A heat kernel signature (HKS) is a feature descriptor for use in deformable shape analysis and belongs to the group of spectral shape analysis methods. For each point in the shape, HKS defines its feature vector representing the point's local and global geometric properties. Applications include segmentation, classification, structure discovery, shape matching and shape retrieval. HKS was introduced in 2009 by Jian Sun, Maks Ovsjanikov and Leonidas Guibas. It is based on the heat kernel, which is a fundamental solution to the heat equation. HKS is one of the many recently introduced shape descriptors which are based on the Laplace–Beltrami operator associated with the shape.
Overview Shape analysis is the field of automatic digital analysis of shapes, e.g., 3D objects. For many shape analysis tasks (such as shape matching/retrieval), feature vectors for certain key points are used instead of using the complete 3D model of the shape. An important requirement of such feature descriptors is for them to be invariant under certain transformations. For rigid transformations, commonly used feature descriptors include shape context, spin images, integral volume descriptors and multiscale local features, among others. HKS allows isometric transformations which generalizes rigid transformations. HKS is based on the concept of heat diffusion over a surface. Given an initial heat distribution u 0 ( x ) {\displaystyle u_{0}(x)} over the surface, the heat kernel h t ( x , y ) {\displaystyle h_{t}(x,y)} relates the amount of heat transferred from x {\displaystyle x} to y {\displaystyle y} after time t {\displaystyle t} . The heat kernel is invariant under isometric transformations and stable under small perturbations to the isometry. In addition, the heat kernel fully characterizes shapes up to an isometry and represents increasingly global properties of the shape with increasing time. Since h t ( x , y ) {\displaystyle h_{t}(x,y)} is defined for a pair of points over a temporal domain, using heat kernels directly as features would lead to a high complexity. HKS instead restricts itself to just the temporal domain by considering only h t ( x , x ) {\displaystyle h_{t}(x,x)} . HKS inherits most of the properties of heat kernels under certain conditions.
Technical details The heat diffusion equation over a compact Riemannian manifold M {\displaystyle M} (possibly with a boundary) is given by
( Δ − ∂ ∂ t ) u ( x , t ) = 0 {\displaystyle \left(\Delta -{\frac {\partial }{\partial t}}\right)u(x,t)=0}
where Δ {\displaystyle \Delta } is the Laplace–Beltrami operator and u ( x , t ) {\displaystyle u(x,t)} is the heat distribution at a point x {\displaystyle x} at time t {\displaystyle t} . The solution to this equation can be expressed as
u ( x , t ) = ∫ h t ( x , y ) u 0 ( y ) d y . {\displaystyle u(x,t)=\int h_{t}(x,y)u_{0}(y)dy.}
The eigen decomposition of the heat kernel is expressed as
h t ( x , y ) = ∑ i = 0 ∞ exp ( − λ i t ) ϕ i ( x ) ϕ i ( y ) {\displaystyle h_{t}(x,y)=\sum _{i=0}^{\infty }\exp(-\lambda _{i}t)\phi _{i}(x)\phi _{i}(y)}
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