KR102461770B1 - Interpolation method within an irregular rectangle and its device - Google Patents

Abstract

Disclosed are an interpolation method within an irregular quadrangle and a device thereof, wherein interpolation coordinates corresponding to an arbitrary position within an irregular quadrangle can be found. The device for interpolation within an irregular quadrangle, which calculates interpolation coordinates inside an irregular quadrangle of which the coordinates of the four corners are defined as (x1, y1), (x2, y2), (x3, y3) and (x4, y4), respectively, comprises: a first derivation part; a first interpolation value calculation part for calculating ξ of interpolation coordinates (ξ, η); a second derivation part; a second interpolation value calculation part (140) for calculating η of the interpolation coordinates (ξ, η); and an interpolation coordinate obtaining part, thereby calculating the interpolation coordinates inside the irregular quadrangle of which the coordinates of the four corners are defined as (x1, y1), (x2, y2), (x3, y3) and (x4, y4), respectively. Accordingly, an interpolation value within an arbitrary irregular quadrangle can be found using the method of finding coordinates (ξ, η) corresponding to an arbitrary position (x, y) within the irregular quadrangle to utilize a known quadrangular finite element interpolation method.

Description

Interpolation method within an irregular rectangle and its device

The present invention relates to a method and apparatus for interpolation within an irregular quadrangle, and more particularly, to an interpolation method and apparatus within an irregular quadrangle that can find interpolation coordinates corresponding to an arbitrary position within an irregular quadrangle.

A 3D Geographic Information System (GIS) is a computer system that can visualize the location and attribute information of natural and artificial objects in 3D space with a computer, and input, edit, and analyze these data. A three-dimensional geographic information system (GIS) collects and utilizes computer hardware and software and geographic data designed to effectively collect, store, update, coordinate, analyze, and present all types of geographically referenceable information.

In a 3D geographic information system, changes in topography are mainly expressed by a Digital Elevation Model (DEM) or an irregular triangular network (TIN), and the 3D structure is a wire-frame model and a surface model. It is expressed using models such as a surface model or a solid model.

In general, a 3D geographic information system expresses topography using a triangular grid-type numerical elevation model (DEM) for effective operation of 3D detail (LOD).

1 is an exemplary diagram of a structure of general three-dimensional topographic information.

The numerical elevation model (DEM) is easy to construct and uses a simple calculation formula when converting the level of detail (LOD), so it is possible to predict performance. In addition, when visualization information is provided at the same level of detail (LOD), the system load and operation speed are constant by reusing the same structure.

A method of expressing the terrain using a numerical elevation model (DEM) includes not only the grid method but also the Triangulated Irregular Network (TIN) method of irregular triangular networks.

The conversion process from TIN to DEM is to obtain the elevation value (z value) for an arbitrary grid inside the triangle by using the interpolation method with the z values of the three points of the triangle.

2 is a diagram for explaining an example of applying an interpolation method using TIN.

As shown in FIG. 2 , the elevation value of the point p' is obtained using interpolation for the z value of the (i, j)-th grid at three points p1, p2, and p3 of a given triangle.

Let the three points be p1(x1, y1, z1), p2(x2, y2, z2), p3(x3, y3, z3), respectively, and the desired point is p'(x', y', z' ) is assumed. Then, x', y' is calculated from the position of the p(i, j) grid. Next, in order to obtain z', a linear interpolation method, that is, a linear equation z=ax+by+c is applied. By substituting the three points p1, p2, and p3 of the triangle, constants a, b, and c are obtained. By substituting (x', y') of p' into the linear equation, the z' value is obtained. The elevation value of the p(i, j) grid is z'.

On the other hand, the numerical elevation model (DEM) has many limitations in expressing the boundary between topographic relief and structures within the resolution. In addition, in terrain where there is not much topographical change, such as open land, the altitude is expressed at regular grid intervals, so it inevitably has a large capacity.

That is, in the case of a general interpolation method, it is difficult to implement, and there is a problem that it takes a long time for interpolation. In addition, since it is necessary to solve a large-scale matrix, prior work such as coordinate system transformation is required.

Korean Patent Registration No. 10-0896712 (May 11, 2009) (System and method for manufacturing numerical elevation models using numerical maps)

Accordingly, the technical problem of the present invention is based on this point, and an object of the present invention is to find interpolation coordinates corresponding to arbitrary positions within an irregular quadrangle in order to utilize a known quadrangular finite element interpolation method within an irregular quadrangle. to provide an interpolation method of

Another object of the present invention is to provide an interpolation device within an irregular quadrangle that performs the above-described interpolation method within an irregular quadrangle.

