Interpolation of a Spatially Distributed Parameter from Discrete Values
Abstract Generated abstract
This paper addresses reconstruction of a continuously distributed spatial parameter inside an object from a limited set of discrete sensor readings, with emphasis on algorithms suitable for use in automatic control. The authors decompose the three-dimensional interpolation problem into planar interpolation from neighboring sensor values and one-dimensional interpolation over height, comparing several simple weighted and bilinear schemes through numerical experiments on 2400 experimental parameter values. The results indicate that a positive-weight planar interpolation formula using four nearby discrete values gives the smallest reduced error among the planar methods considered, while Newton and Gauss parabolic formulas are used for height interpolation. Combining these procedures yields a volume interpolation algorithm whose estimated maximum methodological error for the tested cylindrical object does not exceed about 5 percent.
Full Text
UDC 518.5
INTERPOLATION
OF A SPATIALLY DISTRIBUTED PARAMETER
FROM DISCRETE VALUES
V. N. BEREZA, V. F. DYADIK
(Presented by the scientific seminar of the Faculty of Physics and Technology)
At present, for many automatic-control objects there is a need to determine a spatially distributed parameter at any point of the object from the readings of a limited number of discretely installed sensors, as well as to determine the distribution of the parameter along specified directions—radius, height, length, etc.
In such cases the problem is posed as follows. For a given geometry of the object and a certain number \(N\) of discrete values of the parameter, it is required:
- To determine the arrangement of the sensors (in certain cases it may be given).
- To synthesize an algorithm for reconstructing the continuous quantity from discrete values and, in accordance with it, to compute the values of the parameter at any point of the object’s space, either in a specified set of points or along a specified direction.
- To evaluate the efficiency of the solution.
In the present work we consider the synthesis of interpolation algorithms for reconstructing a parameter that has continuous first and second derivatives and finite values at the boundaries of the object, from 200 discrete values. The use of interpolation algorithms is due to their relative simplicity and the possibility of implementation on a digital computer with small expenditures of machine time, which is important when they are included in the control loop of dynamic objects.
The efficiency of the algorithms was evaluated by means of a numerical experiment. As the evaluation criterion, the maximum reduced error of approximation of the initial ensemble of experimental data described in [1] was adopted.
In order to simplify the synthesis problem, the original problem was decomposed into three parts:
1) synthesis of an algorithm for interpolating the parameter values in the plane of the object from 20 discrete values with a given arrangement;
2) synthesis of an algorithm for interpolation over the height of the object from 5 discrete values;
3) coordination of the algorithms.
Interpolation in the plane may be performed, for example, by the formula of bilinear interpolation obtained for the case of an arbitrary arrangement of sensors (1), as well as by formulas (2) and (3) with positive weight coefficients.
\[ f_A(x,y)=\sum_{i=1}^{n} A_i f_i(x_i,y_i),\quad z=\mathrm{const}, \tag{1} \]
where \(A_1=A_1(x,y,x_1,\ldots,x_n,y_1,\ldots,y_n)\), \(n\) is the number of discrete values,
\(A_n=A_n(x,y,x_1,\ldots,x_n,y_1,\ldots,y_n)\) are coefficients computed for a specific arrangement of sensors and taking both positive and negative values.
\[ f_A(x,y)=\sum_{i=1}^{n} P_i f_i(x_i,y_i),\quad z=\mathrm{const}, \tag{2} \]
where the coefficients \(P_i\) are determined from the system [2]
\[ \begin{cases} \displaystyle \sum_{i=1}^{n} P_i(B_{ij}-B_{i1})-B_{Aj}-B_{A1},\quad (j=2,3\ldots n),\\[6pt] \displaystyle \sum_{i=1}^{n} P_i=1 \end{cases} \]
\[
B_{ij}=\overline{[f(\bar r_i)-f(\bar r_j)]}
\]
is the structural function in general form.
