetary exploration). In this 5-year project, I will also investigate HySure theory/optimization without relying on any HR counterpart, i.e., fully unsupervised single-image super-resolution problem. For HSI, such ambitious direction has never been attempted in prior literature, while its high-impact in remote material identification and classification can be anticipated. |
