JPH09179992A - Method and device for generating splined curve and spline-curved surface - Google Patents

Abstract

PROBLEM TO BE SOLVED: To reduce the influence of an error in inputted numeric data on a shape. SOLUTION: An oversampling means 101 oversamples the numeric data inputted through an input means 105 or the error calculated by an error calculating means 104 and a filtering means 102 performs a filtering process; and an interpolated curve generating means 103 generates a splined curve interpolating generated data and an error calculating means 104 calculates the error based upon numeric data on the generated curves. When this error exceeds a prescribed value, the error is inputted to the oversampling means 101, but when not, an output means 106 outputs a single splined curve in a generated splined curve group.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for generating spline curves and surfaces used in computer graphics or CAD (computer-aided design), and more particularly to a method for generating spline curves and surfaces from numerical data obtained by measurement. And a generating device.

[0002]

2. Description of the Related Art A conventional technique for measuring an analog signal and converting it into a digital signal or the like to convert numerical data into a spline curve or a curved surface will be described with reference to FIGS.

First, an outline will be described with reference to FIGS.

[0004] Numerical data 130 such as observation values such as temperature and pressure, and digital signals obtained by sampling analog signals.
For 1, it is often necessary to find a function that interpolates the numerical data in order to estimate the value 1302 at a position where no value is given (point not sampled) or to graph to see the distribution of the entire value. Get offended.

There is also a demand for measuring a two-dimensional or three-dimensional image of an object using a measuring instrument, converting the obtained point sequence into a curve or a curved surface, and using it for designing a product model.

As an interpolation method that satisfies this requirement most simply, there is a method of connecting two adjacent sampling data (numerical data) with a straight line as shown in FIG. However, in this method, the shape of the obtained function 1401 is not smooth. FIG. 14 shows the numerical data 1301 of FIG.
Are connected by a straight line to show a result 1401.

The input numerical data is often considered to have been sampled from a smooth signal (for example, a continuous analog signal), so the function generated by linear interpolation (1401 in FIG. 14) is the original function. (Function representing the original signal).

On the other hand, a method of interpolating using a polynomial is one method for smoothly connecting sampling data. FIG. 15 shows the numerical data 1301 of FIG.
Are connected by a polynomial. However, in this method, the degree of the polynomial increases as the number of data increases, and the generated polynomial 1501 vibrates violently, so that it is often unsuitable for practical use.

Therefore, a spline function which is a piecewise polynomial in which different polynomials are connected for each section is often used.

At the connection of the polynomial of the spline function,
It has continuity according to the degree of the polynomial used. For example, 3
When the following polynomial is used, the derivatives up to the second order have continuity at the connection part. For this reason, the condition regarding continuity is looser than the case where interpolation is performed by one polynomial, so that a smooth function 1601 with less vibration is obtained.

In order to perform interpolation using a spline function, a polynomial in each section may be determined. However, directly obtaining coefficients of the polynomial is not preferable in terms of calculation errors.

Therefore, normally, the spline function is expressed as a linear combination of B-spline functions, and simultaneous equations regarding the coefficients are solved.

Since the B-spline function has a local support, non-zero elements appear in the coefficient matrix of the simultaneous equations only near the diagonal, and the solution can be obtained stably and at high speed.

Next, a specific calculation method will be described with reference to FIGS.
This will be described with reference to FIG.

Assuming that n is a positive integer, the input numerical data is (x _i , y _i ), x _i <x _{i + 1} , i = 0,..., N (see FIG. 17). The connection of the polynomial spline function is referred to as a "knot" or "nodes", but in the following, each x _i
A method of obtaining a spline function for interpolating input numerical data, with k as a knot, will be described. For convenience of explanation, only the cubic spline function will be described.

