KR100938656B1 - Digital-filtering method for 3-dimensional space data processing - Google Patents

Abstract

PURPOSE: A digital filtering method for 3D space data processing is provided to improve performance using a spectral method using double Fourier series on image data. CONSTITUTION: A base function of double Fourier series comprises a function satisfying an extreme condition. If east and west wave numbers are even, the first differential value of the base function is 0. If the east and west wave numbers are odd, the base function becomes a single sine function. Noise cancellation and smoothing are performed at high speed.

Description

Digital filtering method for three-dimensional spatial data processing {DIGITAL-FILTERING METHOD FOR 3-DIMENSIONAL SPACE DATA PROCESSING}

The present invention relates to a digital filtering method for processing three-dimensional spatial data, and more particularly, to a digital filtering method for processing three-dimensional spatial data using a spectral method using double Fourier series of data.

In recent years, image data to be processed in areas such as weather and climate have been enormous, and these data have to be processed at high speed. In addition, according to this, the digital filtering method for processing three-dimensional spatial data is gradually required to maintain a high processing speed.

In the spherical digital filtering method for three-dimensional spatial data processing, the Regendre spectral method, which uses the Legendre function as a basis function, is commonly used. However, the larger the resolution, the lower the processing speed (computation efficiency) is. Therefore, in order to solve this shortcoming, the Double Fourier Series (DFS) spectral method, which can replace the La Genre spectral method, has been developed and used in the meteorological, climate, and earth science fields.

In the spectral method, one of the solutions of differential equations, an arbitrary function is developed as a series of basis functions that satisfy a boundary condition, and the spatial derivative is represented as the derivative of the basis function. The spectral method is more accurate than other methods, such as the finite difference method, and maintains high efficiency in the calculation of nonlinear terms. Due to these advantages, the spectral method is used in most weather forecasting models and climate models in many countries.

However, the double Fourier spectral method is still not satisfactory in terms of accuracy and computational speed.

An object of the present invention is to provide a digital filtering method for processing three-dimensional spatial data that can increase the processing speed of high resolution data.

In addition, another object of the present invention is to provide a digital filtering method for processing three-dimensional spatial data that can increase the accuracy.

In order to achieve the above object, the present invention is a digital filtering method for processing three-dimensional spatial data by quantizing the arbitrary function of the spherical surface as a double Fourier series, the basis function of the double Fourier series satisfies the extreme conditions And a function, wherein the first derivative is 0 when the east-west wave is even, and the first derivative is 0, and the single sinusoidal function is odd when the east-west wave is odd. Provide a filtering method.

는

이며, 상기

의 경계조건은

이고, 상기

는 경도이고, 상기

는 위도+

이 며, 상기 m은 동서파수이고, 상기

은 복소수의 푸리에 계수이다.Where a double Fourier series of any function of the spherical surface

Is

And said

The boundary condition of

And

Is longitude, said

Is latitude +

M is the east-west wave number

Is a Fourier coefficient of a complex number.

이고, 상기

일 때 깁스 현상을 최소화할 수 있다.Further, the Fourier series in the north-south direction for selecting a function satisfying the boundary condition is

And

Can minimize the cast phenomenon.

이고, 상기

수인 필터차수이며, 상기

는 필터링 대상이 되는 함수이고, 상기

는 필터링된 함수이며, 상기

이다.In addition, a high order harmonic filter using a double Fourier series of any function of the spherical surface

And

Filter order

Is a function to be filtered,

Is a filtered function,

to be.

이고, 상기

은 총 자료수이며, 상기

는 변환 후의 격자점에서의 값이고, 상기

는 변환되기 전의 비격자점의 값이며, 상기

는 객관분석 대상이 되는 격자점으로부터의 거리이고, 상기

는

,

는 비격자 자료간의 평균거리이다.When the spherical surface is non-lattice data, it is converted into grid data by using objective analysis.

And

Is the total number of data,

Is the value at the grid point after conversion, and

Is the value of the non-lattice point before conversion,

Is the distance from the grid point to be subjected to the objective analysis,

Is

,

Is the average distance between non-lattice data.

