Closest Silhouette Point Queries

• Closest silhouette point queries are geometric nearest-point problems defined on boundaries, with definitions depending on implicit surfaces or convex hull representations.
• In the implicit-surface setting, the query is solved via constrained optimization using tangency conditions and interval branch-and-bound, ensuring star-shaped visibility in WoSt.
• For convex hulls, the problem is reformulated as a simplex-constrained least squares task where sketching and gradient projection techniques exploit sparsity for efficient computation.

Searching arXiv for the specified papers and closely related context. arXiv search query: (Huang, 8 Oct 2025) OR "Geometric Queries on Closed Implicit Surfaces for Walk on Stars" OR (Yousefzadeh, 2021) OR "A Sketching Method for Finding the Closest Point on a Convex Hull" Closest silhouette point queries are geometric nearest-point problems whose precise meaning depends on how a boundary is represented. For a smooth closed implicit surface used in Walk on Stars (WoSt), a silhouette point is a boundary point whose outward normal is perpendicular to the viewing direction from a query point $x$, and the operational goal is to find the closest such point so that the star-shaped sampling region contains no silhouette (Huang, 8 Oct 2025). For a finite point set $D\subset \mathbb{R}^d$, the supplied literature uses “silhouette” in the sense of the convex hull $\operatorname{conv}(D)$, and the query becomes the computation of the point on that hull closest to an exterior query point $q$ (Yousefzadeh, 2021). In both formulations, the query is expressed as an optimization problem, but the geometry and solver design differ substantially between the non-convex implicit-surface setting and the convex-hull setting.

1. Definition and geometric meaning

For the closed implicit-surface setting, let

$$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$

be a smooth, closed reflecting boundary with outward normal

$n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$

Given a query point $x\notin \partial \Omega_R$, a silhouette point $z_S$ is any point on $\partial \Omega_R$ satisfying

$n(z_S)\cdot (z_S-x)=0.$

This condition states that the surface normal is perpendicular to the viewing direction $D\subset \mathbb{R}^d$0. In WoSt, the closest such point is required because the resulting radius $D\subset \mathbb{R}^d$1 is used to ensure that the visibility region remains star-shaped (Huang, 8 Oct 2025).

For the convex-hull setting, the silhouette or outline of a finite point set $D\subset \mathbb{R}^d$2 is identified with $D\subset \mathbb{R}^d$3. With a query point $D\subset \mathbb{R}^d$4 lying outside the hull, the problem is

$D\subset \mathbb{R}^d$5

The closest silhouette point is therefore the nearest point on the hull to $D\subset \mathbb{R}^d$6 (Yousefzadeh, 2021).

The two definitions are geometrically related but not identical. In the implicit-surface case, silhouette is defined by a tangency condition involving the normal field. In the convex-hull case, silhouette is the convex outline itself, and the query is a nearest-point problem on a convex set. This suggests that “closest silhouette point query” is best understood as a representation-dependent geometric primitive rather than a single canonical problem.

2. Constrained optimization on closed implicit surfaces

Within WoSt, the closest-silhouette-point query is formulated as a constrained global optimization problem over the closed implicit surface. The objective is the squared Euclidean distance to the query point: $D\subset \mathbb{R}^d$7 Here $D\subset \mathbb{R}^d$8 is the Dirichlet-distance bound found earlier. The optimization therefore minimizes

$D\subset \mathbb{R}^d$9

subject to two scalar equality constraints and a distance bound (Huang, 8 Oct 2025).

A central observation is that WoSt queries can be cast as constrained global optimization or constraint satisfaction problems. For the silhouette query, the two equalities are encapsulated in the vector-valued inclusion function

$\operatorname{conv}(D)$0

No explicit Lagrange-multiplier solver is used. Instead, the problem is handled by interval branch-and-bound because the feasible set is highly non-convex (Huang, 8 Oct 2025).

The role of the inequality $\operatorname{conv}(D)$1 is operationally important. It restricts the search to the box induced by the previously computed Dirichlet-distance and keeps the silhouette search aligned with the WoSt sampling loop. In the terminology of the paper, this is part of the machinery that first enables mesh-free PDE solving via WoSt when boundaries are defined by closed implicit surfaces (Huang, 8 Oct 2025).