First, the knot is expanded right and left. _{That, x -3 = x -2 = x} -1 = x 0, x n + 3 = x n + 2 = x x + 1 = x
_{Set n} . At this time, the knot vectors (x ₋₃ , x ₋₂ ,
.., X _{n + 3} ), the (m−1) -order B-spline function B _{m, j} (x) is defined by the following equations (1) and (2).

[0017]

[Equation 1]

However, when 0 appears in the denominator, the term is set to 0.

Here, a cubic spline function 1801
Since only (see FIG. 18) is handled, the B-spline function B
_{4, j} (x) is abbreviated as " _Bj (x)".

Then, the spline function F (x) becomes B-
As a linear combination of the spline function B _j (x),
It is expressed as

[0021]

[Equation 2]

The spline function to be obtained is determined by obtaining the coefficient p _i in the above equation (3).

Therefore, a simultaneous equation for the coefficient p _i is established. First, it is assumed that the value of the input data is equal to the value of the spline function at each knot (see the following equation (4)).

[0024]

(Equation 3)

While the number of unknowns p _i is n + 3,
Since the number of input data is n + 1, two conditions are insufficient with only the above equation (4). Therefore, the following equation (5), which is an end point condition that the second derivative at both ends is 0,
Add (6).

[0026]

(Equation 4)

By solving the above equations (4), (5) and (6) as simultaneous equations, p _i can be obtained.

In the case of two or more dimensions, it is possible to solve by reducing to one dimension. Since the same applies to higher dimensions, a two-dimensional case will be described.

It is assumed that the input data is arranged in a lattice. That is, x _i , i = 0,..., M and y _j , j =
It is assumed that a value z _ij is given to (x _i , y _j ) for 0,..., N (see FIG. 19).

If knots are expanded in the same manner as in the one-dimensional case in each of the x and y directions, and a cubic spline function is expressed using a B-spline function,
It is given by the following equation (7).

[0031]

(Equation 5)

Where C _j (y) is a B-spline function corresponding to y _j .

At this time, first, one-dimensional interpolation is performed for each i. That is, the function F _i (y) of the variable y
_{Is obtained} for the input data z _{ij as} in the one-dimensional case described above (see FIG. 20).

[0034]

(Equation 6)

With respect to the solution q _ij of the above equation (8), each j
Is subjected to one-dimensional interpolation. That is, the function F _j (x) of the variable x is obtained for q _ij by the one-dimensional method (see FIG. 21).

[0036]

(Equation 7)

P _ij obtained by the above equation (9) is a coefficient of the B-spline function expression of the two-dimensional spline function to be obtained.

[0038]

Problems of the above-mentioned conventional interpolation method using a spline function will be described below with reference to FIGS. 22 and 23. FIG.

In the above-described conventional interpolation method using a spline (B-spline) function, vibration of the generated function is suppressed as compared with interpolation using a polynomial.

However, even in this case, a slight change in the input data may cause a large change.

This phenomenon will be described for a case where data positions are unequal. For example, as shown in FIG. 22, if the values of the input data are all the same, the result of the interpolation is naturally a linear function 2201.

However, if the data function has a variation in a short section, the generated spline function 2301
In FIG. 23, the error is amplified, and a function 2201 (see FIG. 22) is performed in a section where data intervals are wide.
Large difference (error) when comparing the value of
Comes out.

The position of the knot of the spline function is
It is determined by the input numerical data, so that evenly spaced knots are not possible evenly when knots are desired.

If the two-dimensional data is not distributed in a grid pattern, the spline function cannot be uniquely determined.

Therefore, the present invention has been made in view of the above-mentioned problems of the prior art, and when the input data has a small change, the change of the function to be generated is suppressed to a small amount. It is another object of the present invention to provide a method and an apparatus for generating a spline curve or a curved surface, which can freely select a knot of a spline function without depending on input numerical data.

