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<title>Modeling Non-linear Least Squares — Ceres Solver</title>
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Ceres Solver
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1.13
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Why?
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Installation
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Tutorial
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On Derivatives
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Modeling Non-linear Least Squares
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Introduction
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CostFunction
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SizedCostFunction
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AutoDiffCostFunction
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DynamicAutoDiffCostFunction
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NumericDiffCostFunction
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Numeric Differentiation & LocalParameterization
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DynamicNumericDiffCostFunction
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CostFunctionToFunctor
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DynamicCostFunctionToFunctor
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ConditionedCostFunction
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GradientChecker
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NormalPrior
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LossFunction
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Instances
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Theory
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LocalParameterization
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Instances
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AutoDiffLocalParameterization
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Problem
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rotation.h
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Cubic Interpolation
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Solving Non-linear Least Squares
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Covariance Estimation
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General Unconstrained Minimization
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FAQS, Tips & Tricks
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Users
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Contributing
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Version History
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Bibliography
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License
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Ceres Solver
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Docs »
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Modeling Non-linear Least Squares
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Modeling Non-linear Least Squares¶
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Introduction¶
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Ceres solver consists of two distinct parts. A modeling API which
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provides a rich set of tools to construct an optimization problem one
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term at a time and a solver API that controls the minimization
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algorithm. This chapter is devoted to the task of modeling
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optimization problems using Ceres. Solving Non-linear Least Squares discusses
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the various ways in which an optimization problem can be solved using
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Ceres.
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Ceres solves robustified bounds constrained non-linear least squares
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problems of the form:
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(1)\[\begin{split}\min_{\mathbf{x}} &\quad \frac{1}{2}\sum_{i}
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\rho_i\left(\left\|f_i\left(x_{i_1},
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... ,x_{i_k}\right)\right\|^2\right) \\
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\text{s.t.} &\quad l_j \le x_j \le u_j\end{split}\]
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In Ceres parlance, the expression
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\(\rho_i\left(\left\|f_i\left(x_{i_1},...,x_{i_k}\right)\right\|^2\right)\)
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is known as a residual block, where \(f_i(\cdot)\) is a
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CostFunction that depends on the parameter blocks
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\(\left\{x_{i_1},... , x_{i_k}\right\}\).
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In most optimization problems small groups of scalars occur
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together. For example the three components of a translation vector and
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the four components of the quaternion that define the pose of a
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camera. We refer to such a group of scalars as a parameter block. Of
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course a parameter block can be just a single scalar too.
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\(\rho_i\) is a LossFunction. A LossFunction is
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a scalar valued function that is used to reduce the influence of
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outliers on the solution of non-linear least squares problems.
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\(l_j\) and \(u_j\) are lower and upper bounds on the
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parameter block \(x_j\).
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As a special case, when \(\rho_i(x) = x\), i.e., the identity
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function, and \(l_j = -\infty\) and \(u_j = \infty\) we get
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the more familiar unconstrained non-linear least squares problem.
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(2)\[\frac{1}{2}\sum_{i} \left\|f_i\left(x_{i_1}, ... ,x_{i_k}\right)\right\|^2.\]
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For each term in the objective function, a CostFunction is
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responsible for computing a vector of residuals and if asked a vector
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of Jacobian matrices, i.e., given \(\left[x_{i_1}, ... ,
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x_{i_k}\right]\), compute the vector
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\(f_i\left(x_{i_1},...,x_{i_k}\right)\) and the matrices
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\[J_{ij} = \frac{\partial}{\partial
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x_{i_j}}f_i\left(x_{i_1},...,x_{i_k}\right),\quad \forall j
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\in \{1, \ldots, k\}\]
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class CostFunction¶
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public:
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virtual bool Evaluate(double const* const* parameters,
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double* residuals,
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double** jacobians) = 0;
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const vector<int32>& parameter_block_sizes();
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int num_residuals() const;
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protected:
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vector<int32>* mutable_parameter_block_sizes();
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void set_num_residuals(int num_residuals);
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};
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