NonlinearLeastSquares-2.0.0.html

NonlinearLeastSquares (version 2.0.0, 2018-November-9)

NonlinearLeastSquares.py
Version: 2.0.0     Author: Avinash Kak (kak@purdue.edu) Date: 2018-November-9

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View the main module file in your browser View the code for OptimizedSurfaceFit for optimized surface fitting to noisy height data View the code for ProjectiveCamera for camera modeling

`Download Version 2.0.0: gztar`	`Total number of downloads (all versions): 380` `This count is automatically updated at every rotation of the weblogs (normally once every two to four days) Last updated: Fri Apr 19 06:06:01 EDT 2024`

Switch to Version 2.0.1

CHANGES: Version 2.0.0: This version includes sparse bundle adjustment (SBA) to speed up code execution when using nonlinear least-squares for solving structure-from-camera-motion problems with uncalibrated cameras. SBA exploits the sparseness of the Jacobian encountered in such problems. This version also includes improvements to the Levenberg-Marquardt code itself. The most important improvement consists of how the "mu" factor (renamed "alambda" in this module) is initialized. Version 2.0.0 also provides additional methods for the visualization of the results. Version 1.5.0: This version adds a new class, ProjectiveCamera, to the module for demonstrating how nonlinear least-squares can be used for estimating structure of a scene from the data recorded with a calibrated camera in motion. For a simulated demonstration, you can create calibrated cameras from a generic instance of the ProjectiveCamera class that is then subject to various translational and rotational transformations. Version 1.1.1: This version includes constructor options to control the size and the position of the graphics for displaying the results of nonlinear least-squares optimization. Version 1.1: This version fixes a bug in the synthetic data generator function used for illustrating the functionality of the NonlinearLeastSquares class. Changes also made to the information that is printed out when the module is run in the debug mode. Version 1.0: In the form of a class named NonlinearLeastSquares, this module provides a domain agnostic implementation of nonlinear least-squares algorithms (gradient-descent and Levenberg-Marquardt) for fitting a model to observed data. Typically, the 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 another class, OptimizeSurfaceFit, whose job is to fit the best surface (of a specified analytical form) to noisy height data over the XY-plane. The model in this case is the analytical description of the surface and the goal of nonlinear least-squares is to estimate the best values for the parameters in the analytical description. USAGE FOR OPTIMAL SURFACE FITTING: The module includes a domain specific class, OptimizedSurfaceFit, for demonstrating how nonlinear least-squares in the form of the Levenberg-Marquardt algorithm can be used for optimally fitting analytically specified surfaces to noisy data over a plane. For surface fitting, you need to create an instance of the NonlinearLeastSquares class as shown in line (A) below. In addition, you need to create an instance of a domain-specific class such as OptimizedSurfaceFit as is shown at line (B) below. Note that in the example shown in line (B), the domain-specific class has two duties: (1) to synthetically generate noisy height data; and (2) to call on NonlinearLeastSquares to fit an optimized model to the synthetically generated data. optimizer = NonlinearLeastSquares( #(A) max_iterations = 400, delta_for_jacobian = 0.000001, delta_for_step_size = 0.0001, # only needed for GD ) surface_fitter = OptimizedSurfaceFit( #(B) gen_data_synthetically = True, datagen_functional = "7.8*(x - 0.5)**3 + 2.2*(y - 0.5)**2", data_array_size = (16,16), how_much_noise_for_synthetic_data = 0.7, model_functional = "a*(x-b)**3 + c*(y-d)**2", initial_param_values = {'a':2.0, 'b':0.4, 'c':0.8, 'd':0.4}, display_needed = True, ) surface_fitter.set_constructor_options_for_optimizer(optimizer) #(C) result = surface_fitter.calculate_best_fitting_surface('lm') #(D) or result = surface_fitter.calculate_best_fitting_surface('gd') #(E) In line (C), it is the job of the OptimizedSurfaceFit instance as constructed in line (B) to export the domain-related information (after it is packaged under the hood into a domain-agnostic form) to the instance of NonlinearLeastSquares that was constructed in line (A). The information that the OptimizedSurfaceFit instance conveys to the NonlinearLeastSquares instance includes: 1) The observed noisy data vector. This vector is denoted X in the NonlinearLeastSquares class. The same notation is used for the observed data vector in the usage-example class OptimizedSurfaceFit. 2) The initial values to be used for the parameters of the model. 3) A vector of the predicted values for each of the data elements in X, all in functional form involving the parameters of the model. This vector is denoted Fvec in the NonlinearLeastSquares class. 4) The display function to be used for plotting the partial and the final results if at all such results can be displayed in the form of a 2D or 3D graphic with Python's matplotlib library. Finally, the statement in line (D) lets you call on the Levenberg-Marquardt algorithm for estimating the model parameters. If you would rather use the more basic gradient-descent algorithm for that purpose, your call will look like what is shown in line (E). See the script leven_marq.py in the ExamplesOptimizedSurfaceFit subdirectory of the distribution for further information on what calls you need to sequence together for optimally estimating the parameters of an analytically specified surface that you want to fit to noisy data. USAGE FOR ESTIMATING THE SCENE STRUCTURE WITH A CALIBRATED CAMERA IN MOTION: You again need to create an instance of the NonlinearLeastSquares class as shown in line (F) below. In addition, you need to create an instance of a class like ProjectiveCamera as shown at line (G) below. If you are writing your own code for camera modeling, it would be best if you subclass your code from the ProjectiveCamera class in the module. optimizer = NonlinearLeastSquares.NonlinearLeastSquares( #(F) max_iterations = 400, delta_for_jacobian = 0.000001, ) camera = ProjectiveCamera.ProjectiveCamera( #(G) camera_type = 'projective', alpha_x = 1000.0, alpha_y = 1000.0, x0 = 300.0, y0 = 250.0, ) camera.initialize() world_points = camera.make_world_points_random(30) world_points_xformed = camera.apply_transformation_to_generic_world_points(world_points, ..... ) ## ## all_pixels = Move the camera to different positions and ## collect the pixels ## ## Now construct the X vector and the corresponding prediction ## vector (Fvec) for nonlinear least-squares: camera.construct_X_vector( all_pixels ) #(H) camera.construct_parameter_vec_for_calibrated_cameras() #(I) camera.construct_Fvec_for_calibrated_cameras(camera_params_dict) #(J) result = camera.get_scene_structure_from_camera_motion('lm') #(K) where the returned result will include the optimal estimations for the scene parameters. Note that the parameter vector constructed in line (I) defines the hyperplane over which is defined the cost function. The goal of nonlinear least-squares is return that point on the parameter hyperplane where the cost function takes on the least value. Line (J) constructs the prediction vector in terms of the parameters used in line (I). The argument 'lm' in line (K) stands for "Levenberg-Marquardt". See the script sfm_with_calibrated_cameras_translations_only.py