Subpixel-Smooth Projection (SSP)

• SSP is a differentiable binarization method in topology optimization that combines filtered densities with local differential information to model subpixel geometry.
• It unifies density-based and level-set methods by preserving gradient information and ensuring smooth transitions even as designs approach a binary state.
• The method, especially in its SSP2 variant, improves convergence and numerical conditioning by incorporating both gradient and Hessian data during topology changes.

Searching arXiv for the cited SSP-related papers to ground the article in the current literature. Subpixel-Smooth Projection (SSP), also written “subpixel-smoothed projection,” is a differentiable binarization scheme in density-based topology optimization in which the projected density depends not only on the filtered design field $\tilde{\rho}$ but also on local differential structure, originally $\|\nabla \tilde{\rho}\|$ and, in the second-order extension, the Hessian of $\tilde{\rho}$. Its central purpose is to remove the usual trade-off between strong binarization and usable optimization conditioning: SSP behaves like a standard density method at small projection steepness, approaches a level-set-like sharp-interface description as $\beta \to \infty$, and remains suitable for gradient-based optimization in regimes where classical tanh or Heaviside projection becomes ill-conditioned or non-differentiable (Hammond et al., 26 Mar 2025, Romano et al., 8 Jan 2026).

1. Problem setting and rationale

In density-based topology optimization, the design variable is a continuous density field

$\rho(\mathbf{x}) \in [0,1],$

which is first filtered to produce a smooth field $\tilde{\rho}$ and then projected toward a black-and-white design. A standard projection is the smooth Heaviside or tanh map

$$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$

Here, $\beta$ controls binarization strength and $\eta$ is the threshold.

The difficulty is classical. As $\beta \to 0^+$, the projection is approximately the identity and the optimization is well conditioned, but the design remains gray. As $\|\nabla \tilde{\rho}\|$0, the map approaches a Heaviside step, the structure becomes nearly binary, and the objective gradient with respect to the design variables becomes almost everywhere zero while second derivatives become badly behaved. The 2025 SSP formulation identifies four associated drawbacks of traditional projection: a non-differentiable limit, ill-conditioning or stiffness for large finite $\|\nabla \tilde{\rho}\|$1, dependence on problem-specific $\|\nabla \tilde{\rho}\|$2-scheduling, and the absence of a level-set limit compatible with standard adjoint gradients (Hammond et al., 26 Mar 2025).

SSP addresses these issues by replacing the purely pointwise projection $\|\nabla \tilde{\rho}\|$3 with a local mapping

$\|\nabla \tilde{\rho}\|$4

in the first-order version, and with a Hessian-regularized analogue in the second-order version. The key design choice is that the projection depends on both the filtered field and its local spatial variation, thereby analytically modeling subpixel geometry. This allows SSP to act as a drop-in replacement for previous projection schemes while preserving the standard three-field pipeline $\|\nabla \tilde{\rho}\|$5.

2. First-order SSP: local subpixel smoothing from a filtered field

The first-order SSP construction begins from a conceptual exact smoothing step. Let $\|\nabla \tilde{\rho}\|$6. SSP imagines convolving $\|\nabla \tilde{\rho}\|$7 with a compactly supported radially symmetric kernel $\|\nabla \tilde{\rho}\|$8 of radius $\|\nabla \tilde{\rho}\|$9: $\tilde{\rho}$0 At $\tilde{\rho}$1, $\tilde{\rho}$2 is a true step across the interface $\tilde{\rho}$3, so the convolution returns the fraction of the local smoothing ball lying on the material side of the interface.

Direct high-resolution evaluation of this convolution is too expensive. SSP therefore uses a local linearization of the filtered field,

$\tilde{\rho}$4

which makes the nearby interface locally planar. The approximate signed distance from $\tilde{\rho}$5 to that plane is

$\tilde{\rho}$6

With the sign convention used in the formulation, $\tilde{\rho}$7 corresponds to $\tilde{\rho}$8 and $\tilde{\rho}$9 to $\beta \to \infty$0.

In the binary limit $\beta \to \infty$1, the smoothed density becomes a fill-fraction function $\beta \to \infty$2. The formulation imposes several conditions on $\beta \to \infty$3: it is monotone decreasing in $\beta \to \infty$4, symmetric through $\beta \to \infty$5, satisfies $\beta \to \infty$6, is binary outside the smoothing band, and is at least $\beta \to \infty$7. A specific piecewise polynomial choice is

$\beta \to \infty$8

This produces an almost-everywhere binary field with a one-pixel-thick smooth transition layer.

