Intuitive modeling of vapourish objects

International audienceAttempts to model gases in computer graphics started in the late 1970s. Since that time, there have been many approaches developed. In this paper we present a non-physical method allowing to create vapourish objects like clouds or smoky characters. The idea is to create few sketches describing the rough shape of the final vapourish object. These sketches will be used as condensation sets of Iterated Function Systems, providing intuitive control over the object. The advantages of the new method are: simplicity, good control of resulting shapes and ease of eventual object animation

Evolutionary algorithms in artificial intelligence: a comparative study through applications

For many years research in artificial intelligence followed a symbolic paradigm which required a level of knowledge described in terms of rules. More recently subsymbolic approaches have been adopted as a suitable means for studying many problems. There are many search mechanisms which can be used to manipulate subsymbolic components, and in recent years general search methods based on models of natural evolution have become increasingly popular. This thesis examines a hybrid symbolic/subsymbolic approach and the application of evolutionary algorithms to a problem from each of the fields of shape representation (finding an iterated function system for an arbitrary shape), natural language dialogue (tuning parameters so that a particular behaviour can be achieved) and speech recognition (selecting the penalties used by a dynamic programming algorithm in creating a word lattice). These problems were selected on the basis that each should have a fundamentally different interactions at the subsymbolic level. Results demonstrate that for the experiments conducted the evolutionary algorithms performed well in most cases. However, the type of subsymbolic interaction that may occur influences the relative performance of evolutionary algorithms which emphasise either top-down (evolutionary programming - EP) or bottom-up (genetic algorithm - GA) means of solution discovery. For the shape representation problem EP is seen to perform significantly better than a GA, and reasons for this disparity are discussed. Furthermore, EP appears to offer a powerful means of finding solutions to this problem, and so the background and details of the problem are discussed at length. Some novel constraints on the problem's search space are also presented which could be used in related work. For the dialogue and speech recognition problems a GA and EP produce good results with EP performing slightly better. Results achieved with EP have been used to improve the performance of a speech recognition system

Bilinear Fractal Interpolation and Box Dimension

In the context of general iterated function systems (IFSs), we introduce bilinear fractal interpolants as the fixed points of certain Read-Bajraktarevi\'{c} operators. By exhibiting a generalized "taxi-cab" metric, we show that the graph of a bilinear fractal interpolant is the attractor of an underlying contractive bilinear IFS. We present an explicit formula for the box-counting dimension of the graph of a bilinear fractal interpolant in the case of equally spaced data points

The global statistics of return times: return time dimensions versus generalized measure dimensions

We investigate the relations holding among generalized dimensions of invariant measures in dynamical systems and similar quantities defined by the scaling of global averages of powers of return times. Because of a heuristic use of Kac theorem, these latter have been used in place of the former in numerical and experimental investigations; to mark this distinction, we call them return time dimensions. We derive a full set of inequalities linking measure and return time dimensions and we comment on their optimality with the aid of two maps due to von Neumann -- Kakutani and to Gaspard -- Wang. We conjecture the behavior of return time dimensions in a typical system. We only assume ergodicity of the dynamical system under investigation.Comment: Submitted to J. Stat. Phy

Local entropy averages and projections of fractal measures

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: Let m,n be integers which are not powers of the same integer, and let X,Y be closed subsets of the unit interval which are invariant, respectively, under times-m mod 1 and times-n mod 1. Then, for any non-zero t: dim(X+tY)=min{1,dim(X)+dim(Y)}. A similar result holds for invariant measures, and gives a simple proof of the Rudolph-Johnson theorem. Our methods also apply to many other classes of conformal fractals and measures. As another application, we extend and unify Results of Peres, Shmerkin and Nazarov, and of Moreira, concerning projections of products self-similar measures and Gibbs measures on regular Cantor sets. We show that under natural irreducibility assumptions on the maps in the IFS, the image measure has the maximal possible dimension under any linear projection other than the coordinate projections. We also present applications to Bernoulli convolutions and to the images of fractal measures under differentiable maps.Comment: 55 pages. Version 2: Corrected an error in proof Thm. 4.3; some new references; various small correction

Initial distribution spread: A density forecasting approach

Ensemble forecasting of nonlinear systems involves the use of a model to run forward a discrete ensemble (or set) of initial states. Data assimilation techniques tend to focus on estimating the true state of the system, even though model error limits the value of such efforts. This paper argues for choosing the initial ensemble in order to optimise forecasting performance rather than estimate the true state of the system. Density forecasting and choosing the initial ensemble are treated as one problem. Forecasting performance can be quantified by some scoring rule. In the case of the logarithmic scoring rule, theoretical arguments and empirical results are presented. It turns out that, if the underlying noise dominates model error, we can diagnose the noise spread

Triangular Gatzouras-Lalley-type planar carpets with overlaps

We construct a family of planar self-affine carpets with overlaps using lower triangular matrices in a way that generalizes the original Gatzouras--Lalley carpets defined by diagonal matrices. Assuming the rectangular open set condition, Bara\'nski proved for this construction that for typical parameters, which can be explicitly checked, the inequalities between the Hausdorff, box and affinity dimension of the attractor are strict. We generalize this result to overlapping constructions, where we allow complete columns to be shifted along the horizontal axis or allow parallelograms to overlap within a column in a transversal way. Our main result is to show sufficient conditions under which these overlaps do not cause the drop of the dimension of the attractor. Several examples are provided to illustrate the results, including a self-affine smiley, a family of self-affine continuous curves, examples with overlaps and an application of our results to some three-dimensional systems.Comment: 12 figures; v2: improved presentation, updated references, added a three-dimensional example and an Appendix. Results unchange