基于Matlab实现多种算法的曲线拟合

Fitting ellipses, ellipsoids and other quadratic curves and surfaces Copyright 2011-2013 Levente Hunyadi Generating points along an ellipse or ellipsoid, plotting ellipses and ellipsoids in various parametric representations, and fitting ellipses, ellipsoids or other quadratic curves and surfaces to noisy data occur frequently in fields such as computer vision, pattern recognition and system identification. This toolbox provides a fairly comprehensive toolset of estimating quadratic curves and surfaces in an errors-in-variables context, with and without constraints. In addition to classical fitting methods such as least squares (with and without curve or surface normals), Taubin's method, direct ellipse fit by Fitzgibbon et al. [1] and direct ellipsoid fit by Qingde Li and John G. Griffiths [4], the toolbox features an estimation algorithm by the author [2,3], based on and extending the work of István Vajk and Jenő Hetthéssy [5]. The proposed quadratic curve and surface fitting algorithm combines direct fitting with a noise cancellation step, producing consistent estimates close to maximum likelihood but without iterations. REFERENCES [1] Andrew W. Fitzgibbon, Maurizio Pilu and Robert B. Fisher, "Direct Least Squares Fitting of Ellipses", IEEE Trans. PAMI 21, 1999, pp476-480. [2] Levente Hunyadi, "Estimation methods in the errors-in-variables context", PhD dissertation, Budapest University of Technology and Economics, 2013. [3] Levente Hunyadi and István Vajk, "Constrained quadratic errors-in-variables fitting", The Visual Computer, 12 pages, in print, available on-line from October 2013. [4] Qingde Li and John G. Griffiths, "Least Squares Ellipsoid Specific Fitting", Proceedings of the Geometric Modeling and Processing, 2004. [5] István Vajk and Jenő Hetthéssy, "Identification of nonlinear errors-in-variables models", Automatica 39, 2003, pp2099-2107. CONTACT INFORMATION Levente Hunyadi https://hunyadi.info.hu/ Please use my private e-mail address to submit bug reports, which will be addressed upon short notice; reviews, however, are not monitored. Any feedback is most welcome. Quadratic curves and quadric surfaces in implicit form Copyright 2010 Levente Hunyadi Work with general quadratic curves and quadric surfaces given as implicit equation. This submission facilitates working with quadratic curves (ellipse, parabola, hyperbola, etc.) and quadric surfaces (ellipsoid, elliptic paraboloid, hyperbolic paraboloid, hyperboloid, cone, elliptic cylinder, hyperbolic cylinder, parabolic cylinder, etc.) given with the general quadratic equation Q(x) = x' * A * x + b' * x + c = 0 where a pseudo-MatLab notation has been used. A is a symmetric N-by-N matrix (N = 2 or N = 3 not necessarily invertible), b is an N-by-1 column vector, and c is a scalar. The parameter x is an N-by-1 column vector. Those points x that satisfy Q(x) = 0 comprise the quadratic curve or quadric surface. The package comprises of two major components. First, a set of functions is included for quadratic curves that identify the conic section and compute explicit parameters (semimajor axis, semiminor axis, rotation matrix, translation vector) of a conic section given with the general quadratic equation; or plot a conic section, returning a lineseries object (for circles, ellipses and parabolas) or a hggroup object (for hyperbolas). Second, the package contains an algorithm for computing the distance from a point in 2D to a general quadratic curve defined implicitly by a second-degree quadratic equation in two variables or from a point in 3D to a general quadric surface defined implicitly by a second-degree quadratic equation in three variables. Utility functions are included to manipulate matrices of symbolic variables that were used to pre-compute polynomials shipped with the package. CONTACT INFORMATION Levente Hunyadi https://hunyadi.info.hu/ sympoly: Symbolic polynomials Copyright 2009-2011 Levente Hunyadi A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A general multivariate polynomial is captured with the syntax p = sum( c_i * prod( x_j^p_ij ) ) + k where the summation is over i, the product over j, and c_i is the set of polynomial term coefficients, x_ij is a set of symbolic variables, p_ij is the (usually positive integer) exponent of each variable in a term where at least one p_ij is nonzero for a given i, and k is the constant term. sympoly supports regular elementwise and matrix operations like addition, subtraction, multiplication, power and division; transpose and diagonalization; indefinite and definite integration and differentiation w.r.t. a variable; gradient; coefficient extraction; conversion from and to a Symbolic Toolbox sym object and a numeric array; pretty-printing (overloaded disp and display functions); LaTeX and MatLab code generation (a character string that can be passed to eval). In order to get a list of operations supported on a sympoly object, type "methods sympoly" at the command prompt. EXAMPLES % create sympoly objects x = sympoly('x') sympolys y z % combine sympoly objects in arbitrary expressions q = (x-1)^3 + x*y*z - (x+1)*(z-1) % differentiate a sympoly w.r.t. a variable diff(q, x) % create a matrix of symbolic polynomials and take sum of rows sum([ x x+2 ; y+1 z ], 2) % create symbolic polynomials with variables other than strings y0 = sympoly(symvard('y',0)); y1 = sympoly(symvard('y',-1)); y2 = sympoly(symvard('y',-2)); u1 = sympoly(symvard('u',-1)); % create a system equation for a polynomial dynamic system phi = [y0;y1;y2;u1;u1^2;y1*y2;u1*y1] theta = [-1;1.5;-0.7;1;-0.3;-0.05;0.1] G = phi'*theta Further examples are included in the subfolder "demo" in the distribution, reproduced with minor changes from the "Symbolic Polynomial Manipulation" package by John D'Errico. IMPLEMENTATION From an implementation point of view, a scalar sympoly object is a class with read-only properties ConstantValue, Variables, Coefficients and Exponents, where ConstantValue is a numeric scalar, Variables is a 1-by-n row cell vector of strings or a row vector of (subclasses of) symvariable objects, Coefficients is an m-by-1 numeric column vector of polynomial term coefficients, and Exponents is an m-by-n numeric matrix of exponents for each variable in each term. The items in Variables are always sorted in a standard order, e.g. when variables are strings, they are sorted alphabetically. Since sympoly is a new-style class declared with the classdef keyword, you can use inheritance to derive custom classes from sympoly. Most methods of the class sympoly are implemented such that they handle both scalar and array inputs. When invoked on array input, these operations either return a scalar result that applies over all elements, or a result array of the same dimensions as the input array where each element in the result corresponds to an element in the input. COMPARISON In contrast to MatLab's built-in roots function, which deals with univariate polynomials, sympoly can handle polynomials of multiple variables. Since it is restricted to the class of polynomials, it offers better performance and more flexibility than a sym object in the Symbolic Toolbox. sympoly is an extended version of the "Symbolic Polynomial Manipulation" package by John D'Errico using new-style MatLab classes with slight differences in implementation and function signatures. REFERENCES John D'Errico, "Symbolic Polynomial Manipulation", MatLab Central File Exchange, http://www.mathworks.com/matlabcentral/fileexchange/9577 CONTACT INFORMATION Levente Hunyadi https://hunyadi.info.hu/ Please use my private e-mail address to submit bug reports, which will be addressed upon short notice; reviews, however, are not monitored. Any feedback is most welcome.