Automatic Reassembly of Three-Dimensional Jigsaw Puzzles

In this paper, we present an effective algorithm for reassembling three-dimensional apictorial jigsaw puzzles obtained by dividing a curved surface into a finite number of interlocking pieces. As such, our algorithm does not make use of any picture or design that may be painted on the surface; nor does it require a priori knowledge of the overall shape of the original surface. A motivating example is the problem of virtually reconstructing a broken ostrich egg shell. In order to develop and test the algorithm, we also devise a method for constructing synthetic three-dimensional puzzles by randomly distributing points on a compact surface with respect to surface area measure, then determining the induced Voronoi tessellation, and finally curving the Voronoi edges by using Bezier curves with selected control points. Our edge-matching algorithm relies on the method of Euclidean signature curves. The edges of the puzzle pieces are divided into bivertex arcs, whose signatures are directly compared. The algorithm has been programmed in Matlab and is able to successfully reassemble a broad range of artificial puzzles, including those subjected to a reasonable amount of noise. Moreover, significant progress has been made on reassembly of the real-world ostrich egg data.

Valuations and surface area measures

We consider valuations defined on polytopes containing the origin which have measures on the sphere as values. We show that the classical surface area measure is essentially the only such valuation which is

Characterizations of central symmetry for convex bodies in Minkowski spaces

K. Zindler [47] and P. C. Hammer and T. J. Smith [19] showed the following: Let K be a convex body in the Euclidean plane such that any two boundary points p and q of K , that divide the circumference of K into two arcs of equal length, are antipodal. Then K is centrally symmetric. [19] announced the analogous result for any Minkowski plane \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^2$$ \end{document}, with arc length measured in the respective Minkowski metric. This was recently proved by Y. D. Chai — Y. I. Kim [7] and G. Averkov [4]. On the other hand, for Euclidean d -space ℝ d , R. Schneider [38] proved that if K ⊂ ℝ d is a convex body, such that each shadow boundary of K with respect to parallel illumination halves the Euclidean surface area of K (for the definition of “halving” see in the paper), then K is centrally symmetric. (This implies the result from [19] for ℝ 2 .) We give a common generalization of the results of Schneider [38] and Averkov [4]. Namely, let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^d$$ \end{document} be a d -dimensional Minkowski space, and K ⊂ \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{M}^d$$ \end{document} be a convex body. If some Minkowskian surface area (e.g., Busemann’s or Holmes-Thompson’s) of K is halved by each shadow boundary of K with respect to parallel illumination, then K is centrally symmetric. Actually, we use little from the definition of Minkowskian surface area(s). We may measure “surface area” via any even Borel function ϕ: Sd −1 → ℝ, for a convex body K with Euclidean surface area measure dSK ( u ), with ϕ( u ) being dSK ( u )-almost everywhere non-0, by the formula B ↦ ∫ B ϕ( u ) dSK ( u ) (supposing that ϕ is integrable with respect to dSK ( u )), for B ⊂ Sd −1 a Borel set, rather than the Euclidean surface area measure B ↦ ∫ BdSK ( u ). The conclusion remains the same, even if we suppose surface area halving only for parallel illumination from almost all directions. Moreover, replacing the surface are a measure dSK ( u ) by the k -th area measure of K ( k with 1 ≦ k ≦ d − 2 an integer), the analogous result holds. We follow rather closely the proof for ℝ d , which is due to Schneider [38].

One-sided Lebesgue Bernoulli maps of the sphere of degreen2and2n2

We prove that there are families of rational maps of the sphere of degreen2(n=2,3,4,…)and2n2(n=1,2,3,…)which, with respect to a finite invariant measure equivalent to the surface area measure, are isomorphic to one-sided Bernoulli shifts of maximal entropy. The maps in question were constructed by Böettcher (1903--1904) and independently by Lattès (1919). They were the first examples of maps with Julia set equal to the whole sphere.

Section means, integral transforms, and Boolean models

For a stationary particle process X with convex particles in ℝdd ≧ 2, a mean body M(X) can be defined by where h(M,·) denotes the support function of the convex body M, γ the intensity of X, and P0 is the distribution of the typical particle of X (a probability measure on the set of convex bodies with Steiner point at the origin). Replacing the support function h(M,·) by the surface area measure S(M,·) (see Schneider (1993), for the basic notions from convex geometry), we get the Blaschke body B(X) of X, After normalization, the left-hand side represents the mean normal distribution of X. The main problem discussed here is whether B(X) (respectively S(B(X), ·)) is uniquely determined by the mean bodies M(X ∩ E) in random planar sections X ∩ E of X. From more general results in Weil (1995), it follows that the expectation ES(M(X ∩ E), ·) (taken w.r.t. the uniform distribution of two-dimensional subspaces E in ℝd) equals the surface area measure of a section mean B2(B(X)) of B(X). Thus, the formulated stereological question can be reduced to the injectivity of the transform B2 : K ↦ B2(K).

Hörmander's Carleson theorem for the ball

Let denote the unit ball in ℂ2 and let Sdenote its boundary, the unit sphere. For z ∈ B and δ>0, the following non isotropic balls are defined, where A finite positive Borel measure μ, on B is called a Carleson measure if there exists a constant C for whichHere σ denotes normalized surface area measure on S. The following theorem was obtained by Hörmander [6] as a special case of more general variants for strictly pseudoconvex domains in ℂn. Recently Cima and Wogen [3] derived it from a Carleson measure theorem for Bergman spaces of the ball. A different direct approach to the Bergman context, and related settings, is given in Leucking [7].

Local Boundary Behavior of Bounded Holomorphic Functions

Let D ⊂⊂ Cn be a bounded domain with smooth boundary ∂D, and let F be a bounded holomorphic function on D. A generalization of the classical theorem of Fatou says that the set E of points on ∂D at which F fails to have nontangential limits satisfies the condition σ (E) = 0, where a denotes surface area measure. We show in the present paper that this result remains true when σ is replaced by 1-dimensional Lebesgue measure on certain smooth curves γ in ∂D. The condition that γ must satisfy is that its tangents avoid certain directions.