This paper addresses the fundamental computer vision challenge of robust circle detection and fitting in degraded imaging conditions. We present Combinatorial Convolution-based Circle Fitting for Blurry Images (3C-FBI), an algorithm that bridges the gap between circle detection and precise parametric fitting by combining (1) efficient combinatorial edge pixel (edgel) sampling and (2) convolution-based density estimation in parameter space. We evaluate 3C-FBI across three experimental frameworks: (1) real-world medical data from Parkinson's disease assessments (144 frames from 36 videos), (2) controlled synthetic data following established circle-fitting benchmarks, and (3) systematic analysis across varying spatial resolutions and outlier contamination levels. Results show that 3C-FBI achieves state-of-the-art accuracy (Jaccard index 0.896) while maintaining real-time performance (40.3 fps), significantly outperforming classical methods like RCD (6.8 fps) on a standard CPU (i7-10875H). It maintains near-perfect accuracy (Jaccard almost 1.0) at high resolutions (480x480) and reliable performance (Jaccard higher than 0.95) down to 160x160 with up to 20% outliers. In extensive synthetic testing, 3C-FBI achieves a mean Jaccard Index of 0.989 across contamination levels, comparable to modern methods like Qi et al. (2024, 0.991), and surpassing RHT (0.964). This combination of accuracy, speed, and robustness makes 3C-FBI ideal for medical imaging, robotics, and industrial inspection under challenging conditions.