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**1 - 5**of**5**### CONVEX DYNAMICS: DYNAMICAL SYSTEMS WITH DEEP MATHEMATICAL AND TECHNOLOGICAL ROOTS

"... We discuss issues that arise in the geometrical study of some greedy algorithms which are used in particular for digital printing [1] [4] [5] and further examples of scheduling [2] [6]. We consider the case when successively scheduled events can be represented as the elements of a finite subset C of ..."

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We discuss issues that arise in the geometrical study of some greedy algorithms which are used in particular for digital printing [1] [4] [5] and further examples of scheduling [2] [6]. We consider the case when successively scheduled events can be represented as the elements of a finite subset C of a finite dimensional affine space, and C is the set of vertices of a convex polytope P. In digital printing, P would most often be an interval for greyscale image printing, and a 3-cube for color printing, but other polytopes arise. In scheduling, one most often use standard simplices, other polytopes arises as well, specially those polytopes whose corners are corner of some unit cube. In general, considering digital printing on a line as an instance of scheduling, a schedule for us is a sequence of vertices (outputs), i.e., points in the set of corners, that responds to a sequence of inputs (or demands) which are points of the polytope. We judge the quality of a schedule by the closeness in some distance of all the initial segments (terminating at all times) of the cumulated outputs and inputs sequences. The difference of the two cumulated sequences form a sequence of error vectors. A greedy algorithm for a given distance in particular chooses at each step an output so that the distance between the two cumulated sequences (or a norm of the cumulated error) at that step, be as small as possible. Such a process defines a time dependent

### CONVEX DYNAMICS AND APPLICATIONS

"... Abstract. 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 isometries where the pieces are convex. The theorem came to the attention of the authors in connect ..."

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Abstract. 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 isometries 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 consist in obtaining grey 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. 1.

### Animated Visualization of Time-Varying 2D Flows using Error Diffusion

"... This paper presents a fast glyph-placement algorithm for visualization of time-varying 2D flow. The method can be used to place many kinds of glyphs. Here it is applied to two in particular: arrows in a hedgehog diagram and streak lines. It works by overpopulating images with glyphs, and then decima ..."

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This paper presents a fast glyph-placement algorithm for visualization of time-varying 2D flow. The method can be used to place many kinds of glyphs. Here it is applied to two in particular: arrows in a hedgehog diagram and streak lines. It works by overpopulating images with glyphs, and then decimating them. The decimation phase uses error diffusion, but extends this halftoning technique to solve the problem of coloring a collection of shapes which do not lie on a raster grid. Because error diffusion is a greedy algorithm, the method avoids iterative adjustments of glyph positions, and is fast. When used to visualize static flow fields, the resulting images are free of grid and clustering artifacts. It can be extended to visualize time-varying flow fields, by modifying the error diffusion algorithm further to maintain coherence between frames in an animation.

### ERGODIC DYNAMICS IN Σ ∆ QUANTIZATION: TILING INVARIANT SETS AND SPECTRAL ANALYSIS OF ERROR

, 2003

"... Abstract. This paper has two themes that are intertwined: The first is the dynamics of certain piecewise affine maps on R m that arise from a class of analogto-digital conversion methods called Σ ∆ quantization. The second is the analysis of reconstruction error associated to each such method. Σ ∆ q ..."

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Abstract. This paper has two themes that are intertwined: The first is the dynamics of certain piecewise affine maps on R m that arise from a class of analogto-digital conversion methods called Σ ∆ quantization. The second is the analysis of reconstruction error associated to each such method. Σ ∆ quantization generates approximate representations of functions by sequences that lie in a restricted set of discrete values. These are special sequences in that their local averages track the function values closely, thus enabling simple convolutional reconstruction. In this paper, we are concerned with the approximation of constant functions only, a basic case that presents surprisingly complex behavior. An mth order Σ ∆ scheme with input x can be translated into a dynamical system that produces a discrete-valued sequence (in particular, a 0–1 sequence) q as its output. When the schemes are stable, we show that the underlying piecewise affine maps possess invariant sets that tile R m up to a finite multiplicity. When this multiplicity is one (the single-tile case), the dynamics within the tile is isomorphic to that of a generalized skew translation on T m. The value of x can be approximated using any consecutive M elements in q with increasing accuracy in M. We show that the asymptotical behavior of reconstruction error depends on the regularity of the invariant sets, the order m, and some arithmetic properties of x. We determine the behavior in a number of cases of practical interest and provide good upper bounds in some other cases when exact analysis is not yet available. 1.