,

,

및

에 의해 정의되고,

(여기서,

,

,

)이고,

,

,

,

,

,

,

및

이고,

이고, 상기 내삽 연산식은,

에 의해 정의되는 것을 특징으로 한다. In order to realize the object of the present invention, the interpolation method within an irregular quadrangle according to an embodiment is defined as (x1, y1), (x2, y2), (x3, y3) and (x4, y4) Coordinate values of the irregular quadrangle, and a first feature value v1, a second feature value v2, a third feature value v3, and a fourth feature value v4 corresponding to each of the four corners of the irregular quadrangle receiving a coordinate value-characteristic value input unit; obtaining, by a weight obtaining unit, a first weight (N1), a second weight (N2), a third weight (N3), and a fourth weight (N4) of the bilinear quadrilateral element shape function; and obtaining, by an interpolation value obtaining unit, the interpolation value v by substituting the first to fourth weights into an interpolation expression. The first to fourth weights N1, N2, N3, and N4 are

,

,

and

is defined by

(here,

,

,

)ego,

,

,

,

,

,

,

and

ego,

and the interpolation formula,

characterized by being defined by

일 수 있다. In this embodiment, if the condition of A = 0 is satisfied,

can be

일 수 있다. In this embodiment, if the condition of a ₃ =a ₄ =0 is satisfied,

can be

일 수 있다. In this embodiment, if the condition of a ₃ =ξ = 0 is satisfied,

can be

일 수 있다. In this embodiment, if the conditions of a ₃ ≠ 0, a ₄ ≠ 0 and a ₃ +a ₄ ξ=0 are satisfied,

can be

In the present embodiment, the interpolation value v may be a characteristic value including any one of a z value corresponding to the height of the terrain, topography, temperature, humidity, and flow velocity.

을 도출하는 단계 - 여기서,

,

,

,

,

,

,

,

,

,

,

이다; 상기 2차 방정식에서 A가 0인 것으로 체크되면, 제1 내삽값 계산부가

의 수식을 이용하여 ξ를 계산하는 단계; 상기 2차 방정식에서 A가 0이 아닌 것으로 체크되면, 상기 제1 내삽값 계산부가

의 수식을 이용하여 ξ를 계산하는 단계; 제2 도출부가 η에 대한 수식으로

을 도출하는 단계; 상기 η에 대한 수식에서 a₃=a₄=0인 것으로 체크되면, 제2 내삽값 계산부가 According to another embodiment, in order to realize the object of the present invention, each of the coordinates of the four corners is (x1, y1), (x2, y2), (x3, y3) and (x4, y4) are irregular The interpolation method within an irregular rectangle that calculates the interpolation coordinates inside the rectangle, the first derivation unit is a quadratic equation for ξ.

Steps to derive - where,

,

,

,

,

,

,

,

,

,

,

to be; When it is checked that A is 0 in the quadratic equation, the first interpolation value calculation unit

calculating ξ using the formula of ; When it is checked that A is not 0 in the quadratic equation, the first interpolation value calculation unit

calculating ξ using the formula of ; The second derivation part is a formula for η

deriving; When it is checked that a ₃ =a ₄ =0 in the formula for η, the second interpolation value calculation unit

의 수식을 이용하여 η를 계산하는 단계; 상기 η에 대한 수식에서 a₃=0이고 a₄≠0인 것으로 체크되면, 상기 제2 내삽값 계산부가

의 수식을 이용하여 η를 계산하는 단계; 상기 η에 대한 수식에서 a₃≠0, a₄≠0이면서 분모가 0으로 체크되면, 상기 제2 내삽값 계산부가

의 수식을 이용하여 η를 계산하는 단계; 상기 η에 대한 수식에서 a₃≠0, a₄≠0이면서 분모가 0이 아닌 것으로 체크되면, 상기 제2 내삽값 계산부가

의 수식을 이용하여 η를 계산하는 단계; 및 내삽 좌표 획득부가 상기 계산된 ξ값과 η값을 이용하여 내삽 좌표(ξ, η)를 획득하는 단계를 포함한다.

calculating η using the formula of ; When it is checked that a ₃ =0 and a ₄ ≠ 0 in the formula for η, the second interpolation value calculation unit

calculating η using the formula of ; When a ₃ ≠ 0 and a ₄ ≠ 0 in the formula for η and the denominator is checked as 0, the second interpolation value calculation unit

calculating η using the formula of ; When it is checked that a ₃ ≠ 0 and a ₄ ≠ 0 in the formula for η and the denominator is not 0, the second interpolation value calculation unit

calculating η using the formula of ; and obtaining interpolated coordinates (ξ, η) by an interpolation coordinate obtaining unit using the calculated values of ξ and η.

,

,

및

에 의해 정의되고,

(여기서,

,

,

)이고,

,

,

,

,

,

,

및

이고,

이고, 상기 내삽 연산식은,

에 의해 정의되는 것을 특징으로 한다. In order to realize another object of the present invention, the interpolation apparatus within an irregular quadrangle according to an embodiment is defined as (x1, y1), (x2, y2), (x3, y3) and (x4, y4). coordinate values of the irregular quadrangle, and a first characteristic value (v1), a second characteristic value (v2), a third characteristic value (v3), and a fourth characteristic value (v4) corresponding to each of the four corners of the irregular quadrangle a coordinate value-characteristic value input unit receiving an input; a weight obtaining unit obtaining by calculating a first weight (N1), a second weight (N2), a third weight (N3), and a fourth weight (N4) of the bilinear quadrilateral element shape function; and an interpolation value obtaining unit configured to obtain an interpolation value v by substituting the first to fourth weights into an interpolation expression. The first to fourth weights N1, N2, N3, and N4 are

,

,

and

is defined by

(here,

,

,

)ego,

,

,

,

,

,

,

and

ego,

and the interpolation formula,

characterized by being defined by

In the present embodiment, the interpolation value v may be a characteristic value including any one of a z value corresponding to the height of the terrain, topography, temperature, humidity, and flow velocity.

