Canonical Push-Forward SDF Encoder

1. Integration point: add a canonicalization front-end before a 2D CNN, 3D sparse CNN, or point/voxel encoder. The input is an SDF sampled on a fixed reference grid $u\in S_r$; optional intensity or molecular channels are warped with the same map. Predict a displacement field $v_\theta(u)$ and define $\Psi_\theta(u)=u+v_\theta(u)$, followed by differentiable grid sampling of the observed SDF at $\Psi_\theta(u)$. For rigid nuisance removal, predict translation, scale, and rotation first, and compose that transform with the nonrigid map. 2. Pseudocode: sample reference grid $u$; predict $v=\mathrm{DeformNet}(L_S)$; set $x=u+v$; compute $L^*=\mathrm{grid\_sample}(L_S,x)$; encode $z=\mathrm{Encoder}(L^*)$; optimize task loss plus $\lambda_{PF}\|L^*_1-L^*_2\|_1$ for augmented views of the same shape, Jacobian fold penalty $\mathcal L_{diff}$, and optionally inverse consistency $\|\Psi^{-1}(\Psi(u))-u\|^2$. 3. The mathematical quantities computed from the paper are the level-set zero crossing, transported field, Jacobian determinant, and reference-domain comparison. The deformation is estimated by the neural network; estimate Jacobians with autodiff or centered finite differences. Use a smooth SDF target and clamp distances to $[-d_{max},d_{max}]$. 4. First experiment: train a small 2D shape classifier on rotated, reflected, translated, and rescaled MPEG-7 or synthetic silhouettes. Compare a raw-image CNN, a standard spatial transformer, and the PF-T SDF encoder at equal parameter count. Measure accuracy under held-out transformations, embedding cosine consistency between augmented views, and deformation fold rate. A successful result is substantially smaller accuracy degradation under unseen transformations and lower within-class embedding variance without reducing clean accuracy. Then repeat on a small ShapeNet subset with a 3D sparse encoder.