JP2001052161A - Device and method for processing information and recording medium - Google Patents

Abstract

PROBLEM TO BE SOLVED: To remove a noise while keeping an original curve form. SOLUTION: Concerning the device and the method for processing information, a first function for expressing a curve to smoothly approximate the sequence of points is calculated on the basis of the position information of respective points forming the relevant sequence of points, the calculated first function is converted to a second function with a curve length from one end point of the curve as a parameter, the noise block of the curve is detected by analyzing the second function and prescribed noise removing processing is applied for removing a noise in the relevant noise block while preserving the original form of the curve except for the noise block in respect to the second function. Besides, a program for executing such information processing is recorded on the recording medium.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an information processing apparatus and method, and a recording medium, for example, a CAD (Computer Aid).
ed Design) It is suitable for use in an apparatus.

[0002]

2. Description of the Related Art In recent years, in the field of shape design of industrial products and the like, the surface shape of a mock-up made as a prototype is measured by a three-dimensional measuring device, and CAD data is obtained based on positional information of a plurality of measurement points obtained. Generating design data has been done.

[0003] In practice, such CAD design data is obtained by using a predetermined function based on a sequence of points formed by these measuring points (hereinafter referred to as a measuring point sequence) based on positional information of each measuring point. In this case, the curve is generated so as to generate a smooth approximation curve (hereinafter referred to as a smoothed curve).

[0004]

In the process of generating the design data, however, the sequence of measurement points may be disturbed due to a mock-up prototype error or a measurement error by a three-dimensional measuring device. A vibration (hereinafter referred to as noise) occurs in a curve obtained by smoothing the measurement point sequence using the natural spline function in the above-described smooth approximation.

[0005] Such noise can be suppressed by increasing the degree of smoothing at the time of smooth approximation of the sequence of measurement points. However, when the noise is reduced by increasing the degree of smoothing, the curve represented by the natural spline function moves away from each measurement point, and the approximation is reduced.

Further, in the above-mentioned conventional method, there is a problem that noise can be reduced as described above, but the noise cannot be completely removed.

The present invention has been made in view of the above points, and has as its object to propose an information processing apparatus and method and a recording medium capable of generating a high-quality smoothing curve.

[0008]

According to the present invention, there is provided an information processing apparatus for expressing a curve for smooth approximation of a point sequence based on position information of each point forming the point sequence. Arithmetic processing means for calculating the first function, conversion means for converting the calculated first function into a second function having a curve length from one end of the curve as a parameter, and analyzing the second function Noise section detecting means for detecting a noise section of the curve by performing a predetermined noise removal process for removing the noise in the noise section while preserving the original curve shape other than the noise section for the second function. And a noise removing unit to be provided.

As a result, this information processing apparatus can efficiently remove unnecessary noise while preserving the shape of a portion other than the noise section in the curve.

According to the present invention, in the information processing method, a first step of calculating a first function representing a curve that approximates the point sequence smoothly based on position information of each point forming the point sequence. And a second step of converting the calculated first function into a second function using a curve length from one end point of a curve represented by the first function as a parameter; A third step of detecting a noise section of the curve by analysis, and a predetermined noise removal processing for removing the noise of the noise section while preserving the original curve shape other than the noise section for the second function And a fourth step of performing the following.

As a result, according to this information processing method, unnecessary noise can be efficiently removed while preserving the shape of the portion other than the noise section in the curve.

Further, according to the present invention, a first step of calculating a first function expressing a curve that smoothly approximates the point sequence based on the position information of each point forming the point sequence on the recording medium; A second step of converting the calculated first function into a second function using a curve length from one end of the curve as a parameter, and detecting a noise section of the curve by analyzing the second function A third step to perform
And a fourth step of performing a predetermined noise removal process for removing noise in the noise section while preserving the original curve shape in a section other than the noise section. I did it.

As a result, according to the program recorded on the recording medium, unnecessary noise can be efficiently removed while preserving the shape of the portion other than the noise section in the curve.

[0014]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described below in detail with reference to the drawings.

(1) Configuration of CAD apparatus according to the present embodiment In FIG. 1, reference numeral 1 denotes a CA as a whole according to the present embodiment.
D device, CPU (Central Processing Unit)
2. ROM (Read Only Me) in which various programs are stored
mory) 3, RAM (Ra) as work memory of CPU2
ndom Access Memory) 4 and an input / output interface circuit 5 are mutually connected via a CPU bus 6, and the input / output interface circuit 5 is connected to a CRT (Cathode-Ray Tube).
7, a keyboard 8 and a mouse 9 are connected.

In this case, the CPU 2 loads the position data D1 of a plurality of measurement points representing the edge positions of the three-dimensional object provided from the three-dimensional measuring device as an external device into the RAM 4 via the input / output interface circuit 5.

Then, based on the program stored in the ROM 3, the CPU 2 performs a smooth approximation of the edge of the three-dimensional object based on the position data D1 of each measurement point taken into the RAM 4 in accordance with the smoothing curve generation processing procedure RT1 shown in FIG. The generated B-spline curve is displayed on the CRT 7.

A series of processes of the CPU 2 will be described below.

(2) Processing of CPU (2-1) Sorting processing (step SP2) When the CPU 2 takes in each position data D1 of a plurality of measurement points given from outside, the smoothing curve generation processing procedure RT1
Is started in step SP1, and the following step SP2
In (2), the measurement points are sorted in order according to the spatial flow.

That is, the position data D1 of each measurement point provided from the three-dimensional measurement device is not always input in the order of the measurement points along the flow of the edge of the three-dimensional object. Therefore, the CPU 2 sorts the measurement points in order according to the spatial flow using, for example, a method disclosed in Japanese Patent Application No. 10-140997.

(2-2) Density Equalization Processing (Step SP)
3) Subsequently, the CPU 2 executes a density equalization process for equalizing the spatial density of the measurement points for the measurement point sequence obtained by sorting the measurement points as described above. The meaning of such a density uniformization process will be described.

In general, as a method of generating a curve (smoothness) that minimizes the vibration (similarity) while minimizing the distance between each point with respect to a point sequence on the plane, an appropriate order (2m−
1) A method using the following natural spline function is widely used.

In this case, the natural spline function f (x) is
where a _i and c _i are constants,

[0024]

(Equation 1)

The following equation that satisfies

[0026]

(Equation 2)

Is defined as

In the method of generating a smoothing curve using a natural spline function, a point sequence is divided into a plurality of parts (for example, a point sequence is divided into one point), and a smoothing curve is calculated for each divided part. As a whole, a curve obtained by smoothing the point sequence as a whole is generated. In the expressions (1) and (2), x _i is the so-called x-coordinate of a node at this time. Incidentally, in the following, each point of the point sequence is defined as a node.

