Spline (Interpolation)
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The document discusses interpolation methods, focusing on spline interpolation, which utilizes piecewise polynomials for estimating unknown values. It distinguishes between linear, quadratic, and cubic spline methods, highlighting the advantages of cubic splines in providing smooth and visually appealing fits for data in applications like computer graphics and image processing. The best fitting method for specific data in this context is identified as cubic Hermite splines due to their lack of overshooting and ability to preserve data shape.
Spline (Interpolation)
- 1. SPLINE (INTERPOLATION) VIDYASAGAR UNIVERSITY REMOTE SENSING & GIS By Pallab Jana
- 2. Interpolation Interpolation is a statistical method by which related known values are used to estimate an unknown price or potential yield of a security. Interpolation is a method of estimating an unknown price or yield of a security. This is achieved by using other related known values that are located in sequence with the unknown value. Spline Interpolation In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. Spline interpolation is often preferred over polynomial interpolation because the interpolation error can be made small even when using low degree polynomials for the spline. Spline interpolation avoids the problem of Runge's phenomenon, in which oscillation can occur between points when interpolating using high degree polynomials. Originally, spline was a term for elastic rulers that were bent to pass through a number of predefined points ("knots"). These were used to make technical drawings for shipbuilding and construction by hand, as illustrated by Interpolation with cubic splines between eight points. Hand-drawn technical drawings were made for shipbuilding etc. using flexible rulers that were bent to follow pre-defined points INTRODUCTION
- 3. Linear spline models are criticized because of the abrupt change in trend going from one segment to the next, which does not represent what would naturally occur. The true change in trend should be smooth. In mathematical terms this means that the fitting function should have a continuous derivative at each value of the independent variable. This is not possible for linear splines, but it is possible for quadratic splines that are joined segments of parabolas. (Z. Huang and J. Stone, 1999) Quadratic Spline A quadratic spline interpolation method produces a better fit to a continuous function than a linear spline does. A quadratic spline is a continuously differentiable piecewise quadratic function, where quadratic includes all linear combinations of the basic monomials. Quadratic splines are not used in applications as much as natural cubic splines. However, the derivations of interpolating quadratic and cubic splines are similar enough, so that an understanding of the simpler second-degree spline theory allows us to grasp easily the more complicated third-degree spline theory. (Cheney W. and Kincaid D.) Cubic Spline Cubic splines play an important role in fields where smooth interpolation is essential, in modeling, for example, animation and image scaling. In computer graphics interpolating cubic splines are often used to define smooth motion of objects and cameras passing through user-specified positions in a key frame animation system. In image processing splines prove useful in implementing high-quality image magnification. Cubic Hermite Spline Interpolation There are many kinds of piecewise polynomial interpolation. We can try to modify a set of cubic splines with a set of other polynomials such as the cubic Hermite polynomials. These cubic Hermite polynomials allow us to provide not only continuity of the first order derivatives of the curve but also to match the set of chosen slopes. This interpolation is appropriate when both the function and the derivative values are available. In the mathematical field of numerical analysis a cubic Hermite spline, named after Charles Hermite, is a third- degree spline in Hermite form. The Hermite form has two control points and two control tangents for each polynomial. They are simple to calculate but at the same time too powerful. (Grasselli,M., Pelinovsky, D., 2008)
- 4. APPLICATION
- 6. Chemical Engineers deal with a lot of data that are crucial for modeling, process analysis, and process design. Data are often obtained or given for discrete values along a continuum, from which we often require a trend analysis or estimates at intermediate points. If we are developing a mathematical model for a certain process or physical phenomenon, we may also need to obtain some model parameters fitted to experimental data. The procedure that we need for these purposes is called curve fitting. Based on our observations for the data we have used, information from Flow Rate vs Pump Power Table, we conclude: • Linear Spline is a good way of representing spray dryer results. Because it never overshoots the data and preserves monotonicity. The drawback of Linear Spline is, it is not smooth, and it does not represent the real behavior of the function. • Quadratic Splines oscillates and overshoots the data extremely, it does not have any scientific value for our data. • Cubic Splines are visually pleasing. Even though they may oscillate for some data, they produce reasonable fit to the data we used. • For our spray dryer results, Cubic Hermite is showing the best behavior. It does not overshoot, it preserves the shape, it is smooth and monotonic. As a result, for this specific Flow Rate Data, Cubic Hermite is chosen as the best curve fitting approximation CONCLUSION
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