Abstract
Object recognition is one of the topics of artificial intelligence, computer vision, image processing, and machine vision. The classical problem in these areas of computer science is that of determining object via characteristic features. An important feature of the object is its contour. Accurate reconstruction of contour points leads to possibility to compare the unknown object with models of specified objects. The key information about the object is the set of contour points which are treated as interpolation nodes. Classical interpolations (Lagrange or Newton polynomials) are useless for precise reconstruction of the contour. The chapter is dealing with proposed method of contour reconstruction via curves interpolation. First stage consists in computing the contour points of the object to be recognized. Then one can compare models of known objects, given by the sets of contour points, with coordinates of interpolated points of unknown object. Contour points reconstruction and curve interpolation are possible using a new method of Hurwitz-Radon matrices.
TopBackground
The following question is important in mathematics and computer science: is it possible to find a method of curve interpolation and extrapolation in the plane without building the interpolation polynomials? This chapter aims at giving the positive answer to this question. Current methods of curve interpolation are based on classical polynomial interpolation: Newton, Lagrange or Hermite polynomials and spline curves which are piecewise polynomials (Dahlquist & Bjoerck, 1974). Classical methods are useless to interpolate the function that fails to be differentiable at one point, for example the absolute value function f(x) = |x|at x = 0. If point (0;0) is one of the interpolation nodes, then precise polynomial interpolation of the absolute value function is impossible. Also when the graph of interpolated function differs from the shape of polynomials considerably, for example f(x) = 1/x, interpolation is very hard because of existing local extrema of polynomial. We cannot forget about the Runge’s phenomenon: when interpolation nodes are equidistance then high-order polynomial oscillates toward the end of the interval, for example close to -1 and 1 with function f(x) = 1/(1+25x2) (Ralston, 1965). MHR method is free of these bad feature. Computational algorithm is considered and then it is important to talk about time. Complexity of calculations for one unknown point in Lagrange or Newton interpolation based on n nodes is connected with the computational cost of rank O(n2). Proposed method has lower calculation complexity.
A significant problem in risk analysis and decision making is that of appropriate data representation and extrapolation (Brachman & Levesque, 2004). Two-dimensional data can be treated as points on the curve. Classical polynomial interpolations and extrapolations (Lagrange, Newton, Hermite) are useless for data anticipation, because the stock quotations or the market prices represent discrete data and they do not preserve a shape of the polynomial. Also Richardson extrapolation has some weak sides concerning discrete data. This chapter is dealing with the method of data foreseeing and value extrapolation by using a family of Hurwitz-Radon matrices. The quotations, prices or rate of a currency, represented by curve points, consist of information which allows us to extrapolate the next data and then to make a decision (Fagin et al, 1995).
Key Terms in this Chapter
Artificial Intelligence: Intelligence of machines and computers, as a connection of algorithms and hardware, which makes that a man – human being can be simulated by the machines in analyzing risk, decision making, reasoning, knowledge, planning, learning, communication, perception and the ability to move and manipulate objects.
Hurwitz-Radon Matrices: A family of skew – symmetric and orthogonal matrices with columns and rows that create, together with identical matrix, the base in vector spaces of dimensions N = 2, 4 or 8.
MHR Method: The method of curve interpolation and extrapolation using linear (convex) combinations of OHR operators.
OHR Operator: Matrix operator of Hurwitz-Radon built from coordinates of interpolation nodes.
Value Anticipation: Foreseeing next value when last value is known.
Contour Modeling: Calculation of unknown points of the object contour having information about some points of the object contour.
Data Extrapolation: Calculation of unknown values for the points situated outside the ranges of nodes.
Curve Interpolation: Computing new and unknown points of a curve and creating a graph of a curve using existing data points – interpolation nodes.