The last term on the right side of the equation (2) is

[0030]

(Equation 3)

Represents a function called a truncated power function defined by

Here, a sequence of points {P _i } (=
(X _i , y _i ), (0 ≦ i ≦ N−1)), the following expression that can be differentiated by the mth order as a smoothing function to be obtained:

[0033]

(Equation 4)

Considering the natural spline function given by

[0035]

(Equation 5)

In order to minimize and improve the smoothness, the following equation is used.

[0037]

(Equation 6)

Should be minimized.

In the actual calculation, the following equation using a positive weighting constant w _i (0 ≦ i ≦ N−1) set by the user according to the degree of demand for approximation and smoothness, and a positive constant g

[0040]

(Equation 7)

The minimization of the linear sum of the expressions (5) and (6) is performed. That is, in the equation (7), g represents a weight constant relating to the vibration of the curve, and w _i represents a weight constant relating to the closeness of the smoothed curve to the corresponding measurement point P _i . In the following, σ expressed by equation (7) is referred to as an objective function relating to smoothness.

Here, generally,

[0043]

(Equation 8)

It is clear that the natural spline function f (x) when satisfying the condition is an approximation function that minimizes the objective function σ.

Therefore, (N + m) coefficients a _i (0 ≦ i ≦ m) of the natural spline function f (x) given by the equation (2)
-1) and the values of c _i (0 ≦ i ≦ m−1) are formed from the first-order m equations (1) with respect to these coefficients a _i and c _i , and the N equations The desired natural spline function f (x) can be obtained by obtaining from the following equation (8) and substituting these into (2). Then, as described above, the coefficients a _i and c are obtained from the equations (1) and (8).
_{Determining i} can be performed by ordinary matrix calculation.

However, as described above, the edge of the three-dimensional object is divided into a plurality of measurement points p _i (= (x _i , y _i , z _i ), (0 ≦
There are several problems in the case of detecting as the position data of i ≦ N−1)) and generating the smoothed curve using the natural spline function based on the position data of each of the measurement points p _i .

As one of the problems, for example, FIG.
As shown in FIG. 5, there is a case where the measurement points p _i have a spatially non-uniform density. In such a case, generally, in the generation of a smoothing curve applying a natural spline function, the smoothing curve is generated so as to approach each measurement point p _i evenly. There is a problem that the point p _i is biased toward a place with a high spatial density.

Therefore, in the case of the CAD apparatus 1, the CPU 2
Makes the spatial density of the measurement points p _i in the measurement point sequence S uniform in step SP3 of the smoothing curve generation processing procedure RT1.

In practice, the CPU 2 sorts the sequence of measurement points S obtained by sorting the measurement points p _i according to the procedure RT2 shown in FIG. A suitable number n (eg, 1
0), a plurality of partial point sequences S _j (1 ≦ j) composed of measurement points p _i
≦ m) (step SP11).

Next, the CPU 2 calculates the spatial density d _j (1 ≦ j ≦ m) of the measurement point p _i for each partial point sequence S _j (step SP12).

More specifically, as shown in FIG.
Each measurement point p _i (k ≦ i ≦ k + 1) in the partial point sequence S _j
0), the minimum value of each of the x, y, and z coordinates is min X,
and min Y and min Z, x, and set the y and each max X each maximum value of the z coordinate, max Y and max Z smallest rectangular solid 10 _j containing the partial point sequence S _j as the following equation after this

[0052]

(Equation 9)

The cube of the diagonal length l _j of the rectangular parallelepiped 10 _j calculated by the above is calculated as the volume V of the occupied space of the partial point sequence S _j.
j (1 ≦ j ≦ m) and the following equation

[0054]

(Equation 10)

Determining the spatial density d _j partial point sequence S _j by calculating the [0055].

The reason that the normally conceivable volume (ie, the product of the height, width, and height of the rectangular parallelepiped 10 _j ) is not adopted as the volume V _j of the occupied space of the partial point sequence S _j is that the calculated rectangular parallelepiped 10 _This is to prevent the volume of _{j from} largely changing due to a difference in the direction of the partial point sequence S _j with respect to the xyz coordinate axis.

That is, the volume V of the occupied space of the partial sequence of points S _j
When employing the volume of the rectangular parallelepiped 10 _j as _j, for example be a partial point sequence space sequence shown in FIG. 5 S _j same parts point sequence S _j and 'as shown in FIG. 6, since the orientation with respect to the coordinate axes are different The size of the smallest rectangular parallelepiped 10 _j ′ containing the same differs from the size of the rectangular parallelepiped 10 _j shown in FIG. 5, and as a result, the volumes V _j and V _j ′ of the respective rectangular parallelepipeds 10 _j and 10 _j ′ naturally differ greatly. Occurs.

On the other hand, as is clear from FIGS. 5 and 6, the diagonal lengths l _j , l _{j of} each of the rectangular parallelepipeds 10 _j , 10 _j ′.
_j ′ has substantially the same size without being affected so much by the difference in the orientation of the partial point sequences S _j , S _j ′. Therefore the volume V _j of space occupied smallest rectangular solid 10 _j of the diagonal length l _j cubed the partial point sequence S _j containing partial point sequence S _j as described above
By doing so, fair density calculation can be performed without being affected by the direction of the partial point sequence _Sj .

[0059] Based on such a principle CPU2, for each partial point sequence S _j, and calculates the spatial density d _i of the measurement point p _i by sequentially performing operations shown in each (10).

Thereafter, the CPU 2 selects the lowest value from the spatial densities d _i of the measurement points p _i in each of the partial point sequences S _j obtained in this manner, and all the partial point sequences S _j
Unnecessary number of measurement points p _i for other partial point sequences S _j are deleted at random so that the spatial density d _i of this is matched (identified) to this value (step SP13).

Specifically, the CPU 2 calculates the minimum spatial density detected in step SP12 as d _k (1 ≦ k ≦ m),
Let n _{j be} the number of ideal measurement points p _{i in} the space occupied by S _j to make the space density of a certain partial point sequence S _j d _k .
Next formula

[0062]

[Equation 11]

Is calculated to obtain the partial point sequence S _j
Calculate the number DEL _j of the measurement points p _i to be deleted for
Thereafter, using the random number table, DEL _j measurement points p _i are selected from the measurement points p _i included in the partial point sequence S _j and are deleted.

Similarly, the CPU 2 deletes DEL _j measurement points p _i for the other partial point sequences S _j , thereby obtaining the spatial density of the measurement points p _i in the measurement point sequence S. As shown in FIG. 3 (C), the whole is made uniform.

