The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy of the initial hypersurface bounds the entropy at all future singularities. We show here that not only does the round sphere have the lowest entropy of any closed singularity, but there is a gap to the second lowest.

We introduce an interactive technique for manipulating simple 3D shapes based on extracting them from a single photograph. Such extraction requires understanding of the components of the shape, their projections, and relations. These simple cognitive tasks for humans are particularly difficult for automatic algorithms. Thus, our approach combines the cognitive abilities of humans with the computational accuracy of the machine to solve this problem. Our technique provides the user the means to quickly create editable 3D parts¡ª human assistance implicitly segments a complex object into its components, and positions them in space. In our interface, three strokes are used to generate a 3D component that snaps to the shape¡¯s outline in the photograph, where each stroke defines one dimension of the component. The computer reshapes the component to fit the image of the object in the photograph as well as to satisfy various inferred geometric constraints imposed by its global 3D structure. We show that with this intelligent interactive modeling tool, the daunting task of object extraction is made simple. Once the 3D object has been extracted, it can be quickly edited and placed back into photos or 3D scenes, permitting object-driven photo editing tasks which are impossible to perform in image-space.

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The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection ∇ on the TM, such that ∇g is totally symmetric, forming, by definition, a “Codazzi pair” t∇, gu. In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on TM), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of ∇ with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. While reproducing some known results regarding Codazzi transform, conformal-projective transformation, etc., we extend much of these geometric relations, and hence obtain new geometric insights, for the general case of a non-degenerate two-form h (instead of the symmetric g) and an affine connection with possibly non-vanishing torsion. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections.