High-resolution 3D radar data is scarce. Commodity mmWave sensors use small antenna arrays that limit angular resolution to several degrees, and existing datasets provide only 2D range-azimuth maps or sparse point clouds rather than raw analog-to-digital converter (ADC) signals. Hardware scaling is expensive, synthetic-aperture scanning is impractical at fleet scale, and learned synthesis methods are bottlenecked by the very data shortage they aim to address. We present mmIR, an open-source differentiable frequency-modulated continuous-wave (FMCW) radar inverse renderer that fits a physics-based forward model to real captures and re-renders from dense virtual apertures to synthesize high-resolution 3D radar data. Because radar resolution is too coarse to recover geometry directly, mmIR performs LiDAR-assisted inverse rendering: using LiDAR-derived meshes as a geometric scaffold, mmIR optimizes per-vertex International Telecommunication Union (ITU) physics materials, vertex normals, and antenna beam patterns through end-to-end automatic differentiation of a phase-coherent multiple-input multiple-output (MIMO) forward model with multi-bounce propagation, polarization, and free-space diffraction. On seven outdoor and six indoor ColoRadar scenes, mmIR achieves 0.914 mean Pearson correlation on range-azimuth maps versus 0.307 for Sionna-RT. Scenes trained on a cascaded imaging radar transfer to a co-located single-chip radar without re-training (0.554 correlation), and dense virtual arrays (100x100 elements) produce single-frame 3D occupancy validated against LiDAR. Project page: https://mmwave-inverse-rendering.github.io/

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