(2-3) Weighting Optimization Process (Step SP4) On the other hand, as described above, the edge of the three-dimensional object is detected as position data of a plurality of measurement points p _i , and the position data of each of these measurement points p _i is detected. Another problem in the case of generating a smoothed curve using a natural spline function on the basis of the above is an error caused by the shape accuracy of a three-dimensional object or the measurement accuracy of a three-dimensional measuring device.

That is, in the generation of a smoothed curve using a natural spline function, as described above, a smoothed curve is generated so as to uniformly approach each measurement point p _i . For this reason, when there is a measurement point p _i having a large error, there is a problem that the generated smoothed curve is unnecessarily vibrated due to the influence of the measurement point p _i .

For example, in FIG. 7, a smoothing curve obtained through a weighting optimization process according to the present embodiment described later becomes a curve K1, whereas the conventional method measures points p _{k + 1} , p _k Due to the influence of _{k + 3} , p _{k + 7} and p _{k + 11} , the smoothed curve is generated in a state of oscillating like the curve K2.

[0068] As one method for solving such a problem, a weight constant g in the smoothness of the representative (7) of natural spline function as described above, is set larger method of relative weight constant w _i thought Can be However, according to this method, there is a problem that the generated smoothed curve departs from every measurement point p _i and the approximation is reduced.

Therefore, in this embodiment, each partial point sequence S _j that has been subjected to the above-described density uniformization processing (step SP3)
(Hereinafter, these are respectively referred to as a partial point sequence S _j ′), the degree of influence of each measurement point p _{i on} the expected vibration of the smoothed curve is predicted, and the degree of influence on the vibration is determined based on the prediction result. by the large value of the weight constant w _i of the measurement point p _i of smaller than the value of the weight constant w _i with respect to other measurement points p _i, thereby suppressing the vibration of the smoothing curve.

Here, as a method of predicting the degree of influence of each measurement point p _i on the vibration of the smoothed curve, a straight line L _j (1 ≦ j ≦ m) approximating the expected smoothed curve is partially used. A method is used in which each point sequence S _j ′ is calculated, and based on the distance from each measurement point p _i to the corresponding straight line L _j , the greater the distance between the measurement points p _{i, the} greater the influence on vibration. Can be. And such a straight line L
_j can be obtained as follows.

That is, when a point cloud is defined in a three-dimensional space, the spatial flow (correlation) of the point cloud is divided into a plurality of small spaces, and a linear multiple regression model That is, it can be represented by a plane on which the sum of the squares of the distances to each point is minimum.

In particular, when the point group is arranged so as to follow a curve in a three-dimensional space, the point group in each small space can be obtained by changing the viewpoint (phenomenon variable) as shown in FIG. Can be obtained as the intersection of the linear multiple regression model obtained.

Specifically, {(x _i , y _i , z _i )}
(1 ≦ i ≦ n) is a spatial point group, x and y are explanatory variables, z
Is a regression plane _Sz with

[0074]

(Equation 12)

Is given as

Here, the regression coefficients α and β are expressed by the following equations.

[0077]

(Equation 13)

Can be calculated as the solution of the simultaneous equations given by The regression constant K is given by

[0079]

[Equation 14]

As described above, the average values x _AV , y of x _i , y _i , z _i (1 ≦ i ≦ n) are respectively _assigned to x, y, and z of S _z.
It can be obtained by substituting _AV, the z _AV.

Similarly, for example, y and z are explained functions, x
Is a regression plane S _x with

[0082]

(Equation 15)

Is as follows. And the regression coefficients β ', γ
Can be obtained as a solution of a simultaneous equation similar to the equation (13), and the regression constant K 'can be obtained from an equation similar to the equation (14).

Here, if the point group is arranged in a curved line close to a straight line in space, it can be approximated by any of different regression planes generated by changes in different viewpoints (phenomenon functions or explanatory functions). It is thought that. That is, it is considered that the straight line L for predicting the spatial arrangement of such a point group is obtained as the intersection line S _x ∩S _z of the regression plane from different viewpoints.

Therefore, such a straight line L is expressed by the following equation.

[0086]

(Equation 16)

Can be calculated as (16)
In the equation, θ _{1 to} θ ₄ are as follows:

[0088]

[Equation 17]

Can be calculated as

Based on this principle, the CPU 2 determines in FIG. 9 in step SP4 of the smoothing curve generation processing procedure RT1.
Accordance weighting optimization procedure RT3 shown in, First, the 'one partial point sequence S _j of the' respective parts point sequence S _j, calculates (16) the straight line L given by formula (= L _j) (step SP21).

Subsequently, as shown in FIG. 9, the CPU 2 calculates the distance q _i from the straight line L (= L _j ) calculated in step SP21 for each measurement point p _i forming the partial point sequence S _j ′. calculated as above influence, respectively calculates the value of the weight constant w _i with respect to the respective measurement points p _i based on the calculation result (step SP22).

In this case, the weighting constant w _i needs to be smaller as the measurement point p _i is farther from the straight line L (= L _j ).

[0093]

(Equation 18)

As described above, the value of the weight constant w _i for each measurement point p _i is calculated.

Then, the CPU 2 performs the same processing for the other partial point sequence S _j ′ (step SP 21 and step S 21).
P22), and a sequence of measurement points S ′ (FIG. 3 (c)) composed of the sequence of measurement points S subjected to the density uniformization process.
The values of the weight constants w _i are determined to be appropriate values for all the measurement points p _i .

(2-4) Smoothing Curve Generation Processing (Step SP5) Subsequently, the CPU 2 calculates the value of the weighting constant w _i as described above, based on the position data D1 of each measurement point p _i . Using a natural spline function, a sequence of measurement points S ′
Is generated. This will be described.

In general, the natural spline function f (x) is
As described above, it is defined as Expression (2) that satisfies Expression (1). However, the natural spline function f (x) defined in this way has the following two restrictions.

That is, as a first constraint, in equation (2), an equation cannot be established unless the value of y is uniquely determined for the value of x in the natural spline function f (x) defining the curve. Therefore, y depends on how to set the coordinate axes that define the curve.
F (x) cannot be calculated for a multi-valued function that takes a plurality of values.

As a second restriction, when the point sequence is not on a plane, the smoothing curve is not necessarily on the plane.
Equation (2) cannot represent such a curve.

Therefore, in the case of the CAD apparatus 1, the 2
A smoothing curve is generated using the method disclosed in Japanese Patent Application No. 11-72612 in order to escape the two constraints. That is, this C
The AD device 1 expresses the natural spline function as a vector with the same parameters for each of the x, y, and z coordinates (hereinafter, such a natural spline function is referred to as a parametric natural spline function), and the x, y
, And each parametric natural spline function of the z coordinate is determined independently so as to minimize the respective objective function.

