Bruce Kitchens

The mathematics of halftoning

This paper describes some mathematical aspects of halftoning in digital printing. Halftoning is the technique of rendering a continuous range of colors using only a few discrete ones. There are two major classes of methods: dithering and error diffusion. Some discussion is presented concerning the method of dithering, but the main emphasis is on error diffusion.

Periodic points for onto cellular automata

Summary Let φ be a one-dimensional surjective cellular automaton map. We prove that if φ is a closing map, then the configurations which are both spatially and temporally periodic are dense. (If φ is not a closing map, then we do not know whether the temporally periodic configurations must be dense.) The results are special cases of results for shifts of finite type, and the proofs use symbolic dynamical techniques.

State splitting for variable-length graphs (Corresp.)

The state splitting algorithm of Adler, Coppersmith, and Hassner for graphs with edges of fixed length is extended to graphs with edges of variable lengths. This has the potential to improve modulation code construction techniques. Although the ideas of state splitting come from dynamical systems, completely graph-theoretic terms are used.

Eventual factor maps and compositions of closing maps

We prove some results related to the question of the existence of factor maps and eventual factor maps between shifts of finite type. Our main result is that if A and B are integral eventually positive (IEP) matrices, and A eventually factors finite-to-one onto B , then there exists an IEP matrix C such that A eventually factors onto C by left closing maps and C eventually factors onto B by right closing maps. This answers the question of the existence of finite-to-one eventual factor maps when the spectrum of A is simple. As a corollary, if in addition to the above hypothesis, χ* A =χ* B , (where χ* A is the characteristic polynomial of A modulo x ), then A is shift equivalent to B .

Isomorphism rigidity of irreducible algebraic ℤd-actions

Abstract.An irreducible algebraic ℤd-actionα on a compact abelian group X is a ℤd-action by automorphisms of X such that every closed, α-invariant subgroup Y⊊X is finite. We prove the following result: if d≥2, then every measurable conjugacy between irreducible and mixing algebraic ℤd-actions on compact zero-dimensional abelian groups is affine. For irreducible, expansive and mixing algebraic ℤd-actions on compact connected abelian groups the analogous statement follows essentially from a result by Katok and Spatzier on invariant measures of such actions (cf. [4] and [3]). By combining these two theorems one obtains isomorphism rigidity of all irreducible, expansive and mixing algebraic ℤd-actions with d≥2.

Metrics and entropy for non-compact spaces

We investigate Bowen’s metric definition of topological entropy for homeomorphisms of non-compact spaces. Different equivalent metrics may assign to the homeomorphism different entropies. We show that the infimum of the metric entropies is greater than or equal to the supremum of the measure theoretic entropies. An example shows that it may be strictly greater. If the entropy of the homeomorphism can vary as the metrics vary we see that the supremum is infinity.

An invariant for continuous factors of Markov shifts

Let XA and XB be subshifts of finite type with Markov measures (p, P) and (q, Q). It is shown that if there is a continuous onto measure-preserving factor map from XA to XB, then the block of the Jordan form of Q with nonzero eigenvalues is a principal submatrix of the Jordan form of P. If XA and XB are irreducible with the same topological entropy, then the same relationship holds for the matrices A and B. As a consequence, gB(t)/gA(t), the ratio of the zeta functions, is a polynomial. From this it is possible to construct a pair of equalentropy subshifts of finite type that have no common equal-entropy continuous factor of finite type, and a strictly sofic system that cannot have an equal-entropy subshift of finite type as a continuous factor.

Automorphisms of one-sided subshifts of finite type

We prove that the automorphism group of a one-sided subshift of finite type is generated by elements of finite order. For one-sided full shifts we characterize the finite subgroups of the automorphism group. For one-sided subshifts of finite type we show that there are strong restrictions on the finite subgroups of the automorphism group.

Convex dynamics and applications

This paper proves a theorem about bounding orbits of a time dependent dynamical system. The maps that are involved are examples in convex dynamics, by which we mean the dynamics of piecewise isometrics where the pieces are convex. The theorem came to the attention of the authors in connection with the problem of digital halftoning. Digital halftoning is a family of printing technologies for getting full-color images from only a few different colors deposited at dots all of the same size. The simplest version consists in obtaining gray-scale images from only black and white dots. A corollary of the theorem is that for error diffusion, one of the methods of digital halftoning, averages of colors of the printed dots converge to averages of the colors taken from the same dots of the actual images. Digital printing is a special case of a much wider class of scheduling problems to which the theorem applies. Convex dynamics has roots in classical areas of mathematics such as symbolic dynamics, Diophantine approximation, and the theory of uniform distributions.

Error bounds for error diffusion and related digital halftoning algorithms

We study error bounds of error diffusion and related digital halftoning algorithms. We define a large class of error diffusion algorithms and give sufficient and necessary conditions for the existence of an error diffusion algorithm with bounded error. In particular, we show that there exists an error diffusion algorithm with bounded errors if and only if the input colors lie in the convex hull of the output colors. We discuss boundedness of a human visual system based error. In addition, we discuss the relationship between digital halftoning and some classical mathematical problems such as the chairman assignment problem.