[0046]

In order to achieve the above-mentioned object, the present invention is a method for generating a spline curve from numerical data obtained by measuring an analog signal, wherein a spline curve for interpolating numerical data is used. The generation is performed by repeating the step of generating an approximate curve of the input numerical data to bring the error from the numerical data closer to 0, and in the step of generating the approximate curve, the numerical data or the error is oversampled. Then, further filtering is performed to generate new data, and a spline curve that interpolates the generated data is obtained.

As the numerical data, data represented as a scalar value for a point in a one-dimensional space can be taken.

Alternatively, the numerical data may be data represented as vector values for points in a one-dimensional space.

In the spline curve generation method of the present invention, oversampling can be performed by interpolating two adjacent points with a linear function.

Further, in the spline curve generating method of the present invention, the value of the B-spline function at the knot can be used for filtering.

The method of generating a spline curved surface of the present invention is a method of generating a spline curved surface from numerical data obtained by measuring an analog signal, and generating a spline curved surface for interpolating numerical data The step of generating an approximate curved surface for data is repeated to bring the error from the numerical data closer to 0. In the step of generating the approximate curved surface, the numerical data or the error is oversampled and further filtered. Then, new data is generated, and a spline curved surface that interpolates the generated data is obtained.

As the numerical data, data expressed as a scalar value for a point in the two-dimensional space can be taken.

Alternatively, the numerical data may be data represented as vector values for points in a two-dimensional space.

In the method for generating a spline surface according to the present invention, oversampling can be performed by interpolating values for three adjacent points with a linear function.

Further, in the method for generating a spline surface according to the present invention, the value of the B-spline function at the knot can be used for filtering.

The spline curve generating apparatus according to the present invention comprises: an input means for inputting numerical data obtained by measuring an analog signal; and an oversampling means for oversampling the numerical data input by the input means to generate new data. Filtering means for filtering the data generated by the oversampling means; interpolation curve generating means for generating a spline curve for interpolating the data filtered by the filtering means; and spline generated by the interpolation curve generating means Error calculation means for calculating an error of the curve with respect to the numerical data, and output means for converting a spline curve group generated by the interpolation curve generation means into a single spline curve and outputting the result.

The input means can input a scalar value for a point in a one-dimensional space as numerical data.

Alternatively, the input means may input a vector value for a point in a one-dimensional space as numerical data.

In the spline curve generating apparatus according to the present invention, the oversampling means can perform oversampling by interpolating values for two adjacent points with a linear function.

Further, in the spline curve generating device according to the present invention, the filtering means can use the value of the B-spline function at the knot for filtering.

An apparatus for generating a spline surface according to the present invention comprises an input means for inputting numerical data obtained by measuring an analog signal, and an oversampling means for oversampling the numerical data input by the input means to generate new data. Filtering means for filtering the data generated by the oversampling means; interpolated surface generation means for generating a spline surface for interpolating the data filtered by the filtering means; and spline generated by the interpolated surface generation means Error calculation means for calculating an error of the surface with respect to the numerical data, and output means for converting the spline surface group generated by the interpolation surface generation means into a single spline surface and outputting the same.

The input means can input a scalar value for a point in a two-dimensional space as numerical data.

Alternatively, the input means can input a vector value for a point in a two-dimensional space as numerical data.

In the spline curved surface generating apparatus according to the present invention, the oversampling means can perform oversampling by interpolating values of three adjacent points by a linear function.

Further, in the spline surface generating apparatus of the present invention, the filtering means can use the value of the B-spline function at the knot for filtering.

[0066]

The principle and operation of the present invention will be described below in detail with reference to FIGS.

First, oversampled data is generated from the input data (see FIG. 7) (see FIG. 8). In general, oversampling refers to subdividing the interval between sampling points (sampling points) of input data (sampling data) to create data at the subdivided points. Here, the sampling interval of input data is used. Is called oversampling, not only when the sampling intervals are uneven, but also when the data is created in an even layout when the sampling intervals are uneven.

There are various methods for oversampling. Here, a method of calculating a data value at a desired position after interpolating between data by a linear function will be described as an example.

