Maks Ovsjanikov, Mirela Ben-Chen, Justin Solomon, Adrian Butscher and Leonidas Guibas (2012), ACM Transactions on Graphics 31(4), 1–11.
Key takeaways
From the reading, NOTE-822.
- Maps as operators. A bijection T: M → N is equivalent to the linear map f ↦ f∘T⁻¹ on functions, a matrix C in chosen bases (Remarks 4.1–4.3). General matrices C are “functional maps” that need not come from any point map.
- Orthogonality (Theorem 5.1, proof checked). With volume preservation, or discretely orthonormal bases, C is orthogonal, so the inverse map is Cᵀ.
- Eigenspaces. An isometry commutes with the Laplace–Beltrami operator, so C links only eigenfunctions of equal eigenvalue. Truncated eigenspaces are stable under near-isometries given a spectral gap, a result cited to Kato, not shown.
- Linear constraints. Descriptor, landmark and segment preservation (Ca = b) and commutativity with operators (R_F C = C S_F) are linear in C, so inference is a least-squares solve.
- Results. About 30–40 eigenfunctions represent ground-truth maps between 27.8K-vertex shapes; the method beat Blended Intrinsic Maps on SCAPE and TOSCA; maps compose by matrix multiplication across collections.
- Scope. Near-isometries only; all structure assumes a bijection.
Standing in the record
Filed 2026-10-12 for QUESTION-052 and read in full on 2026-10-14 (NOTE-822). It is a worked case of a correspondence as an operator between two spaces, truncated to low Laplacian eigenspaces. It supports the question’s eigenspace framing: an isometry preserves eigenspaces, not eigenvectors (CLAIM-164). It does not reach the directed case: its structure needs a bijection, and it gives no construction for one-way maps. THEORY-297, filed from it, holds the orthogonality and eigenspace results; it is not filed as an answer to QUESTION-052 because it says nothing about the directed half.