Skip to content

Deformable Registration

deformable_registration: Non-rigid point cloud alignment using Coherent Point Drift (CPD) and Gaussian deformation fields. Handles elastic deformations between shapes.
Methods cpd_deformable, gaussian_deformation
Input files .ply, .pcd, .txt, .xyz
Bundled datasets to try open3d_bunny, open3d_icp_demo, kitti_raw_2011_09_26_drive_0001, medshapenet, c3vd_endoscopy_splats
CLI pardesline run deformable_registration.<method> input.ext [--param value]

How it works

Deformable registration finds a non-rigid transformation that aligns a source shape to a target. CPD models source points as GMM centroids and fits them via EM optimization.

The displacement function is:

\[ \mathbf{v}(\mathbf{x}) = \sum_{m=1}^{M} w_m \phi(\|\mathbf{x} - \mathbf{y}_m\|) \]

where \(\phi\) is a Gaussian kernel and \(w_m\) are optimized coefficients.

Methods

cpd_deformable

CPD treats alignment as probability density estimation. The E-step computes correspondence probabilities, the M-step updates the transformation with regularization: $\(E(W) = \frac{1}{2\sigma^2} \sum_n \|T(y_n) - x_n\|^2 + \frac{\lambda}{2} \text{tr}(W^T G W)\)$

import pardesline as pl

fixed, moving = pl.load("fixed.ply"), pl.load("moving.ply")
result = fixed.run("deformable_registration.cpd_deformable", moving=moving)  # catalog defaults
result.provenance[-1]  # the step, every parameter used, duration, version

Output: .ply. Script: python -m pardesline.ops.deformable_registration.cpd_deformable --input ... --output ....

Defaults validated on: generated clouds in tests/test_ops_smoke.py; the point-cloud-registration skill documents its use after a rigid stage

Parameter Type Default Range What it does Why this default, when to change it
max_iterations int 100 10 to 500 Maximum EM iterations. 100 EM iterations: CPD converges in 30-80 on organ-sized clouds; the tolerance below usually stops it first.
tolerance float 0.001 0.0001 to 0.01 Convergence tolerance. 1e-3 relative change of the objective: tighter costs iterations without visible change in the displacement field.
beta (advanced) float 100.0 0.1 to 200.0 CPD Gaussian-kernel width (smoothness). 100 is pycpd's default Gaussian kernel width in file units, i.e. a very smooth, almost global deformation for unit-scale clouds; lower it to about the size of the structures you expect to move independently (10-20 mm for organs).
lambda_val (advanced) float 2.0 0.1 to 20.0 CPD regularization weight. 2.0 regularisation weight: the pycpd default, balancing fit against smoothness; raise it when the field folds.
w (advanced) float 0.0 0.0 to 0.99 Outlier/noise weight. 0 assumes no outliers; set 0.1-0.3 when the fixed cloud has clutter the moving cloud should not reach.
low_rank (advanced) bool false Use low-rank Gaussian-kernel acceleration. Off below about 10k points; on above, where the full kernel matrix no longer fits in memory.
num_eig (advanced) int 100 10 to 500 Eigenvectors when low_rank=True. 100 eigenvectors keep the low-rank approximation close to the full kernel at a fraction of the cost.

gaussian_deformation

A localized Gaussian deformation: $\(\mathbf{d}(\mathbf{p}) = d_{\max} \cdot \exp\left(-\frac{\|\mathbf{p} - \mathbf{c}\|^2}{2\sigma^2}\right) \cdot \hat{\mathbf{n}}\)$

import pardesline as pl

obj = pl.load("input.ply")
result = obj.run("deformable_registration.gaussian_deformation")  # catalog defaults
result.provenance[-1]  # the step, every parameter used, duration, version

Output: .ply. Script: python -m pardesline.ops.deformable_registration.gaussian_deformation --input ... --output ....

Defaults validated on: generated clouds in tests/test_ops_smoke.py

Parameter Type Default Range What it does Why this default, when to change it
displacement float 0.02 0.001 to 0.1 Maximum displacement at pressure center. 0.02 in file units: a visible but plausible dent on a unit-scale object, used to synthesise test cases for non-rigid registration.
radius float 0.1 0.01 to 0.5 Radius of influence. 0.1 in file units: the dent spans 10 % of a unit object.
sigma_factor (advanced) float 0.5 0.05 to 1.0 Gaussian falloff sigma as a fraction of radius. 0.5: the Gaussian falls to about 13 % at the radius, so the dent fades smoothly instead of leaving a crease.

Generated from src/pardesline/catalog/data/deformable_registration.json and _defaults.json on 2026-10-10.