COS2A Algorithm for Hyperspectral Imaging

• COS2A is a computational imaging algorithm that converts 12‑band Sentinel‑2 scenes into 172‑band AVIRIS‑like hyperspectral cubes with mixed spatial resolutions.
• It combines deep unfolding with a convex low‑rank reconstruction framework under the CODE paradigm to address a challenging small‑data inverse problem.
• The method leverages a spectral–spatial duality theorem to recast spectral super‑resolution as a coupled‑NMF spatial problem, enhancing interpretability and performance.

Searching arXiv for the cited COS2A paper and the similarly named but unrelated matrix-functions paper. COS2A, short for Conversion from Sentinel‑2 to AVIRIS, is a computational imaging algorithm for reconstructing an AVIRIS‑like hyperspectral cube from a single multi‑resolution Sentinel‑2 scene. In the formulation reported in "COS2A: Conversion from Sentinel-2 to AVIRIS Hyperspectral Data Using Interpretable Algorithm With Spectral-Spatial Duality" (Lin et al., 9 Jul 2025), the input is a 12-band Sentinel‑2 image with mixed spatial resolutions of 10/20/60 m, and the target is a 172-band VNIR–SWIR hyperspectral image at 10 m resolution. The method is explicitly designed for the very small‑data regime and combines a deep unfolding stage with a convex low-rank reconstruction stage under the authors’ CODE (convex/deep) learning framework. A central theoretical claim is a spectral–spatial duality theorem that recasts the spectral super-resolution problem into a coupled-NMF spatial super-resolution form, permitting the use of an existing convex solver (Lin et al., 9 Jul 2025). A common source of confusion is the unrelated matrix-function literature: "Computing the Action of Trigonometric and Hyperbolic Matrix Functions" (Higham et al., 2016) does not define any algorithm named COS2A.

1. Problem definition and sensing model

COS2A addresses a remote-sensing inverse problem in which Sentinel‑2 provides broad spatial coverage but only 12 usable spectral bands in the reported setup, whereas AVIRIS provides a much denser spectral sampling. The reported target is an AVIRIS-like subset of 172 bands obtained after removing water-vapor absorption and corrupted channels from the original AVIRIS acquisition (Lin et al., 9 Jul 2025). In the notation of the paper, the Sentinel‑2 observation is represented by

$Y_S \in \mathbb{R}^{12 \times L},$

and the hyperspectral target by

$Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$

A distinctive complication is that Sentinel‑2 bands are natively acquired at multiple resolutions: 4 bands at 10 m, 6 bands at 20 m, and 2 bands at 60 m in the configuration described in the paper (Lin et al., 9 Jul 2025). To place all inputs on a common grid, lower-resolution bands are stored as if they were on the 10 m lattice by replication: each 20 m pixel is copied 4 times and each 60 m pixel 36 times. This produces a unified $12 \times L$ representation without eliminating the underlying multi-resolution character of the observation model (Lin et al., 9 Jul 2025).

The paper emphasizes that this problem is substantially more ill-posed than the usual spectral super-resolution settings studied on CAVE- or Harvard-type benchmarks. The difficulty arises from the 12 $\to$ 172 spectral lift, the mixed-resolution input, the scarcity of real overlapping AVIRIS–Sentinel‑2 pairs, and the absence of official spectral response functions linking AVIRIS to Sentinel‑2 in the reported setup (Lin et al., 9 Jul 2025). This suggests that COS2A is positioned not as a generic hyperspectral reconstruction pipeline, but as a sensor-pair-specific inverse method whose design is closely tied to these structural constraints.

2. Forward model, low-rank factorization, and CODE objective

The convex phase of COS2A uses only the four high-resolution Sentinel‑2 bands, denoted

$\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$

together with a scene-adaptive spectral response matrix

$\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$

The associated forward model is

$\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$

with data fidelity

$$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$

The paper states that the full 12-band multi-resolution observation is used by the deep stage, while the convex data term is restricted to the four 10 m bands, with lower-resolution information entering through regularization design (Lin et al., 9 Jul 2025).