For finite $\beta \to \infty$9, SSP introduces two side values

$\rho(\mathbf{x}) \in [0,1],$0

and defines

$\rho(\mathbf{x}) \in [0,1],$1

The mapping is therefore standard projection away from interfaces and a subpixel-aware blend inside the interface band. The smoothing radius $\rho(\mathbf{x}) \in [0,1],$2 is chosen on the scale of one grid spacing; the practical guidance given in the 2025 formulation is $\rho(\mathbf{x}) \in [0,1],$3, with the example $\rho(\mathbf{x}) \in [0,1],$4 (Hammond et al., 26 Mar 2025).

3. Limiting behavior, conditioning, and the density–level-set bridge

The principal significance of SSP lies in its asymptotic behavior. When $\rho(\mathbf{x}) \in [0,1],$5, the scalar projection becomes the identity, $\rho(\mathbf{x}) \in [0,1],$6, and SSP degenerates to the usual filtered density. In that regime it behaves like a conventional density-based method, with easy topology changes and no strong binarization. When $\rho(\mathbf{x}) \in [0,1],$7, SSP yields a field that is binary away from interfaces and transitions smoothly across a thin layer $\rho(\mathbf{x}) \in [0,1],$8. As the grid spacing tends to zero and $\rho(\mathbf{x}) \in [0,1],$9 shrinks, this non-binary layer becomes vanishingly thin. The resulting behavior is level-set-like: geometry is represented by a moving implicit interface while the projected field remains differentiable with respect to the design degrees of freedom (Hammond et al., 26 Mar 2025).

This is why the 2025 paper presents SSP as a unification of density-based and level-set methods. Density-based optimization is retained in the low-$\tilde{\rho}$0 regime, where topology can change easily. Shape evolution in an almost-everywhere binarized structure is recovered in the $\tilde{\rho}$1 regime, but without the vanishing gradients and severe Hessian stiffness associated with the classical tanh limit.

The chain rule is correspondingly modified. Standard density-based optimization differentiates through $\tilde{\rho}$2. SSP instead requires differentiation through $\tilde{\rho}$3, producing sensitivity terms with respect to both $\tilde{\rho}$4 and $\tilde{\rho}$5. Because $\tilde{\rho}$6 and $\tilde{\rho}$7 are linear operators in the design field, the mapping is well suited to automatic differentiation and adjoint pipelines. A key analytical result reported in the 2025 formulation is that second derivatives of $\tilde{\rho}$8 with respect to the design degrees of freedom remain bounded even as $\tilde{\rho}$9 and $$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$0, avoiding the usual ill-conditioning induced by steep tanh projection (Hammond et al., 26 Mar 2025).

SSP is not, however, a substitute for regularization of the design space itself. Filtering, minimum length-scale control, and other fabrication constraints remain necessary. The method regularizes the projection step, not the underlying admissible geometry.

4. Second-order SSP (SSP2) and differentiability through topology changes

The original SSP, retrospectively called SSP1, has an important limitation. Its signed distance

$$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$1

is undefined where $$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$2. This occurs at exactly the kinds of events that matter for topology changes, such as the merging of two nearby interfaces or the closing of a narrow neck. In such cases SSP1 falls back to a Heaviside-like behavior at the critical point, and the projected density can jump discontinuously as the filtered field crosses the threshold. The 2026 analysis shows this explicitly for a 1D quadratic example and for 2D Cassini ovals: at the merging point, SSP1 exhibits a jump where a smooth continuation is required for $$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$3 optimization theory (Romano et al., 8 Jan 2026).

SSP2 remedies this by regularizing the signed distance with the Hessian of the filtered field. If $$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$4 denotes the Hessian and $$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$5 its Frobenius norm, SSP2 defines

$$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$6

At points where the gradient vanishes, the Hessian term prevents division by zero and keeps the signed distance finite, provided the Hessian is nonzero, which is the typical case at a local extremum. SSP2 also modifies the internal sampling points: $$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$7 with $$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$8, and then uses the same fill-factor structure as SSP1.

The consequence is precise. SSP2 is twice differentiable with respect to the design variables even at merging interfaces, while remaining quasi-binary. Away from merging interfaces, and in the limit $$P_{\beta,\eta}(\tilde{\rho}) = \frac{\tanh\big(\beta \eta\big) + \tanh\big(\beta(\tilde{\rho} - \eta)\big)} {\tanh\big(\beta \eta\big) + \tanh\big(\beta(1 - \eta)\big)}.$$9, SSP2 reduces to SSP1. The additional grayness introduced by the Hessian regularization is reported as empirically negligible, with “<1% difference in loss,” and the area of gray pixels decays with resolution. This matters algorithmically because many optimizers used in topology optimization, including CCSA/MMA and interior-point methods, assume $\beta$0 objective and constraint functions. SSP2 restores those regularity assumptions in connectivity-dominant cases, while keeping the implementation close to that of SSP1 (Romano et al., 8 Jan 2026).