The data generated by the oversampling is filtered. In the filtering process, for example, new data is obtained by obtaining a weighted average of surrounding data from original data. The data generated by the filtering process has undulations more gentle than the original data (see FIG. 9).

A value at a point corresponding to a knot of a spline function used for interpolation is selected from the filtered data (see FIG. 10). Then, a spline function is generated using the method described as the above-described conventional technique (see FIG. 11).

The undulations of the data used for the interpolation are made gentle by passing through the processing of oversampling and filtering, and the generated spline function does not show a large variation.

However, the data used when generating the spline function is data obtained by oversampling and filtering the input data, and is not the input data itself, so the original input data (sampling data) And a difference (error) in the value of the spline function at that position (sampling point) (see FIG. 12).

The difference is again subjected to oversampling and filtering.

At this time, if the processing result appears as a value at the knot, the error is interpolated to improve the accuracy of the spline function. On the other hand, if it does not appear as a value in knots, reduce the knot interval,
Generate spline functions.

According to the present invention, the data obtained by interpolation and the original input data (sampling data)
Finally, a desired interpolation for the input data is achieved by reducing the error step by step.

[0076]

Embodiments of the present invention will be described in detail below with reference to the drawings.

[0077]

Embodiment 1 A first embodiment of the present invention will be described with reference to FIG. As shown in FIG. 1, the present embodiment employs an input unit 1 for inputting a scalar value for a point in a one-dimensional space.
05, an oversampling unit 101 for oversampling the numerical data input via the input unit 105, a filtering unit 102 for filtering the data output from the oversampling unit 101, and an output from the filtering unit 102. Interpolation curve generating means 103 for generating a spline function as interpolation for the new data, and error calculating means 104 for calculating an error of the spline curve generated by the interpolation curve generating means 103 with respect to the input numerical data.
And output means 106 for converting the spline curve group generated by the interpolation curve generation means 103 into a single spline curve and outputting the single spline curve. The value of the input data obtained by the error calculation means 104 and the sample points thereof Is supplied to the oversampling means 101.
The oversampling means 101 oversamples this error and outputs it.

Next, the processing operation of this embodiment will be described with reference to the flowcharts shown in FIGS. Hereinafter, the input one-dimensional numerical data is set to (t _i , x _i ), i = 0,..., N, where n is a predetermined positive integer. This indicates that the sample value at the sampling point t _i is x _i , for example. The spline curve generated by the interpolation curve generating means 103 has knots at equal intervals, and the knot interval is d. For the sake of explanation,
Here, the initial value of d is set to 1.

The oversampling means 101 determines the value at each knot by interpolation of the input data using a linear function (step 201). That is, when k is an integer, the value a _k for the knot t = kd is _defined as t _j ≦ t <t _{j + 1}
Is determined by the following equation (10).

[0080]

(Equation 8)

[0081] For convenience of _{explanation, 0 ≦ t 0, t n} ≦ n d d (n d
Is a positive integer). Also, for a t <t ₀ t = kd and a _k = x _n for _{a k = x 0, t ≧} t n a is t = kd.

The filtering means 102 performs a digital filtering process of preferably performing a product-sum operation (convolution operation) with a filter coefficient on the value sequence {a _k } for each knot kd obtained by the oversampling means 101. (Step 202). The filter uses the value of the B-spline function for the knots. In particular,
A new value b _k is calculated by the following equation (11). In the following equation (11), coefficients 1/6, 2/3, and 1/6 by which the input data strings a _k-1 , a _k , and a _{k + 1} are multiplied indicate filter coefficients.

[0083]

[Equation 9]

Next, the magnitude of the value b _k obtained by the filtering means 102 is checked (step 203).

If the value of b _k is equal to or smaller than a predetermined value (specified value), the accuracy of approximation is not further improved in a spline curve having a knot interval of d. d / 2 (Step 20)
4).

On the other hand, if it is larger than the specified value, an interpolation spline curve is generated by the interpolation curve generation means 103 (step 205).

