Residual-Distribution-Aware Capacity Analysis for Reversible Data Hiding in Encrypted 3D Medical Volumes with Integer-Valued Voxels

Reversible data hiding in encrypted media enables auxiliary information to be embedded into protected content while allowing exact recovery of the processed data. Although this technique has been extensively investigated for images and surface-based 3D models, its application to 3D medical volumes remains insufficiently explored. This study presents a residual-distribution-aware capacity analysis for reversible data hiding in encrypted 3D medical volumes with integer-valued voxels. The current framework is restricted to finite, numerically integer-valued voxel samples stored in single-precision format, which can be mapped losslessly to a nonnegative integer domain without quantization. Rather than treating prediction-error minimization as the sole objective, we analyze how residual occupancy across power-of-two bit-length intervals governs leading-zero-count-based embedding capacity. Because the leading-zero count changes only when a residual crosses an interval boundary, reducing the prediction error within the same interval does not necessarily increase embedding capacity. Experiments on nine categories of the Medical Segmentation Decathlon show that pure embedding capacity is jointly determined by representation-derived redundancy, residual-dependent redundancy, and auxiliary information overhead. The benefit of inter-slice prediction is dataset-dependent rather than universal, while homogeneous zero-valued regions can contribute substantially to favorable residual distributions. End-to-end validation at 99% of the available pure embedding capacity achieved error-free secret-data extraction and exact recovery of the processed voxel arrays at both the numerical and bit levels. These results provide a capacity-oriented interpretation of encrypted volumetric reversible data hiding based on residual distribution rather than prediction-error magnitude alone.