in the ExamplesStructureFromCameraMotion subdirectory of the distribution for further information on what calls you need to sequence together for estimating the structure of a scene that is being viewed with a moving camera. USING SPARSE BUNDLE ADJUSTMENT FOR ESTIMATING THE SCENE STRUCTURE WITH AN UNCALIBRATED CAMERA IN MOTION: You would again need to construct an instance of the NonlinearLeastSquares and an instance of the ProjectiveCamera class as shown in lines (F) and (G) above. But now you would need to replace the code in lines (H) through (K) by: camera.construct_X_vector_for_bundle_adjustment( all_pixels ) #(L) camera.construct_parameter_vec_for_uncalibrated_cameras_with_rodrigues_rotations() #(M) camera.construct_Fvec_for_bundle_adjustment() #(O) result = camera.get_scene_structure_from_camera_motion_with_bundle_adjustment() #(P) The construction of the observation vector X in line (L) is specific to sparse bundle adjustment --- in the sense that you first want to list the pixels in all the cameras for the first world point, then you want to list the pixels in all the cameras for the second world point, and so on. Ordering the observed data in this fashion is necessary to create the sort of block sparsity in the Jacobian that is exploited in SBA. Note that we assume that the intrinsic parameters of the camera are known already. So the goal of the bundle adjustment is to estimate the six extrinsic parameters, the three for translation of the camera and the three (in the form of Rodrigues rotations) for the rotation. See the script bundle_adjust_sfm_with_uncalibrated_cameras_translations_only.py in the ExamplesStructureFromCameraMotion subdirectory of the distribution for further information on what calls you need to sequence together for estimating both the camera parameters and the scene structure for a scene that is being viewed with a moving uncalibrated camera. INTRODUCTION: Nonlinear least-squares is a widely used set of algorithms for fitting a model to noisy data through the minimization of a cost function. The observed data is represented as a vector (denoted X in this module) and how the model would predict each element of X by another vector that is denoted Fvec. As you would expect, each element of Fvec is a function of the parameters of the model. The goal of nonlinear least-squares is to find the best possible values for the parameters, best in the sense of minimizing a cost function that is almost always the square of the vector norm of the difference between X and Fvec. The simplest approach to solving a nonlinear least-squares problem is Gradient Descent (GD). The mental imagery that best explains GD is to think of all of the model parameters as constituting a hyperplane and the cost function as a surface over the hyperplane. Gradient descent consists of starting at some point in the hyperplane, looking straight up at the surface, and walking in the direction of the steepest descent on the surface. At each point on the hyperplane, you make the size of your next step proportional to the value of the gradient on the cost-function surface. Assuming there are no local minima to trap your progress, GD will eventually take you to the point on the hyperplane that is directly below the global minimum for the cost function. Even when getting trapped in a local minimum is not an issue (because, let's say, you made a good choice for the starting point), the basic gradient-descent algorithm suffers from the shortcoming that the closer you get to the minimum, the smaller your steps will be, and, therefore, the slower your progress toward the destination. One way to get round this problem with gradient-descent is to use the Gauss-Newton formula to make a direct jump to the minimum --- assuming you are already sufficiently close to it. However, should you inadvertently try to make a Gauss-Newton jump when still far from the minimum, your algorithm could become numerically unstable and crash. The algorithm that does a good job of combining the numerical robustness of gradient-descent with the speed of Gauss-Newton, while making sure that the latter is not invoked too early, is the Levenberg-Marquardt (LM) algorithm. Given a start point on the hyperplane spanned by the model parameters, LM starts by taking a GD step. In subsequent iterations, before committing itself to a step, LM checks whether or not that step can safely be taken with GN. If not, it steers the path toward GD. However, if the condition for taking the GN step safely is satisfied, it goes ahead with that. In this manner, LM can get to the minimum in a small number of steps, often under 10, whereas for the same problem and the same start point, GD might take more than a hundred. The NonlinearLeastSquares class in this module provides implementations for both the basic Gradient Descent (GD) algorithm and the more efficient Levenberg-Marquardt algorithm for getting to the minimum of a cost function. Starting with Version 2.0.0. this class also includes an SBA (Sparse Bundle Adjustment) variant of the Levenberg-Marquardt algorithm for optimal computation of both the structure and the camera parameters with the data collected by an uncalibrated camera in motion. From a programming standpoint, the most notable feature of NonlinearLeastSquares is that it is domain agnostic. That is, you should be able to use NonlinearLeastSquares for solving any problem that requires a GD or an LM solution for finding the optimum values for a set of model parameters through the minimization of a cost function. The fact that NonlinearLeastSquares is generic implies that you have to write a class of your own for the specific domain in which you wish to use NonlinearLeastSquares. The domain specific class that you create must export to NonlinearLeastSquares values for the following options in its constructor: -- The observed data vector. This vector is denoted X in the NonlinearLeastSquares class. -- The initial values to be used for the parameters of the model. -- A vector of the predicted values for each of the data elements in X, all in functional form involving the parameters of the model. This vector is denoted Fvec in the NonlinearLeastSquares class. -- The display function to be used for plotting the partial and the final results if at all such results can be displayed in the form of a 2D or 3D graphic with Python's matplotlib library. And, if you wish for NonlinearLeastSquares to use your analytically specified partial derivatives in the Jacobian matrix that it needs for the next-step calculations, your domain-specific class must also export that matrix to NonlinearLeastSquares. In the absence of a user-supplied Jacobian matrix, NonlinearLeastSquares estimates it numerically. [See the implementation code for the OptimizedSurfaceFit class supplied with this module --- that class is an example of a domain-specific class that uses NonlinearLeastSquares for carrying out the needed optimization --- for how your own domain-specific class can construct a Jacobian matrix in a functional form and supply it to NonlinearLeastSquares.] The NonlinearLeastSquares class provides several setter methods that your own domain-specific class can use to convey the above-mentioned information to NonlinearLeastSquares. To illustrate how you may wish to write your domain specific class, this module also comes with two additional classes named OptimizedSurfaceFit and ProjectiveCamera, the first for fitting surfaces to noisy data over an xy-plane and the second for estimating the structure of a 3D scene from the data