That is, in step 205, the interpolation curve generation means 103 generates a spline curve using the filter operation result b _k for the knot {kd}. At that time, a spline curve F a (t), the {kd} having a knot B- expressed by the following equation as a linear combination of the spline function _{B k (t) (12)} , simultaneous equations for coefficients p _k (13 ), (14) and (15) are solved.

[0088]

(Equation 10)

The coefficient {p _i } of the B-spline function corresponding to the same knot interval d is sequentially added. on the other hand,
When the knot interval d has changed (the processing of step 204)
Is stored in another array (Array).

The error calculating means 104 calculates the difference d (x) between the value F (t _i ) at each t _i of the generated spline curve F (t) and x _i.
x _i is calculated (step 206).

[0091] Then, examine the magnitude of the difference dx _i (step 207).

In the determination processing of step 207, the difference d
If x _i is preset predetermined value (predetermined value) or less, since the input data can be interpolated to output the spline curve by the procedure described below by the output unit 106 (step 209). Otherwise, the error dx _i is input to the oversampling means 101 as new data x _i (step 208).

The output means 106 outputs the interpolation curve generating means 10
, D = 1, 1/2, 1/2 ² , ...
The spline curves corresponding to 1/2 ^l, spacing knot is converted to a spline curve is a 1/2 ^l, B- added coefficients p _i spline function creates one spline curve, and outputs (step 209).

[0094]

Embodiment 2 A second embodiment of the present invention will be described below. FIG. 3 is a block diagram showing a configuration of the second exemplary embodiment of the present invention.

Referring to FIG. 3, in the present embodiment,
Input means 305 for inputting a vector value V _i for a point t _i in one-dimensional space. Oversampling means 30
1. Filtering means 302, interpolation curve generating means 30
3. The processing of each of the error calculation means 304 and the output means 306 is based on the value of each component of the vector value (that is, the scalar value). The same processing as the means 103, the error calculation means 104, and the output means 106 is performed. Therefore, the description of the processing of these means is omitted.

[0096]

Third Embodiment A third embodiment of the present invention will be described below. FIG. 4 is a block diagram showing a configuration of the third exemplary embodiment of the present invention.

Referring to FIG. 4, in the third embodiment of the present invention, input means 405 for inputting a scalar value for a point in a two-dimensional space, numerical data input by the input means 405, or error calculation. Oversampling means 401 for oversampling the error calculated by means 404 as input data, filtering means 402 for filtering the data calculated by oversampling means 401, and new data calculated by filtering means 402 are interpolated. Interpolation surface generating means 403 for generating a spline function for
There is provided an error calculation unit 404 for calculating an error between the spline function generated in step 03 and the data, and an output unit 406 for outputting data of the generated spline surface.

The operation of the fourth embodiment of the present invention will be described below in detail with reference to the flowcharts of FIGS.

Here, for convenience of explanation, it is assumed that the input numerical data is arranged in a lattice (two-dimensional matrix). That is, u _i , i = 0,..., M and v _j , j =
It is assumed that a grid point (u _i , v _j ) is given a value x _ij for 0,..., _N.

Then, the four points (u _i , v _j ), (u _{i + 1} ,
v _j ), (u _i , v _{j + 1} ), and (u _{i + 1} , v _{j + 1} ), (u _{i + 1} , v _j ) and (u _i , v _{j + It is} assumed that a diagonal line connecting ₁ ) is drawn and triangulated.

Even if they are not arranged in a lattice, the same processing can be performed by performing triangulation such as Delaunay triangulation, so that the details when they are not arranged in a lattice are omitted.

The knot interval of the spline function is d in both the u and v directions. The initial value of d is 1.

The oversampling means 401 obtains a value at each knot grid point by interpolation using a linear function of the input data (step 501). That is, when k and l are integers, knot lattice point (u, v) = (kd,
The value c _kl in ld) is a value obtained by interpolating a value at a vertex of a triangle including (u, v) with a linear function.

