feat(invdes): smoothed projection

Author: mahlau-flex

CHANGELOG.md

@@ -39,6 +39,7 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 - Added warning if port mesh refinement is incompatible with the `GridSpec` in the `TerminalComponentModeler`.
 - Various types, e.g. different `Simulation` or `SimulationData` sub-classes, can be loaded from file directly with `Tidy3dBaseModel.from_file()`.
 - Added `interp_spec` in `EMEModeSpec` to enable faster multi-frequency EME simulations. Note that the default is now `ModeInterpSpec.cheb(num_points=3, reduce_data=True)`; previously the computation was repeated at all frequencies.
+- Added `smoothed_projection` for topology optimization of completely binarized designs.
 
 ### Breaking Changes
 - Edge singularity correction at PEC and lossy metal edges defaults to `True`.

docs/api/plugins/autograd.rst

@@ -84,4 +84,5 @@ Inverse Design
     tidy3d.plugins.autograd.invdes.make_filter_and_project
     tidy3d.plugins.autograd.invdes.ramp_projection
     tidy3d.plugins.autograd.invdes.tanh_projection
+    tidy3d.plugins.autograd.invdes.smoothed_projection
 

tests/test_plugins/autograd/invdes/test_projections.py

@@ -0,0 +1,101 @@
+from __future__ import annotations
+
+import autograd
+import numpy as np
+
+from tidy3d.plugins.autograd.invdes.filters import ConicFilter
+from tidy3d.plugins.autograd.invdes.projections import smoothed_projection, tanh_projection
+
+
+def test_smoothed_projection_beta_inf():
+    nx, ny = 50, 50
+    arr = np.zeros((50, 50), dtype=float)
+
+    center_x, center_y = 25, 25
+    radius = 10
+    x = np.arange(nx)
+    y = np.arange(ny)
+    X, Y = np.meshgrid(x, y)
+    distance = np.sqrt((X - center_x) ** 2 + (Y - center_y) ** 2)
+
+    arr[distance <= radius] = 1
+
+    filter = ConicFilter(kernel_size=5)
+    arr_filtered = filter(arr)
+
+    result = smoothed_projection(
+        array=arr_filtered,
+        beta=np.inf,
+        eta=0.5,
+    )
+    assert not np.any(np.isinf(result) | np.isnan(result))
+    assert np.isclose(result[center_x, center_y], 1)
+    assert np.isclose(result[0, -1], 0)
+    assert np.isclose(result[0, 0], 0)
+    assert np.isclose(result[-1, 0], 0)
+    assert np.isclose(result[-1, -1], 0)
+
+    # fully discrete input should lead to fully discrete output
+    discrete_result = smoothed_projection(
+        array=arr,
+        beta=np.inf,
+        eta=0.5,
+    )
+    assert np.all(np.isclose(discrete_result, 0) | np.isclose(discrete_result, 1))
+
+
+def test_smoothed_projection_beta_non_inf():
+    nx, ny = 50, 50
+    arr = np.zeros((50, 50), dtype=float)
+
+    center_x, center_y = 25, 25
+    radius = 10
+    x = np.arange(nx)
+    y = np.arange(ny)
+    X, Y = np.meshgrid(x, y)
+    distance = np.sqrt((X - center_x) ** 2 + (Y - center_y) ** 2)
+
+    arr[distance <= radius] = 1
+
+    # fully discrete input should still be fully discrete output
+    discrete_result = smoothed_projection(
+        array=arr,
+        beta=1.0,
+        eta=0.5,
+    )
+    assert np.all(np.isclose(discrete_result, 0) | np.isclose(discrete_result, 1))
+
+    filter = ConicFilter(kernel_size=11)
+    arr_filtered = filter(arr)
+
+    smooth_result = smoothed_projection(
+        array=arr_filtered,
+        beta=1.0,
+        eta=0.5,
+    )
+    # for sufficiently smooth input, the result should be the same as tanh projection
+    tanh_result = tanh_projection(
+        array=arr_filtered,
+        beta=1.0,
+        eta=0.5,
+    )
+    assert np.isclose(smooth_result, tanh_result, rtol=0, atol=1e-4).all()
+
+
+def test_smoothed_projection_initialization():
+    # test that for initialization at eta=0.5, projection returns simply 0.5
+    arr = np.zeros((5, 5), dtype=float) + 0.5
+    result = smoothed_projection(array=arr, beta=1.0, eta=0.5)
+    assert np.all(np.isclose(result, 0.5))
+
+
+def test_projection_gradient():
+    # test that gradient is finite
+    arr = np.zeros((5, 5), dtype=float) + 0.5
+
+    def _helper_fn(x):
+        return smoothed_projection(array=x, beta=1.0, eta=0.5).mean()
+
+    val, grad = autograd.value_and_grad(_helper_fn)(arr)
+    assert val == 0.5
+    assert np.all(~(np.isnan(grad) | np.isinf(grad)))

tidy3d/plugins/autograd/invdes/__init__.py

@@ -16,7 +16,7 @@
     make_filter_and_project,
 )
 from .penalties import ErosionDilationPenalty, make_curvature_penalty, make_erosion_dilation_penalty
-from .projections import ramp_projection, tanh_projection
+from .projections import ramp_projection, smoothed_projection, tanh_projection
 