3. Interval analysis and branch-and-bound realization

The implicit-surface solver adopts the MINIMIZE skeleton from Snyder (1992). Its first ingredient is an interval extension $\operatorname{conv}(D)$2 for a real function $\operatorname{conv}(D)$3, defined so that $\operatorname{conv}(D)$4 contains $\operatorname{conv}(D)$5. The construction uses standard interval-arithmetic rules for sum, product, and composition, together with dual-intervals for automatic differentiation. For intervals $\operatorname{conv}(D)$6 and $\operatorname{conv}(D)$7,

$\operatorname{conv}(D)$8

These extensions are built for $\operatorname{conv}(D)$9, $q$0, and $q$1 (Huang, 8 Oct 2025).

The second ingredient is a tri-valued constraint predicate $q$2. It is positive if both intervals contain $q$3, negative if either interval excludes $q$4, and unknown otherwise. The branch-and-bound procedure starts from the root box

$q$5

always extracts the box $q$6 with the smallest lower bound $q$7 on $q$8, and subdivides along its longest edge into two children boxes of half-width (Huang, 8 Oct 2025).

Pruning follows two tests. A box is discarded if $q$9 is negative, and it is also discarded if the lower bound $$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$0 exceeds the current best upper bound $$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$1. A box is accepted as containing a valid approximate minimizer when its diameter falls below the tolerance and $$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$2 is positive. The convergence check is

$$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$3

The output is obtained by extracting centroids of accepted boxes (Huang, 8 Oct 2025).

The reported guarantees and costs are specific. In the worst case, the number of boxes grows exponentially in $$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$4. In practice, for $$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$5 and smooth $$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$6, the query typically resolves in a few thousand to ten-thousand boxes. Branch-and-bound with interval analysis guarantees finding a global minimizer up to the prescribed box-diameter tolerance. In the Julia implementation, the tolerance is set to

$$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$7

where $$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$8 is the width of the Roberts–Muller shell used by WoSt; a smaller tolerance yields a tighter $$\partial \Omega_R=\{z\in \mathbb{R}^d \mid f_R(z)=0\}$$9 but more boxes (Huang, 8 Oct 2025).

4. Role inside Walk on Stars and reported numerical behavior

The closest-silhouette-point query is not an isolated geometric subroutine in the WoSt setting. It is part of the step-size control mechanism for Robin-WoSt. After the Dirichlet distance $n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$0 is computed by the closest-point query, the silhouette query returns $n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$1, and the star radius is set to

$n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$2

The algorithm then continues as in Miller et al.’s Robin-WoSt extension. The purpose is to ensure that the random “star” step is bounded so that the visibility region remains star-shaped (Huang, 8 Oct 2025).

The paper states that, to the best of its knowledge, it is the first to study closest silhouette point queries and Robin radius bound queries on closed implicit surfaces. This is tied directly to the broader claim that the formulations and methods first enable mesh-free PDE solving via WoSt when boundaries are defined by closed implicit surfaces (Huang, 8 Oct 2025).

The numerical experiment highlighted for this query is Figure 1, in which the method is tested on a closed harmonic-RBF implicit surface. The measured $n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$3 matches the analytic silhouette radius to within $n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$4 relative error when $n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$5 is set to $n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$6. Typical runtimes on a modern CPU are reported to be on the order of tens of milliseconds per query in 3D, which makes repeated calls inside WoSt feasible (Huang, 8 Oct 2025).

These results are specifically tied to the implicit-surface formulation. A plausible implication is that the main practical bottleneck is not the abstract definition of the silhouette condition, but rather the availability of reliable global geometric queries over the chosen boundary representation.

5. Convex-hull formulation as simplex-constrained least squares

For a finite point set, the nearest-hull version of the problem is written in matrix form by stacking the data points as the rows of an $n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$7 matrix $n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$8. Any convex combination of rows is $n(z)=\frac{\nabla f_R(z)}{\|\nabla f_R(z)\|}.$9, where $x\notin \partial \Omega_R$0 lies in the probability simplex

$x\notin \partial \Omega_R$1

The problem becomes

$x\notin \partial \Omega_R$2

Equivalently, with $x\notin \partial \Omega_R$3, $x\notin \partial \Omega_R$4, and $x\notin \partial \Omega_R$5, it takes the standard form

$x\notin \partial \Omega_R$6

(Yousefzadeh, 2021).

