Abstract
Integral invariants have been proven to be useful for shape matching and recognition, but fundamental mathematical questions have not been addressed in the computer vision literature. In this article we are concerned with the identifiability and numerical algorithms for the reconstruction of a star-shaped object from its integral invariants. In particular we analyse two integral invariants and prove injectivity for one of them. Additionally, numerical experiments are performed.
Highlights
Integral invariants and corresponding signatures have been introduced by Manay et al [4] as a tool for shape matching and classification
It is natural to regard the invariant of Ω as a mapping on the sphere as well. It is shown in Lemma 3.1 that, up to a nonlinear transformation, the cone area integral invariant for star-shaped objects is equivalent to computing integrals of the radial function over spherical caps
To substantiate the applicability of the presented theoretical results, we present at the end of the corresponding section the output of the numerical experiments for each discussed integral invariant
Summary
Integral invariants and corresponding signatures have been introduced by Manay et al [4] as a tool for shape matching and classification (see [10] for further applications). It is natural to regard the invariant of Ω as a mapping on the sphere as well It is shown in Lemma 3.1 that, up to a nonlinear (but rather trivial) transformation, the cone area integral invariant for star-shaped objects is equivalent to computing integrals of the radial function over spherical caps. The question, whether a function on Sn−1 is uniquely determined by its integrals over all spherical caps of a given aperture ε has been treated in [8], where the uniqueness is proven for all but countably many apertures 0 < ε < 2π (cf Theorem 3.3) This result shows that the required injectivity holds in the case of the cone area integral invariant, new problems arise by closer investigation of the properties of the invariant. In the following we refer to a function f satisfying above conditions as kernel function
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