2 | | Interpolation/smoothing is an important aspect of PIV. Linear interpolation is the simplest option but it cannot smooth noise, and for 2D (or 3D) fields it does not extrapolate data beyond the convex hull of the initial data set. Global polynomial interpolation generally produce excessive smoothing, for low order polynomials, or spurious oscillations, for high order polynomials. Cubic spline is an optimum general method to interpolate a 1D data series $f_{i}$ measured at a set of abscissa $x_{i},i=1,\ldots,N.$ The spline interpolation function $f(x)$ is made of piecewise cubic polynomials matched at the measurement abscissa $x_{i}$ by the condition $f(x_{i})=f_{i} $, with the continuity of the first derivative. It can be shown that this spline function minimizes the global curvature $\int_{x_{1}}^{x_{N}}f''(x)''^''{2}dx$ . The name 'spline' refers to a flexible wooden strip used in ship design to draw a smooth curved line pinned on a set of reference points. Its equilibrium shape indeed minimizes the elastic energy. This is proportional to the integral of $f''^{2}$ if it represents a small transverse displacement from the absissa line. ^ |
| 2 | Interpolation/smoothing is an important aspect of PIV. Linear interpolation is the simplest option but it cannot smooth noise, and for 2D (or 3D) fields it does not extrapolate data beyond the convex hull of the initial data set. Global polynomial interpolation generally produce excessive smoothing, for low order polynomials, or spurious oscillations, for high order polynomials. Cubic spline is an optimum general method to interpolate a 1D data series $f_{i}$ measured at a set of abscissa $x_{i},i=1,\ldots,N.$ The spline interpolation function $f(x)$ is made of piecewise cubic polynomials matched at the measurement abscissa $x_{i}$ by the condition $f(x_{i})=f_{i} $, with the continuity of the first derivative. It can be shown that this spline function minimizes the global curvature $\int_{x_{1}}^{x_{N}}f''(x)^{2}dx$ . The name 'spline' refers to a flexible wooden strip used in ship design to draw a smooth curved line pinned on a set of reference points. Its equilibrium shape indeed minimizes the elastic energy. This is proportional to the integral of $f''^{2}$ if it represents a small transverse displacement from the absissa line. |