Interpolation package

This package supplies linear and spline interpolation functions.

Linear Interpolation

lerp(src_val, dst_val, ratio) (for scalar interpolation) interpolates src_val and dst_val with ratio. This will be replaced with std::lerp(src_val, dst_val, ratio) in C++20.

lerp(base_keys, base_values, query_keys) (for vector interpolation) applies linear regression to each two continuous points whose x values arebase_keys and whose y values are base_values. Then it calculates interpolated values on y-axis for query_keys on x-axis.

Spline Interpolation

spline(base_keys, base_values, query_keys) (for vector interpolation) applies spline regression to each two continuous points whose x values arebase_keys and whose y values are base_values. Then it calculates interpolated values on y-axis for query_keys on x-axis.

Evaluation of calculation cost

We evaluated calculation cost of spline interpolation for 100 points, and adopted the best one which is tridiagonal matrix algorithm. Methods except for tridiagonal matrix algorithm exists in spline_interpolation package, which has been removed from Autoware.

Method	Calculation time
Tridiagonal Matrix Algorithm	0.007 [ms]
Preconditioned Conjugate Gradient	0.024 [ms]
Successive Over-Relaxation	0.074 [ms]

Spline Interpolation Algorithm

Assuming that the size of base_keys ($x_i$) and base_values ($y_i$) are $N + 1$, we aim to calculate spline interpolation with the following equation to interpolate between $y_i$ and $y_{i+1}$.

\[Y_i(x) = a_i (x - x_i)^3 + b_i (x - x_i)^2 + c_i (x - x_i) + d_i \ \ \ (i = 0, \dots, N-1)\]

Constraints on spline interpolation are as follows. The number of constraints is $4N$, which is equal to the number of variables of spline interpolation.

\[\begin{align} Y_i (x_i) & = y_i \ \ \ (i = 0, \dots, N-1) \\ Y_i (x_{i+1}) & = y_{i+1} \ \ \ (i = 0, \dots, N-1) \\ Y'_i (x_{i+1}) & = Y'_{i+1} (x_{i+1}) \ \ \ (i = 0, \dots, N-2) \\ Y''_i (x_{i+1}) & = Y''_{i+1} (x_{i+1}) \ \ \ (i = 0, \dots, N-2) \\ Y''_0 (x_0) & = 0 \\ Y''_{N-1} (x_N) & = 0 \end{align}\]

According to this article, spline interpolation is formulated as the following linear equation.

\[\begin{align} \begin{pmatrix} 2(h_0 + h_1) & h_1 \\ h_0 & 2 (h_1 + h_2) & h_2 & & O \\ & & & \ddots \\ O & & & & h_{N-2} & 2 (h_{N-2} + h_{N-1}) \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ v_3 \\ \vdots \\ v_{N-1} \end{pmatrix}= \begin{pmatrix} w_1 \\ w_2 \\ w_3 \\ \vdots \\ w_{N-1} \end{pmatrix} \end{align}\]

where

\[\begin{align} h_i & = x_{i+1} - x_i \ \ \ (i = 0, \dots, N-1) \\ w_i & = 6 \left(\frac{y_{i+1} - y_{i+1}}{h_i} - \frac{y_i - y_{i-1}}{h_{i-1}}\right) \ \ \ (i = 1, \dots, N-1) \end{align}\]

The coefficient matrix of this linear equation is tridiagonal matrix. Therefore, it can be solve with tridiagonal matrix algorithm, which can solve linear equations without gradient descent methods.

Solving this linear equation with tridiagonal matrix algorithm, we can calculate coefficients of spline interpolation as follows.

\[\begin{align} a_i & = \frac{v_{i+1} - v_i}{6 (x_{i+1} - x_i)} \ \ \ (i = 0, \dots, N-1) \\ b_i & = \frac{v_i}{2} \ \ \ (i = 0, \dots, N-1) \\ c_i & = \frac{y_{i+1} - y_i}{x_{i+1} - x_i} - \frac{1}{6}(x_{i+1} - x_i)(2 v_i + v_{i+1}) \ \ \ (i = 0, \dots, N-1) \\ d_i & = y_i \ \ \ (i = 0, \dots, N-1) \end{align}\]

Tridiagonal Matrix Algorithm

We solve tridiagonal linear equation according to this article where variables of linear equation are expressed as follows in the implementation.

\[\begin{align} \begin{pmatrix} b_0 & c_0 & & \\ a_0 & b_1 & c_2 & O \\ & & \ddots \\ O & & a_{N-2} & b_{N-1} \end{pmatrix} x = \begin{pmatrix} d_0 \\ d_2 \\ d_3 \\ \vdots \\ d_{N-1} \end{pmatrix} \end{align}\]