Differentiable Extrusion Optimization

Updated 30 October 2025

Differentiable extrusion optimization is an approach that combines geometric modeling, optimization theory, and automatic differentiation for precise design and reverse engineering of extrusions.
It leverages neural reconstruction, analytic parameter recovery, and differentiable CAD programs to enable gradient-based adjustments, achieving superior axis, center, and fitting metrics.
The framework supports real-time shape editing, inverse rendering, and process control in manufacturing and 3D printing through effective sensitivity analysis and differentiable rendering techniques.

Differentiable extrusion optimization is an interdisciplinary paradigm at the intersection of geometric modeling, optimization theory, and automatic differentiation, enabling the precise and gradient-based design, analysis, and reverse engineering of extrusion processes and geometries. This area covers neural reconstruction of CAD primitives, analytic parameter recovery, inverse editing of CAD programs, and the integration of optimization solvers into gradient-based pipelines. The differentiable formulations ensure compatibility with end-to-end learning, sensitivity analysis, and engineering design tasks.

1. Geometric Foundations and Problem Scope

Extrusion modeling is a canonical approach in computer-aided design (CAD), wherein a 2D sketch is extended along a defined axis to produce a 3D object—an extrusion cylinder. The parameterization typically involves:

Axis ( $e \in \mathbb{S}^2$ )
Center ( $c \in \mathbb{R}^3$ )
2D sketch/profile (closed loop, normalized and scaled by $s$ )
Extents ( $r^{\text{min}}, r^{\text{max}}$ )

Differentiable extrusion optimization extends this representation by coupling the geometric parameterization with differentiable computation, allowing optimization algorithms to adjust extrusion parameters using gradient information to minimize objectives or match target shapes. This is central in scenarios requiring automatic reverse engineering, real-time shape editing, and physical manufacturing process optimization (Uy et al., 2021).

2. Differentiable Extrusion Parameter Estimation from Point Clouds

The Point2Cyl framework (Uy et al., 2021) exemplifies a differentiable pipeline for reconstructing extrusion cylinders from noisy or incomplete point clouds:

The neural network (PointNet++ encoder) predicts per-point segmentation, probabilistic base/barrel labels, and normals as geometric proxies.
Base/barrel labels are determined via the point normal’s alignment with the extrusion axis:

$\begin{mcases} b_i = 0 : n_i \perp e \implies n_i^\top e = 0 \ b_i = 1 : n_i \parallel e \implies n_i^\top e = \pm 1 \end{mcases}$

Axis estimation uses a closed-form eigenvalue problem, with weighted matrices for soft assignments:

$\hat{e} = \arg\min_{e, \|e\|=1} e^\top H_w e$

where $H_w = \mathbf{N}^\top \mathbf{W}_\text{barrel} \mathbf{N} - \mathbf{N}^\top \mathbf{W}_\text{base} \mathbf{N}$ .

Center, scale, and extents are estimated via weighted (differentiable) averages/max/min projections.
The entire recovery process is differentiable, enabling backpropagation with reconstruction and proxy (segmentation, label, normal) losses.

This approach outperforms heuristic approaches (Hough voting, direct regression) on segmentation IoU, normal error, axis error, center error, and fitting loss metrics.

Method	Axis Error (deg)	Center Error	Fitting Loss
Hough Voting	58.9–59.8	0.12–0.04	0.15–0.17
Direct Prediction	30.2–48.8	0.14–0.07	1.41–0.49
Point2Cyl	7.9–8.1	0.05–0.03	0.07–0.03

3. Differentiable CAD Programs and Bidirectional Optimization

CAD modeling as differentiable programs (Cascaval et al., 2021) introduces a domain-specific language wherein each geometric operation (including extrusion) is a differentiable function of its parameters. The system represents the output mesh and all intermediate operations as a computation graph amenable to reverse-mode automatic differentiation.

Inverse editing is formulated as a constrained nonlinear optimization:

$E_\text{edit}(P) = \sum_{v_i \in V'} \|V_i(P) - \bar{v}_i\|_2^2$

with program constraints $g_j(P) \geq 0$ , and composition with further heuristic energies.