을 도출하는 제1 도출부 - 여기서,

,

,

,

,

,

,

,

,

,

,

이다; 상기 2차 방정식에서 A가 0인 것으로 체크되면,

의 수식을 이용하여 ξ를 계산하고, 상기 2차 방정식에서 A가 0이 아닌 것으로 체크되면,

의 수식을 이용하여 ξ를 계산하는 제1 내삽값 계산부; η에 대한 수식으로

을 도출하는 제2 도출부; 상기 η에 대한 수식에서 a₃=a₄=0인 것으로 체크되면,

의 수식을 이용하여 η를 계산하고, 상기 η에 대한 수식에서 a3=0이고 a₄≠0인 것으로 체크되면,

의 수식을 이용하여 η를 계산하고, 상기 η에 대한 수식에서 a₃≠0, a₄≠0이면서 분모가 0으로 체크되면,

의 수식을 이용하여 η를 계산하고, 상기 η에 대한 수식에서 a₃≠0, a₄≠0이면서 분모가 0이 아닌 것으로 체크되면,

의 수식을 이용하여 η를 계산하는 제2 내삽값 계산부; 및 상기 계산된 ξ값과 η값을 이용하여 내삽 좌표(ξ, η)를 획득하는 내삽 좌표 획득부를 포함한다. According to another embodiment, each of the coordinates of the four corners is defined as (x1, y1), (x2, y2), (x3, y3) and (x4, y4) An interpolation device within an irregular rectangle that calculates interpolation coordinates inside an irregular rectangle is a quadratic equation for ξ.

A first derivation unit for deriving

,

,

,

,

,

,

,

,

,

,

to be; If A is checked as 0 in the quadratic equation,

Calculate ξ using the formula of and if A is checked as non-zero in the quadratic equation,

a first interpolation value calculation unit for calculating ξ using the equation of ; as a formula for η

a second derivation unit for deriving If it is checked that a ₃ =a ₄ =0 in the formula for η,

Calculate η using the formula of , and if it is checked that a3 = 0 and a ₄ ≠ 0 in the formula for η,

η is calculated using the formula of , and if a ₃ ≠ 0 and a ₄ ≠ 0 in the formula for

_η is calculated using _the formula of

a second interpolation value calculation unit for calculating η using the formula of ; and an interpolation coordinate obtaining unit for obtaining interpolated coordinates (ξ, η) using the calculated values of ξ and η.

According to the interpolation method and the device within such an irregular quadrangle, a method of finding the coordinates (ξ, η) corresponding to an arbitrary position (x, y) in an irregular quadrangle in order to utilize the known quadrangular finite element interpolation method It is possible to find the interpolation value inside an arbitrary irregular rectangle.

1 is an exemplary diagram of a structure of general three-dimensional topographic information.
2 is a diagram for explaining an example of applying an interpolation method using TIN.
3 is a block diagram for explaining a system in which the interpolation apparatus within an irregular quadrangle according to the present invention can be used.
4A is a diagram for explaining a general representation of a topography with a triangular grid. Figure 4b is a diagram for explaining the representation of the topography in a rectangular grid according to the present invention.
5 is a diagram for describing a u vector and a v vector.
6 is a block diagram illustrating an interpolation apparatus within an irregular quadrangle according to an embodiment of the present invention.
7 is a block diagram illustrating an interpolation apparatus within an irregular quadrangle according to another embodiment of the present invention.
8A and 8B are flowcharts illustrating an interpolation method within an irregular quadrangle according to an embodiment of the present invention.

Hereinafter, the present invention will be described in more detail with reference to the accompanying drawings. Since the present invention can have various changes and can have various forms, specific embodiments are illustrated in the drawings and described in detail in the text. However, this is not intended to limit the present invention to the specific disclosed form, it should be understood to include all modifications, equivalents and substitutes included in the spirit and scope of the present invention.

In describing each figure, like reference numerals have been used for like elements. In the accompanying drawings, the dimensions of the structures are enlarged than the actual size for clarity of the present invention.

Terms such as first, second, etc. may be used to describe various elements, but the elements should not be limited by the terms. The above terms are used only for the purpose of distinguishing one component from another. For example, without departing from the scope of the present invention, a first component may be referred to as a second component, and similarly, a second component may also be referred to as a first component. The singular expression includes the plural expression unless the context clearly dictates otherwise.

In the present application, terms such as "comprise" or "have" are intended to designate that a feature, number, step, operation, component, part, or a combination thereof described in the specification exists, but one or more other features It is to be understood that it does not preclude the possibility of the presence or addition of numbers, steps, operations, components, parts, or combinations thereof.