In the simplest case \(B_{ij}=|\bar r_i-\bar r_j|\)
\[ f_A(x,y)=\sum_{i=1}^{n} P_i f_i(x_i,y_i),\quad z=\mathrm{const}, \tag{3} \]
where
\[
P_i=\frac{L}{C|\bar r_A-\bar r_i|^p};
\]
\(B_i(\bar r_A)\) is the structural function.
\[ L=\frac{B_i(\bar r_A)}{B_i(\bar r_A)}; \]
for \(B_i(\bar r_A)=|\bar r_A-\bar r_i|\), \(C=1\), \(p=1\), \(n=4\), we obtain
\[ P_1=\frac{B_2B_3B_4}{\Delta};\quad P_2=\frac{B_1B_3B_4}{\Delta};\quad P_3=\frac{B_1B_2B_4}{\Delta}; \]
\[ P_4=\frac{B_1B_2B_3}{\Delta};\quad \Delta=[(B_1+B_2)B_3B_4]+[(B_3+B_4)B_1B_2]. \]
Table 1
| Interpolation algorithm | Reduced error | Number of discretes | Complexity—number of operations | Complexity—number of operations | Complexity—number of operations | Noise immunity coefficient | Additional error | |
|---|---|---|---|---|---|---|---|---|
| In the plane | 1 | 4.6 | 3 | 24 | 28 | — | 1.9 | 5.7 |
| In the plane | 2 | 3.3 | 3 | 52 | 40 | 6 | 0.9 | 2.4 |
| In the plane | 3 | 2.7 | 4 | 11 | 15 | 3 | 0.5 | 1.5 |
| By height | 4 and 5 | 2.2 | 3 | 11 | 9 | — | 0.7 | 2.1 |
As interpolation algorithms by height, on the basis of studies of the approximation error of experimental curves of the vertical distribution of the parameter, it is proposed to use formula 1.
Newton’s formula on even interpolation intervals (4) and Gauss’s second formula (5) on odd ones,
\[ f_A(z)=f(z_0)+(z-z_0)f(z_0,z_1)+(z-z_0)(z-z_1)f(z_0,z_1,z_2), \tag{4} \]
\[ z_0<z<z_1 \]
\[ f_A(z)=f(z_0)+(z-z_0)f(z_0,z_1)+(z-z_0)(z-z_1)f(z_{-1},z_0,z_1), \tag{5} \]
\[ z_0<z<z_1 \]
where \(f(z_0,z_1,z_2)\) and \(f(z_{-1},z_0,z_1)\) are divided differences.
The results of calculations of the interpolation error on the BESM-4 computer at control points of the object for a set of experimental data (2400 parameter values within the volume of the object) are given in Table 1. For the calculation, discrete values recorded in the neighborhood of the control point and introduced automatically during the computation were used.
The quantity \(\delta_{\mathrm{dop}}\) in Table 1 is the additional error caused by the inaccuracy of measuring the quantity \(f_i\) (3%), and characterizes the noise immunity of the algorithm.
From an analysis of the results in Table 1 it follows that, among the interpolation algorithms considered in the plane, algorithm 3 is the best.
Coordinating the algorithms of planar and height interpolation for computing the values of the parameter at various points of the object (specified, for example, by the operator or automatically for control of the technological process) leads to a combined algorithm of volume interpolation and is carried out as follows:
Fig. 1. On the concept of a combined algorithm of volume interpolation
Fig. 2. Block diagram of the volume-interpolation algorithm
- For point \(A\) with coordinates \(x_A, y_A, z_A\) (Fig. 1), the planes \(z=z_{\mathrm{I}},\ z=z_{\mathrm{II}},\ z=z_{\mathrm{III}}\) are chosen, where, from the measured values
\(f_i^{I}, f_i^{II}\) and \(f_i^{III},\ i = 1, 2 \ldots 4\), the values \(f_A^{I}, f_A^{II}\) and \(f_A^{III}\) are computed by algorithm (3).
- From \(f_A^{I}, f_A^{II}\) and \(f_A^{III}\), using formulas (4) and (5), the desired value \(f_A = f(\bar r_A)\) is calculated.
The maximum possible error in this case is
\[ \bar{\Delta}_{\max}^{V} = \bar{\Delta}_{\max}^{x,y} + \bar{\Delta}_{\max}^{z} = 2.7 + 2.2 = 4.9\%. \]
A flowchart of the spatial interpolation algorithm is given in Fig. 2.
Conclusions
-
The possibility of applying simple interpolation formulas to determine the values of a parameter continuously distributed in the volume of an object from its discrete values has been considered.
-
A decomposition of the spatial problem into parts has been carried out; their solution and matching have been performed; the final algorithm is a combination of planar and one-dimensional parabolic interpolation.
-
An estimate has been made of the methodological error of elevation, planar, and spatial interpolation for a cylindrical object with a specified parameter distribution; the maximum value of the latter for the set of control points does not exceed 5%.
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