Centerness-Aware Projection (CAP)

• The paper introduces CAP, an innovative projection mechanism that integrates class weight adjustments to resolve many-to-one conflicts in 3D-to-2D mapping.
• CAP computes an adjusted score by dividing point depth by user-defined class weights, ensuring semantically critical points are prioritized in LiDAR data.
• Efficient implementation in range-view projection and weighted ℓ1 regularization demonstrates CAP’s practical benefits in enhancing target class performance and sparse representations.

Class-Weighted-Aware Projection (CWAP) is a class-prioritization mechanism in selection and regularization tasks, particularly relevant to many-to-one mappings in feature projection and to optimization under structured $\ell_1$ constraints. It is deployed in two principal domains: (1) range-view projection of 3D LiDAR point clouds, where semantic information is used to guide pixel selection, and (2) projected gradient methods for sparse learning, where class weights enter the penalty geometry. CWAP equips pipelines with explicit control over the influence of semantic classes or feature groups via weight vectors, enabling practitioners to prioritize or suppress particular classes during the projection process.

1. CWAP in Range-View Projection for 3D Point Clouds

The canonical use case for CWAP in point cloud processing is in transforming a 3D LiDAR scan into a 2D range image for 2D deep learning. Conventional mapping from 3D to 2D pixels faces many-to-one conflicts: several points may project to the same pixel, thereby requiring a selection rule. The baseline approach selects the point with the minimal depth, meaning the closest to the LiDAR origin. However, this depth-centric rule disregards object semantics and local structure, often eliminating contextually or semantically important points.

CWAP resolves these ambiguities by introducing user-defined class weights into the selection criterion. Instead of selecting the point with minimal raw depth $d_i$, CWAP computes for each candidate point $i$ an adjusted score: $s_i = \frac{d_i}{w_i + \varepsilon}$ where $w_i$ denotes the class weight assigned to the semantic class of point $i$, and $\varepsilon>0$ stabilizes the denominator. The point with the minimal $s_i$ is selected for each pixel. Assigning large positive $w_c$ to class $c$ increases its selection probability, while negative $w_c$ guarantees dominance over positive or zero-weighted competitors (Mousavi et al., 26 Jan 2026).

2. General Mathematical Formulation and Algorithmic Implementation

CWAP's core is the mapping from input features (or points) and their associated classes to a selection or projection criterion governed by class weights. In range-view projection, the adjusted depth $d_i' = d_i/(w_i + \varepsilon)$ directly guides pixel selection.

The CWAP selection algorithm follows:

1. For each 3D point, compute raw depth and project to image pixel;
2. Lookup the class-specific weight $w_i$;
3. Calculate $s_i$;
4. For each pixel, select the point with the minimum $s_i$;
5. Assign selected point's depth and attributes to the pixel.

This procedure incurs minimal additional runtime and memory burden compared to depth-based projection—one extra weight lookup and division per point (Mousavi et al., 26 Jan 2026).

3. CWAP in Weighted $\ell_1$ Ball Projection and Sparse Learning

CWAP emerges in optimization literature as a weighted $\ell_1$ ball projection, incorporating class/group weights to control sparsity patterns or class representation. Given $y\in\mathbb{R}^d$ (input vector), $w\in\mathbb{R}_+^d$ (coordinate weights), and $a>0$ (radius), the canonical projection is: $$x^* = \arg\min_x \|x-y\|_2^2 \quad \text{subject to} \quad \sum_{i=1}^d w_i |x_i| \leq a$$ The solution follows a “soft-threshold” rule parameterized by a unique $\lambda^*\ge0$: $$x_i = \text{sign}(y_i)\cdot\max\{|y_i| - w_i\lambda^*,\,0\}$$ where $\lambda^*$ enforces the weighted $\ell_1$ constraint. Efficient projection algorithms exploit radix bucketing, pivoting, or sorting. The w-bucket algorithm, in particular, achieves linear empirical runtime for high-dimensional vectors (e.g., $d=10^7$ in 8 ms on commodity hardware) (Perez et al., 2020).

Class weights enter as scaling factors applied to groups or semantic indices. For feature $i$ assigned to class $c(i)$,

$w_i \leftarrow \alpha_{c(i)}w_i^{\text{base}}$

where $\alpha_{c(i)}$ reflects the desired class penalty or priority. This paradigm applies unchanged to all weighted $\ell_1$ projection algorithms (Perez et al., 2020, Wang, 2015).