The present invention can provide a digital filtering method for processing three-dimensional spatial data that can improve the performance while reducing the Gibbs phenomenon using the spectral method using a double Fourier series.

In addition, the present invention can provide a digital filtering method for processing three-dimensional spatial data capable of performing noise removal and smoothing at high speed with accuracy.

Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings.

However, the present invention is not limited to the embodiments disclosed below, but will be implemented in various forms, and only the embodiments are intended to complete the disclosure of the present invention, and to those skilled in the art to fully understand the scope of the invention. It is provided to inform you. Like reference numerals in the drawings refer to like elements.

1 is a view for explaining the grid data conversion of the three-dimensional spatial data processing digital filtering method according to the present invention, Figure 2 is a view showing the boundary conditions of the three-dimensional spatial data processing digital filter according to the present invention, Figure 3 is a schematic diagram showing the difference between the polar boundary conditions according to the east-west wave number of the three-dimensional spatial data processing digital filter according to the present invention.

In the digital filtering method for processing 3D spatial data according to the present invention, referring to FIGS. 1 to 3, a step (S ₁ ) of converting non-lattice data into lattice data and spectrally converting the lattice data into a double Fourier series and a transient equation The step S ₂ includes the process of forming a column.

Step S ₁ of converting the non-lattice data into the grid data converts the non-lattice data into the grid data through objective analysis as shown in FIG. 1. Of course, if the data used is grid data, the step of converting non-lattice data into grid data may be omitted.

The step of spectralizing the lattice data into the double Fourier series and the transient equation (S ₂ ) includes the step of expanding the spherical coordinate system lattice data into the Fourier series in the longitudinal direction (S _2-1 ) and spectralizing the lattice data by the transient equation. Step S _2-2 is included.

Expanding the spherical coordinate system grid data into the Fourier series in the longitudinal direction (S _{2 -1} ) expands the spherical coordinate system grid data converted from the non-lattice data into the Fourier series in the longitudinal direction.

라 할 때, 동서방향으로는 주기적이므로 푸리에 변환이 가능하며, 이를 수식으로 나타내면 수식 1과 같다. 이때,

는 경도이며, φ는 위도+π로 정의된다.First of all, let's take an arbitrary continuous function

In this case, the Fourier transform is possible because it is periodic in the east-west direction. At this time,

Is longitude and φ is defined as latitude + π.

[Equation 1]

은 복소수의 푸리에 계수를 의미한다. 이때,

의 경계조건은 동서파수가 짝수일 때 짝함수(Parity Function)를 이용하여 1차 미분이 0이 되는 조건을 만족해야 하며, 동서파수가 홀수일 때 단일의 정현함수로 구성된다. 또한, 이와 같은

의 경계조건은 수식 2와 같다.Where m is the east-west wave,

Denotes a complex Fourier coefficient. At this time,

The boundary condition of must satisfy the condition that the first derivative is 0 by using the Parity Function when the east-west wave is even and consists of a single sinusoid when the east-west wave is odd. Also, like this

The boundary condition of is shown in Equation 2.

[Formula 2]

의 경계조건이 수식 2를 만족할 때

는 수식 3과 같이 남북방향으로 반구간의 정현 또는 여현함수로 나타낼 수 있다.Also,

When boundary condition of satisfies Equation 2

Can be expressed as a sine or cosine function of the hemisphere in the north-south direction as shown in Equation 3.

[Equation 3]

는 경계조건을 만족시키기 위해 도입된 짝함수(Parity Function)이다. 이때, 동서파수가 홀수이거나 0인 경우 반구간의 고속 푸리에 변환(Fast Fourier Transform, FFT)이 가능하며, 동서파수가 짝수인 경우

를 먼저 구하면 고속 푸리에 변환의 사용이 가능하여 효율적인 계산이 가능하다.Referring to Equation 3,

Is a Parity Function introduced to satisfy the boundary condition. In this case, if the east-west wave is odd or 0, fast Fourier transform (FFT) of the hemisphere is possible, and if the east-west wave is even

First, we can use fast Fourier transform, which allows efficient computation.