recorded by a camera in motion. METHODS: Constructing an instance of NonlinearLeastSquares: optimizer = NonlinearLeastSquares( X = None, Fvec = None, num_measurements = None, num_parameters = None, initial_params_dict = None, initial_param_values_file = None, jacobian_functionals_array = None, delta_for_jacobian = 0.000001, delta_for_step_size = 0.0001, max_iterations = 200, ) In most usage scenarios, though, you are likely to call NonlinearLeastSquares directly with just the following constructor options and let your own domain specific class set the other options at run time. That is, in your own code, you will first create an instance of NonlinearLeastSquares through the following call to its constructor: optimizer = NonlinearLeastSquares( max_iterations = 400, delta_for_jacobian = 0.000001, ) and subsequently have your own domain-specific class call the various setter methods of NonlinearLeastSquares for giving values to its other constructor options. As to which setter methods your own class should call is presented in the rest of this section. CONSTRUCTOR OPTIONS: debug, debug2: When set to True, it prints out a lot of information at each iteration of the nonlinear least-squares algorithm invoked. display_function: If the problem you are trying to solve with nonlinear least-squares allows for the result of optimization to be visualized, use this constructor option to supply a reference to the function you would like to be used for such visualization. Fvec: Fvec_BA: These must be set to a numpy vector (actually a numpy matrix with just one column) whose elements are the "predictors" for the corresponding observed values in the X and the X_BA vectors, respectively. (We use the symbol 'X' to denote the vector of measured data, as you will soon see. X_BA does the same for the bundle-adjustment variant of the basic Levenberg-Marquardt algorithm.) Each element of Fvec and Fvec_BA is, in general, a function of all of the model parameters. initial_params_dict: This is set to a dictionary whose keys are the model parameters and whose values the initial values for the model parameters. The initial values for the model parameters specify the point in the parameter hyperplane where you want to start the descent to the minimum. initial_param_values_file: If your problem involves a very large number of parameters, it may be more convenient to place all their initial values in a text file. Each parameter and its initial value must be in a line all by itself. See the file "initial_params_gd2.txt" in the Examples directory for how this file needs to be formatted. jacobian_functionals_array: If you wish to specify the partial derivatives of the functional elements in the Fvec vector with respect to the model parameters, you can supply those as a numpy chararray through this constructor option. num_measurements: This is simply the number of data values (meaning the number of measurements) in X. num_parameters: This is the number of model parameters that you wish to calculate with nonlinear least-squares. X: X_BA: This variable must be set to a numpy vector (actually a numpy matrix with just one column) whose elements constitute the observed data. The number of elements in X would equal the number of observed data elements that you are using for calculating the optimum values of the model parameters. Whereas the variable X stores the observations for any regular application of the Levenberg-Marquardt algorithm, the variable X_BA stores the observed data for the case when you want to use the bundle-adjustment variant of Levenberg-Marquardt. Whereas the generic Levenberg-Marquardt places no constraints on how the observations are arranged in the vector X, that is not the case with the bundle-adjustment variant of the same algorithm. Both X and M_BA must be a numpy vector, meaning a numpy matrix with just one column. METHODS: (1) grad_descent(): This is the implementation of the basic gradient-descent algorithm. (2) leven_marq(): This is the implementation of the Levenberg-Marquardt algorithm mentioned in the Introduction section. (3) set_debug(): This allow your own domain-specific class to set the 'debug' attribute of an instance of NonlinearLeastSquares. (4) set_display_function(): Some problem domains allow the result of a nonlinear least-squares calculation to be displayed with 2D or 3D graphics. For example, if the goal is to fit an analytical form (in the form of, say, a polynomial) to noisy height data over a flat plane and you use the nonlinear least-squares algorithm to calculate the best values for the parameters of the analytical form, you should be able to visualize the quality of your results by displaying both the original noisy data and the model you fit to the data. When such a visualization of the results is possible, you can pass the definition of the your display function through this setter function. (5) set_Fvec(): set_Fvec_BA() Each of these methods expects for its main argument a numpy matrix with a single column whose each element must be a functional form that predicts the corresponding element in the measurement vector X or X_BA. These functional forms will be functions of the model parameters. The first of the two methods is for the regular implementation of the Levenberg-Marquardt algorithm and the second for the bundle-adjustment variant of the same. (6) set_initial_params(): This method expects for its main argument a dictionary of <key,value> pairs in which the keys are the model parameters and the the corresponding values the initial values to be given to those parameters. (7) set_jacobian_functionals_array(): This method expects for its argument an Nxp matrix of functionals for the partial derivatives needed for the Jacobian matrix. N is the number of data elements in the X vector and p is the number of parameters in the model. To elaborate, if you are using nonlinear least-squares to fit an optimal surface to noisy height values over the xy-plane, your X vector will be a single-column numpy matrix and each row of this vector would correspond to one height value at some (x,y) point. The corresponding row in the argument jacobian_functionals_array contains p functionals, with each functional being a partial derivative of the model functional (with its x and y set according to where the height was recorded) with respect to the parameter corresponding to the column index. (8) set_num_measurements(): This method sets the number of data elements in the X vector in the instance of the NonlinearLeastSquares on which the method is invoked. (9) set_num_parameters(): This method sets the number of model parameters that will be used in the nonlinear least-squares optimization. (10) set_X(): set_X_BA(): These two methods set the data vector, in the form of a numpy matrix consisting of only one column, in an instance of NonlinearLeastSquares. The first of these for the regular implementation of the Levenberg-Marquardt algorithm and the second for the bundle-adjustment version of the same. This is the data to which you which you want to fit a given model and you want NonlinearLeastSquares to estimate the best values for the parameters of the model. OPTIMIZED SURFACE FITTING: This section presents a class named OptimizedSurfaceFit to illustrate how you can use the functionality of NonlinearLeastSquares in your own code. The goal of OptimizedSurfaceFit is to fit a model surface to noisy height data over the xy-plane in the xyz-coordinate frame, with the model surface being described by a polynomial function. Here are some examples of such polynomials: "a*(x-b)**2 + c*(y-d)**2 + e" "a*x**2 + c*y**2" "a*x + b*y + c" where the value returned by the polynomial for given values of the coordinate pair (x,y) is the height above the xy-plane at that point. Given the sort of a model surface shown above, the problem becomes one of optimally estimating the value for the model parameters from the noisy observed data. If vector X represents the measured data over a set of (x,y) points in the form of a vector of observations, we can now write for the cost function: d^2 = || X - Fvec ||^2 where Fvec is a vector of the predictions as dictated by the model at each of the (x,y) point. That is, each element of the Fvec vector is a prediction for the corresponding element of the measurement vector X. The quantity d^2 is the square of the vector norm of the prediction error, meaning the difference between the observations in X and the predictions in Fvec. Given X and Fvec vectors, We can call on NonlinearLeastSquares to help us find the best values for the parameters of the model surface. A typical call to OptimizedSurfaceFit's constructor looks like: surface_fitter = OptimizedSurfaceFit( gen_data_synthetically = True, datagen_functional = "7.8*(x - 0.5)**3 + 2.2*(y - 0.5)**2", data_array_size = (16,16), how_much_noise_for_synthetic_data = 0.7, model_functional = "a*(x-b)**3 + c*(y-d)**2", initial_param_values = {'a':2.0, 'b':0.4, 'c':0.8, 'd':0.4}, display_needed = True, display_size = (12,8), display_position = (500,300), debug = True, ) or, if you wish to also supply the partial derivatives of the model functional that can be used by OptimizedSurfaceFit for specifying the the Jacobian matrix to NonlinearLeastSquares, like surface_fitter = OptimizedSurfaceFit( gen_data_synthetically = True, datagen_functional = "7.8*(x - 0.5)**3 + 2.2*(y - 0.5)**2", data_array_size = (16,16), how_much_noise_for_synthetic_data = 0.5, model_functional = "a*(x-b)**3 + c*(y-d)**2", initial_param_values = {'a':2.0, 'b':0.4, 'c':0.8, 'd':0.4}, partials_for_jacobian = {'a':'(x-b)**2', 'b':'-2*a*(x-b)', 'c':'(y-d)**2', 'd':'-2*c*(y-d)'}, display_needed = True, display_size = (12,8), display_position = (500,300), ) With regard to the constructor option 'partials_for_jacobian', it is important to realize that what is passed to OptimizedSurfaceFit's constructor is not directly the Nxp Jacobian matrix (where N is the number of observations in X and p the number of parameters in the model). Instead, it is a set of partial derivatives of the model functional with respect to the parameters of the functional. However, OptimizedSurfaceFit knows how to translate into an Nxp numpy chararray of the functionals needed for the Jacobian. The constructor options for the OptimizedSurfaceFit class: data_array_size: The synthetic height that is generated by OptimizedSurfaceFit is over a unit square in the xy-plane. Both the x and the y coordinates of this square range over the interval 0.0 to 1.0. If you set this constructor option to, say, (16,16), the unit square will be sampled over a 16x16 grid. datagen_functional: When gen_data_synthetically is set to True, you must supply an algebraic expression in the form of a string that OptimizedSurfaceFit can use to generate the height data. Here is an example of what such a string looks like: "7.8*(x - 0.5)**3 + 2.2*(y - 0.5)**2" You can call any function inside the string that the Python math library knows about. debug: This flag is passed on to NonlinearLeastSquares. When set to True, it causes that class to display useful information during each iteration of the nonlinear least-squares algorithm. display_needed: Fitting optimal surfaces to height data lends itself well to 3D visualization. So if you'd like to see the the surfaces that correspond to the optimal values for the model parameters, set this constructor option to True. display_position: It is set to a tuple of two integers, with the first integer specifying the horizontal coordinate and the second the vertical coordinate of the upper left corner of the display. These two coordinate values are with respect to the upper left corner of your terminal screen. Horizontal coordinates are positive to the right and vertical coordinates positive pointing down. Setting this constructor parameter is optional. If not set, matplotlib will use its default values. display_size: It is set to a tuple of two integers, with the first integer specifying the width and the second the height of the display. Setting this constructor parameter is optional. If not set, matplotlib will use its default values. gen_data_synthetically: If set to True, OptimizedSurfaceFit can generate the measurement height data for your synthetically according to the function you specify through the constructor option 'datagen_functional'. how_much_noise_for_synthetic_data: This option controls the amount of noise that is added to the height data generated according to the datagen_functional. The best way to give a meaningful value to this construction option is to set to some fraction of the largest coefficient in the datagen_functional. initial_param_values: Through this option, you can transmit to OptimizedSurfaceFit your initial guesses for the values of the parameters in the model functional. If you cannot think of a guess, you try setting all the parameters to zero. OptimizedSurfaceFit conveys your initial values for the parameters to the NonlinearLeastSquares class. You express the initial values in the form of a dictionary, whole keys are the name of the parameters and whose values the initial values to be given to those parameters. initial_param_values_file: If the number of parameters in the problem you are addressing is large, it may be more convenient to supply the initial values for the parameters through a text file. measured_data_file: If the amount of measured data is large, it may be more convenient to feed it into the module through a text file. model_functional: This is the algebraic expression that we want to fit to the noisy height data. OptimizedSurfaceFit will call on NonlinearLeastSquares to estimate the best values for the parameters of this algebraic expression. For example, if the model_functional is "a*(x-b)**3 + c*(y-d)**2", the NonlinearLeastSquares will find the best possible values for the parameters a, b, c, and d --- best in the sense of minimizing the cost function described previously. model_functional_file: If the model functional is too long and/or too complex to be specified as an option directly in a call to the constructor, you can also place it in a text file through the model_functional_file option. optimizer: Through this constructor option, you can have an instance variable of the same name to hold a reference to an instance of NonlinearLeastSquares. partials_for_jacobian: Although the NonlinearLeastSquares class can numerically estimate the partial derivatives of the element of the Fvec vector with respect to the model parameters, with this option you can supply your own analytical forms for the partial derivatives that OptimizedSurfaceFit can convert into a Jacobian matrix before transmitting it to NonlinearLeastSquares. Here are the methods defined for OptimizedSurfaceFit: (1) construct_Fvec(): This method constructs the Fvec vector that NonlinearLeastSquares needs for comparing with the measurement vector X. Each element of Fvec is a prediction for the corresponding element of X and this prediction is a functional form involving model parameters. (2) construct_jacobian_array_in_functional_form(): This method is used only when the user supplies analytical forms for the partial derivatives