Ultra-Low Dose Computed Tomography with 90% Radiation Reduction: High-Fidelity Deterministic Regularized Recovery via Idempotent Metric Projectors and Proximal Resolvents

Abstract: Ionizing radiation exposure from diagnostic Computed Tomography (CT) presents a well-documented lifetime risk of secondary radiation-induced malignancies, posing an acute dilemma in pediatric imaging, recurring oncological monitoring, and trauma assessment. Reducing the projection count by 90% via prospective angular under-sampling (e.g., collecting only 18 angular views across 180° via pulsed grid-controlled or multi-source beam gating) drastically curtails radiation dose under the ALARA (As Low As Reasonably Achievable) mandate. However, classical Filtered Backprojection (FBP) fails catastrophically under this severe sub-Nyquist sampling, producing dense streak and star aliasing artifacts that obliterate pathology. While recent deep learning architectures have attempted sparse CT inversion, they are hindered by stochastic hallucinations, unverified synthetic lesions, lack of physical conservation guarantees, and massive GPU computing overhead. In this paper, we establish an algebraic, deterministic, non-deep-learning alternative founded on idempotent metric projection operators (Π² = Π), sequential Kaczmarz hyperplanes, and Krasnoselskii-Mann averaged operator splitting for Cyclic Projection Onto Convex Sets (POCS) regularized via proximal resolvents. We constrain the ill-posed inverse Radon problem via five closed physical sets and operators: (1) sequential Radon measurement hyperplane metric projections ΠHθ, (2) physical non-negativity Π+, (3) spatial field-of-view support bounds ΠFOV, (4) anatomical tissue attenuation upper-bounds Πbox, and (5) isotropic Total Variation (TV) proximal Moreau resolvents proxλ TV. Experimental evaluation is conducted across a cohort of N = 20 anatomical configurations derived from authentic clinical patient CT Digital Imaging and Communications in Medicine (DICOM) scans from a GE Healthcare scanner (10 in-silico affine-augmented thoracic configurations and 10 anatomical sub-region skeletal windows) under prospective 18-angle sampling with realistic Poisson photon counting statistics and electronic detector noise. Across the patient cohort, the proposed IdemSolver engine achieves PSNR = 22.96 ± 4.87 dB (reaching 26.79 dB on primary thoracic anatomy, a +15.29 dB boost over Sparse FBP at 11.50 dB and +2.90 dB over SART-TV at 23.89 dB), and cohort-leading structural similarity SSIM = 0.9619 ± 0.0158 (up to 0.9656, an absolute gain of +48.2% over FBP). Detailed 1D anatomical line profiles confirm sharp preservation of cortical bone margins (μ ≈ 0.98) and smooth paraspinal/pulmonary transitions without TV cartooning or staircasing artifacts. We position 128 × 128 reconstruction as an ultra-fast (178.0 ms) intra-procedural scout and real-time bedside screening preview, while proving through information-theoretic analysis that full diagnostic evaluation on 512 × 512 grids requires proportional angular scaling (36–64 views) to preserve structural similarity (SSIM > 0.85) under 82%–90% dose reduction. Operating on a single commodity CPU core with linear memory scaling (< 25 MB RAM) and certified zero generative false-positive hallucination risk, IdemSolver provides a mathematically rigorous, energy-efficient engine for bedside, intra-operative, and portable point-of-care CT systems.