및 짝수파수에 대해서

일 경우 깁스(Gibbs) 현상이 발생할 수 있으므로,

의 값이 되도록 하여 깁스 현상을 감소시킴과 동시에 성능을 향상시키도록 한다.The present invention also relates to an odd east-west wave number m.

And even frequency

May cause Gibbs,

To reduce the Gibbs phenomenon to improve the performance.

The present invention is applied to the global meteorological observation data as follows.

값을 나타낸 것이다. 또한, powero가 0이고 powere가 1.0인 자료는 기존 기술에 따른 깁스 현상을 나타낸 것이다. 또한, 해상도는 1024ㅧ512 격자이며, 최대파수는 340이고, 비교모수는 홀수파수에 대한

이 0, 0, 0.5, 0, 0, 0이며 짝수파수에 대한

이 1, 0.5, 0.5, 1/128, 0.001, 0.0001이다.FIG. 4 shows the cast phenomenon that occurs when a small component is removed by adding a vortex field of typhoon scale arbitrarily to the vortex of the global meteorological observation data, and FIG. It is a graph showing the cross section. Where powero and powere correspond to odd and even frequencies, respectively.

The value is shown. In addition, data with powero of 0 and powere of 1.0 indicates the Gibbs phenomenon according to the existing technology. Also, the resolution is 1024 ㅧ 512 grating, the maximum frequency is 340, and the comparison parameter is for the odd frequency

Is 0, 0, 0.5, 0, 0, 0 and for even frequency

These are 1, 0.5, 0.5, 1/128, 0.001, and 0.0001.

의 값이 0.01보다 작을 경우에는 큰 차이가 없으나, 홀수파수에 대한

이 0이며 짝수파수에 대한

이 0.0001일 때 깁스 현상이 줄어드는 것을 알 수 있다. 즉, 도 4에서 중심부에 발생된 깁스 현상이 본 발명을 적용했을 때 감소되는 것을 볼 수 있으며, 도 5에서 그래프의 상하 변화가 클수록 깁스 현상이 큰 것을 의미하므로 이 역시 본 발명에서 깁스 현상이 감소되는 것을 알 수 있다.4 and 5,

If the value of is less than 0.01, there is no big difference.

Is 0 and for even frequency

At 0.0001, the cast phenomenon decreases. That is, it can be seen that the cast phenomenon generated in the center in Figure 4 is reduced when applying the present invention, the larger the vertical change in the graph in Figure 5 means that the larger the cast phenomenon, this also reduces the Gibbs phenomenon in the present invention It can be seen that.

Spectralizing the lattice data with the transient equation (S _2-2 ) is performed by applying the double Fourier series function of the step (S _2-1 ) of the spherical coordinate system lattice data into the Fourier series in the longitudinal direction to the basic differential operator. Apply the solution to the differential equation.

First, the transient equation and its spectral form in the sphere can be expressed relatively simply by using the spherical coordinate system, and when the time and length of all variables are dimensionless with the inverse of the Earth's angular velocity and the radius of the Earth, Equation 4

[Equation 4]

는

이고,

는

이며,

는

이다. 또한, 여기서,

는 동서방향의 속도이고,

는 남북방향의 속도이며,

는 행성와도이고,

는 상대와도를 의미한다.At this time,

Is

ego,

Is

,

Is

to be. Also, here,

Is the speed in the east-west direction,

Is the speed in the north-south direction,

Is also with the planet,

Means also your opponent.

In addition, the speed and the eddy are represented by wired functions, and the speed is expressed in spectral form as shown in Equation 5.

[Equation 5]

In this case, Equation 5 is a differential operator in the north-south direction.

In addition, the Laplacian operator is the same as Equation 6.

[Equation 6]

라 하면,

Say,

Also, the advection operator is the same as Equation 7.