The hyperspectral image is assumed to obey the standard low-rank unmixing model

$Y_H = AS,$

where

$$A \in \mathbb{R}^{172 \times N}, \qquad S \in \mathbb{R}^{N \times L},$$

with $Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$0 the endmember matrix and $Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$1 the abundance matrix. The final reconstruction is

$Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$2

The paper treats $Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$3 as the model order and reports using $Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$4 in the convex phase across experiments (Lin et al., 9 Jul 2025).

Under the authors’ CODE principle, the generic objective is

$Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$5

with a quadratic regularizer centered at a rough deep estimate: $Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$6 Specialized to the endmember–abundance parameterization, the COS2A objective is written as

$Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$7

The paper reports $Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$8 and $Y_H \in \mathbb{R}^{172 \times L}, \quad M:=172.$9 (Lin et al., 9 Jul 2025).

Two additional priors complete the formulation. The first is a minimum-volume surrogate,

$12 \times L$0

which the paper connects to Craig’s minimum-volume unmixing criterion. The second is the $12 \times L$1 penalty on $12 \times L$2, intended to promote sparse abundances (Lin et al., 9 Jul 2025). In this architecture, the deep prior contributes a rough scene-adaptive estimate, while the convex phase imposes low-rank, minimum-volume, and sparsity structure.

3. Deep unfolding stage and rough hyperspectral prior

The rough estimate $12 \times L$3 is produced by a small unrolled ADMM network with a Deep-Image-Prior-type objective. The reported optimization is

$12 \times L$4

which is rewritten using an auxiliary variable $12 \times L$5 as

$12 \times L$6

The scaled augmented Lagrangian is

$12 \times L$7

where $12 \times L$8 is the dual variable and $12 \times L$9 is trainable (Lin et al., 9 Jul 2025).

The ADMM decomposition consists of three updates. The $\to$0-update is a proximal denoising step,

$\to$1

interpreted in the paper as a deep denoiser. The $\to$2-update is a quadratic subproblem. The paper first writes the closed form

$\to$3

and then uses the Woodbury identity to obtain the implemented form

$\to$4

with

$\to$5

The dual update is

$\to$6

After $\to$7 stages, the network outputs

$\to$8

The paper reports $\to$9 (Lin et al., 9 Jul 2025).

The unfolded network has two principal modules. Model 1 realizes the denoising/proximal block using a residual-in-residual CNN denoiser. Model 2 realizes the linear operator in the $\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$0-update by combining a skip connection with downsampling, a symmetric fully connected layer, and upsampling, corresponding to a “project to multispectral, process, then back-project” operator (Lin et al., 9 Jul 2025). Stage 1 includes an initial spectral upsampling from 12 bands to 172 bands; intermediate stages execute the ADMM pattern; the final stage outputs $\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$1. The paper describes the network as lightweight, with approximately 0.75M parameters, and states that this small size is intentional because the CODE framework only requires a rough prior rather than a high-capacity end-to-end predictor (Lin et al., 9 Jul 2025).

A plausible implication is that the deep module is not intended to solve the inverse problem by itself. Instead, it supplies a structured initialization around which the convex stage imposes physically motivated constraints.

4. $\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$2-quadratic regularization and spectral–spatial duality

A defining feature of COS2A is the use of a quadratic regularizer with a nontrivial PSD weighting matrix $\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$3: $\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$4 The paper designs $\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$5 from a spatial blur operator. Let

$\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$6

with blurring factor $\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$7. Then

$\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$8

Because $\widetilde{Y}_S \in \mathbb{R}^{4 \times L},$9 acts on the spectral dimension, the blur encoded by $\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$0 is purely spatial (Lin et al., 9 Jul 2025).

The associated interpretation is that the regularizer enforces agreement between the unknown hyperspectral cube and a blurred version of the deep prior at the level of spatial regions. The paper states the equivalent identity

$\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$1

derived via vectorization and Kronecker-product algebra (Lin et al., 9 Jul 2025). This reformulation is central because it permits the deep prior to enter the convex stage as a pseudo coarse-resolution hyperspectral observation.