For this purpose, first, the vertices of a triangle including (u, v) are defined as (u _i , v _j ), (u _{i + 1} , v _j ),
When (u _i , v _{j + 1} ), the barycentric coordinates b ₀ , b ₁ , b ₂ are obtained from the following equations (16), (17), (18).

B ₀ u _i + b ₁ u _{i + 1} + b ₂ u _i = u (16) b ₀ v _j + b ₁ v _j + b ₂ v _{j + 1} = v (17) b ₀ + b ₁ + b ₂ = 1… (18)

The barycentric coordinates b ₀ , b ₁ , thus obtained,
Using b ₂ , the value c _kl is obtained as in the following equation (19).

C _kl = b ₀ x _ij + b ₁ x _{i + 1, j} + b ₂ x _{i, j + 1} (19)

For convenience of explanation, 0 ≦ u ₀ , u _m ≦ _md d, 0
_{≦ v 0, v n ≦ n} d d (m d, n d is a positive integer) is denoted by.

Outside the grid of the input data, if the u coordinate is within the range of the grid, the nearest outermost side is selected and the u coordinate is set to the value of the same point. The v coordinate is handled in the same manner.

If both the u coordinate and the v coordinate are outside the range of the grid, the value is set to the closest one of the four vertices on the outermost periphery of the grid.

The filtering means 402 calculates the grid point (k) of each knot obtained by the oversampling means 401.
A two-dimensional filter operation (sum-of-products operation of the filter coefficient and the array {c _kl }) is performed on the array of values {c _kl } for (d, ld) (step 502). The filter uses the product of the values of the B-spline function for the knot grid points. Specifically, a new value {e _kl } is calculated by the following equation (20).

[0112]

[Equation 11]

Next, the magnitude of the value e _kl obtained by the filtering means 402 is checked (step 503). if,
If the magnitude of the value e _kl is equal to or less than a predetermined value (specified value), the knot interval d is reduced to 1 / 2d because the approximation accuracy is not further improved by the spline function having the knot interval d. (Step 504). On the other hand, if it is larger than the specified value, in step 505, the interpolation surface generating unit 403 generates an interpolation spline surface.

The interpolation surface generating means 403 calculates (kd, l
Generate a spline function using e _kl for d).
Expressing the cubic spline surface F (u, v) using a B-spline function results in the following equation (21).

[0115]

(Equation 12)

However, B _i (u) and C _j (v) are B-spline functions whose knots correspond to {kd} and {ld}, respectively.

At this time, first, the interpolation in the one-dimensional case is performed for each i. That is, the function F _i (v) of the variable v
_{Is obtained} for the input data e _{ij as} in the above-described one-dimensional case.

[0118]

(Equation 13)

For the solution q _ij of the above equation (22), one-dimensional interpolation is performed for each j this time. That is, the function F _j (u) of the variable u is obtained with respect to q _ij by the one-dimensional method.

[0120]

[Equation 14]

P _ij obtained by the above equation (23) is a value to be obtained.

The coefficients {p _ij } of the B-spline curved surface corresponding to the same knot interval d are sequentially added. If d has changed, it is stored in another array.

The error calculating means 404 calculates a value F (u, v) for each (u _i , v _j ) of the generated spline surface F (u, v).
The difference dx _ij (error) between _i , v _j ) and x _ij is calculated (step 506).

Then, the value of dx _ij is checked (step 5).
07). If dx _ij is equal to or less than the specified value, the input data has been successfully interpolated, and the process proceeds to the output unit 406. If not, the error dx _ij is input to the oversampling means 401 as new data x _ij (step 508).

The output means 406 outputs the interpolation curve generation means 40
3, d = 1, 1/2, 1/2 ² ,..., 1 /
A group of spline surfaces corresponding to 2 ^l is defined as having a knot interval of 1
/ 2 ^l to a spline surface, add a coefficient p _ij of a B-spline function to create one spline surface,
Output (Step 509).