In addition, unless otherwise defined, all terms used herein, including technical or scientific terms, have the same meaning as commonly understood by one of ordinary skill in the art to which the present invention belongs. Terms such as those defined in a commonly used dictionary should be interpreted as having a meaning consistent with the meaning in the context of the related art, and should not be interpreted in an ideal or excessively formal meaning unless explicitly defined in the present application. does not

3 is a block diagram for explaining a system in which the interpolation apparatus within an irregular quadrangle according to the present invention can be used.

Referring to FIG. 3 , in the exemplary system, reconstruction of the surface topography is performed by processor 100 loaded with an appropriate program. The program may be stored in the persistent storage device 500 such as a hard disk, and may be executed from a random access memory such as DRAM (not shown).

Processor 100 operates on measurements received via input module 200 . The input module 200 may be received from the measurement system 300 . The measurement system 300 may be external to the system of the present invention, in which case the measurements may be received via an appropriate communication system (eg, a wired or wireless LAN or a wide area network such as the Internet).

The measurement system 300 is preferably part of a system for reconstructing a surface. Advantageously, the measurement system 300 may be a deflectometry system. For the sake of detail, it is assumed that measurements are provided for a two-dimensional grid of measurement points, where for each point, the slopes in the x and y directions are given. The method works equally well for other suitable coordinate systems using, for example, r, phi grids. The grid does not need to be neatly arranged, e.g. it is not necessary for the measurement points in two directions to form an equidistant x, y grid, and a connection can be established between the measurement points (similar to nodes belonging to finite elements in an FEM mesh). Just being able is enough.

The input module 200 may receive an input from a user. As devices of the input module 200 , a mouse 400 and a keyboard 450 are illustrated. Output to the user may be supplied via an output device 600 , which may include a display. In particular, the above-described input module 200 may allow the user to indicate an area of the surface (and/or grid) for which an accurate reconstruction of the surface is required. For other areas, conventional reconstructions may occur. Such conventional reconstruction can be used to reconstruct the edges of the selected region.

Hereinafter, a reconstruction method according to the present invention will be described.

4A is a diagram for explaining a general representation of a topography with a triangular grid. Figure 4b is a diagram for explaining the representation of the topography in a rectangular grid according to the present invention.

Referring to FIGS. 4A and 4B , in the case of an irregularly shaped topography disposed on the xy plane, it can be expressed as a triangular grid or a square grid, which are standardized standard grids.

A triangular or rectangular grid is expressed by the ξ axis, which is a horizontal axis, and the η axis, which is a vertical axis, which are converted into a standard function.

In the case of an irregular quadrangle, the internal interpolation value (v) is as shown in Equation 1 below.

[Formula 1]

Here, the interpolation value v may be a coordinate value or a characteristic value. The above-mentioned coordinate value may be an x value or a y value on a plane coordinate. In addition, the characteristic value may be a z value, a temperature value, a humidity value, a flow velocity value, etc. corresponding to the height of the terrain.

In Equation 1, each of N1, N2, N3, and N4 is the same as Equation 2 below.

[Equation 2]

Equation 2 above defines a bilinear quadrilateral element shape function.

When the x-coordinates (x1, x2, x3, x4) of the four corner points of the irregular quadrangle are given using the above-described bilinear quadrilateral element shape function, the interpolated x-coordinate of an internal arbitrary point is as shown in Equation 3 below.

[Equation 3]

If the above Equation 3 is rearranged, it is as Equation 4 below.

[Equation 4]

,

,

및

이다. here,

,

,

and

to be.

Since the y-coordinates (y1, y2, y3, y4) of the four corner points of the irregular quadrangle are derived in the same way, the interpolated y-coordinate of an internal arbitrary point is as shown in Equation 5 below.

[Equation 5]

,

,

및

이다. here,

,

,

and

to be.

The present invention is to find the interpolation coordinates (ξ, η) when an arbitrary point (x, y) inside an irregular rectangle is given by using the above-described interpolation process in reverse.

Then, a process of finding the interpolated coordinates (ξ, η) using a quadratic equation will be described below.

The x-coordinate of an arbitrary point can be defined as in Equation 6 below.

[Equation 6]

If Equation 6 is arranged as η, it is as Equation 7 below.

[Equation 7]

Also, the y-coordinate of an arbitrary point may be defined as in Equation 8 below.

[Equation 8]

By rearranging Equation 8, a quadratic equation for ξ can be obtained as shown in Equation 9 below.

[Equation 9]

The quadratic equation for ξ and the solution to ξ can be expressed briefly as in Equation 10.

[Equation 10]

,

,

이다. 또한,

,

,

,

이다.here,

,

,

to be. In addition,

,

,

,

to be.

If A = 0 in Equation 10, it becomes a linear equation, so it is as Equation 11 below.

[Equation 11]

If A = 0, the following Equation 12 is shown.

[Equation 12]

To rearrange this, Equation 13 is given below.

[Equation 13]

5 is a diagram for describing a u vector and a v vector.

