LDDMM: influence of control points location

import shutil
from pathlib import Path

import numpy as np
import pyvista as pv
from matplotlib import pyplot as plt

import polpo.lddmm as plddmm
from polpo.mesh.geometry import centroid2farthest_vertex, vertexwise_euclidean
from polpo.mesh.surface import PvSurface
from polpo.mesh.varifold.tuning import SigmaFromLengths
from polpo.plot.pyvista import RegisteredMeshesColoredPlotter
from polpo.preprocessing.load.pregnancy.deformetrica import get_two_random_meshes

[KeOps] Warning : CUDA was detected, but driver API could not be initialized. Switching to CPU only.

RECOMPUTE = False

STATIC_VIZ = True
VIZ = 1

if STATIC_VIZ:
    pv.set_jupyter_backend("static")

STRUCT_NAME = "L_Hipp"


OUTPUTS_DIR = Path.home() / ".polpo/results" / f"lddmm_control_points_{STRUCT_NAME}"
REGISTRATION_DIR = OUTPUTS_DIR / "registration"


if OUTPUTS_DIR.exists() and RECOMPUTE:
    shutil.rmtree(OUTPUTS_DIR)

meshes = get_two_random_meshes(
    OUTPUTS_DIR,
    mesh_names=("source", "target"),
    target_reduction=0,
)

if VIZ > 1:
    pl = pv.Plotter(border=False)

    for mesh in meshes.values():
        pl.add_mesh(mesh.as_pv(), show_edges=True, opacity=0.6)

    pl.show()

We select the varifold kernel using characteristic lengths.

sigma_search = SigmaFromLengths(
    ratio_charlen_mesh=2.0,
    ratio_charlen=0.25,
)

sigma_search.fit(meshes.values())

metric = sigma_search.optimal_metric_

sigma_search.sigma_

np.float64(5.589180490389398)

Following LDDMM: how to register a mesh to a template?.

registration_kwargs = dict(
    kernel_width=4.0,
    regularisation=1.0,
    max_iter=2000,
    freeze_control_points=False,
    metric="varifold",
    tol=1e-16,
    attachment_kernel_width=sigma_search.sigma_,
)

fmt = "%0.6f"
delimiter = " "

registration_dirs = []
mesh_filenames = list(meshes.keys())

registration_dir = REGISTRATION_DIR / "uniform"
registration_dirs.append(registration_dir)


if not registration_dir.exists():
    # ~420 cps
    plddmm.registration.estimate_registration(
        mesh_filenames[0],
        mesh_filenames[1],
        output_dir=registration_dir,
        **registration_kwargs,
    )

Using mesh vertices as control points.

registration_dir = REGISTRATION_DIR / "vertices_full"
registration_dirs.append(registration_dir)


if not registration_dir.exists():
    registration_dir.mkdir(parents=True, exist_ok=True)

    cps_name = registration_dir / "ControlPoints.txt"
    np.savetxt(
        cps_name,
        list(meshes.values())[0].vertices,
        fmt=fmt,
        delimiter=delimiter,
    )

    # ~1002 cps
    plddmm.registration.estimate_registration(
        mesh_filenames[0],
        mesh_filenames[1],
        output_dir=registration_dir,
        initial_control_points=cps_name,
        **registration_kwargs,
    )

Using randomly sampled mesh vertices as control points.

# TODO: use farthest point sampling?
# TODO: use curvature based selection

registration_dir = REGISTRATION_DIR / "vertices_random"
registration_dirs.append(registration_dir)


if not registration_dir.exists():
    registration_dir.mkdir(parents=True, exist_ok=True)

    vertices = list(meshes.values())[0].vertices
    cps_idx = np.random.choice(range(vertices.shape[0]), size=420, replace=False)

    cps_name = registration_dir / "ControlPoints.txt"
    np.savetxt(
        cps_name,
        vertices[cps_idx],
        fmt=fmt,
        delimiter=delimiter,
    )

    # ~420 cps
    plddmm.registration.estimate_registration(
        mesh_filenames[0],
        mesh_filenames[1],
        output_dir=registration_dir,
        initial_control_points=cps_name,
        **registration_kwargs,
    )

Comparison

reconstructed = {}
cps = {}

for registration_dir in registration_dirs:
    reconstructed[registration_dir.name] = PvSurface(
        plddmm.io.load_deterministic_atlas_reconstruction(registration_dir, as_pv=True)
    )
    cps[registration_dir.name] = plddmm.io.load_cp(registration_dir)

_, target = meshes.values()

{
    name: metric.dist(target, reconstructed_)
    for name, reconstructed_ in reconstructed.items()
}

{'uniform': np.float64(3.8427535440006966),
 'vertices_full': np.float64(3.9352090397352595),
 'vertices_random': np.float64(4.156589211729441)}

ref_dist = centroid2farthest_vertex([target])[0]

for name, reconstructed_ in reconstructed.items():
    cps_ = cps[name]

    pl = RegisteredMeshesColoredPlotter()

    pl.add_meshes(
        target.as_pv(),
        reconstructed_.as_pv(),
        ref_dist=ref_dist,
        show_edges=True,
        opacity=0.8,
        name="vertex diffs / ref",
    )

    pl.add_title(name, font_size=8.0)

    pl.add_points(pv.PolyData(cps_), color="red")

    pl.show()

_, ax = plt.subplots()

for name, reconstructed_ in reconstructed.items():
    vals = vertexwise_euclidean(target, reconstructed_) / ref_dist

    ax.hist(
        vals,
        weights=1 / len(vals) * np.ones_like(vals),
        label=name,
        alpha=0.6,
    )

ax.set_xlabel("Vertexwise euclidean distances / Ref dist")
ax.set_ylabel("Frequency")

ax.legend(bbox_to_anchor=(1.02, 1), loc="upper left");

Conclusion: uniform grid is at least as good as other naive control points selection strategy.

Further reading

• LDDMM: importance of the data attachment term

• LDDMM: influence of regularisation

• LDDMM: influence of time points