Gradients $\nabla F = \frac{\partial F(P)}{\partial P}$ (mesh vertex positions with respect to parameters) are compiled for high-performance optimization, supporting real-time editing and design space exploration.
This method excels for highly-parametric models but requires static mesh topology for differentiability during optimization; discrete topology changes must be performed by direct program edits.

4. Differentiable Optimization via Model Transformations

Differentiable optimization frameworks such as DiffOpt.jl (Besançon et al., 2022) generalize differentiation through the solution of parametrized optimization problems. Extrusion optimization tasks (e.g., optimal feed rate, shape, temperature for 3D printing or manufacturing) can be modeled as constrained problems in JuMP/MOI, with parameters present in objectives or constraints.

Differentiation utilizes the Implicit Function Theorem:

$\frac{ds}{dp} = -[D_x f(p, s(p))]^{-1} D_p f(p, s(p))$

For QPs and conic programs, sensitivities w.r.t. arbitrary problem parameters enable backpropagation from shape/production objectives to control inputs.
Model transformations via MOI bridges (e.g., quadratic constraints to rotated second order cones using Cholesky decomposition) ensure that non-standard models are mapped into differentiable forms, and derivatives are propagated through to the original parametrization.
Full integration with differentiable programming (ChainRulesCore.jl, ML pipelines) and robust support for sensitivity analysis, hyperparameter tuning, and gradient-based shape design.

Feature	Supported in DiffOpt.jl
Parameter differentiation (objective/constraints)	Yes
Non-standard/model transformations	Yes
Mixed convex problems (QCQP, etc.)	Yes

5. Differentiable Rendering and Optimization of SDF-Based Extrusions

Differentiable rendering using signed distance fields (SDFs) (Wang et al., 14 May 2024) is essential in silhouette-driven shape and extrusion optimization, especially when the geometry is implicitly defined (e.g., for neural or sampled representations).

The rendering equation derivative decomposes into an interior integral and a lower-dimensional boundary integral:

$\frac{\mathrm{d}I}{\mathrm{d}\theta} = \int_{\mathcal{S}} \frac{\mathrm{d}f(\omega)}{\mathrm{d}\theta} d\omega + \int_\mathcal{B} v_n(\omega)\, \Delta f(\omega)\, d\ell(\omega)$

The “relaxed boundary integral” expands the boundary $\mathcal{B}$ into a thin, easily sampled band $\mathcal{A}$ defined by SDF proximity:

$I_{\mathcal{A}} = \frac{1}{\epsilon} \int_{\mathcal{A}} v_n(\mathbf{y})\, \Delta f(\omega)\, d\omega$

with $0 < \mathrm{SDF}(\mathbf{y}) < \epsilon$ , $\mathrm{SDF}'(\mathbf{y}) = 0$ .

Bias-variance tuning via $\epsilon$ provides robust, low-variance silhouette gradients, suitable for geometry/material optimization in extrusion-based design.
Empirical results demonstrate accuracy, robustness, and performance competitive with more complicated guiding/reparameterization methods (Wang et al., 14 May 2024).

6. Application Domains and Limitations

Differentiable extrusion optimization finds direct application in reverse engineering (reconstructing CAD-ready primitives from scans), geometric design (gradient-based adjustment of extrusion parameters in CAD systems), process engineering (manufacturing, 3D printing shape and process control), and inverse rendering (matching shadow/silhouette objectives).

Notwithstanding its generality, several frameworks require static mesh topology (for CAD program optimization), and the boundary sampling relaxation introduces controlled bias in SDF-based rendering derivatives. Topology changes during optimization currently require explicit program edits and cannot be handled via parameter gradients alone (Cascaval et al., 2021).

7. Significance and Outlook

Differentiable extrusion optimization establishes a unified foundation for integrating advanced geometric reasoning, optimization, and differentiable programming. The confluence of neural decomposition, analytic closed-form estimation, inverse procedural modeling, and solver-integrated differentiation furnishes robust, scalable solutions for real-world geometric, engineering, and manufacturing design problems. The paradigm’s scope is likely to expand further as optimization frameworks become more expressive and SDF- and neural-based geometry representations are adopted as first-class primitives in CAD and engineering analysis.