The main theoretical statement is Theorem 1, called Spectral-Spatial Duality. The paper states that the COS2A spectral super-resolution problem is mathematically equivalent to a coupled-NMF spatial super-resolution problem widely used in optical remote sensing (Lin et al., 9 Jul 2025). The reference CNMF objective is written as

$\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$2

where

$\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$3

The equivalence works by identifying the COS2A data term

$\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$4

with the high-resolution multispectral term, and the regularizer

$\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$5

with the coarse hyperspectral term through the relation

$\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$6

The paper then interprets $\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$7 as a pseudo low-resolution hyperspectral image and $\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$8 as the high-resolution multispectral image (Lin et al., 9 Jul 2025). This suggests that the primary theoretical contribution is not merely a new objective, but an inverse-problem transformation that makes an otherwise unusual spectral SR problem compatible with a mature spatial SR solver class.

5. End-to-end pipeline and optimization workflow

The reported COS2A workflow proceeds in six steps (Lin et al., 9 Jul 2025). First, the deep unfolding network produces the rough estimate $\widetilde{\Phi} \in \mathbb{R}^{4 \times 172}.$9. Second, the scene-adaptive spectral response $\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$0 is estimated from $\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$1 and the four 10 m Sentinel‑2 bands using nonnegative ridge regression: $\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$2 with $\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$3. The paper states that the nonnegativity constraint enforces physical meaning, while the ridge term discourages excessive sparsity in the response across bands (Lin et al., 9 Jul 2025).

Third, the method instantiates the convex objective using $\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$4, the $\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$5 matrix defined from the blur operator, and the estimated $\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$6. Fourth, it applies the spectral–spatial duality theorem to convert the spectral SR problem into the coupled-NMF spatial SR form. Fifth, instead of introducing a new ad hoc optimizer, the paper states that it calls an existing convex algorithm for CNMF spatial super-resolution, specifically COCNMF, to recover $\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$7 and $\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$8 (Lin et al., 9 Jul 2025). Sixth, it reconstructs the final AVIRIS-like image via

$\widetilde{Y}_S \approx \widetilde{\Phi} Y_H,$9

The paper presents this pipeline as an interpretable composition of three elements: explicit deep-unfolding iterations, a quadratic regularizer with a blur-based $$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$0, and a convex NMF solver with minimum-volume and sparsity penalties (Lin et al., 9 Jul 2025). In that sense, interpretability is tied to modular correspondence between each computational block and a stated optimization component, rather than to post hoc explainability.

6. Training protocol, evaluation setup, and reported performance

Two training and evaluation regimes are reported. The first is a simulation study based on 646 AVIRIS tiles of size $$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$1, retaining 172 bands after removing absorption channels. Sentinel‑2 observations are simulated by constructing a $$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$2 spectral response matrix from published central wavelengths and bandwidths, keeping the 10 m bands at full resolution, uniformly blurring the 20 m bands with factor 2, uniformly blurring the 60 m bands with factor 6, and replicating lower-resolution pixels to the 10 m grid. The split is 15:1:1 for train/validation/test (Lin et al., 9 Jul 2025).

The second regime uses 526 real AVIRIS/Sentinel‑2 pairs, cropped to $$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$3 after AVIRIS Level‑2 and Sentinel‑2 Level‑2A preprocessing, AVIRIS resampling to 10 m, orientation correction, co-registration via ENVI, and removal of cloud and black-border regions. The split is 8:1:1 (Lin et al., 9 Jul 2025).

The deep unfolding network is trained end-to-end on $$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$4 patches with batch size 8, for 30 epochs, using 20,000 overlapping patches for training+validation and data augmentation via random flips and rotations. The training loss is the robust $$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$5 loss

$$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$6

The paper notes that the spectral response $$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$7 is known in simulation but estimated by ridge regression on real data (Lin et al., 9 Jul 2025).

For comparison, the paper introduces a divide-and-conquer baseline based on MST++, splitting the 12 $$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$8 172 mapping into four disjoint 3 $$\text{DF}(Y_H \mid \widetilde{\Phi}, \widetilde{Y}_S)=\|\widetilde{Y}_S-\widetilde{\Phi}Y_H\|_F^2.$$9 43 branches and concatenating their outputs. Evaluation uses PSNR, SAM, RMSE, and SSIM on four land-cover categories: coastline/lake, mountain, farm, and city, with 10 test images per category (Lin et al., 9 Jul 2025).

The reported quantitative results are as follows.