In this case, expressing the natural spline function as a vector with the same parameters for each of the x, y, and z coordinates means that the natural spline function is expressed as x, y, and z.
expressed using a parameter t independent of y, and z
It means to expand on a three-dimensional space including the axis.

As a result, the natural spline function defined by the above equations (1) and (2) is

[0103]

[Equation 19]

The following equation that satisfies

[0105]

(Equation 20)

It is rewritten as follows. This (19)
In the expression and the expression (20), ^t (*) means the transposed matrix or transposed vector of the matrix or vector (*), and the same applies to the following.

Determining the parametric natural spline functions f (t) of the x, y and z coordinates independently so as to minimize the respective objective functions means that the desired parametric natural spline function f (t) is Optimal value,
This means that it can be solved by minimizing each objective function of x, y and z coordinates independently. Here, each objective function of x, y and z coordinates is expressed by the following equation.

[0108]

(Equation 21)

Is σ defined by

Then, the parametric natural spline function f (t) for the point sequence {(P _i )} on the three-dimensional space is
By simply minimizing each of the objective functions σ (= ^t (σ _x , σ _y , σ _z )), it is possible to define a curve that satisfies approximation and smoothness for the point sequence {(P _i )}. Yes (Japanese Patent Application
11-72612).

The expression form of the parametric natural spline function f (t) is the same as that of a non-parametric natural spline function. Therefore, the objective function σ (= ^t
(Σ _x , σ _y , σ _z )) is the minimum of the objective function σ represented by the expression (7) of the natural spline function f (x) given by the expression (2) which is not parametric. Similarly to the case where the values are realized by the coefficients a _i and c _i obtained as the solutions of the simultaneous equations given by the equations (1) and (8),

[0112]

(Equation 22)

Can be calculated.

Accordingly, the parametric natural spline function f (t) for the x, y, and z coordinates for the point sequence {(P _i )} is obtained by combining the equations (19) and (22) and setting the coefficient groups a _i , c It can be obtained by finding the value of _i and substituting this into equation (20).

In practice, such coefficients a _i , c _i
The value of

[0116]

(Equation 23)

Since the following holds, theoretically, the inverse matrix M ^-1 of the matrix M on the left side is obtained and multiplied by the matrix Q on the right side.
It can be easily obtained.

[0118] CPU2 Based on such principle, calculated in the step SP5 of the smoothing curve generation procedure RT1, the measurement point sequence S ', the position data D1 of each measurement point p _i, each weighting optimization process (step SP4) The values of the coefficients a _i and c _i are obtained based on the weight constant w _i of each of the measured points p _i and equation (23), and are substituted into equation (20) to obtain x, y and z. parametric natural spline function f _x for the coordinate _{(t), f y (t} ), f
_z (t) is obtained.

[0119] (2-5) conversion processing (step SP6) Subsequently CPU2, each parametric natural spline function f of the obtained x, y and z-coordinate in the step SP5 (t) (= f _x (t), f _y (t), f _z
(T)) is converted into a parametric natural spline function (hereinafter, referred to as an s parametric function) using the curve length s as a parameter. The meaning will be described.

As described above, the edges of the three-dimensional object are detected as the positions of the plurality of measurement points p _i , and a smoothing curve is generated using the natural spline function based on the position data D1 of these measurement points p _i. In this case, even if the density equalization process (step SP3) and the weighting optimization process (step SP4) as described above are performed in advance, the generated smoothed curve has finely divided vibrations (hereinafter referred to as noise). )
Occurs.

As a method of removing such noise, for example, as shown in FIG. 11, the “section where the inflection point frequently occurs” in the smoothing curve K3 as shown in FIG. section referred to as a _N) and to define and locate such noise interval a _N in smooth curves K3, modify the curve shape of the noise interval a _N a smooth curve shape as shown by a one-dot chain line in FIG. 11 A method of performing a noise removal process is considered.

As a method for detecting the “section where inflection points frequently occur” in the smoothed curve, the smoothed curve K
3, a “curve section in which an inflection point sequence in which the distance to an adjacent inflection point is sufficiently short” may be found.

However, even if the parametric natural spline function f (t) defined by the equation (20) is subjected to the analysis processing (for example, the calculus processing for the parameter t), the vibration frequency and the curve length of the smoothing curve are obtained. Cannot be calculated based on the distance concept.

Thus, in the case of this embodiment,
The parametric natural spline function f (t) defined by the equation (20) is converted into an s parametric function using a curve length (distance) s from one end of a smoothed curve represented by the parametric natural spline function f (t) as a parameter. To be redefined as

By doing so, it is possible to extract the features of the smoothed curve based on the concept of distance.
For example, it is possible to easily find out, for example, a “curve section in which an inflection point sequence in which the distance to an adjacent inflection point is sufficiently short” exists.

Here, the ordinal numbers of the n nodes 0, 1,...
The (2m-1) -dimensional parametric natural spline function f (t) represented by the parameter t having n-1 as a value at the node can be defined by the equation (20) as described above.

The parametric natural spline function f (t) is expressed by the following equation using the curve length s as a parameter.

[0128]

(Equation 24)

The s parametric function f represented by
^* Can be converted to (s). Note that a ₀ and c _i in the equation (24) are the property of the equation (24) as a natural spline function (equation (1)) and n nodes (in this embodiment, the measurement point sequence S ′ Measurement points p _i )
Can be determined from passing through.

Then, this s parametric function f
^* The curve length s from the start node (t = 0) of (s) to any point t is given by

[0131]

(Equation 25)

It can be defined as a function of the parameter t as shown below.

Here, in order to actually perform such an integral calculation and to improve the calculation efficiency of the curve length s at each node,
A curve length s from the start node to an arbitrary parameter value t (i-1 <t ≦ i) in an arbitrary node section [i-1, i].
The formula for calculating (t, i-1, i) is

[0134]

(Equation 26)

Is rewritten as follows. The integrand Δs (t, i−1, i) in equation (26) represents the differential quotient of the curve length s.

Here, when calculating the curve length s at a plurality of points, it is assumed that the calculation is performed in order from the closest to the starting node.
Part of the first term on the right side of the first equation of equation (6) can be reduced in calculation because it has been obtained by the calculation of the previous node, and the second term on the right side of the first equation of the equation (26) is Since the integration is performed within the adjacent node section, the integrand Δs (t, i−1,
i) can be expanded to a polynomial of a power function without a cutting condition. Actually, Δs (t, i−1, i) is given by the following equation.

[0137]

[Equation 27]

It can be obtained as follows.

The first-order derivative D1 ⁽¹⁾ Δs (t, i) for the parameter t in Δs (t, i−1, i)
−1, i) and the second-order derivative D1 ⁽²⁾ Δs (t, i−
1, i) are given by

[0140]

[Equation 28]

[0141]

(Equation 29)

Is as follows.

