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Mike Vitus
43 lines
2.1 KiB
Markdown
43 lines
2.1 KiB
Markdown
Sampled Functions
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--
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It is common to not have an analytical representation of the optimization
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problem but rather a table of values at specific inputs. This commonly occurs
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when working with images or when the functions in the problem are expensive to
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evaluate. To use this data in an optimization problem we can use interpolation
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to evaluate the function and derivatives at intermediate input values.
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There are many libraries that implement a variety of interpolation schemes, but
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it is difficult to use them in Ceres' automatic differentiation framework.
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Instead, Ceres provides the ability to interpolate one and two dimensional data.
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The one dimensional interpolation is based on the Cubic Hermite Spline. This
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interpolation method requires knowledge of the function derivatives at the
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control points, however we only know the function values. Consequently, we will
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use the data to estimate derivatives at the control points. The choice of how to
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compute the derivatives is not unique and Ceres uses the Catmull–Rom Spline
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variant which uses `0.5 * (p_{k+1} - p_{k-1})` as the derivative for control
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point `p_k.` This produces a first order differentiable interpolating
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function. The two dimensional interpolation scheme is a generalization of the
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one dimensional scheme where the interpolating function is assumed to be
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separable in the two dimensions.
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This example shows how to use interpolation schemes within the Ceres automatic
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differentiation framework. This is a one dimensional example and the objective
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function is to minimize `0.5 * f(x)^2` where `f(x) = (x - 4.5)^2`.
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It is also possible to use analytical derivatives with the provided
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interpolation schemes by using a `SizedCostFunction` and defining the
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``Evaluate` function. For this example, the evaluate function would be:
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```c++
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bool Evaluate(double const* const* parameters, double* residuals, double** jacobians) const {
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if (jacobians == NULL || jacobians[0] == NULL)
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interpolator_.Evaluate(parameters[0][0], residuals);
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else
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interpolator_.Evaluate(parameters[0][0], residuals, jacobians[0]);
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return true;
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}
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```
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