 __all__ = [
     "CircularFilter",
@@ -34,5 +34,6 @@
     "make_filter_and_project",
     "make_gaussian_filter",
     "ramp_projection",
+    "smoothed_projection",
     "tanh_projection",
 ]

tidy3d/plugins/autograd/invdes/projections.py

@@ -70,3 +70,114 @@ def tanh_projection(
     num = np.tanh(beta * eta) + np.tanh(beta * (array - eta))
     denom = np.tanh(beta * eta) + np.tanh(beta * (1 - eta))
     return num / denom
+
+
+def smoothed_projection(
+    array: NDArray,
+    beta: float = BETA_DEFAULT,
+    eta: float = ETA_DEFAULT,
+    scaling_factor=1.0,
+) -> NDArray:
+    """
+    Apply a subpixel-smoothed projection method.
+    The subpixel-smoothed projection method is discussed in [1]_ as follows:
+
+    This projection method eliminates discontinuities by applying first-order
+    smoothing at material boundaries through analytical fill factors. Unlike
+    traditional quadrature approaches, it works with maximum projection strength
+    (:math:`\\beta = \\infty`) and derives closed-form expressions for interfacial regions.
+
+    Prerequisites: input fields must be pre-filtered for continuity (for example
+    using a conic filter).
+
+    The algorithm detects whether boundaries intersect grid cells. When interfaces
+    are absent, standard projection is applied. For cells containing boundaries,
+    analytical fill ratios are computed to maintain gradient continuity as interfaces
+    move through cells and traverse pixel centers. This enables arbitrarily large
+    :math:`\\beta` values while preserving differentiability throughout the transition
+    process.
+
+    .. warning::
+        This function assumes that the device is placed on a uniform grid. When using
+        ```GridSpec.auto``` in the simulation, make sure to place a ``MeshOverrideStructure`` at
+        the position of the optimized geometry.
+
+    .. warning::
+        When using :math:`\\beta = \\infty` the function will produce NaN values if
+        the input is exactly equal to ``eta``.
+
+    Parameters
+    ----------
+    array : np.ndarray
+        The input array to be projected.
+    beta : float = BETA_DEFAULT
+        The steepness of the projection. Higher values result in a sharper transition.
+    eta : float = ETA_DEFAULT
+        The midpoint of the projection.
+    scaling_factor: float = 1.0
+        Optional scaling factor to adjust dx and dy to different resolutions.
+
+    Example
+    -------
+        >>> import autograd.numpy as np
+        >>> from tidy3d.plugins.autograd.invdes.filters import ConicFilter
+        >>> arr = np.random.uniform(size=(50, 50))
+        >>> filter = ConicFilter(kernel_size=5)
+        >>> arr_filtered = filter(arr)
+        >>> eta = 0.5  # center of projection
+        >>> smoothed = smoothed_projection(arr_filtered, beta=np.inf, eta=eta)
+
+    .. [1] A. M. Hammond, A. Oskooi, I. M. Hammond, M. Chen, S. E. Ralph, and
+       S. G. Johnson, "Unifying and accelerating level-set and density-based topology
+       optimization by subpixel-smoothed projection," arXiv:2503.20189v3 [physics.optics]
+       (2025).
+    """
+    # sanity checks
+    if array.ndim != 2:
+        raise ValueError(f"Smoothed projection expects a 2d-array, but got shape {array.shape=}")
+
+    # smoothing kernel is circle (or ellipse for non-uniform grid)
+    # we choose smoothing kernel with unit area, which is r~=0.56, a bit larger than (arbitrary) default r=0.55 in paper
+    dx = dy = scaling_factor
+    smooth_radius = np.sqrt(1 / np.pi) * scaling_factor
+
+    original_projected = tanh_projection(array, beta=beta, eta=eta)
+
+    # finite-difference spatial gradients
+    rho_filtered_grad = np.gradient(array)
+    rho_filtered_grad_helper = (rho_filtered_grad[0] / dx) ** 2 + (rho_filtered_grad[1] / dy) ** 2
+
+    nonzero_norm = np.abs(rho_filtered_grad_helper) > 1e-10
+
+    filtered_grad_norm = np.sqrt(np.where(nonzero_norm, rho_filtered_grad_helper, 1))
+    filtered_grad_norm_eff = np.where(nonzero_norm, filtered_grad_norm, 1)
+
+    # distance of pixel center to nearest interface
+    distance = (eta - array) / filtered_grad_norm_eff
+
+    needs_smoothing = nonzero_norm & (np.abs(distance) < smooth_radius)
+
+    # double where trick
+    d_rel = distance / smooth_radius
+    polynom = np.where(
+        needs_smoothing, 0.5 - 15 / 16 * d_rel + 5 / 8 * d_rel**3 - 3 / 16 * d_rel**5, 1.0
+    )
+    # F(-d)
+    polynom_neg = np.where(
+        needs_smoothing, 0.5 + 15 / 16 * d_rel - 5 / 8 * d_rel**3 + 3 / 16 * d_rel**5, 1.0
+    )
+
+    # two projections, one for lower and one for upper bound
+    rho_filtered_minus = array - smooth_radius * filtered_grad_norm_eff * polynom
+    rho_filtered_plus = array + smooth_radius * filtered_grad_norm_eff * polynom_neg
+    rho_minus_eff_projected = tanh_projection(rho_filtered_minus, beta=beta, eta=eta)
+    rho_plus_eff_projected = tanh_projection(rho_filtered_plus, beta=beta, eta=eta)
+
+    # Smoothing is only applied to projections
+    projected_smoothed = (1 - polynom) * rho_minus_eff_projected + polynom * rho_plus_eff_projected
+    smoothed = np.where(
+        needs_smoothing,
+        projected_smoothed,
+        original_projected,
+    )
+    return smoothed