[0126]

Fourth Embodiment A fourth embodiment of the present invention will be described with reference to FIG. In the fourth embodiment of the present invention, the input means 605 inputs a vector value V _ij for a point (u _i , v _j ) in a two-dimensional space.

The processing of each of the oversampling means 601, the filtering means 602, the interpolation surface generating means 603, the error calculating means 604, and the output means 606 corresponds to the oversampling means corresponding to each component of the vector value in the third embodiment. Since the same processing as that of 401, filtering means 402, interpolation surface generation means 403, error calculation means 404, and output means 406 is performed,
Details are omitted.

[0128]

As described above, according to the method and apparatus for interpolating numerical data according to the present invention, the spline is applied to the value obtained by filtering the oversampled data with respect to the input numerical data. Due to the configuration for interpolation, there is an effect that, for a small change, the change of the function obtained as a result can be suppressed to a small value. Therefore, a smoother interpolation function can be obtained than in the conventional method.

[Brief description of the drawings]

FIG. 1 is a diagram showing a configuration of a first exemplary embodiment of the present invention.

FIG. 2 is a flowchart for explaining the operation of the embodiment shown in FIG. 1;

FIG. 3 is a diagram showing a configuration of a second exemplary embodiment of the present invention.

FIG. 4 is a diagram showing a configuration of a third exemplary embodiment of the present invention.

FIG. 5 is a flowchart for explaining the operation of the embodiment shown in FIG. 4;

FIG. 6 is a diagram showing a configuration of a fourth embodiment of the present invention.

FIG. 7 is a diagram for explaining the principle and operation of the present invention.

FIG. 8 is a diagram for explaining the principle and operation of the present invention.

FIG. 9 is a diagram for explaining the principle and operation of the present invention.

FIG. 10 is a diagram for explaining the principle and operation of the present invention.

FIG. 11 is a diagram for explaining the principle and operation of the present invention.

FIG. 12 is a diagram for explaining the principle and operation of the present invention.

FIG. 13 is a diagram for explaining a conventional technique.

FIG. 14 is a diagram for explaining a conventional technique.

FIG. 15 is a diagram for explaining a conventional technique.

FIG. 16 is a diagram for explaining a conventional technique.

FIG. 17 is a diagram for explaining a conventional technique.

FIG. 18 is a diagram for explaining a conventional technique.

FIG. 19 is a diagram for explaining a conventional technique.

FIG. 20 is a diagram for explaining a conventional technique.

FIG. 21 is a diagram for explaining a conventional technique.

FIG. 22 is a diagram for explaining a problem of the conventional technique.

FIG. 23 is a diagram for explaining a problem of the conventional technique.

[Explanation of symbols]

 101 oversampling means 102 filtering means 103 interpolation curve generation means 104 error calculation means 105 input means 106 output means 301 oversampling means 302 filtering means 303 interpolation curve generation means 304 error calculation means 305 input means 306 output means 401 oversampling means 402 filtering Means 403 Interpolation surface generation means 404 Error calculation means 405 Input means 406 Output means 601 Oversampling means 602 Filtering means 603 Interpolation surface generation means 604 Error calculation means 605 Input means 606 Output means 1301 Input numerical data 1302 Obtain corresponding value Points 1401 Piecewise linear function 1501 Polynomial 1601 Spline function 1801 Third-order B-sp In function 2201 spline function 2301 spline function

Claims (22)