of the model functional with respect to each of the model parameters. (When the user does not supply such partial derivatives, NonlinearLeastSquares estimates the Jacobian through numerical approximations.) (3) display_function() The problem addressed by OptimizedSurfaceFit lends itself well to visualization of the quality of the results returned by NonlinearLeastSquares. With the definition for this method that is provided, you can see how well the model parameters estimated by NonlinearLeastSquares fit the noisy height data. (4) gen_data(): This method generates the height data over the xy-plane according to the analytical form that is supplied to it as its main argument. We refer to this analytical form as the 'model functional'. (5) get_initial_params_from_file(): If you want to use model functions that have a large number of parameters, it might be easier to place their values in a text file and have OptimizedSurfaceFit get it from the file by using this method. (6) get_measured_data_from_text_file(): If you would like to supply the height data through a text file (rather than have the class generate it automatically for you), then this is the method to call for reading in the data from the file. The method assumes that that individual data elements are separated by whitespace characters (space, tab, newline, etc.). Since OptimizedSurfaceFit knows about the size of the array both along the x-coordinate and along the y-coordinate, it knows how to interpret the data in the text file. (7) get_model_functional_from_file(): If the model functional is too long, you can get OptimizedSurfaceFit to read it from a text file by using this option. (8) set_constructor_options_for_optimizer() The responsibility of this method is to take all of the user supplied information and reconstitute it into a form that is needed by NonlinearLeastSquares taking into account the peculiarities of your domain. ESTIMATING SCENE STRUCTURE WHEN USING A CALIBRATED CAMERA IN MOTION: This section presents a class named ProjectiveCamera to illustrate how you can use the functionality of NonlinearLeastSquares in your own code for estimating the structure of a 3D scene from the data recorded by a calibrated camera in motion. To create a simulated structure-from-camera-motion demonstration with this module, you must first create an instance the ProjectiveCamera class. A typical call to ProjectiveCamera's constructor looks like: camera = ProjectiveCamera.ProjectiveCamera( camera_type = 'projective', alpha_x = 1000.0, alpha_y = 1000.0, x0 = 300.0, y0 = 250.0, ) camera.initialize() This returns a camera whose optic axis is aligned with the world-Z axis and whose image plane is parallel to the world-XY plane. The parameters 'alpha_x' and 'alpha_y' are for the focal length of the camera in terms of the image sampling intervals along the x-axis and along the y-axis, respectively. The parameters 'x0' and 'y0' are for the coordinates of the point in the camera image plane where the optic axis penetrates the image plane with respect to the origin in the image plane (which is usually a corner of the image). world_points = camera.make_world_points_for_triangle() world_points_xformed = camera.apply_transformation_to_generic_world_points( world_points, (0,0,0), (0.0,0.0,5000.0), 1.0) which generates a triangle defined by its three vertices from a method defined for the ProjectiveCamera class and then moves the scene triangle along the optic axis of the camera (the world-Z axis) by 5000 units. After the transformation, the three vertices are at the coordinates (3000,3000,5000), (4000,3000,5000), and (4000,5000,5000). Subsequently, you must move the camera to different positions and orientations and use the camera matrix constructed by the ProjectiveCamera instance to project the world triangle into the camera images. You are going to need the following two methods defined for the ProjectiveCamera class for these camera motions: rotate_previously_initialized_camera_around_x_axis( theta_x_delta ) translate_a_previously_initialized_camera( (0.0,y_motion_delta,0.0) ) At each camera position/orientation achieved with the above two methods, you can record the pixels with the following call: pixels = camera.get_pixels_for_a_sequence_of_world_points( world_points_xformed ) Subsequently, you must make the following call: construct_X_vector( all_pixels ) where 'all_pixels' is the set of all the pixel recorded in all the positions of the camera. You would also need to create a Prediction Vector, Fvec, for the observed data whose elements are predictor functionals in terms of the scene parameters that need to be estimated. This is achieved with a call like: construct_Fvec_for_calibrated_cameras( camera_params_dict ) Now you are ready to call the following method: get_scene_structure_from_camera_motion('lm') which will invoke the Levenberg-Marquardt method on the NonlinearLeastSquares class to estimate the scene structure. The constructor options for the ProjectiveCamera class: camera_type: You can only set this constructor option to 'projective' in the current version of the module. Eventually, I intend to include 'orthographic' as another possibility for this option. alpha_x: alpha_y: These options are for the focal length in terms of the image sampling intervals used along the image x-axis and along the image y-axis, respectively. x0: y0: These options are for the coordinates of the point in the camera image plane where the optic axis penetrates the image plane with respect to the origin in the image plane (which is usually a corner of the image). camera_rotation: Using the (roll,pitch,yaw) convention you can specify the rotation for the camera in the constructor itself. However, for experimenting with structure-from-camera-motion experiments, it is easier to first construct a camera in its generic pose and to then call the rotate and translate methods on it in order to move to a different position and orientation. camera_translation: Using a triple to indicate displacements along the world-X, world-Y, and world-Z, you can specify a translation for the camera in the constructor itself. However, for experimenting with structure-from-camera-motion experiments, it is easier to first construct a camera in its generic pose and to then call the rotate and translate methods on it in order to move to a different position and orientation. Here are the methods defined for ProjectiveCamera: (1) add_new_camera_to_list_of_cameras(): You will find this utility method useful for enumerating all the different camera positions you will be using in a simulated structure-from-camera-motion experiment. (2) apply_transformation_to_generic_world_points() After you have constructed a scene object (typically just a simple shape like a triangle or a tetrahedron), you can call on this method to change its position and the pose in the world frame. The method takes FOUR arguments: (1) The scene structure in the form of a list of homogeneous coordinates for the world points on the object. (2) The first is a triple that specifies the rotation using the (roll,pitch,yaw) convention. (3) The second argument is a triple for the displacement along the world-X, world-Y, and world-Z coordinates. (4) The scale factor by which you want to expand or shrink the scene object. (3) construct_Fvec_for_calibrated_cameras(camera_params_dict) This method constructs the prediction vector Fvec vector that NonlinearLeastSquares needs for comparing with the measurement vector X. Each element of Fvec is a prediction for the corresponding element of X and this prediction is a