fit

fit is a python library for fitting shapes like lines, circles, and polygons to data points. I wrote this library to help with a machine vision problem of identifying many circular objects of arbitrary scale in an image. To fit a single shape to data points I use well-known algorithms, but to identify best fits for multiple shapes (an NP-hard problem in most cases), I use a stochastic approximation algorithm that I wrote.

Documentation

class Shape

Contains methods for fitting particular shapes to data. This class is not used directly. Other classes, such as Circle and Line, inherit from Shape and are used for fitting particular kinds of shapes to data with specified algorithms.

The following functions can be used on any Shape:

• Shape.single(self, points) Fits a single copy of the shape to the data.
• Shape.many(self, points, numShapes) Fits multiple copies of the shape to the data.

To inherit from Shape (e.g. when implementing your own kind of shape fitter), you must implement the following functions:

• Shape._single(self, points) An algorithm for fitting a single copy of the particular shape to the data defined by numpy.matrix points (where the columns are each individual point and rows contain their coordinates). Returns a numpy.matrix column vector of parameters that define the fitted shape.
• Shape.get(self, theta) Takes a numpy.matrix column vector theta of parameters that define the shape and returns the parameters in a more programmer-friendly format (e.g. a tuple).
• Shape.getMany(self, thetas) Takes a numpy.matrix of column vectors thetas of parameters that defines all the shapes that fit some data, and returns them in a more programmer-friendly format (e.g. a tuple of lists).
• Shape.error_function(self, p, theta) Some sort of measure of error that defines how well a particular point p fits on the shape defined by parameter vector theta.
• Shape.initial_theta_function(self, points, numShapes) A function that stochastically generates a numpy.matrix of parameter column vectors that serve as the initial theta values in the algorithm to fit multiple shapes to data.
• Shape.initial_lambda_function(self, points, numShapes) A function that computes the best value to initialize lambda with, which is often computed based on properties such as how spread apart the points are.

class Circle

Class for fitting circles to data in a scale- and translation-invariant way.

circle = Circle.__init__(algorithm='bullock', solve='exact')

Creates a circle fitter that fits a circle to a set of points. Radius and center of the circle are allowed to vary, and the algorithm for fitting a single circle and the strategy for doing so can be chosen.

• algorithm (string) Algorithm used to compute the best fit circle

  • 'algebraic' Calculates a solution that minimizes the regression error, where points are defined by the algebraic equation of a circle, ax^Tx + b^T*x + c = 0
  • 'bullock' Calculates a solution that minimizes the geometric distance, defined by sum_i(g(u_i)^2), where g(u_i) = ||u_i - x||^2 - r^2

• solve (string) Technique used to find the optimum defined by the algorithm

  • 'exact' Computes the exact solution using linear regression
  • 'approximate' Computes an approximate solution using gradient descent

• Raises a ValueError if the algorithm or the solve parameters are not one of the ones listed above

x, r = Circle.single(self, points)

Parameters:

• self (Circle) Circle with the algorithm and solve parameters defined
• points (dxn np.matrix) A matrix of column vectors containing the coordinates of each point to fit

Outputs:

• x (dx1 np.matrix) Coordinates of the center of the circle
• r (float) Radius of the circle

Failure:

• Raises a NotImplementedError is solve is 'approximate' -- gradient descent solutions haven't been implemented yet
• Running singleCircle on an empty matrix of points or fewer points than required to define a circle in the given dimensionality results in undefined behavior

xs, rs = Circle.many(self, points, numCircles)

Parameters:

• self (Circle) Circle with the algorithm and solve parameters defined
• points (dxn np.matrix) A matrix of column vectors containing the coordinates of each point to fit
• numCircles (int) Number of circles to fit to the points

Outputs:

• xs (dxk np.matrix) Column vectors containing the coordinates of the centers of each fitted circle
• rs (1xk np.matrix) Radius of each fitted circle

Failure:

• Check the failure conditions for Circle.single