Then, Δs (t, i−1, i) is calculated by using the equations (28) and (29) to obtain the interval (i−1, i−1, i).
At any point h in 1), the following equation

[0144]

[Equation 30]

A quadratic approximation function Δs ^* (t,
i-1, i, h). Note that Δs (t,
Since i−1, i) is a substantially quadratic function with respect to the parameter t, a polynomial approximation of order 3 or higher is meaningless.

The approximation function Δs ^* (t, i−1,
By using (i, h), the integration procedure (calculation of the coefficient of the cubic function) can be set in advance, and can be described by an ordinary programming language.

However, since the accuracy of the polynomial approximation decreases as the distance from the vicinity of the expansion reference point h decreases, the node section [i−1,
When calculating the curve length in [i], it is necessary to calculate and integrate an approximation function at a sufficient number of division points. That is, the curve length s (i-1, i) in the node section [i-1, i] is given by the following equation.

[0148]

(Equation 31)

Approximate calculation is possible.

Further, using this equation (31), each node p _i (= (x _i , y _i , z _i ), i = 3) from the start node of the parametric function represented by the parameter t as in equation (20)
The curve length s _i up to 0, 1, 2,...

[0151]

(Equation 32)

Can be calculated by

The resulting curve length s _{i of the} node is
S ₀ , s ₁ ,..., S _n−1 are connected to nodes (x _i , y _i , z
_i ), i = 0, 1,..., n−1, and a natural spline function that passes through all these nodes (degree of approximation = 1) is calculated by a conventional natural spline function calculation procedure. The parametric natural spline function f (t) given by the original equation (20) can be converted into an s parametric function using the curve length s as a parameter.

[0154] CPU2 based on such principles, (32) sequentially obtains the curve length s _i to each node, each made at the measurement point p _i on the basis of the equation, the column of the curve length s _i of the obtained node s _0, s _1, ......, a s _n-1 as the data string of the nodes of the curve length s, passes all of these nodes a (w _i = 1) to s parametric function f ^* (s) calculation method of conventional natural spline function Is calculated using

(2-6) Noise Removal Processing (Step SP)
7) Subsequently, the CPU 2 performs the above-described conversion processing (step SP6).
In accordance with the noise removal processing procedure RT4 shown in FIG. 12, unnecessary noise generated in the smoothing curve represented by the s parametric function f ^* (s) is removed from the s parametric function f ^* (s) obtained in. To remove noise. This will be described below along the procedure.

In the following, each s parametric function f ^* _x (s), f ^* _{y for} x, y and z coordinates
Since the processing for (s) and f ^* _z (s) is the same,
S parametric function f ^* _x (s), f ^* _y
(S) and f ^* _z (s) are represented as D0 (s).

(2-6-1) Noise Section Detection Processing (Step SP31) First, the CPU 2 analyzes the s parametric function D0 (s) obtained by the conversion processing (step SP6) to obtain “adjacent inflections”. A noise section A _N (FIG. 13A), which is a curve section in which an inflection point sequence having a sufficiently short distance from a point exists.

In this case, as a method of detecting such a noise section A _N , analysis processing using differentiation can be applied. That is, when the smoothing curve K10 as shown in FIG. 13A is represented by the s-parametric function D0 (s), the first-order derivative D1 (s) of the s-parametric function D0 (s) is represented by FIG. Curve K as shown in B)
11 is a function representing

[0159]

[Equation 33]

The derivative D2 (s) of the second order given by
When = 2, it is a function representing a broken line K12 as shown in FIG.

Here, the intersection of the broken line K12 in FIG. 13C and the horizontal axis corresponds to each inflection point in the smoothed curve K10 in FIG. 13A. Therefore, this broken line K
By sequentially detecting the coordinates of the intersection of the intersection 12 and the horizontal axis, based on the coordinates of each of these intersections, the “curve where there is an inflection point sequence whose distance to an adjacent inflection point is sufficiently short” exists. Section ”can be detected.

Based on this principle, the CPU 2 calculates the first-order derivative D1 (s) and the second-order derivative D2 (s) of the s parametric function D0 (s) obtained by the conversion process (step SP6), respectively. At the same time, the value of the parameter s at which the value of the obtained second-order derivative D2 (s) becomes “0” is sequentially detected.

Then, based on the value of the parameter s obtained in this way, the CPU 2 determines the “section in which the inflection point sequence whose distance to the adjacent inflection point is sufficiently short” exists in the noise section A Detect as _N.

(2-6-2) New Second Derivative Function Generation Processing (Step SP32) Subsequently, the CPU 2 reduces the noise of each noise section A _N while preserving the original smoothed curve K10 as much as possible. Generate a new second derivative D2a (s) for elimination. This will be described below.

First, if the noise section A _N can be detected as described above, the second-order derivative D2 (s) is set so that the number of inflection points in this noise section A _N is set to “0”. )
Starting point of the noise interval A _N in (s _st, D2
(S _st )) and a new second-order derivative D2a (s) as shown in FIG. 13D by replacing the end point (s _ed , D2 (s _ed )) with a straight line, etc. By performing a double integration process on this, a new s-parametric function D0a (s) from which noise has been removed as shown in FIG.
Can be obtained. Actually, also in the present embodiment, the noise removal processing is performed according to such a procedure as described later.

However, for example, each value D2 of the second-order derivative D2 (s) at the start point and end point of the noise section A _N
When 2 (s _st ) and D2 (s _ed ) have different codes, in order to preserve as much as possible the original curve shape for the portion other than the noise section A _N in the s parametric function D0 (s), the noise section it is necessary to present a meaningful inflection point one in a _N.

[0167] Thus the new second derivative D2a (s) policy when generating a (hereinafter, this is referred to as an improved policy), 2 for each noise interval A _N, at the start and end points, respectively Each value D2 of the derivative D2 (s) of the order
(S _st ) and D2 (s _ed )
2 (s _st ) and D2 (s _ed ) have the same sign, the number of inflection points in the noise section A _N is set to “0”, and D2
When (s _st ) and D2 (s _ed ) have different codes, it can be seen that the number of inflection points in the noise section A _N may be set to “1”.

As a method for setting the number of inflection points included in the noise section A _N to “0” or “1” as described above,
A method of replacing the noise section A _N in the derivative D2 (s) of the order with a function of some straight line is conceivable.

However, if the noise section A _N in the second-order derivative D2 (s) is replaced by an appropriate straight line function without considering the functions before and after the noise section A _N , the smoothed curve K10 has a portion other than the noise section A _N. An adverse effect occurs and the noise section A _N
There is a problem that the original curve shape cannot be preserved in portions other than.

