高光譜張量補全（Hyperspectral Tensor Completion, HTC）是高光譜影像（HSI）預處理中的核心任務，旨在從受損、條帶遮擋或隨機缺失的觀測資料中恢復完整的三維張量資訊。隨著高光譜遙測應用對解析度與即時性的要求不斷增加，而張量分解方法因其能捕捉多維結構特徵而受到廣泛關注。然而，基於張量分解之補全方法高度依賴複雜的多因子非凸交替尋優，計算複雜度極高且記憶體消耗巨大；而基於幾何光譜解混的輕量化方法，則依賴現實中難以滿足的純淨像素假設，以及可用的完整像元光譜向量來構造端元幾何結構。
為應對這些限制，本研究提出一種基於最大體積內切橢球（Maximum Volume Inscribed Ellipsoid, MVIE）的低秩高光譜張量補全演算法（MVIE-TC）。巧妙利用最大內切橢球的幾何拓撲特性，將數據純度要求放寬至更符合實際的寬鬆限界，同時將端元尋優與張量重構任務架構於嚴格的凸錐優化準則，得以在多項式時間內完成全域最優解的計算。該互補設計結合最大內切橢球對極端缺失的抵抗力，與凸幾何架構的強大收斂性。實驗結果顯示，所提出的方法在多組真實與仿真遙測數據集上，相較於現有十二種基準方法兼顧計算效率的同時，在定量與定性中均取得優秀表現，且在下遊土地覆蓋分類應用實驗中優於基準方法，證明了所提出算法作為高光譜數據預處理算法，在實際工程應用中之高度可用性。

Hyperspectral imaging technology integrates spatial visualization with high-resolution spectral analysis, providing rich multidimensional continuous spectral profiles for remote sensing, environmental monitoring, agricultural assessment, and target identification. A hyperspectral image is naturally structured as a three-dimensional tensor, where two dimensions represent spatial coordinates and the third dimension corresponds to continuous spectral wavelength bands. Despite its widespread utility, real-world hyperspectral data acquisition is frequently compromised by sensor hardware constraints, atmospheric interference, cloud cover, and random signal transmission losses. These degradation factors yield severely incomplete observed tensors characterized by dead lines, missing spectral bands, stripe noise, and isolated missing entries. Hyperspectral Tensor Completion is thus a critical preprocessing pipeline designed to recover the complete three-dimensional tensor from corrupted or partially observed inputs. While classical matrix completion treats spatial and spectral dimensions in isolation, modern tensor-based approaches aim to exploit global spatial-spectral correlations and low-rank structures simultaneously. However, existing methodologies encounter severe theoretical and operational bottlenecks when deployed in real-time or high-resolution scenarios.
Existing completion techniques can be broadly categorized into two main frameworks, both of which possess intrinsic limitations. Tensor decomposition-based methods, which rely on Canonical Polyadic decomposition, Tucker decomposition, or Tensor Train formats, capture multidimensional spatial-spectral dependencies by solving low-rank approximation problems. However, these formulations inherently depend on multi-factor, non-convex alternating optimization frameworks that suffer from high computational complexity, heavy memory overhead, slow convergence rates, and a susceptibility to local minima. As image dimensions and spectral band numbers scale, their runtime becomes unacceptable for practical applications. On the other hand, lightweight geometric spectral unmixing algorithms exploit physical mixing models, assuming that each pixel spectrum is a linear combination of pure constituent endmember spectra weighted by fractional abundances. While computationally efficient and physically interpretable, conventional geometric unmixing heavily relies on the strict pure-pixel assumption, which is rarely satisfied in real-world environments dominated by mixed pixels. Furthermore, when scenes suffer from severe or continuous band-wise missing values, valid convex hulls cannot be formed, causing standard geometric endmember extraction to fail completely.
To resolve the fundamental tradeoff between computational efficiency, structural robustness, and physical interpretability, this study introduces a novel low-rank hyperspectral tensor completion framework termed Maximum Volume Inscribed Ellipsoid Tensor Completion. The proposed algorithm operates through a streamlined three-stage pipeline that reformulates tensor recovery into a computationally tractable convex optimization framework. To prevent geometric optimization failures caused by contiguous missing spectral entries, the initial stage establishes a robust spatial-spectral pre-restoration mechanism. Leveraging the strong continuity between adjacent narrow-band channels and local spatial neighborhoods, a rapid nearest-neighbor spectral search and mapping procedure fills missing values temporarily to reconstruct a well-conditioned approximate global data convex hull. This linear-complexity step requires only the weak condition that each spatial pixel retains at least one valid spectral observation. This operator only traverses the index set once, incurring virtually no floating-point overhead and minimizing the preprocessing impact on overall algorithm efficiency. In the second stage, the algorithm extracts pure endmember spectra by fitting a maximum volume inscribed ellipsoid inside the constructed convex hull, relaxing strict pure-pixel requirements to a realistic purity bound. By embedding endmember search within a standard convex cone optimization framework, the endmember matrix is derived in closed form via direct algebraic transformation, eliminating iterative updates and guaranteeing a globally optimal solution in polynomial time. Finally, the corresponding abundance tensor is estimated via parallel non-negative least squares, allowing the complete three-dimensional hyperspectral tensor to be reconstructed directly through tensor modal multiplication at unprecedented execution speed.
The performance of the proposed algorithm was rigorously evaluated on multiple benchmark hyperspectral datasets, including the Washington DC Mall and Pavia University scenes, across various simulated and real-world degradation patterns against twelve representative benchmark completion algorithms. In terms of reconstruction fidelity, the proposed method achieved superior quantitative performance across all standard metrics, attaining peak signal-to-noise ratios between thirty-two and thirty-three decibels, structural similarity indices exceeding zero point ninety-seven, and spectral angle mapper values well below three degrees. Visually, recovered bands preserved crisp spatial edges, fine spatial textures, and smooth, natural spectral curves without artificial artifacts or color distortions. In terms of computational efficiency, owing to its closed-form geometric solution and convex formulation, the algorithm completed tensor recovery within seven to eleven seconds on standard hardware configurations, representing an execution speedup of one hundred to three hundred times compared to conventional non-convex tensor decomposition algorithms. Furthermore, in downstream land-cover classification evaluations using the recovered data, the algorithm achieved an overall accuracy exceeding ninety-nine percent, outperforming all baseline completion methods and proving that it successfully preserves subtle spectral signatures critical for discriminating visually similar material classes.
The proposed Maximum Volume Inscribed Ellipsoid Tensor Completion algorithm successfully overcomes the long-standing trade-off between computational speed and physical fidelity in hyperspectral image recovery. By combining spatial-spectral pre-restoration with convex geometric unmixing, the methodology provides a mathematically sound, ultra-fast, and robust solution for hyperspectral tensor completion, establishing high practical utility for real-time remote sensing applications and automated Earth observation systems. Although the proposed MVIE-TC algorithm demonstrates excellent completion performance, future research will focus on integrating adaptive weights or sparse priors into the convex geometric framework, enabling autonomous missing-pattern identification and tensor completion under extreme conditions without valid observation mask priors.

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