As shown in FIG. 5 , if the u vector and the v vector are considered, the following Equation 14 is obtained.

[Equation 14]

Therefore, Equation 13 can be rewritten as a vector component as shown in Equation 15 below.

[Equation 15]

If Equation 15 is rearranged, Equation 16 below is obtained.

[Equation 16]

This means that u × v = 0, that is, when the two vectors are parallel.

Substitute ξ obtained by Equation 10 or Equation 11 into Equation 7 to obtain η.

Here, if the denominator of Equation 7 is zero, the value cannot be obtained. Let's take a closer look at the case where the denominator is zero.

First, a ₃ =a ₄ =0. Substituting this into Equation 6, it corresponds to the case where x = a ₁ +a ₂ ξ. Therefore, ξ=(x - a ₁ )/a ₂ , and substituting it into Equation 8 and rearranging it, Equation 17 is as follows.

[Equation 17]

Second, a ₃ =ξ = 0, and substituting this into Equation 6, it is when x = a ₁ . By substituting ξ=0 into Equation 8 and rearranging it, Equation 18 is the same.

[Equation 18]

Third, a ₃ ≠ 0 and a ₄ ≠ 0, but a ₃ +a ₄ ξ=0. When ξ=-a ₃ /a ₄ is substituted into Equation 6, it corresponds to the case when x=a ₁ -(a ₂ a ₃ )/a ₄ . Substituting ξ into Equation 8 and rearranging it, Equation 19 is as follows.

[Equation 19]

In order to help understanding when the denominator is 0 in Equation 7, if Equation 6 is arranged with respect to ξ, Equation 20 is shown below.

[Equation 20]

Substituting this into Equation 8 and rearranging it, a quadratic equation for η as shown in Equation 21 below can be obtained.

[Equation 21]

Similarly, when A' = 0, a linear equation is formed, and thus a solution such as Equation 22 below can be obtained.

[Equation 22]

A case where A' = 0 is shown in Equation 23.

[Equation 23]

If the p and q vectors are considered as in FIG. 5 , a result as shown in Equation 24 below can be obtained.

[Equation 24]

Equation 24 means p × q = 0 , that is, when two vectors are parallel.

Based on the above results, the process of finding the interpolated coordinates (ξ, η) using the solution of the quadratic equation is summarized as follows. The process of finding the interpolated coordinates (ξ, η) described below may be implemented programmatically.

The quadratic equation for ξ is as follows.

If A≠0 in the quadratic equation for ξ, find the value of ξ using the root formula below.

If A=0 in the quadratic equation for ξ, obtain the value of ξ using the following equation.

Meanwhile, η is basically obtained using the following equation.

In this case, a problem arises when the denominator is 0. When a ₃ and a ₄ are written as p and q components in FIG. 5 , the following equation is obtained.

If the p vector and the q vector are parallel to each other and parallel to the vertical axis, then p ₁ =q ₁ =0, so a ₃ =a ₄ =0. In this case, η is calculated using the following formula.

If the components p _{1 and} q ₁ of the p vector and the q vector are equal in magnitude and opposite to each other, then a ₃ =0. At this time, if a point with x=a ₁ is given, ξ=0, and in this case, η is calculated using the following equation.

If a ₃ ≠ 0, a ₄ ≠ 0 and the denominator is 0, calculate η using the following formula.

6 is a block diagram illustrating an interpolation apparatus within an irregular quadrangle according to an embodiment of the present invention.

Referring to FIG. 6 , an interpolation apparatus within an irregular quadrangle according to an embodiment of the present invention includes a coordinate value-characteristic value input unit 110 , a weight acquirer 112 , and an interpolation value acquirer 114 , Interpolated coordinates are calculated inside the irregular quadrilateral where the coordinates of the four corners of the irregular quadrangle are defined by (x1, y1), (x2, y2), (x3, y3), and (x4, y4), respectively. In the present embodiment, the interpolation apparatus has been described as consisting of the coordinate value-characteristic value input unit 110, the weight obtaining unit 112, and the interpolation value obtaining unit 114, but this is logically divided for convenience of explanation only. It is not objectively segregated. That is, each of the coordinate value-characteristic value input unit 110 , the weight acquisition unit 112 , and the interpolation value acquisition unit 114 illustrated in the present embodiment may be implemented in separate software or may be implemented in integrated software. may be Meanwhile, the interpolation apparatus described in FIG. 6 may be implemented and mounted programmatically or software-wise in the processor 100 illustrated in FIG. 3 .

The coordinate value-characteristic value input unit 110 receives the coordinate values of the irregular quadrangle defined by (x1, y1), (x2, y2), (x3, y3) and (x4, y4), and the four corners of the irregular quadrangle. A first characteristic value v1 , a second characteristic value v2 , a third characteristic value v3 , and a fourth characteristic value v4 corresponding to each are received. Each of the above-described characteristic values v1, v2, v3, v4 may be a z value corresponding to the height of the terrain, or may be a temperature value, a humidity value, or a flow velocity value.