5. Integration into topology-optimization workflows and demonstrated applications

Algorithmically, SSP alters only the projection stage. A typical pipeline is: design variables $\beta$1; filtering to obtain $\beta$2; evaluation of $\beta$3, $\beta$4, and, for SSP2, $\beta$5 at simulation points; computation of $\beta$6 via SSP; interpolation of material properties, for example

$\beta$7

primal and adjoint PDE solves; backpropagation through SSP and the filter; and a gradient-based update step. The 2025 paper emphasizes that the only structural change to a standard three-field code is the replacement of $\beta$8 by $\beta$9, and the 2026 extension states that SSP2 adds minimal complexity relative to SSP or traditional projection schemes (Hammond et al., 26 Mar 2025, Romano et al., 8 Jan 2026).

Formulation	Computational setting	Representative demonstrations
SSP	finite-difference, Fourier-modal, finite-element methods	waveguide crossing, dielectric metagrating, plasmonic near-field focusing
SSP2	thermal and photonic problems	thermal metamaterial, photonic wavelength demultiplexer

The reported empirical behavior is consistent with the formulation’s stated goals. In a 2D silicon waveguide crossing, the 2025 paper compares the standard projection and SSP through the gradient norm and iteration-level convergence. For the standard projection, $\eta$0 tends to zero as $\eta$1; for SSP, the gradient norm converges to a nonzero value, enabling continued optimization in the binary regime. At $\eta$2, SSP converges significantly faster than standard projection, and at $\eta$3 standard projection cannot make progress while SSP still converges almost as fast as at $\eta$4. Across the broader set of photonics inverse-design problems, the method is described as exhibiting “both faster convergence and greater simplicity” (Hammond et al., 26 Mar 2025).

The 2026 paper sharpens this picture for topology-change-dominant cases. In a porous thermal metamaterial problem with $\eta$5 from the outset and 100 random initializations, the convergence counts were: both SSP1 and SSP2 converged, 71; SSP2 only, 24; SSP1 only, 1; neither, 4. Thus SSP2 converged in 95 cases and SSP1 in 72. By contrast, in a composite thermal metamaterial where both phases conduct, and in a photonic wavelength demultiplexer, both methods converged in all runs and their convergence curves were very similar. This supports the narrower claim made in the paper: SSP2 is especially advantageous in connectivity-dominant problems, where frequent interface merging makes the differentiability of topology changes algorithmically decisive (Romano et al., 8 Jan 2026).

6. Scope, related uses, and acronym disambiguation

The term SSP is not unique across fields. In the topology-optimization literature discussed above, SSP denotes Subpixel-Smooth Projection or subpixel-smoothed projection. In contrast, “Selective Vision-Language Subspace Projection” is a training-free CLIP alignment method that uses local image features, SVD, and orthogonal projection to reduce the modality gap in few-shot classification; despite sharing the acronym, it is unrelated in formulation, objectives, and application domain (Zhu et al., 2024).

A second source of ambiguity comes from subpixel prediction in medical image segmentation. The 2021 paper “Small Lesion Segmentation in Brain MRIs with Subpixel Embedding” does not use the SSP name, but its architecture combines 2$\eta$6-resolution subpixel prediction with a learnable downsampler that aggregates four subpixel predictions into each coarse-grid output by a content-aware, normalized weighted average. The paper itself describes this as a learnable downsampler or projection operator, and a plausible implication is that it exemplifies a broader design pattern—subpixel prediction followed by smooth, learned projection—rather than the specific topology-optimization method formalized as SSP (Wong et al., 2021).

This distinction matters because the shared vocabulary of “subpixel” and “projection” can obscure substantial differences in meaning. In topology optimization, SSP is a differentiable binarization method for mapping filtered densities to physical material fields. In the MRI segmentation setting, the comparable idea is an architectural strategy for predicting on a denser lattice and projecting back to the original grid. In CLIP adaptation, the same acronym refers to projection in feature space rather than in physical or image space. The commonality is therefore structural only at a very high level; the technical content is domain-specific.

In its strictest current usage, Subpixel-Smooth Projection refers to the topology-optimization formulation in which subpixel interface geometry is analytically encoded through local differential information of the filtered field. SSP1 provides a first-order, quasi-binary, level-set-like projection that remains well conditioned at large $\eta$7, and SSP2 extends that construction with Hessian regularization to preserve twice differentiability through topology changes.