Underdetermined Blind Source Separation via Weighted Simplex Shrinkage Regularization and Quantum Deep Image Prior

Abstract

As most optical satellites remotely acquire multispectral images (MSIs) with limited spatial resolution, multispectral unmixing (MU) becomes a critical signal processing technology for analyzing the pure material spectra for high-precision classification and identification. Unlike the widely investigated hyperspectral unmixing (HU) problem, MU is much more challenging as it corresponds to the underdetermined blind source separation (BSS) problem, where the number of sources is larger than the number of available multispectral bands. In this article, we transform MU into its overdetermined counterpart (i.e., HU) by inventing a radically new quantum deep image prior (QDIP), which relies on the virtual band-splitting task conducted on the observed MSI for generating the virtual hyperspectral image (HSI). Then, we perform HU on the virtual HSI to obtain the virtual hyperspectral sources. Though HU is overdetermined, it still suffers from the ill-posed issue, for which we employ the convex geometry structure of the HSI pixels to customize a weighted simplex shrinkage (WSS) regularizer to mitigate the ill-posedness. Finally, the virtual hyperspectral sources are spectrally downsampled to obtain the desired multispectral sources. The proposed geometry/quantum-empowered MU (GQ-$μ$) algorithm can also effectively obtain the spatial abundance distribution map for each source, where the geometric WSS regularization is adaptively and automatically controlled based on the sparsity pattern of the abundance tensor. Simulation and real-world data experiments demonstrate the practicality of our unsupervised GQ-$μ$ algorithm for the challenging MU task. Ablation study demonstrates the strength of QDIP, not achieved by classical DIP, and validates the mechanics-inspired WSS geometry regularizer.

Figures (7)

• Figure 1: Graphical illustration of the proposed weighted simplex shrinkage (WSS) regularization. Unlike existing volume-minimization types of regularization that blindly promote the volume minimization without considering the sparsity pattern of the abundances, the proposed WSS regularization further takes the sparsity level into consideration. Intuitively, when an endmember ${\bm a}_n^{k}$ corresponds to a highly sparse abundance map at the $k$th algorithmic iteration, this endmember is geometrically very far away from the data center ${\bm c}$. So, the endmember requires a larger force (i.e., a larger weight $w_3$) to draw it to be closer to ${\bm c}$ at the $(k+1)$th iteration. By contrast, for the relatively dense abundance map (low sparsity), the corresponding endmember just needs to be slightly pushed to the center ${\bm c}$, thus requiring a smaller weight (e.g., $w_1$). A larger weight $w_n$ will be associated with more orange arrows (denoting a higher force).
• Figure 2: Graphical illustration of the proposed quantum deep image prior (QDIP), composed of a core quantum module, reshape operator, inverse quantum collapse (QC) module, and softmax operator. First, we use the core quantum module to obtain the quantum features. Specifically, the module is developed by the quantum full-expressibility (FE) layer "$R_X-XX-R_Z-XX-R_X$", with trainable real-valued parameters (i.e., $\alpha_i, \beta_i, \delta_i$, and $\rho_i$). With the provable FE, it can perform any valid quantum operators (cf. Theorem \ref{['theorem: FE']}). Also, we use the Toffoli (i.e., CCNOT) entangled layers for additional entanglement, and we employ the Pauli-Z measurement (cf. Supplementary Table I) to obtain the quantum statistics. Once the core quantum module holds the desired quantum states, the inverse-QC module is skillfully leveraged to prevent the valuable quantum information from being vanished during the reading-out stage. Eventually, the softmax operator guarantees that the output of QDIP satisfies the abundance sum-to-one property. More detailed configurations are summarized in Supplementary Table II.
• Figure 3: False-color compositions (bands 25, 12, and 6 as RGB) of NASA's HSIs captured by the AVIRIS sensor for different locations and landscapes.
• Figure 4: The qualitative comparisons over the Boulder dataset. The corresponding multispectral endmembers (i.e., M-endmember) of the $n$th abundance are shown in the $n$th subfigure within the rightmost column, where the horizontal and vertical axes represent the spectral bands and reflectance values, respectively.
• Figure 5: The qualitative comparisons over the Ottawa dataset. The corresponding multispectral endmembers (i.e., M-endmember) of the $n$th abundance are shown in the $n$th subfigure within the rightmost column, where the horizontal and vertical axes represent the spectral bands and reflectance values, respectively.

Theorems & Definitions (1)

• Theorem 1