,

,

및

에 의해 정의된다. 이때,

이다. 여기서,

,

,

이다. 또한

,

,

,

,

,

,

및

이고,

이다. The weight obtaining unit 112 calculates and obtains the first weight N1 , the second weight N2 , the third weight N3 , and the fourth weight N4 of the bilinear quadrilateral element shape function. The first to fourth weights N1, N2, N3, and N4 are

,

,

and

is defined by At this time,

to be. here,

,

,

to be. In addition

,

,

,

,

,

,

and

ego,

to be.

에 의해 정의된다. 상기 내삽값(v)은 지형의 높이에 해당하는 z값을 포함할 수 있다. 다른 한편, 상기 내삽값(v)은 온도값, 습도값, 유속값 중 어느 하나를 포함할 수 있다. The interpolation value obtaining unit 114 obtains the interpolation value v by substituting the first to fourth weights N1, N2, N3, and N4 into an interpolation expression. The interpolation formula is,

is defined by The interpolation value v may include a z value corresponding to the height of the terrain. On the other hand, the interpolation value v may include any one of a temperature value, a humidity value, and a flow rate value.

7 is a block diagram illustrating an interpolation apparatus within an irregular quadrangle according to another embodiment of the present invention.

Referring to FIG. 7 , an interpolation apparatus within an irregular quadrangle according to another embodiment of the present invention includes a first derivation unit 120 and a first interpolation value calculation unit 122 for calculating ξ of interpolation coordinates (ξ, η). ), a second derivation unit 124, a second interpolation value calculation unit 126 for calculating η of the interpolation coordinates (ξ, η), and an interpolation coordinate obtaining unit 128, each of the coordinates of the four corners is Calculate the interpolated coordinates inside an irregular rectangle defined by (x1, y1), (x2, y2), (x3, y3) and (x4, y4). In the present embodiment, the interpolation apparatus includes the first derivation unit 120 , the first interpolation value calculation unit 122 , the second derivation unit 124 , the second interpolation value calculation unit 126 , and the interpolation coordinate obtaining unit 128 . ), but it is logically divided for convenience of explanation, not hardware. That is, the first derivation unit 120 , the first interpolation value calculation unit 122 , the second derivation unit 124 , the second interpolation value calculation unit 126 , and the interpolation coordinate acquirer 128 , which are illustrated in the present embodiment. Each may be implemented as separate software, or may be implemented as integrated software. On the other hand, the interpolation apparatus according to another embodiment of the present invention described with reference to FIG. 7 may be implemented and mounted programmatically or software-wise in the processor 100 illustrated in FIG. 3 .

을 도출한다. 여기서,

,

,

,

,

,

,

,

,

,

,

이다. The first derivation unit 120 is a quadratic equation for ξ.

to derive here,

,

,

,

,

,

,

,

,

,

,

to be.

의 수식을 이용하여 ξ를 계산하고, 상기 2차 방정식에서 A가 0이 아닌 것으로 체크되면,

의 수식을 이용하여 ξ를 계산한다. When the first interpolation value calculation unit 122 is checked that A is 0 in the quadratic equation,

Calculate ξ using the formula of and if A is checked as non-zero in the quadratic equation,

Calculate ξ using the formula of

을 도출한다. The second derivation unit 124 is an equation for η.

to derive

의 수식을 이용하여 η를 계산하고, 상기 η에 대한 수식에서 a3=0이고 a₄≠0인 것으로 체크되면,

의 수식을 이용하여 η를 계산하고, 상기 η에 대한 수식에서 a₃≠0, a₄≠0이면서 분모가 0으로 체크되면,

의 수식을 이용하여 η를 계산하고, 상기 η에 대한 수식에서 a₃≠0, a₄≠0이면서 분모가 0이 아닌 것으로 체크되면,

의 수식을 이용하여 η를 계산한다. If the second interpolation value calculation unit 126 is checked that a ₃ =a ₄ = 0 in the equation for η,

Calculate η using the formula of , and if it is checked that a3 = 0 and a ₄ ≠ 0 in the formula for η,

η is calculated using the formula of , and if a ₃ ≠ 0 and a ₄ ≠ 0 in the formula for

_η is calculated using _the formula of

Calculate η using the formula in

The interpolated coordinate obtaining unit 128 obtains interpolated coordinates (ξ, η) by using the calculated values of ξ and η.

8A and 8B are flowcharts illustrating an interpolation method within an irregular quadrangle according to an embodiment of the present invention.

을 도출한다(단계 S102). 여기서,

,

,

,

,

,

,

,

,

,

,

이다. 상기한 ξ에 대한 2차 방정식의 도출은 제1 도출부(120)에 의해 수행될 수 있다. 7, 8A and 8B, as a quadratic equation for ξ of interpolated coordinates (ξ, η)

is derived (step S102). here,

,

,

,

,

,

,

,

,

,

,

to be. The derivation of the quadratic equation for ξ may be performed by the first derivation unit 120 .

It is checked whether or not the condition in which A is 0 in the quadratic equation derived in step S102 is satisfied (step S104). A check as to whether or not the condition A is 0 may be performed by the first interpolation value calculation unit 122 .

