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.[1] It is based on 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.[2]
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
u0(x)
ht(x,y)
x
y
t
ht(x,y)
ht(x,x)
M
\left(\Delta-
| \partial | |
| \partialt |
\right)u(x,t)=0
\Delta
u(x,t)
x
t
u(x,t)=\intht(x,y)u0(y)dy.
ht(x,y)=
| infty | |
| \sum | |
| i=0 |
\exp(-λit)\phii(x)\phii(y)
λi
\phii
ith
\Delta
T:M → N
M
N
ht(x,y)=ht(T(x),T(y))
T
ht(x,x)=
| infty | |
| \sum | |
| i=0 |
\exp(-λit)
| 2(x). | |
| \phi | |
| i |
\Delta
M
N
\exp(-λit)
λi
Since
ht(x,x)
| \{h | |
| t1 |
| (x,x),\ldots,h | |
| tn |
(x,x)\}
t1,\ldots,tn
In most applications, the underlying manifold for an object is not known. The HKS can be computed if a mesh representation of the manifold is available, by using a discrete approximation to
\Delta
L=A-1W
A
A(i,i)
i
W
L
L=\PhiΛ\PhiTA
Λ
L
\Phi
Kt=\Phi\exp(-tΛ)\PhiT.
kt(i,j)
i
j
t
The main property that characterizes surfaces using HKS up to an isometry holds only when the eigenvalues of the surfaces are non-repeating. There are certain surfaces (especially those with symmetry) where this condition is violated. A sphere is a simple example of such a surface.
The time parameter in the HKS is closely related to the scale of global information. However, there is no direct way to choose the time discretization. The existing method chooses time samples logarithmically which is a heuristic with no guarantees[4]
The discrete heat kernel requires eigendecomposition of a matrix of size
n x n
n
n
The performance guarantees for HKS only hold for truly isometric transformations. However, deformations for real shapes are often not isometric. A simple example of such transformation is closing of the fist by a person, where the geodesic distances between two fingers changes.
Source:
The (continuous) HKS at a point
x
ht(x,x)
s(x)
ht(x,x)=
| 1 | |
| 4\pit |
+
| s(x) | |
| 12\pi |
+O(t).
x
t
The WKS follows a similar idea to the HKS, replacing the heat equation with the Schrödinger wave equation,
\left(i\Delta+
| \partial | |
| \partialt |
\right)\psi(x,t)=0
\psi(x,t)
x
p(x)=
| infty | |
| \sum | |
| i=0 |
| 2(λ | |
| f | |
| i) |
| 2(x) | |
| \phi | |
| i |
f
fi(x)
\{
| p | |
| f1 |
| (x),\ldots,p | |
| fn |
(x)\}
Similar to the HKS, the GPS[5] is based on the Laplace-Beltrami operator. GPS at a point
x
x
SGWS[6] provides a general form for spectral descriptors, where one can obtain HKS by specifying the filter function. SGWS is a multiresolution local descriptor that is not only isometric invariant, but also compact, easy to compute and combines the advantages of both band-pass and low-pass filters.
Even though the HKS represents the shape at multiple scales, it is not inherently scale invariant. For example, the HKS for a shape and its scaled version are not the same without pre-normalization. A simple way to ensure scale invariance is by pre-scaling each shape to have the same surface area (e.g. 1). Using the notation above, this means:
\begin{align} s&=\sumjAj\\ A&=A/s\\ λi&=sλiforeachi\\ \phii&=\sqrt{s}\phiiforeachi\\ \end{align}
Alternatively, scale-invariant version of the HKS can also be constructed by generating a Scale space representation.[7] In the scale-space, the HKS of a scaled shape corresponds to a translation up to a multiplicative factor. The Fourier transform of this HKS changes the time-translation into the complex plane, and the dependency on translation can be eliminated by considering the modulus of the transform.. An alternative scale invariant HKS can be established by working out its construction through a scale invariant metric, as defined in.[8]
The HKS is defined for a boundary surface of a 3D shape, represented as a 2D Riemannian manifold. Instead of considering only the boundary, the entire volume of the 3D shape can be considered to define the volumetric version of the HKS.[9] The Volumetric HKS is defined analogous to the normal HKS by considering the heat equation over the entire volume (as a 3-submanifold) and defining a Neumann boundary condition over the 2-manifold boundary of the shape. Volumetric HKS characterizes transformations up to a volume isometry, which represent the transformation for real 3D objects more faithfully than boundary isometry.
The scale-invariant HKS features can be used in the bag-of-features model for shape retrieval applications.[10] The features are used to construct geometric words by taking into account their spatial relations, from which shapes can be constructed (analogous to using features as words and shapes as sentences). Shapes themselves are represented using compact binary codes to form an indexed collection. Given a query shape, similar shapes in the index with possibly isometric transformations can be retrieved by using the Hamming distance of the code as the nearness-measure.