functional form involving the structure parameters. (4) construct_parameter_vec_for_calibrated_cameras() This method constructs an ordered list of the SYMBOLIC NAMES to be used for each of the coordinates for the scene points that need to be estimated. This list looks like "['X_0', 'Y_0', 'Z_0', 'X_1', 'Y_1', 'Z_1', 'X_2' ......]" (5) construct_structure_ground_truth() This method packages the scene world points in a way that makes it convenient to output in your terminal window the estimated coordinates for the scene points, the ground-truth value for those coordinates, and the initial guesses supplied for them. (6) construct_X_vector(all_pixels) As mentioned in the Introduction, we use the notation X to represent a vector of all the observed data. For a structure-from-camera-motion problem, the observed data consists of all the pixels in all of the camera positions. This method orders the x- and the y-coordinates of all the recorded pixels in the same fashion as the order given to the scene points in world-3D. (7) display_structure() This method is used to display in the form of a Matplotlib figure the following three things simultaneously: the estimated scene structure, the actual world points used for the scene object, and the initial guesses supplied for those coordinates to the nonlinear least-squares algorithm. The three parameters for this method are named 'structure_points_estimated', 'world_points_xformed', and 'initial_values_supplied'. (8) get_all_cameras() This utility method is convenient for getting hold of all the cameras that supplied the data for solving the structure-from- camera-motion problem. We consider an instance of ProjectiveCamera at each of its positions in world-3D as a distinct camera. So if you move the camera to, say, 20 different locations, you are in effect using 20 cameras. (9) get_pixels_for_a_sequence_of_world_points() For any given camera position, this method applies the corresponding camera matrix to each world point, which must be in homogeneous coordinates, in the sequence of world points supplied to the method as its argument. (10) get_scene_structure_from_camera_motion('lm') get_scene_structure_from_camera_motion_with_bundle_adjustment() You must call the first of these two methods for estimating the scene structure for the calibrated cameras case after you have collected all the pixel data from all the different positions of the camera. Obviously, before you can call this method, you would need to construct the observation vector X from the pixel data the predictor vector Fvec from the parameters of the cameras at each of their positions. And you call the second method for doing the same for the case of uncalibrated cameras if you want to use the bundle-adjustment version of the Levenberg-Marquardt algorithm. (11) initialize() This method packs the constructor options supplied to the ProjectiveCamera constructor in the form of the camera's intrinsic parameter matrix K. If a translation and/or a rotation is specified for the camera through the constructor, those are also incorporated in the 3x4 camera matrix P put together by this method. (12) make_world_points_for_triangle() This method returns a scene object that consists of a triangle in world 3D. I have found a triangle defined by its three world points to be convenient for testing the basic logic of the algorithm for solving a structure-from-camera-motion problem. The triangle returned by this method can be subject to any orientation changing and position changing transformation. (13) make_world_points_from_tetrahedron_generic() Like the previous method, this method returns world points on a tetrahedron in world 3D that you can subsequently use for your simulated structure-from-moving-camera experiment. (14) print_camera_matrix() This utility method is convenient for displaying the 3x4 camera matrix for any or all of the positions of the camera. (15) rotate_previously_initialized_camera_around_world_X_axis() This method incrementally rotates the camera clockwise around the world-X axis by an angle 'theta' in degrees that is supplied to the method as its argument. (16) rotate_previously_initialized_camera_around_world_Y_axis() This method incrementally rotates the camera clockwise around the world-Y axis by an angle 'theta' in degrees that is supplied to the method as its argument. (17) set_constructor_options_for_optimizer( optimizer ) set_constructor_options_for_optimizer_BA( optimizer ) A ProjectiveCamera instance uses these method to pass on to NonlinearLeastSquares all the information needed by the latter (such as the observed data vector X or X_BA and the prediction vector Fvec or Fvec_BA) for constructing an optimum estimate of the scene structure. The argument 'optimizer' that this method takes is an instance of NonlinearLeastSquares. (18) set_initial_values_for_params() Every nonlinear least-squares algorithm needs a starting guess for whatever it is that is being estimated. In most cases, you would construct a random guess for the parameters and supply those values to this method in the form of a dictionary in which each key is the symbolic name of one of the parameters being estimated and the value a random guess for the parameter. (19) set_params_list( params_arranged_list ) You will use this method to pass on to the instance of ProjectiveCamera an ordered list of the parameters you want estimated with nonlinear least-squares. (20) translate_a_previously_initialized_camera() This method incrementally displaces the camera by 'translation' that is supplied to it as its argument. The argument 'translation' consists of a triple of real numbers that stand for a displacement along the world-X, along the world-Y, and along the world-Z. ESTIMATING SCENE STRUCTURE AND CAMERA PARAMETERS WHEN USING AN UNCALIBRATED CAMERA IN MOTION: The problem of scene construction becomes a lot more challenging when the camera in motion is uncalibrated. When I say uncalibrated, I mean uncalibrated with respect to its extrinsic pararameters. We assume in all cases in this document that the intrinsic parameters of the camera are known. What makes structure estimation more complicated in this case is that each new camera position adds 6 additional variables to the overall parameter space, 3 for translation and 3 for rotation as expressed by the Rodrigues parameters. Let's say you have M camera positions and N structure points. That would call for (6*M + 3*N) parameters to be estimated by the Levenberg-Marquardt algorithm because each camera has six external pose parameters associated with it and each structure point is defined by its three world coordinates. Assuming that you can see all of the structure points in all the cameras, all of the pixels recorded in all of the camera position would constitute a 2*N*M dimensional observation vector X. Therefore, your Jacobian will be of size [(2*N*M) x (6*M+3*N)]. For an example, say you have 20 camera positions and 100 structure points, your Jacobian will be of size 4000x420. Calculating a Jacobian of this size and multiplying it by its transpose could take many more than several minutes on a run of the mill machine. And having to calculate the Jacobian multiple times in the iterative framework of nonlinear least-squares estimation could test your patience as you are waiting for the results. Fortunately, this otherwise long execution time can be significantly shortened if you take advantage of the sparsity of the Jacobian when using uncalibrated cameras. As to why the Jacobian is sparse, consider the fact that each row of the Jacobian is a partial