의 수식을 이용하여 ξ를 계산한다(단계 S106). 상기한 단계 S106에서 수행되는 ξ의 계산은 제1 내삽값 계산부(122)에 의해 수행될 수 있다. If it is checked that A satisfies the condition of 0 in step S104,

Calculate ξ using the formula of (step S106). The calculation of ξ performed in step S106 may be performed by the first interpolation value calculation unit 122 .

의 수식을 이용하여 ξ를 계산한다(단계 S108). 상기한 단계 S108에서 수행되는 ξ의 계산은 제1 내삽값 계산부(122)에 의해 수행될 수 있다.If it is checked in step S104 that the condition that A is 0 is not satisfied,

Calculate ξ using the formula of (step S108). The calculation of ξ performed in step S108 may be performed by the first interpolation value calculation unit 122 .

을 도출한다(단계 S110). 상기한 η에 대한 수식의 도출은 제2 도출부(124)에 의해 수행될 수 있다. Following step S106 or step S108, as an expression for η of the interpolated coordinates (ξ, η)

is derived (step S110). The derivation of the above-described equation for η may be performed by the second derivation unit 124 .

In the equation for η derived in step S110, it is checked whether the condition of a ₃ =a ₄ =0 is satisfied (step S112). A check as to whether the condition of a ₃ =a ₄ = 0 is satisfied may be performed by the second interpolation value calculator 126 .

의 수식을 이용하여 η를 계산한다(단계 S114). 상기한 단계 S114에서 수행되는 η의 계산은 제2 내삽값 계산부(126)에 의해 수행될 수 있다. If it is checked that the condition of a ₃ =a ₄ = 0 is satisfied in step S112,

Calculate η using the formula of (step S114). The calculation of η performed in step S114 may be performed by the second interpolation value calculation unit 126 .

If it is checked in step S112 that the condition of a ₃ =a ₄ =0 is not satisfied, it is checked whether the condition of a ₃ =0 and a ₄ ≠ 0 is satisfied in the equation for η (step S116). The operation of step S116 may be performed by the second interpolation value calculator 126 .

의 수식을 이용하여 η를 계산한다(단계 S118). 상기한 단계 S118에서 수행되는 η의 계산은 제2 내삽값 계산부(126)에 의해 수행될 수 있다. If it is checked in step S116 that a ₃ = 0 and a ₄ ≠ 0 condition is satisfied,

Calculate η using the formula of (step S118). The calculation of η performed in step S118 may be performed by the second interpolation value calculation unit 126 .

If the condition for a ₃ =0 and a ₄ ≠ 0 is not satisfied in step S116, it is checked whether the condition for a ₃ ≠ 0, a ₄ ≠ 0 and the denominator is 0 in the formula for η is satisfied (step S116) S120). The operation of step S120 may be performed by the second interpolation value calculator 126 .

의 수식을 이용하여 η를 계산한다(단계 S122). 상기한 단계 S122에서 수행되는 η의 계산은 제2 내삽값 계산부(126)에 의해 수행될 수 있다.If it is checked that the condition that a ₃ ≠ 0, a ₄ ≠ 0 and the denominator is 0 in step S120 is satisfied,

Calculate η using the formula of (step S122). The calculation of η performed in step S122 may be performed by the second interpolation value calculation unit 126 .

의 수식을 이용하여 η를 계산한다(단계 S124). 상기한 단계 S124에서 수행되는 η의 계산은 제2 내삽값 계산부(126)에 의해 수행될 수 있다. If it is checked that a ₃ ≠ 0, a ₄ ≠ 0 and the denominator is 0 in step S120 is not satisfied,

Calculate η using the formula of (step S124). The calculation of η performed in step S124 may be performed by the second interpolation value calculation unit 126 .

Following step S114, step S118, step S122, and step S124, the interpolated coordinates (ξ, η) are obtained using the calculated values of ξ and η (step S126). In step S126, the interpolation coordinates (ξ, η) may be obtained by the interpolation coordinate obtaining unit 128 .

As described above, according to the present invention, in order to utilize the known rectangular finite element interpolation method, arbitrary You can find the interpolation value inside an irregular rectangle.

In addition, according to the present invention, a grid-type DEM can be generated using an irregular rectangular mesh and interpolation method at irregular elevation points extracted by a numerical map.

Although the above has been described with reference to the embodiments, those skilled in the art can variously modify and change the present invention within the scope without departing from the spirit and scope of the present invention described in the claims below You will understand.