[Claims] 1. A method of generating a spline curve from input numerical data, comprising the step of generating an approximate curve of the input numerical data when generating a spline curve for interpolating numerical data. By repeating, the error between the input numerical data and the approximate curve is controlled so as to approach 0, and at the time of generating the approximate curve, the input numerical data or the input numerical data is Oversampling the error from the approximate curve, generating new data by filtering the oversampled value, and obtaining a spline curve that interpolates the new data. Generation method. 2. The method according to claim 1, wherein the input numerical data is represented as a scalar value for a point in a one-dimensional space. 3. The spline curve generating method according to claim 1, wherein the input numerical data is represented as a vector value with respect to a point in a one-dimensional space. 4. The oversampling is performed by interpolating two adjacent points of the input numerical data with a predetermined linear function. Spline curve generation method. 5. The spline curve generation method according to claim 1, wherein a value of a B-spline function at the knot is used for the filtering process. 6. A method for generating a spline curved surface from input numerical data, comprising the step of generating an approximate curved surface of the input numerical data when generating a spline curved surface for interpolating numerical data. By repeating, the error between the input numerical data and the approximate curved surface is controlled so as to approach 0, and at the time of generating the approximate curved surface, the input numerical data or the input numerical data is Oversampling the error with the approximated curved surface, filtering the oversampled value to generate new data, and obtaining a spline curved surface that interpolates the new data. Generation method. 7. The spline surface generating method according to claim 6, wherein said input numerical data is represented as a scalar value for a point in a two-dimensional space. 8. A method according to claim 6, wherein said input numerical data is represented as a vector value for a point in a two-dimensional space. 9. The spline according to claim 6, wherein the oversampling is performed by interpolating three adjacent points of the input numerical data with a predetermined linear function. How to generate a surface. 10. The method for generating a spline curved surface according to claim 6, wherein a value of a B-spline function at the knot is used for a filtering process. 11. Oversampling means for oversampling input data input via input means to generate and output new data, and inputting data output from said oversampling means and applying predetermined data to said data. Filtering means for performing a filtering process, interpolation curve generating means for generating a spline curve for interpolating data output from the filtering means, and error of the spline curve generated by the interpolation curve generating means with respect to the input data An spline curve generation device comprising: an error calculation unit that calculates the spline curve group; and an output unit that converts the spline curve group generated by the interpolation curve generation unit into a single spline curve and outputs the result. 12. The spline curve generating apparatus according to claim 11, wherein said input data input via said input means comprises a scalar value for a point in a one-dimensional space. 13. The spline curve generating device according to claim 11, wherein the input data input via the input means is a vector value for a point in a one-dimensional space. 14. The apparatus according to claim 11, wherein said oversampling means performs oversampling by interpolating two adjacent points of said input data with a predetermined linear function. The described spline curve generator. 15. The spline curve generating apparatus according to claim 11, wherein said filtering means performs a filtering process using a value of a B-spline function at the knot as a filter coefficient. 16. Oversampling means for oversampling input data input via input means to generate and output new data, and inputting data output from said oversampling means and applying predetermined data to the data. Filtering means for performing a filtering process; interpolation surface generating means for generating a spline surface for interpolating data output from the filtering means; and an error of the spline curve generated by the interpolation surface generating means with respect to the input data. And an output unit that converts the group of spline surfaces generated by the interpolated surface generation unit into a single spline surface and outputs the single spline surface. 17. The spline surface generating apparatus according to claim 16, wherein said input data input via said input means comprises a scalar value for a point in a two-dimensional space. 18. The spline surface generating apparatus according to claim 16, wherein said input data input via said input means comprises vector values for points in a two-dimensional space. 19. The oversampling means for performing oversampling by interpolating three mutually adjacent points of the input data input via the input means with a predetermined linear function. Item 19. An apparatus for generating a spline curved surface according to any one of Items 16 to 18. 20. The spline surface generating apparatus according to claim 16, wherein said filtering means performs a filtering process using a value of a B-spline function at the knot as a fill coefficient. 21. A spline curve according to claim 11, wherein said input means inputs, as said input data, numerical data consisting of a digital signal obtained by sampling an analog signal. Generator. 22. The spline surface according to claim 16, wherein said input means inputs numerical data consisting of a digital signal obtained by sampling an analog signal as said input data. Generator.