Scan-based Compressed Terahertz Imaging and Real-Time Reconstruction via the Complex-valued Fast Block Sparse Bayesian Learning Algorithm

Compressed Sensing based Terahertz imaging (CS-THz) is a computational imaging technique. It uses only one THz receiver to accumulate the random modulated image measurements where the original THz image is reconstruct from these measurements using compressed sensing solvers. The advantage of the CS-THz is its reduced acquisition time compared with the raster scan mode. However, when it applied to large-scale two-dimensional (2D) imaging, the increased dimension resulted in both high computational complexity and excessive memory usage. In this paper, we introduced a novel CS-based THz imaging system that progressively compressed the THz image column by column. Therefore, the CS-THz system could be simplified with a much smaller sized modulator and reduced dimension. In order to utilize the block structure and the correlation of adjacent columns of the THz image, a complex-valued block sparse Bayesian learning algorithm was proposed. We conducted systematic evaluation of state-of-the-art CS algorithms under the scan based CS-THz architecture. The compression ratios and the choices of the sensing matrices were analyzed in detail using both synthetic and real-life THz images. Simulation results showed that both the scan based architecture and the proposed recovery algorithm were superior and efficient for large scale CS-THz applications.

💡 Research Summary

This paper addresses the challenges of large‑scale two‑dimensional terahertz (THz) imaging using compressed sensing (CS). Traditional CS‑THz systems employ a single detector and an N × N random mask, requiring N² measurements to reconstruct an N × N complex‑valued image. As N grows beyond 100, the required sensing matrix becomes prohibitively large, leading to excessive memory consumption, high computational complexity, and practical difficulties in fabricating and switching masks.

To overcome these limitations, the authors propose a scan‑based CS‑THz architecture that compresses the image column‑by‑column. A small sensing matrix Φ of size M × N (M ≪ N) is used repeatedly for each column, so only N different mask patterns are needed instead of N². The measurement process can be realized with either a single detector that moves a 1‑D mask across the object or with a line array of N detectors that acquire an entire row simultaneously, further reducing acquisition time by a factor of N. Mathematically, the overall sensing matrix is the Kronecker product I_N ⊗ Φ, preserving the same compression ratio definition CR = (N − M)/N while dramatically shrinking the dimensionality of the problem from N² to N·M.

The key algorithmic contribution is a complex‑valued Block Sparse Bayesian Learning (BSBL) method tailored to the scan‑based setting, called BSBL‑FM‑MMV (Fast Marginalized – Multiple Measurement Vector). The image X∈ℂ^{N×N} is partitioned into g blocks X_i of size d_i × N. Each block is modeled as a matrix‑variate Gaussian with zero mean and covariance γ_i I_{d_i} ⊗ I_N, where γ_i is a hyper‑parameter representing the block’s average variance. This hierarchical prior automatically promotes block‑sparsity: blocks with small γ_i are effectively pruned during learning.

Measurements are modeled as Y = ΦX + E, where E is i.i.d. complex Gaussian noise with precision β. Using standard Gaussian identities, the posterior p(X|Y;γ,β) and the marginal likelihood L(γ,β) are derived. To avoid the O(N³) cost of directly inverting the large matrix C = β^{-1}I_M + ΦΓΦ^H, the authors apply the Fast Marginalized Likelihood Maximization (FMLM) technique. By separating C into contributions from each block and employing the Woodbury matrix identity, the cost function can be expressed as a sum of block‑wise terms L(i). Each γ_i admits a closed‑form update:

γ_i = (1/(d_i N)) Tr