Local Features and Keypoints

A single local feature is rarely unique to one class, so the problem is how to tell a power line from a young beech tree when both read around $0.95$ on linearity. The same difficulty recurs across the catalogue: a point on the shaft of a stone column and a point on the smooth back face of a capital may both score above $0.9$ on planarity. What separates these near-collisions in the final label is the radius at which the feature was computed and the context the classifier reads alongside it, and in my experience choosing the right feature catalogue, at the right scales, is more consequential than choosing the classifier.

Features are where the geometric half of the book hands over to the semantic half. The machinery they rest on is already in place: the principal component analysis of Chapter 7 supplies the covariance eigenvalues from which every feature in the next section is built, and the spatial indexes of Chapter 4 answer the millions of neighbourhood queries that feature extraction issues. Downstream, segmentation (Chapter 11) reads planarity and curvature to group points into regions, classification (Chapter 12) reads the full feature vector to assign each point a semantic label, and the descriptors of Chapter 13 extend the same machinery into rotation-invariant histograms for matching and registration. The chapter builds the vocabulary in three steps: the eigenvalue-based dimensionality features, the scalar features (curvature, roughness, density, verticality, height variance) that complement them, and the multi-scale computation that frees both families from the choice of a single radius.

Why local geometric features?

A point cloud is an unstructured collection of coordinates. None of the analytical chapters that follow can operate directly on those coordinates, and they all operate instead on the summaries of local shape formalised at the start of this chapter. A region-growing segmenter (Chapter 11) reads the surface normal and the local curvature to decide whether two adjacent points belong to the same surface. A Random Forest classifier reads a vector of geometric, radiometric, and contextual features to assign each point to a semantic class. A registration descriptor reads the same eigenvalue features to construct a rotation-invariant signature for matching. Even a deep network operating directly on raw points learns its own features in its first few layers, and what those layers learn is closely related to what this chapter computes by hand.

Because they are the input to every analytical operation that follows, the quality of every downstream result depends on the quality of these features. A feature that is sensitive to noise, biased by point density, or computed at the wrong scale will degrade every classifier, every segmenter, and every descriptor that consumes it.