100: processor 110: coordinate value-characteristic value input unit
112: weight acquisition unit 114: interpolation value acquisition unit
120: first derivation unit 122: first interpolation value calculation unit
124: second derivation unit 126: second interpolation value calculation unit
128: interpolation coordinate acquisition unit 200: input module
300: measuring system 400: mouse
450: keyboard 500: persistent storage
600: output device

Claims (10)

Coordinate values of an irregular quadrangle defined by (x1, y1), (x2, y2), (x3, y3), and (x4, y4), and a first characteristic value ( v1), the second characteristic value (v2), the third characteristic value (v3), and the fourth characteristic value (v4) receiving the coordinate value-characteristic value input unit;
obtaining, by a weight obtaining unit, a first weight (N1), a second weight (N2), a third weight (N3), and a fourth weight (N4) of the bilinear quadrilateral element shape function; and
and obtaining, by an interpolation value obtaining unit, the interpolation value v by substituting the first to fourth weights into an interpolation expression, wherein the first to fourth weights N1, N2, N3, N4 include,

,

,

and

is defined by

(here,

,

,

)ego,

,

,

,

,

,

,

and

ego,

ego,
The interpolation formula is,

An interpolation method within an irregular rectangle, characterized in that defined by According to claim 1, If the condition of A = 0 is satisfied,

An interpolation method, characterized in that According to claim 1, If the condition of a ₃ =a ₄ =0 is satisfied,

Interpolation method within an irregular rectangle, characterized in that The method according to claim 1, wherein if the condition of a ₃ =ξ = 0 is satisfied,

Interpolation method within an irregular rectangle, characterized in that The method according to claim 1, wherein a ₃ ≠ 0, a ₄ ≠ 0 and a ₃ +a ₄ ξ=0 if the conditions are satisfied,

Interpolation method within an irregular rectangle, characterized in that The interpolation method according to claim 1, wherein the interpolation value (v) is a characteristic value including any one of a z value corresponding to the height of the terrain, topography, temperature, humidity, and flow velocity. Coordinate values of an irregular quadrangle defined by (x1, y1), (x2, y2), (x3, y3), and (x4, y4), and a first characteristic value (v1) corresponding to each of the four corners of the irregular quadrangle ), a second characteristic value (v2), a third characteristic value (v3), and a fourth characteristic value (v4), a coordinate value-characteristic value input unit;
a weight obtaining unit obtaining by calculating a first weight (N1), a second weight (N2), a third weight (N3), and a fourth weight (N4) of the bilinear quadrilateral element shape function; and
and an interpolation value obtaining unit that obtains an interpolation value v by substituting the first to fourth weights into an interpolation expression, wherein the first to fourth weights N1, N2, N3, and N4 include:

,

,

and

is defined by

(here,

,

,

)ego,

,

,

,

,

,

,

and

ego,

ego,
The interpolation formula is,

Interpolation device within an irregular rectangle, characterized in that defined by. The interpolation apparatus according to claim 7, wherein the interpolation value (v) is a characteristic value including any one of z-value corresponding to the height of the terrain, topography, temperature, humidity, and flow velocity. In the interpolation method within an irregular rectangle, which calculates the interpolation coordinates inside the irregular rectangle where the coordinates of the four corners are each defined as (x1, y1), (x2, y2), (x3, y3) and (x4, y4) ,
The first derivation is a quadratic equation for ξ

Steps to derive - where,

,

,

,

,

,

,

,

,

,

,

to be;
When it is checked that A is 0 in the quadratic equation, the first interpolation value calculation unit

calculating ξ using the formula of ;
When it is checked that A is not 0 in the quadratic equation, the first interpolation value calculation unit

calculating ξ using the formula of ;
The second derivation part is a formula for η

deriving;
When it is checked that a ₃ =a ₄ =0 in the formula for η, the second interpolation value calculation unit

calculating η using the formula of ;
When it is checked that a ₃ =0 and a ₄ ≠ 0 in the formula for η, the second interpolation value calculation unit

calculating η using the formula of ;
When a ₃ ≠ 0 and a ₄ ≠ 0 in the formula for η and the denominator is checked as 0, the second interpolation value calculation unit

calculating η using the formula of ;
When it is checked that a ₃ ≠ 0 and a ₄ ≠ 0 in the formula for η and the denominator is not 0, the second interpolation value calculation unit

calculating η using the formula of ; and
and acquiring interpolated coordinates (ξ, η) using the calculated ξ and η values by an interpolation coordinate obtaining unit. In an interpolator within an irregular rectangle that calculates the interpolation coordinates inside an irregular rectangle where the coordinates of the four corners are each defined by (x1, y1), (x2, y2), (x3, y3) and (x4, y4) ,
As a quadratic equation for ξ

A first derivation unit for deriving

,

,

,

,

,

,

,

,

,

, and

to be;
If A is checked as 0 in the quadratic equation,

Calculate ξ using the formula of and if A is checked as non-zero in the quadratic equation,

a first interpolation value calculation unit for calculating ξ using the equation of ;
as a formula for η

a second derivation unit for deriving
If it is checked that a ₃ =a ₄ =0 in the formula for η,

Calculate η using the formula of , and if it is checked that a3 = 0 and a ₄ ≠ 0 in the formula for η,

η is calculated using the formula of , and if a ₃ ≠ 0 and a ₄ ≠ 0 in the formula for

_η is calculated using _the formula of

a second interpolation value calculation unit for calculating η using the formula of ; and
and an interpolation coordinate obtaining unit for obtaining interpolated coordinates (ξ, η) by using the calculated values of ξ and η.

Images