derivative of the prediction for one observation with respect to ALL the parameters. Obviously, when you take the partial derivative of the prediction function for a pixel in one specific camera with respect to the parameters of all other cameras, all those partial derivatives will be zero. The implementations of the Levenberg-Marquardt algorithm that take advantage of the sparsity of the Jacobian are commonly referred to as "Sparse Bundle Adjustment" or just "Bundle Adjustment". This logic that this module uses for exploiting the sparsity of the Jacobian is based on the paper "SBA: A Software Package for Generic Sparse Bundle Adjustment" by Manolis Lourakis and Antonis Argyros that appeared in ACM Transactions on Mathematical Software, March 2009. In honor of these two authors, I have named all of the block Jacobian submatrices that take partial derivatives of the predictors with respect to the camera parameters as the Argyros matrices. And I have named all of the block Jacobian matrices that do the same with respect to the structure variables as Lourakis matrices. In this module, the "bundle-adjustment" version of Levenberg-Marquardt algorithm can be found in the method bundle_adjust() of the NonlinearLeastSquares class. As you will notice, the observation vector that I called X in the implementation of the method leven_marq() is now called X_BA in the implementation of the method bundle_adjust(). The reason has to do with the fact that the basic Levenberg-Marquardt algorithm that you see in leven_marq() does not care how the observed data is arranged in the vector X. Typically, in my personal computer vision code for multi-camera situations, I arrange the data by the camera. That is, I group together all of the pixels recorded in one camera, followed by all of the pixels in the second camera, and so on. That is how the observed data (and, therefore, also the prediction vector Fvec) is arranged in the method leven_marq(). Unfortunately, that ordering of the data does not work for sparse bundle adjustment. To maximally exploit sparse bundle adjustment, you must order the observation vector so that your first group together the pixels for what we may refer to as the first structure point, to be followed by the pixels in all the cameras for the second structure point, and so on. Because this ordering is critical to the SBA algorithm described in the paper by Lourakis and Argyros, I have denoted the observation vector X_BA in the method bundle_adjust(). The order in which the predictor functionals are placed in the predictor vector Fvec must correspond to the order used in X_BA. So the prediction vector in bundle_adjust() is named Fvec_BA. About how to invoke the bundle-adjustment variant of Levenberg-Marquardt in your own code, note the following methods of the ProjectiveCamera class that are specially meant for that purpose: (1) construct_X_vector_for_bundle_adjustment() It is this method's job to arrange the observed data (meaning the pixels in the camera images) in the special order that is needed by the bundle-adjustment algorithm. (2) construct_parameter_vec_for_uncalibrated_cameras_with_rodrigues_rotations() When using uncalibrated cameras, the parameter vector must include three translational and three rotational parameters for each camera. These are in addition to the three (X,Y,Z) parameters for each structure point being tracked in the cameras. The job of this method is to synthesize this parameter vector. (3) construct_Fvec_for_bundle_adjustment() The ordering used for the observed data (meaning the pixels in the cameras) in the X_BA vector must also be used in the prediction vector when you use the bundle-adjustment variant of the basic Levenberg-Marquardt algorithm. This method creates the needed prediction vector. (4) get_scene_structure_from_camera_motion_with_bundle_adjustment() It is this method of the ProjectiveCamera class that invokes the bundle_adjust() method of the NonlinearLeastSquares class. THE ExamplesOptimizedSurfaceFit DIRECTORY: See the 'ExamplesOptimizedSurfaceFit' directory in the distribution for examples of how you can use the NonlinearLeastSquares class for solving optimization problems. These examples are based on the domain specific class OptimizedSurfaceFit that knows about fitting model surfaces to noisy height data over a flat plane. You will see the following four scripts in this directory: leven_marq.py grad_descent.py leven_marq_with_partial_derivatives.py grad_descent_with_partial_derivatives.py For the first two scripts, the NonlinearLeastSquares instance used will estimate the needed Jacobian matrix through appropriate numerical approximation formulas applied to the elements of the Fvec vector. On the other hand, for the third and the fourth scripts, your own domain-specific class must construct the Jacobian matrix, in the form of an array of functions. In the case of the domain-specific class OptimizedSurfaceFit that comes with this module, this Jacobian matrix is constructed from the user-supplied partial derivatives for the model functional. In order to become familiar with the NonlinearLeastSquares class, you might wish to play with the four scripts listed above by: -- Trying different functional forms for the 'datagen_functional' for different shaped surfaces. When you change the algebraic form of 'datagen_functional' for the OptimizedSurfaceFit class, make sure that you also change the algebraic form supplied for 'model_functional'. Note that nonlinear least-squares can only calculate the parameters of a model functional that best fit the noisy height data; it cannot conjure up a new mathematical form for the surface. So the basic mathematical form of the 'model_functional' must be the same as that of the 'datagen_functional'. -- Trying different degrees of noise. As mentioned elsewhere, when you supply a numerical value for the constructor option 'how_much_noise_for_synthetic_data' for the OptimizedSurfaceFit class, the number you enter should be in proportion to the largest numerical coefficient in the 'datagen' functional. Change this numerical value and see what happens to the quality of the final results. -- Try different values for the initial values of the model parameters. Since, depending on where the search for the optimum solution is started, all nonlinear least-squares methods can get trapped in a local minimum, see what happens when you change these initial values. -- Try different algebraic expressions for the 'model_functional' constructor option for the OptimizedSurfaceFit class. But note that if you change the algebraic form of this functional, you must also change the algebraic form of the 'datagen_functional' option. -- Try running the example with and without the partial derivatives that are supplied through the 'partials_for_jacobian' option for the OptimizedSurfaceFit class. THE ExamplesStructureFromCameraMotion DIRECTORY: See the 'ExamplesStructureFromCameraMotion' directory in the distribution for the following three example scripts that show how you can use the NonlinearLeastSquares module for estimating the structure of a 3D scene from the images recorded by a moving camera. Version 1.5 of this module: sfm_with_calibrated_cameras_translations_only.py sfm_with_uncalibrated_cameras_translations_only.py bundle_adjust_sfm_with_uncalibrated_cameras_translations_only.py where the string "sfm" stands for "structure from motion". The first script listed above is for estimating the scene structure with a calibrated camera in motion. As you play with this method, make sure you change the level of noise in