Category	DAC baseline	COS2A
Coastline/lake	PSNR 27.70, SAM 8.41°, RMSE 0.0264, SSIM 0.9285	PSNR 35.89, SAM 2.45°, RMSE 0.0080, SSIM 0.9704
Mountain	PSNR 28.29, SAM 4.93°, RMSE 0.0180, SSIM 0.9317	PSNR 33.23, SAM 2.03°, RMSE 0.0092, SSIM 0.9512
Farm	PSNR 27.10, SAM 9.05°, RMSE 0.0427, SSIM 0.8896	PSNR 35.07, SAM 2.15°, RMSE 0.0113, SSIM 0.9527
City	PSNR 28.08, SAM 8.47°, RMSE 0.0408, SSIM 0.8230	PSNR 35.84, SAM 3.38°, RMSE 0.0175, SSIM 0.9209
Overall average	PSNR 27.79, SAM 7.71°, RMSE 0.0320, SSIM 0.8932	PSNR 35.01, SAM 2.50°, RMSE 0.0115, SSIM 0.9488

The runtime reported for a $Y_H = AS,$0 image is approximately 0.066 s for the pure deep DAC baseline and approximately 14.6 s for COS2A, with the paper attributing the overhead mainly to the convex CNMF stage (Lin et al., 9 Jul 2025). Qualitatively, the paper states that COS2A preserves fine edges and textures more effectively and produces spectral curves for representative pixels that align more closely with AVIRIS references, including subtle NIR structure (Lin et al., 9 Jul 2025).

For real Sentinel‑2 $Y_H = AS,$1 AVIRIS validation, the paper introduces an illumination calibration scalar

$Y_H = AS,$2

where $Y_H = AS,$3 is the 12-dimensional subvector of an AVIRIS pixel aligned to Sentinel‑2 band centers. The reported outcome is that COS2A reconstructions visually and spectrally resemble real AVIRIS observations over coastline, mountain, farm, and city scenes, and in some cases fit Sentinel‑2 samples better than raw AVIRIS because of illumination differences (Lin et al., 9 Jul 2025).

7. Interpretation, limitations, and related uses

COS2A is presented as an interpretable hybrid convex/deep method rather than a purely learned black-box predictor. In the paper’s decomposition, the ADMM-unrolled network corresponds to a rough prior generator, $Y_H = AS,$4 encodes a specific spatial blur model, and the final inference stage is a convex coupled-NMF solver with explicit endmember-volume and abundance-sparsity priors (Lin et al., 9 Jul 2025). This architecture is tailored to the small-data setting, where large-scale paired training corpora are unavailable.

Several limitations and assumptions are explicitly stated. The method depends on accurate co-registration and preprocessing for real data. Its latent scene model is linear, relying on $Y_H = AS,$5, linear spectral response estimation, and a uniform blur model for the lower-resolution content. The $Y_H = AS,$6 construction uses $Y_H = AS,$7 region-level averaging, which the paper presents as an approximation to the way 20 m and 60 m bands are represented on the 10 m grid, rather than a detailed optical PSF model (Lin et al., 9 Jul 2025). The convex stage is substantially more expensive than a pure deep forward pass, which limits throughput for real-time use. The method is also sensor specific in the sense that the reported formulation is tuned to Sentinel‑2 inputs and AVIRIS-like outputs, with scene-adaptive response estimation and training choices calibrated to that pair (Lin et al., 9 Jul 2025).

The paper proposes several downstream applications for the reconstructed AVIRIS-like products, including land cover and material classification, vegetation and crop monitoring, change detection and anomaly detection, and mangrove dynamics and environmental monitoring (Lin et al., 9 Jul 2025). It also suggests extending the general strategy to other sensor-pair mappings and to more resource-constrained deployment settings (Lin et al., 9 Jul 2025). A plausible implication is that the broader methodological significance of COS2A lies less in any one architectural block than in the combination of three ideas: scene-adaptive deep priors, a blur-structured quadratic regularizer, and an inverse-problem transformation that reuses established coupled-NMF machinery.

A final clarification is useful because of naming ambiguity. The 2016 paper "Computing the Action of Trigonometric and Hyperbolic Matrix Functions" (Higham et al., 2016) develops algorithms such as trigmv and trighmv for matrix-function actions and explicitly does not use the label “COS2A.” Accordingly, in current arXiv usage, COS2A refers to the Sentinel‑2-to-AVIRIS conversion algorithm of 2025 rather than to any matrix cosine action method (Higham et al., 2016).