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Packit	ea1746	`<title>Interfacing with Automatic Differentiation — Ceres Solver</title>`
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Packit	ea1746	`Ceres Solver`
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Packit	ea1746	`1.13`
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Packit	ea1746	`Why?`
Packit	ea1746	`Installation`
Packit	ea1746	`Tutorial`
Packit	ea1746	`On Derivatives`
Packit	ea1746	`Spivak Notation`
Packit	ea1746	`Analytic Derivatives`
Packit	ea1746	`Numeric derivatives`
Packit	ea1746	`Automatic Derivatives`
Packit	ea1746	`Interfacing with Automatic Differentiation`
Packit	ea1746	`A function that returns its value`
Packit	ea1746	`A function that returns its value and derivative`
Packit	ea1746	`A function that is defined as a table of values`
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Packit	ea1746	`Modeling Non-linear Least Squares`
Packit	ea1746	`Solving Non-linear Least Squares`
Packit	ea1746	`Covariance Estimation`
Packit	ea1746	`General Unconstrained Minimization`
Packit	ea1746	`FAQS, Tips & Tricks`
Packit	ea1746	`Users`
Packit	ea1746	`Contributing`
Packit	ea1746	`Version History`
Packit	ea1746	`Bibliography`
Packit	ea1746	`License`
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Packit	ea1746	`Ceres Solver`
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Packit	ea1746	`Docs »`
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Packit	ea1746	`On Derivatives »`
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Packit	ea1746	`Interfacing with Automatic Differentiation`
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Packit	ea1746	`Interfacing with Automatic Differentiation¶`
Packit	ea1746	`Automatic differentiation is straightforward to use in cases where an`
Packit	ea1746	`explicit expression for the cost function is available. But this is`
Packit	ea1746	`not always possible. Often one has to interface with external routines`
Packit	ea1746	`or data. In this chapter we will consider a number of different ways`
Packit	ea1746	`of doing so.`
Packit	ea1746	`To do this, we will consider the problem of finding parameters`
Packit	ea1746	`\(\theta\) and \(t\) that solve an optimization problem of the`
Packit	ea1746	`form:`
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Packit	ea1746	`\[\begin{split}\min & \quad \sum_i \left \\|y_i - f\left (\\|q_{i}\\|^2\right) q_i`
Packit	ea1746	`\right \\|^2\\`
Packit	ea1746	`\text{such that} & \quad q_i = R(\theta) x_i + t\end{split}\]`
Packit	ea1746	`Here, \(R\) is a two dimensional rotation matrix parameterized`
Packit	ea1746	`using the angle \(\theta\) and \(t\) is a two dimensional`
Packit	ea1746	`vector. \(f\) is an external distortion function.`
Packit	ea1746	`We begin by considering the case, where we have a templated function`
Packit	ea1746	`TemplatedComputeDistortion that can compute the function`
Packit	ea1746	`\(f\). Then the implementation of the corresponding residual`
Packit	ea1746	`functor is straightforward and will look as follows:`
Packit	ea1746	`template <typename T> T TemplatedComputeDistortion(const T r2) {`
Packit	ea1746	`const double k1 = 0.0082;`
Packit	ea1746	`const double k2 = 0.000023;`
Packit	ea1746	`return 1.0 + k1 * y2 + k2 * r2 * r2;`
Packit	ea1746	`}`
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Packit	ea1746	`struct Affine2DWithDistortion {`
Packit	ea1746	`Affine2DWithDistortion(const double x_in[2], const double y_in[2]) {`
Packit	ea1746	`x[0] = x_in[0];`
Packit	ea1746	`x[1] = x_in[1];`
Packit	ea1746	`y[0] = y_in[0];`
Packit	ea1746	`y[1] = y_in[1];`
Packit	ea1746	`}`
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Packit	ea1746	`template <typename T>`
Packit	ea1746	`bool operator()(const T* theta,`
Packit	ea1746	`const T* t,`
Packit	ea1746	`T* residuals) const {`
Packit	ea1746	`const T q_0 = cos(theta[0]) * x[0] - sin(theta[0]) * x[1] + t[0];`
Packit	ea1746	`const T q_1 = sin(theta[0]) * x[0] + cos(theta[0]) * x[1] + t[1];`
Packit	ea1746	`const T f = TemplatedComputeDistortion(q_0 * q_0 + q_1 * q_1);`
Packit	ea1746	`residuals[0] = y[0] - f * q_0;`
Packit	ea1746	`residuals[1] = y[1] - f * q_1;`
Packit	ea1746	`return true;`
Packit	ea1746	`}`
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Packit	ea1746	`double x[2];`
Packit	ea1746	`double y[2];`
Packit	ea1746	`};`

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Interfacing with Automatic Differentiation¶