CAIR IIT Mandi • Theoretical & Computational Foundation

Methodology & Mathematical Engine

Interactive breakdown of the radiation-free spatial mapping operator from external patient surface manifolds to internal 3D organ centroids.

8 Architectural Stages•Sex-Aware 121-Organ Model

Theoretical Foundation•Stage 1 of 8

Problem Formulation & Continuous Spatial Manifold

Radiation-Free Continuous External-to-Internal Coordinate Mapping

Joint Centroid & Uncertainty Mapping OperatorKaTeX Typeset • Exact Formulation

\mathcal{F}_\theta: (\mathcal{S}, \mathcal{T}, g) \longmapsto \left\{ (\hat{\mathbf{c}}_k, \hat{\mathbf{\Sigma}}_k) \in \mathbb{R}^3 \times \mathbb{S}_{++}^3 \;\Big|\; k = 1, \dots, K \right\}

Variable Definitions & Dimensionality:

\mathcal{S}

:External patient surface point cloud (N = 4,096 points on ∂Ω_body)

\mathcal{T}

:Set of K = 121 anatomical queries (visceral organs, vascular & skeletal)

g

:Biological sex prior (Female: g = 0, Male: g = 1)

\hat{\mathbf{c}}_k

:Predicted 3D Euclidean centroid coordinate [X, Y, Z] in mm

\hat{\mathbf{\Sigma}}_k

:Calibrated 3x3 epistemic spatial covariance matrix

Unlike conventional ionizing CT and MRI modalities that acquire direct volumetric slices, Surface2Anatomy solves the inverse boundary value problem: estimating the exact 3D internal metric centroids of internal anatomical organs strictly from the geometric topology of the external body surface.

Zero Radiation DoseSafe for continuous intraoperative tracking, pediatrics, and repeated radiation oncology setups.

Real-Time InferenceSub-50ms forward pass latency enables real-time guidance for robotic ultrasound and laparoscopic arms.

121 Anatomical OrgansExpanded sex-aware atlas covering cranial, thoracic, abdominal, vascular, and pelvic reproductive anatomy.

Stage 01 / 08