Therefore, in the present embodiment, a new second-order derivative D2a (s) is generated by the following method in order to remove noise while preserving the original smoothing curve K10 as much as possible. To do it.

That is, the values D2 (s _st ) and D2 (s _ed ) at the start point and end point of the noise section A _N of the second derivative D2 (s) are defined as variables α and β, respectively, and the noise section A _N New second-order derivative D2a before and after
(S) is

[0172]

(Equation 34)

The function is defined as shown in FIG.

In the equation (34), ε is a distance from the outside of the noise section A _N to a region where the second-order derivative D2 (s) is not changed. In this case, to remove noise as a smooth curve from the original smoothed curve K10, a new second-order derivative D2a
Since (s) also needs to be a (smooth) continuous function that does not change abruptly, it is necessary to set ε to a degree that does not significantly increase the range of curve shape change, and at the same time, to some extent. There is.

The s parametric function D0a expressing a new noise-removed new smoothing curve K10 'by double integrating the new second-order derivative D2a (s) defined as in the equation (34). In the case of generating (s), in order to preserve the curve shape of the original smoothed curve K10 as much as possible, at s = _sed + ε and s = _sst− ε, the original s
The parametric function D0 (s) completely matches the new s-parametric function D0a (s), and the first-order derivative D1 (s) of the original s-parametric function D0 (s)
What is necessary is that the first order derivative D1a (s) of the new s-parametric function D0 (s) also completely match.

When such a condition is satisfied, D
From the last condition of the definition of 2a (s),

[0177]

(Equation 35)

The simultaneous equations are established, and the values of the variables α and β can be obtained from the simultaneous equations. Accordingly, by solving the simultaneous equations of equation (35) to obtain α and β, and substituting these into equation (34), a new second-order derivative D2a (s) can be obtained.

However, in this case, it is necessary to determine whether α and β obtained by solving the simultaneous equations of equation (35) are in accordance with the above-described improvement policy.

[0180] Such determination is based on the fact that α and β obtained by solving the simultaneous equations of equation (35) are as follows:

[0181]

[Equation 36]

Can be determined by determining whether or not the above condition is satisfied.

If α and β do not satisfy the expression (36), the range of the noise section A _N is gradually increased within a range allowable as the amount of deformation of the curve shape.
When the values of α and β satisfying the expression (36) are obtained, the values are substituted into the expression (34) to obtain a desired new 2
The derivative of the order D2a (s) can be obtained.

On the basis of this principle, the CPU 2 firstly sets a new second-order derivative D2a (s) for one noise section A _{N in} equation (34) in step SP33 of the noise removal processing procedure RT4 (FIG. 12). Define
Α and β are obtained by solving the simultaneous equations of equation (35).

Further, the CPU 2 determines that α and β are (36)
It is determined whether or not the expression is satisfied, and if a positive result is obtained, a new second-order derivative D2a (s) is determined by substituting α and β into Expression (34).

On the other hand, if a negative result is obtained, the CPU 2 slightly decreases the value s _st of the value s _st at the start point of the noise section A _N and slightly increases the value s s of the value s _ed at the end point. So that the noise section A _N is expanded.

Then, the CPU 2 obtains α and β by solving the equation (35) again.
And β satisfy the equation (36). Then, the CPU 2 repeats the same processing as long as α and β satisfying the expression (36) are obtained within a range allowable as the amount of deformation of the curved shape.

Then, the CPU 2 eventually obtains α and β satisfying the expression (36) within this range, and then obtains α and β.
Into the equation (34), thereby generating a new second-order derivative D2a (s) for the noise section A _N.
The CPU2, if not got to α and β satisfying the expression (36) within this range, stops the noise removal processing for the noise interval A _N.

Further, the CPU 2 sequentially executes the same processing for the other noise sections A _N , thereby generating a new second-order derivative D2a (s) for each noise section A _N.

(2-6-3) New Curve Generation Process (Step SP33) Subsequently, for each noise section A _N , the CPU 2 calculates the new second-order derivative D2a (s) obtained as described above. By performing double integration processing, the s parametric function D0 (s) shown in FIG. 13 (F) is obtained by removing noise from the original s parametric function D0 (s) shown in FIG. 13 (A).
0a (s) is generated.

In this case, a new second-order derivative D2a (s)
S parametric function D0 among the primitive functions of
(S) and the first derivative D1a (s) of the same new s-parametric function D0a (s) except for the noise section A _N
Is, of course, completely identical to the original first-order derivative D1 (s).

[0192]

(37)

The definition can be made as follows.

Since the new s-parametric function D0a (s) identical to the original curve except for the noise section A _N naturally completely matches the original s-parametric function D0 (s), the following equation is obtained.

[0195]

(38)

Is defined as follows.

On the basis of this principle, the CPU 2 substitutes the new first-order derivative D2a (s) defined by the equation (34) into the equation (37) in step SP34 to thereby obtain a new first-order derivative. D1a (s) is obtained and further substituted into the equation (38) to calculate the s-parametric function D0a (s).

[0198] (2-6-4) Specific examples of the noise removal processing noted below, a series of steps as described above after the detection of the noise interval A _N (step SP32 and step SP3
A specific example of the noise removal processing according to 3) will be described.

[0199] Here, in this example, in order to simplify the process, s the order of the limit of the parametric function D0 (s), provided three restriction restriction on the limits and noise interval A _N about the curve length s _i between nodes Shall be.

That is, the s parametric function D0 (s)
Is set to 3 as a limit of the order of. In this case, degree 3
Is the minimum order of the spline function that can control the curvature of the curve. By setting the order to 3, s
Second order derivative D2 of parametric function D0 (s)
(S) becomes a polygonal line K12 as shown in FIG. 13 (C), and the presence or absence of an inflection point can be determined only by the sign determination of the value of the second-order derivative D2 (s) at the node.

[0201] As limitations on the curve length s _i between nodes, it is assumed the curve length s _i between each node are equal. Further, the distance [s _n -s _n-1 ] between arbitrary nodes is set to ε.

Further, as a restriction on the noise section A _N , it is assumed that nodes are both end points. In addition, the distance ε to the curve part where the curve shape is stored is set to one node.

As a result of such a restriction, the second-order derivative values α and β at the end points of the noise section A _N [s _st , s _ed ] from which noise has been removed as described above are expressed by the following equations, respectively.

[0204]

[Equation 39]

[0205]

(Equation 40)

The following is obtained.

Here, κ ₁ to κ ₄ are represented by the following equations, respectively.

[0208]

[Equation 41]

[0209]

(Equation 42)

[0210]

[Equation 43]

[0211]

[Equation 44]

Is given by Ε is the length of the curve between equally spaced nodes.

