One 、Akima Introduce

There are various methods for interpolation of measurement data , Such as linear interpolation 、 Polynomial interpolation 、 Spline interpolation, etc . But here it is. Akima Interpolation method has unique advantages .

• Linear interpolation Just consider the influence of two nearby points ;
• Polynomial interpolation when , Low order polynomials have fewer parameters , The interpolation accuracy is very low , The use of higher-order polynomials will make the solution unstable , appear “ Runge ” The phenomenon , That is, the interpolation function agrees well with the actual data at the interpolation point , And there is a large deviation outside the interpolation point ;
• Therefore, researchers have developed on the basis of polynomials Piecewise polynomial , namely Spline function . Spline function not only maintains the characteristics of simple polynomial operation , It also avoids the disadvantage of numerical instability when the polynomial order is high , So it has been widely used . But in spline interpolation , Determine the polynomial on any cell , Consider the impact of all data points on it . This not only expands the range of error propagation , It also increases a lot of work . Sometimes, only a few data points near the interpolation point are used as control points to interpolate .

Akima Interpolation method Like cubic spline function, the effect of factor derivative value is considered , Therefore, the whole interpolation curve is smooth . Cubic spline interpolation method has minimum modulus 、 The characteristics of optimal approximation and convergence , and Aikma The curve obtained by interpolation method is more smooth than spline interpolation curve , A more natural . The common disadvantage of the two is that there will be bulges at the strong thermocline . under these circumstances , Their results can be replaced by linear interpolation or preferably three-point parabolic interpolation .Akima Another advantage of interpolation is ： In engineering applications, it is usually necessary to interpolate a series of observed data points into a smooth curve , When the calculated or measured data is large , It is impossible and unnecessary to wait until all data points are provided before interpolation , available Akima Interpolation provides data points for interpolation .

Akima Detailed mathematical derivation of interpolation method 、 Theoretical proof 、 application , If you are interested, you can refer to the original paper ࿱