Learned Dictionaries with Total Variation and Non-Negativity for Single-Cell Microscopy: Convergence Theory and Deterministic Multi-Channel Cell Feature Unification

Abstract

We introduce a variational dictionary learning algorithm with hybrid penalization for single-cell microscopy signals. The cost functional couples least-squares data fidelity with total-variation (TV) regularization and a non-negativity constraint, promoting edge-preserving, physically meaningful reconstructions. The learning task is formulated with an explicit unitary constraint on the dictionary, ensuring well-conditioned representations. The optimization is solved by an alternating proximal-gradient scheme; we prove PDHG iterates converge to the regularized minimizer under an explicit step-size condition (tau*sigma < 1/8), and that under a variational source condition (VSC) the regularized solution converges to the true solution at the optimal O(delta) rate with lambda proportional to delta. Beyond reconstruction, we address multi-channel cell feature unification: given five imaging channels of the BSCCM dataset (DPC Left, Right, Top, Bottom, Brightfield), we learn a family of per-channel unitary dictionaries, each adapted to its channel's optical physics, and concatenate the per-channel sparse codes into a single channel-agnostic cell descriptor. This deterministic approach is mathematically transparent, reproducible, and compatible with clinical AI auditability requirements. On BSCCM-tiny (N=1000 cells, K=512 atoms) the framework reaches reconstruction fidelities of 97.06-97.54% on DPC channels and 94.79% on Brightfield, with bit-identical iterates across runs. Biological validation yields unsupervised lymphoid-vs-myeloid separation at ARI=0.575, NMI=0.471 (permutation p<0.0001).