Expanding the objective yields the quadratic program

$x\notin \partial \Omega_R$7

With Lagrange multipliers $x\notin \partial \Omega_R$8 for the equality constraint and $x\notin \partial \Omega_R$9 for nonnegativity, the Lagrangian is

$z_S$0

where $z_S$1 and $z_S$2. The KKT conditions are

$z_S$3

$z_S$4

Solving these conditions yields the unique optimal $z_S$5, and $z_S$6 is the desired silhouette point (Yousefzadeh, 2021).

The sketching paper emphasizes sparsity of the solution support. By Carathéodory’s theorem, the optimal $z_S$7 has at most $z_S$8 nonzeros. This observation motivates algorithms that avoid treating all $z_S$9 points as equally relevant when $\partial \Omega_R$0 (Yousefzadeh, 2021).

6. Sketching, gradient projection, and computational regimes

The sketching framework proposed for the convex-hull problem exploits the expected sparsity of the optimal support. The data matrix $\partial \Omega_R$1 is partitioned into $\partial \Omega_R$2 subsets $\partial \Omega_R$3, either by sorting rows in increasing distance to $\partial \Omega_R$4 or at random. A working sketch $\partial \Omega_R$5 is initialized with $\partial \Omega_R$6. The simplex-constrained least-squares problem is solved on $\partial \Omega_R$7 by a gradient-projection method, the resulting coefficients are warm-started when $\partial \Omega_R$8 is appended, and the process continues until all $\partial \Omega_R$9 parts are included, at which point $n(z_S)\cdot (z_S-x)=0.$0 in the exact scheme (Yousefzadeh, 2021).

The solver builds on the classical two-phase gradient-projection algorithm described in Nocedal–Wright, §16.7. At iteration $n(z_S)\cdot (z_S-x)=0.$1, it computes the projected gradient direction through the Cauchy point, identifies active constraints, performs an inexact subspace minimization over free coordinates, and updates the iterate until the KKT residual is below tolerance $n(z_S)\cdot (z_S-x)=0.$2. In the formulation given,

$n(z_S)\cdot (z_S-x)=0.$3

the projected point is

$n(z_S)\cdot (z_S-x)=0.$4

the active set is

$n(z_S)\cdot (z_S-x)=0.$5

and subspace minimization is carried out over the free indices $n(z_S)\cdot (z_S-x)=0.$6 (Yousefzadeh, 2021).

The convergence discussion is classical and conditional on the convex quadratic structure and the polyhedral feasible set. The Cauchy step ensures sufficient decrease, subspace minimization reduces the objective further, finite identification of the true active set occurs in finitely many iterations, and once the active set is correct, the method behaves as a projected Newton method in the reduced space and converges superlinearly. No explicit acceleration such as Nesterov’s momentum is given, although warm-starting from the previous sketch is described as a practical speedup (Yousefzadeh, 2021).

The complexity statements are also explicit. At sketch stage $n(z_S)\cdot (z_S-x)=0.$7, each gradient evaluation costs $n(z_S)\cdot (z_S-x)=0.$8 and each projection costs $n(z_S)\cdot (z_S-x)=0.$9. The total time is roughly

$D\subset \mathbb{R}^d$00

which is stated to be much smaller than

$D\subset \mathbb{R}^d$01

when the sketches remain small. The paper as presented does not include a detailed empirical section, but it reports datasets with $D\subset \mathbb{R}^d$02 up to $D\subset \mathbb{R}^d$03 and $D\subset \mathbb{R}^d$04 in the hundreds to thousands, lists CVX, Gurobi, and NLopt as off-the-shelf solvers, and states that the sketching method outpaces them by factors of $D\subset \mathbb{R}^d$05–$D\subset \mathbb{R}^d$06 while achieving identical objective value (Yousefzadeh, 2021).

Taken together, the two lines of work show that closest silhouette point queries admit sharply different computational treatments depending on whether the boundary is an implicit non-convex surface or a convex hull. The common structure is optimization over a geometric boundary; the decisive difference is whether the feasible set is governed by non-convex equalities requiring certified global search, or by convex hull membership expressible as simplex-constrained least squares.