Then, the equations (41) to (44) are replaced with the equation (39).
Α and β obtained by substituting into equations (40)
Satisfies the equation (36), a new s from which noise is removed
The parametric function D0a (s) is expressed by the following equation using α and β.

[0214]

[Equation 45]

Is as follows.

Note that ν _{1 to} ν ₁₂ are represented by the following equations, respectively.

[0219]

[Equation 46]

[0218]

[Equation 47]

[0219]

[Equation 48]

[0220]

[Equation 49]

[0221]

[Equation 50]

[0222]

(Equation 51)

[0223]

(Equation 52)

[0224]

(Equation 53)

[0225]

(Equation 54)

[0226]

[Equation 55]

[0227]

[Equation 56]

[0228]

[Equation 57]

Are represented by

It should be noted that equation (45) is effective only when α and β calculated from equations (39) and (40) satisfy equation (36). It is.
If the expression (36) is not satisfied, the noise section A _N may be extended by a distance ε on the right and left sides within an appropriate range.
By using the equations (46) to (57) while taking such incidental considerations into account, the noise-removed smoothing curve K1
0 'can be generated.

Actually, when the above-described noise removal processing is performed on the s parametric function D0 (s) expressing the curve K20 as shown in FIG. 14A on the xy plane, for example, a new s obtained is obtained. Parametric function D0a (s)
A curve K20 'as shown in FIG.

Therefore, FIGS. 14A and 14B
As is clear from the above, it was confirmed that by performing the above-described noise removal processing, a curve as an improved version in which noise was removed from the original smoothed curve could be generated.

(2-7) B-Spline Curve Generation Processing (Step SP8) Subsequently, in step SP8 of the smoothing curve generation processing procedure RT1, the CPU 2 executes the noise removal processing generated as described above. S for the, y and z coordinates
Based on the parametric function D0a (s), the coordinates of auxiliary points between each measurement point (node) p _i on the smoothed curve K10 ′ expressed by the s parametric function D0a (s) are randomly determined, and these are determined. B- passing over the auxiliary point
Calculate the spline curve.

Then, the CPU 2 calculates such a B-spline curve for each measurement point p _i ,
After displaying each of the obtained B-spline curves on the CRT 7, the process proceeds to step SP9, where the smoothed curve generation processing procedure RT1 is completed.

(3) Operation and Effect of the Embodiment In the above configuration, the CPU 2 of the CAD device 1
Based from an external device 3 dimensional measuring device in the position data D1 of each measurement point p _i representing the edges of a three-dimensional object supplied, firstly to sort each measurement point p _i along the spatial stream (step SP2).

Subsequently, the CPU 2 equalizes the spatial density of the measurement points p _{i in} the obtained measurement point sequence S (step S).
Together with P3), the value of the weighting constant w _i corresponding to the position of each measurement point p _i of the measurement point sequence S ′ thus obtained is calculated (step SP4), and thereafter the position of each measurement point p _i is calculated. data, based on the value or the like of the weight constant w _i,
A parametric natural spline function f (t) is calculated using t as a parameter that smoothly approximates the measurement point sequence S ′ (step SP5).

Next, the CPU 2 converts the parametric natural spline function f (t) into an s parametric function f ^* (s) using the curve length s of the smoothed curve as a parameter (step SP6).

Further, the CPU 2 thereafter performs a noise removal process for removing a noise component from the s parametric function f ^* (s) (step SP7), and performs B-B based on the obtained new s parametric function. A spline function is calculated, and a curve based on the B-spline function is displayed on the CRT 7 as a curve obtained by smoothing the edge measured by the three-dimensional measuring device (step SP
8).

In the CAD apparatus 1, as the noise removal processing, the s parametric function D0 (s) is analyzed to detect a noise section A _N, and the second derivative of the s parametric function f ^* (s) is obtained. The noise section A _N in D2 (s) is replaced with a function of a straight line that satisfies a predetermined condition for preserving the original curve shape other than the noise section,
The new second-order derivative D2a (s) thus obtained is expressed as 2
Since the multiple integration is performed, unnecessary noise can be effectively removed while preserving the curve shape of the smoothed curve K10 represented by the original s parametric function D0 (s).

Further, in this CAD apparatus, since the parametric function (s parametric function D0 (s)) is used as a function for detecting the noise section A _N based on the concept of distance, the direction of the coordinates is limited. The analysis processing for detecting the noise section A _N as described above can be performed without any need.

According to the above arrangement, the original smoothing curve K1
The s parametric function f ^* (s) expressing 0 is analyzed to detect the noise section A _N of the smoothed curve K10, and the second-order derivative D2 (s) of the s parametric function f ^* (s) is detected. the noise interval a _N in), the noise interval a _N in the noise upstairs new by replacing the linear function that satisfies the predetermined condition for removing the while preserving as much as possible the original curved shape Derivative function D2a
(S) is generated, and this is double-integrated to generate a new noise-removed smooth approximation curve K10 ', so that the shape of the smoothed curve K10 other than the noise section A _N is changed. Unnecessary noise can be efficiently removed while storing, and a CAD apparatus that can generate a high-quality smoothing curve can be realized.

(4) Other Embodiments In the above-described embodiment, a case has been described in which the present invention is applied to the CAD apparatus 1. However, the present invention is not limited to this. The present invention can be widely applied to various other information processing apparatuses configured to generate a smoothed curve of a point sequence based on position information of each point forming the sequence. In this case, the position information of each point in the point sequence may be provided from an external device other than the three-dimensional measuring device.

In the above-described embodiment, the parametric natural spline function f representing the smoothed curve of the measurement point sequence S based on the position data D1 of each measurement point p _i.
Arithmetic processing means for calculating (t) and conversion means for converting the parametric natural spline function f (t) into an s parametric function f ^* (s) having a curve length s from one end of the smoothed curve as a parameter A noise section detecting means for analyzing the s parametric function f ^* (s) and detecting a noise section A _N in the smoothed curve; and an element other than the noise section A _N for the s parametric function f ^* (s). And the noise removing means for performing the noise removing process for removing the noise in the noise section A _N because the curve shape does not depend on the single CPU 2 has been described. However, the present invention is not limited to this. These may be constituted by two or more CPUs.

Further, in the above embodiment, the recording medium for recording the program according to the present invention is RO
Although the case where the M3 is applied has been described, the present invention is not limited to this, and various other recording media such as a disk-shaped recording medium such as a hard disk and a CD-ROM, and a tape-shaped recording medium such as a magnetic tape. Can be widely applied.

Further, in the above embodiment, the case where the sequence of measurement points S is in a three-dimensional space has been described. However, the present invention is not limited to this. Can be applied. In this case, the parametric natural spline function is calculated as a parametric natural spline function for each coordinate defining the "plane" instead of the "space", and is converted into a parametric function having the curve length s as a parameter. Good.