the initial values supplied for the structure parameters to be estimated. As you will see, the method works even when the initial values for the parameters are far from their true values. Note that the ProjectiveCamera class makes it easy to specify calibrated cameras. The constructor of the class first gives you a camera for which you can specify the internal and the external parameters through the constructor options. Subsequently, you can apply translational and rotational transformations to the camera to move it to different locations in world 3D. Since the 3x4 camera matrices for all these positions of the camera are known, you end up with a set of fully calibrated cameras for experimenting with structure-from-motion simulations. The second and the third scripts listed above are for the case of uncalibrated cameras, with the former a straightforward application of the Levenberg-Marquardt algorithm and the latter a bundle-adjustment variant of the same. Logically, both these methods must return identical answers. (If you encounter a case when the two do not return the same answer, please send a bug report to me. I'd appreciate that very much.) Just to give you an idea of the speed-up you will get with bundle-adjustment, when I run the second script listed above on my laptop, it takes about 15 minutes for the number of structure points and the number of camera positions used in that script. For exactly the same number of structure points and the camera positions, the third script takes only a couple of minutes. You can only imagine the speed-up you will get with a C-based library for bundle adjustment --- such as the "sba" library mentioned in the paper by Lourakis and Argyros that I mentioned earlier in this documentation page. CAVEAT Note the bundle-adjustment variant of the Levenberg-Marquardt algorithm that you see in the bundle-adjust() method of the NonlinearLeastSquares module is meant for just educational purposes. Being pure Python, it cannot compete with highly optimized C-based code you will see in, say, the "sba" library mentioned in the article by Lourakis and Argyros that I have cited earlier in this documentation. So if your needs for nonlinear least-squares for estimating the structure and the camera parameters are primarily of the production variety, go directly to those publicly available libraries. INSTALLATION: The NonlinearLeastSquares class was packaged using setuptools. For installation, execute the following command-line in the source directory (this is the directory that contains the setup.py file after you have downloaded and uncompressed the package): sudo python setup.py install and/or sudo python3 setup.py install On Linux distributions, this will install the module file at a location that looks like /usr/local/lib/python2.7/dist-packages/ and for Python3 at a location like /usr/local/lib/python3.5/dist-packages/ If you do not have root access, you have the option of working directly off the directory in which you downloaded the software by simply placing the following statements at the top of your scripts that use the NonlinearLeastSquares class: import sys sys.path.append( "pathname_to_NonlinearLeastSquares_directory" ) To uninstall the module, simply delete the source directory, locate where the NonlinearLeastSquares module was installed with "locate NonlinearLeastSquares" and delete those files. As mentioned above, the full pathname to the installed version is likely to look like /usr/local/lib/python2.7/dist-packages/NonlinearLeastSquares* If you want to carry out a non-standard install of the NonlinearLeastSquares module, look up the on-line information on Disutils by pointing your browser to http://docs.python.org/dist/dist.html BUGS: Please notify the author if you encounter any bugs. When sending email, please place the string 'NonlinearLeastSquares' in the subject line. ABOUT THE AUTHOR: The author, Avinash Kak, recently finished a 17-year long "Objects Trilogy Project" with the publication of the book "Designing with Objects" by John-Wiley. If interested, visit his web page at Purdue to find out what this project was all about. You might like "Designing with Objects" especially if you enjoyed reading Harry Potter as a kid (or even as an adult, for that matter). For all issues related to this module, contact the author at kak@purdue.edu If you send email, please place the string "NonlinearLeastSquares" in your subject line to get past the author's spam filter. COPYRIGHT: Python Software Foundation License Copyright 2018 Avinash Kak @endofdocs Imported Modules glob itertools math numpy os re scipy sys Classes __builtin__.object NonlinearLeastSquares class NonlinearLeastSquares(__builtin__.object) Methods defined here: __init__(self, *args, **kwargs) constructor bundle_adjust(self, *args, **kwargs) grad_descent(self) leven_marq(self) leven_marq_v1_5(self) set_Fvec(self, Fvector) This method supplies the NonlinearLeastSquares class with the prediction vector whose each element is a functional form of the prediction in the observed data vector X. Note that Fvec is a column vector --- meaning a numpy matrix with just one column. set_Fvec_BA(self, Fvector_BA) You need to call this method for providing the NonlinearLeastSquares class with the prediction vector if you are going to be using the bundle-adjustment capabilities of the class. set_X(self, X) set_X_BA(self, X_BA) set_debug(self, debug) set_display_function(self, display_function) set_initial_params(self, initial_params_dict) set_jacobian_functionals_array(self, jacobian_functionals_array) This method expects for its argument an Nxp matrix of functionals for the partial derivatives needed for the Jacobian matrix. N is the number of measurements in the X vector and p is the number of parameters in the model. If you are using nonlinear least-squares to fit optimal surfaces to noisy measurements over the xy-plane, each element of the X vector would correspond to one such measurement at some (x,y) coordinates. And an element the argument jacobian_functionals_array chararray would correspond to the partial derivative of the model functional that already has incorporated the (x,y) coordinates corresponding to that row and that is a partial derivative of the model with respect to the parameter corresponding to the column. set_num_measurements(self, how_many_measurements) set_num_parameters(self, how_many_parameters) set_params_arranged_list(self, params_list) set_params_ordered_list(self, params_list) set_problem(self, prob) If you are using this module to solve structure from camera motion (sfm) problems, use this method to set 'self.problem' to 'sfm_N' where 'N' is the number of world points you are tracking. This is needed because sfm needs the specialized display function defined for the ProjectiveCamera class. Data descriptors defined here: __dict__ dictionary for instance variables (if defined) __weakref__ list of weak references to the object (if defined) Data __author__ = 'Avinash Kak (kak@purdue.edu)' __copyright__ = '(C) 2018 Avinash Kak. Python Software Foundation.' __date__ = '2018-November-9' __url__ = 'https://engineering.purdue.edu/kak/distNonlinearLeastSquares/NonlinearLeastSquares-2.0.0.html' __version__ = '2.0.0' Author Avinash Kak (kak@purdue.edu)

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		__author__ = 'Avinash Kak (kak@purdue.edu)' __copyright__ = '(C) 2018 Avinash Kak. Python Software Foundation.' __date__ = '2018-November-9' __url__ = 'https://engineering.purdue.edu/kak/distNonlinearLeastSquares/NonlinearLeastSquares-2.0.0.html' __version__ = '2.0.0'