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A Python module for solving optimization problems with nonlinear least-squares

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Consult the module API page at

for all information related to this module, including information regarding the latest changes to the code. The page at the URL shown above lists all of the module functionality you can invoke in your own code.

With regard to the basic purpose of this module, it provides a domain agnostic implementation of nonlinear least-squares algorithms (gradient-descent and Levenberg-Marquardt) for fitting a model to observed data. Typically, a model involves several parameters and each observed data element can be expressed as a function of those parameters plus noise. The goal of nonlinear least-squares is to estimate the best values for the parameters given all of the observed data. In order to illustrate how to use the NonlinearLeastSquares class, the module also comes with two additional classes: OptimizedSurfaceFit and ProjectiveCamera.

The job of OptimizedSurfaceFit is to fit the best surface to noisy height data over an XY-plane. The model in this case would be an analytical expression for the height surface and the goal of nonlinear least-squares would be to estimate the best values for the parameters in the model.

And the job of ProjectiveCamera is to demonstrate how nonlinear least-squares can be used for estimating scene structure from camera motion. The underlying ideas is that you take multiple images of a scene with a camera — something that you can simulate with ProjectiveCamera. You feed the pixels thus recorded into the NonlinearLeastSquares class to estimate the coordinates of the scene structure points and, when using uncalibrated cameras, to also estimate the extrinsic parameters of the camera at each of its positions.

Starting with Version 2.0.0, the module includes code for the bundle-adjustment variant of the Levenberg-Marquardt algorithm.

Typical usage syntax for invoking the domain-agnostic NonlinearLeastSquares through your own domain-specific class such as OptimizedSurfaceFit or ProjectiveCamera is shown below:

optimizer =  NonlinearLeastSquares(
                 max_iterations = 200,
                 delta_for_jacobian = 0.000001,
                 delta_for_step_size = 0.0001,

surface_fitter = OptimizedSurfaceFit(
                     gen_data_synthetically = True,
                     datagen_functional = "7.8*(x - 0.5)**4 + 2.2*(y - 0.5)**2",
                     data_dimensions = (16,16),
                     how_much_noise_for_synthetic_data = 0.3,
                     model_functional = "a*(x-b)**4 + c*(y-d)**2",
                     initial_param_values = {'a':2.0, 'b':0.4, 'c':0.8, 'd':0.4},
                     display_needed = True,
                     debug = True,


result = surface_fitter.calculate_best_fitting_surface('lm')
result = surface_fitter.calculate_best_fitting_surface('gd')


optimizer =  NonlinearLeastSquares.NonlinearLeastSquares(
                                     max_iterations = 400,
                                     delta_for_jacobian = 0.000001,
                                     delta_for_step_size = 0.0001,

camera = ProjectiveCamera.ProjectiveCamera(
                     camera_type = 'projective',
                     alpha_x = 1000.0,
                     alpha_y = 1000.0,
                     x0 = 300.0,
                     y0 = 250.0,

world_points = camera.make_world_points_for_triangle()
world_points_xformed = camera.apply_transformation_to_generic_world_points(world_points, ..... )

##  Now move the camera to different positions and orientations and then

result = camera.get_scene_structure_from_camera_motion('lm')


result = camera.get_scene_structure_from_camera_motion_with_bundle_adjustment()

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