Varifold Moment Invariants for Sustainable and Explainable Contour Feature Extraction

Much of the literature on structured image recognition has disproportionately focused on the comparison of classification algorithms. Rather than investigating which classifier performs best, this paper instead asks: what should a classifier know before it ever makes a prediction? In structured vision problems such as gesture recognition, facial expression categorization, and medical image analysis, discriminative information lies less in individual pixels and more in spatial relationships between semantic parts. Raw pixel spaces are high-dimensional, sensitive to nuisance variation, and often obfuscate the geometric structures that make visual tasks interpretable. Landmark extraction provides one form of dimension reduction, but it does not by itself determine the information preserved. This paper studies the post-landmark feature map as the central object of analysis and proposes a systematic framework for constructing and interpreting landmark-derived representations as an, informed, feature-based "dimension reduction" step. Using static hand gesture recognition as a case study, we evaluate coordinate, distance, angle, and hybrid representations through perturbation and ablation experiments. The results show that visually variable data exposes substantial gaps between raw coordinate features and their geometrically invariant counterparts, while hybrid representations achieve the strongest overall performance by combining complementary geometric components. These findings frame feature construction as a fundamental modeling decision and ultimately suggests that the question of what representation should a classifier learn from is one worth asking. The code used for feature construction and evaluation is available at https://github.com/ShivMaureeCWRU/Feature_based_dimension_reduction

This work introduces mathematical morphology-an established visual computing theory-into machine learning to exploit shape and density aspects often overlooked by standard techniques. We propose a fast clustering algorithm based on morphological reconstruction that accurately preserves cluster shapes and density. This scheme offers unique features: an intrinsic sense of maximal clusters, cost-free noise removal, and diverse growth patterns controlled by structuring elements.Additionally, we propose a novel distance metric combining Minkowski and Chebyshev distances, highly efficient for morphological dilations. In $Z^2$ discrete neighbourhood iterations, it is roughly 1.3 times faster than Manhattan and 329.5 times faster than Euclidean distances. When evaluated using a k-Nearest Neighbours (k-NN) classifier across 33 UCI datasets against 14 other distances, our metric achieved above-average accuracies most frequently (26 of 33 cases) and the best overall accuracy in 9 cases.Finally, we introduce novel morphological classifiers. Unlike current literature, this proposal uniquely models shape, density, and fractal information in datasets.

We introduce PLACE (Persistence-Landmark Analytic Classification Engine), a closed-form pipeline for classifying point clouds and graphs through their persistent-homology signatures. Three quantitative guarantees -- a margin-based excess-risk rate, a closed-form descriptor-selection rule, and a per-prediction certificate -- are derived from training labels alone, with no learned weights or held-out calibration. The embedding sums Mitra-Virk single-point coordinate functions over a sparse landmark grid; the closed-form weight rule $w_k^2 \propto (d_{k+1}^2 - d_k^2)/R_k^2$ maximizes the distortion slope in Mitra-Virk's affine certificate under $ν$-coherence. (i) An $O(kR/(Δ\sqrt{m_{\min}}))$ margin bound, driven by class-mean separation $Δ$ and embedding radius $R$, matched in the sample-starved regime $m \lesssim R/Δ$ by a Le Cam minimax lower bound. (ii) The Mahalanobis margin under Ledoit-Wolf-shrunk covariance is the strongest closed-form ranker on a 64-descriptor chemical-graph pool (mean Spearman $ρ= +0.56$ across 11 benchmarks, positive on 10 of 11); the isotropic surrogate $Δ/\sqrt{\ell}$ admits a closed-form selection-consistency rate on the homogeneous protein/social pools. (iii) A training-time-decided certificate, with no per-prediction overhead, in three concrete radii (Pinelis, Gaussian plug-in, and variance-aware Pinelis-Bernstein). Empirically, PLACE is the strongest diagram-based method on Orbit5k and matches the strongest topology-based baseline within statistical noise on MUTAG and COX2; remaining gaps fall into two diagnosable regimes (descriptor blindness on NCI1/NCI109; pool-coverage limits elsewhere). The Pinelis-Bernstein radius fires on 8 of the 12 benchmarks; on MUTAG the empirical and population nearest-centroid rules agree on every one of 940 held-out test predictions, validating the certificate's mechanism.