4. Selection and Tuning of Class Weights

CWAP's flexibility derives from tunable class weights. Guidance for weight selection:

• Zero weight ($w_c=0$): pure depth or unweighted selection.
• Positive weights: increase selection frequency for class $c$; doubling $w_c$ approximately doubles its priority.
• Negative weights: guarantee class $c$ dominance in selection within each pixel's candidate set.

Typical deployments assign zero weights to non-target classes and moderate positive or negative weights to classes of interest. For critical classes such as “pedestrian” or “motorcycle,” $w_c=-1$ forces their retention in highly ambiguous pixels during training (Mousavi et al., 26 Jan 2026). Over-weighting a class can produce slight degradations in other classes’ performance due to the hard exclusivity imposed by the pixel mapping (Mousavi et al., 26 Jan 2026).

5. Empirical Performance, Use Cases, and Limitations

Empirical evaluations of CWAP on the SemanticKITTI semantic segmentation benchmark demonstrate significant gains for targeted classes with negligible impact on non-target categories and overall performance. For example, using $w_{\text{truck}}=-1$, truck IoU rose from 56.8 to 77.5; similar gains appeared for “other-veh” and “motorcycle.” Effects on non-target “stuff” classes were typically $\pm0.1$–$0.5$ points (Mousavi et al., 26 Jan 2026).

CWAP is applied only during training—using ground-truth labels for class weights—since labels are not available at inference, where projection reduces to baseline depth-based selection. Excessively large positive weights ($>2$–$3\times$) may yield diminishing returns or even minor reductions in overall mIoU (Mousavi et al., 26 Jan 2026). In projection-based gradient descent, normalized class weights can be used so that the scaled ball radius is invariant across different weightings (Perez et al., 2020).

Use cases span LiDAR segmentation, projected-sparse feature selection, class-proportion estimation, and group-aware model regularization. CWAP is compatible with range-view pipelines, sparse learning frameworks, and any architecture requiring class-prioritized projection under hard resource or mapping constraints.

6. Algorithmic Complexity and Practical Recommendations

CWAP selection via weighted depth mapping requires only a table lookup and floating-point division per input. For weighted $\ell_1$ ball projection, algorithmic complexity scales as follows:

• w-sort: $O(d\log d)$ time, $O(d)$ space;
• w-pivot$^f$: worst-case $O(d^2)$ but $O(d)$ average;
• w-bucket$^f$: $O(d)$ worst/average due to radix partitioning (Perez et al., 2020).

Practical tips include preallocation of arrays, avoidance of inner-loop dynamic memory allocation, partitioning using bitwise key extraction, and double-precision arithmetic in cases of large dynamic range. On real-world high-dimensional data, w-bucket$^f$ outperforms alternatives by $5$–$10\times$ in runtime, achieving 20–50% improvements on sparse-support tasks (Perez et al., 2020).

7. Formal Derivation, Correctness, and Related Projection Operators

CWAP’s solution structure in penalized projection is established via KKT conditions. For the weighted $\ell_1$ projection with sum constraint: $$\min_{x \in \mathbb{R}^n} \frac{1}{2}\|x - y\|_2^2 + \sum_{i=1}^n w_i x_i, \quad \text{s.t.} \sum_{i=1}^n x_i = \tau, \; x_i \ge 0$$ The closed-form is: $x_i = \max\{y_i - w_i - \alpha,\,0\}$ with $\alpha$ justified by the unique threshold ensuring $\sum_{i=1}^n x_i = \tau$. Sorting $u_i = y_i - w_i$, forming prefix sums, and identifying the largest $\rho$ such that $u_{(\rho)} > (S(\rho) - \tau)/\rho$ yields $\alpha = (S(\rho) - \tau)/\rho$. The operator admits a proof of correctness by strict convexity and threshold-characterized monotonicity (Wang, 2015).

This framework directly specializes to class-weighted projections, facilitating learning-to-rank, calibrated classification, and class-proportion estimation tasks with explicit prioritization. CWAP thus encompasses both selection-driven projection to range images and penalized optimization for sparse representations.