  This article presents a new distance for measuring shape dissimilarity
between objects. Recent publications introduced the use of eigenvalues of the
Laplace operator as compact shape descriptors. Here, we revisit the eigenvalues
to define a proper distance, called Weighted Spectral Distance (WESD), for
quantifying shape dissimilarity. The definition of WESD is derived through
analysing the heat-trace. This analysis provides the proposed distance an
intuitive meaning and mathematically links it to the intrinsic geometry of
objects. We analyse the resulting distance definition, present and prove its
important theoretical properties. Some of these properties include: i) WESD is
defined over the entire sequence of eigenvalues yet it is guaranteed to
converge, ii) it is a pseudometric, iii) it is accurately approximated with a
finite number of eigenvalues, and iv) it can be mapped to the [0,1) interval.
Lastly, experiments conducted on synthetic and real objects are presented.
These experiments highlight the practical benefits of WESD for applications in
vision and medical image analysis.