Further, in the above embodiment, the second-order derivative D2 (s) of the s-parametric function f ^* (s)
Has been described in which the noise section A _N is replaced with a function of a straight line that satisfies the predetermined condition. However, the present invention is not limited to this, and the predetermined condition is satisfied according to the s parametric function f ^* (s). The function may be replaced with a curve function.

[0247]

As described above, according to the present invention, in the information processing apparatus and method, based on the position information of each point forming a point sequence, a first curve representing a smooth approximation of the point sequence is represented. Calculating a function, converting the calculated first function into a second function using the curve length from one end of the curve as a parameter,
A predetermined noise removal process for detecting a noise section of a curve by analyzing the second function, and removing the noise of the noise section for the second function while keeping the original curve shape other than the noise section The information processing apparatus and method can effectively remove unnecessary noise while preserving the shape of the portion other than the noise section in the curve, and can generate a high-quality smoothed curve. realizable.

In the recording medium, based on the position information of each point forming the point sequence, a first step of calculating a first function expressing a curve that approximates the point sequence smoothly, A second step of converting the function of the curve into a second function using a curve length from one end point of the curve as a parameter, a third step of detecting a noise section of the curve by analyzing the second function, A fourth step of performing predetermined noise removal processing for removing noise in the noise section while preserving the original curve shape other than the noise section for the second function. By recording, according to this program, unnecessary noise can be effectively removed while preserving the original curve shape of the portion other than the noise section in the curve, and The recording medium which can be adapted to produce a smoothed curve of high quality can be realized.

[Brief description of the drawings]

FIG. 1 is a block diagram illustrating a configuration of an image processing apparatus according to an embodiment.

FIG. 2 is a flowchart illustrating a smoothing curve generation processing procedure;

FIG. 3 is a schematic diagram used for describing a density equalization process.

FIG. 4 is a flowchart illustrating a procedure of a density equalization process.

FIG. 5 is a schematic diagram used for describing a density equalization process.

FIG. 6 is a schematic diagram used for explaining a density uniformization process.

FIG. 7 is a schematic diagram for explaining a weighting optimization process;

FIG. 8 is a conceptual diagram for explaining a method of generating a predicted straight line from a multiple regression model.

FIG. 9 is a flowchart illustrating a weighting optimization process procedure;

FIG. 10 is a schematic diagram for explaining a weighting optimization process;

FIG. 11 is a schematic diagram used for describing noise removal processing.

FIG. 12 is a flowchart illustrating a noise removal processing procedure.

FIG. 13 is a graph showing shapes of curves and polygonal lines represented by functions and derivatives.

FIG. 14 is a schematic diagram illustrating a specific example of noise removal.

[Explanation of symbols]

1 CAD apparatus, 2 CPU, 3 ROM, 10
_j: rectangular parallelepiped, A _N: noise section, D1: position data, d _j: spatial density, K10: smoothed curve, g:
Weight constant, p _i ... Measurement point, S... Measurement point sequence, S _j.
Partial point sequence, s, s _i ... Curve length, w _i.
T1... Smoothing curve generation processing procedure, RT2... Density uniformization processing procedure, RT3... Weighting optimization processing procedure, RT
4. Noise removal processing procedure.

Claims (12)

[Claims] 1. An arithmetic processing means for calculating a first function representing a curve which approximates a point sequence smoothly based on position information of each point forming the point sequence, and The first function is
Converting means for converting a curve length from one end point of the curve represented by the first function into a second function having a parameter as a parameter; analyzing the second function, generating noise in the curve; A noise section detecting means for detecting a noise section which is a section where the noise is present, and a predetermined noise for removing the noise in the noise section while preserving the original curve shape other than the noise section with respect to the second function. An information processing apparatus comprising: a noise removing unit that performs a removing process. 2. The calculation means calculates a parametric function in which a parameter other than the curve length is the same parameter for each coordinate defining a plane or a space, as the first function. The information processing apparatus according to claim 1, wherein the processing unit converts the first function into the second function using the curve length as a parameter. 3. The noise section detecting means calculates a second derivative of the second function, and detects the noise section based on the calculated second derivative. Item 2. The information processing device according to item 1. 4. The noise elimination means includes: as the noise elimination processing, storing the noise interval in the second derivative of the second function and the shape of the original curve other than the noise interval. Generating a new second-order derivative by substituting a third function that represents a curve or a straight line that satisfies the predetermined condition of, and performing a double integration process on the new second-order derivative. The information processing apparatus according to claim 1. 5. A first step of calculating a first function expressing a curve that approximates the point sequence smoothly based on position information of each point forming the point sequence; A second step of converting a function into a second function using a curve length from one end of the curve represented by the first function as a parameter, analyzing the second function, A third step of detecting a noise section that is a section where noise is occurring; and, for the second function, the noise of the noise section while preserving the shape of the original curve other than the noise section. And a fourth step of performing a predetermined noise removal process for removing the noise. 6. In the first step, a parametric function is calculated as the first function, wherein a parameter other than the curve length is the same parameter for each coordinate defining a plane or space. The information processing method according to claim 5, wherein in the step (b), the first function is converted into the second function. 7. In the third step, a second derivative of the second function is calculated, and the noise section is detected based on the calculated second derivative. Item 6. The information processing method according to Item 5. 8. In the fourth step, as the noise removal processing, the noise section in the second derivative of the second function is stored, and the shape of the original curve other than the noise section is stored. Generating a new second-order derivative by substituting a third function representing a curve or a straight line that satisfies a predetermined condition for performing the second integration with the new second-order derivative. The information processing method according to claim 5, wherein 9. A first step of calculating a first function representing a curve that approximates the point sequence smoothly based on the position information of each point forming the point sequence; A second step of converting a function into a second function using a curve length from one end of the curve represented by the first function as a parameter, analyzing the second function, A third step of detecting a noise section that is a section where noise is occurring; and, for the second function, the noise of the noise section while preserving the shape of the original curve other than the noise section. A program for executing information processing including a fourth step of performing a predetermined noise removal process for removing noise. 10. In the first step, a parametric function in which parameters other than the curve length are the same for each coordinate defining a plane or space is calculated as the first function. 10. The recording medium according to claim 9, wherein in the step (c), the first function is converted to the second function. 11. The third step of calculating a second derivative of the second function, and detecting the noise interval based on the calculated second derivative. Item 10. The recording medium according to Item 9. 12. In the fourth step, as the noise removal processing, the noise section in the second derivative of the second function is stored, and the shape of the original curve other than the noise section is stored. Generating a new second-order derivative by substituting a third function that represents a curve or a straight line that satisfies the predetermined condition of, and performing a double integration process on the new second-order derivative. The recording medium according to claim 9, wherein