[Formula 7]

On the other hand, the Laplacian operator can be calculated using a triple diagonal matrix.

In addition, for example, the solution of the elliptic differential equation including the Helmholtz equation may be applied to the above-described solution corresponding to the Laplacian operator, which is the same as Equation 8.

Equation 8

Where is a positive coefficient and the filter coefficient.

In addition, if the above-described Laplacian arithmetic equation is applied to a higher-order Laplacian, a harmonic filter may be configured, which is the same as Equation 9.

[Equation 9]

는 양수로서 필터차수이며,

는 주어진 강제함수 즉, 필터링 대상이 되는 함수이고,

는 필터링 즉, 역산된 함수를 의미한다. 이와 같은 본 발명에 따른 고차수 조화함수 필터를 영상자료의 복원 및 노이즈 제거와 지형자료의 평활화 등에 적용할 수 있다.here,

Is a positive order filter order,

Is the given coercion function, that is, the function to be filtered

Means filtering, or inverse function. Such a high-order harmonic filter according to the present invention can be applied to reconstruction of image data, noise removal, and smoothing of terrain data.

Next, the application of the above-described digital filtering method for processing 3D spatial data according to the present invention to restore 3D image data will be described with reference to the accompanying drawings. Among the contents to be described later, overlapping descriptions of the above-described digital filtering method for processing 3D spatial data of the present invention will be omitted or briefly described.

FIG. 6 is a non-lattice data showing a human head, FIG. 7 is a grid data obtained by converting FIG. 6 to a grid data, FIG. 8 is a grid data with noise added to the grid data of FIG. Lattice data with noise removed from Lattice data.

First, the non-lattice data of FIG. 6 is converted into grid data through objective analysis. In this case, converting the non-lattice data as shown in FIG. 6 into a grid data is as shown in FIG. 7, and the conversion from the non-lattice data to the grid data is implemented by introducing an objective analysis such as Equation 10.

Equation 10

은 총 자료수이고,

는 변환 후의 격자점에서의 값이며,

는 변환되기 전의 비격자점의 값이고,

는 객관분석 대상이 되는 격자점으로부 터의 거리이며,

는

,

는 비격자 자료간의 평균거리를 나타낸다.here,

Is the total number of data,

Is the value at the grid point after conversion,

Is the value of the non-lattice point before conversion,

Is the distance from the grid point subject to objective analysis,

Is

,

Represents the average distance between non-lattice data.

이며, 해상도는 512×512 격자이다.When the data converted into the lattice data is distorted due to the addition of random noise as shown in FIG. 8, the noise can be removed using a spherical coordinate high-order harmonic filter according to the present invention. In this case, the order of the filter is 12, and the filter coefficient is

The resolution is 512 x 512 grids.

As described above, when the digital filtering method for processing 3D spatial data according to the present invention is applied to the restoration of 3D image data, the noise can be removed as shown in FIG. It is almost identical to the grid data of.

Next, a case in which the digital filtering method for processing 3D spatial data according to the present invention is applied to smoothing of high resolution terrain data will be described with reference to the accompanying drawings. Among the contents to be described later, overlapping descriptions of the digital filtering method for processing 3D spatial data according to the present invention described above will be omitted or briefly described.

FIG. 10 is uneven smoothed terrain data having a lattice resolution of 2048 × 1024, and FIG. 11 is smoothed terrain data of FIG.

의 연산을 통해 처리할 수 있는 것을

의 연산으로도 가능하게 할 수 있다. 예를 들어, 위도 및 경도의 해상도가 각각 0.1도인 경우 격자 자료의 개수는 약 5400000개가 되는데, 본 발명을 이에 적용할 경우 기존의 방법보다 계산의 효율이 약 수십 배 내지 백배까지 향상된다. 또한, 본 발명은 푸리에 변환을 사용하기 때문에 정확도는 해상도에 관계없이 매우 높은 수준으로 유지된다.The development of satellite observation systems has made it possible to use very high resolution terrain data. These topographical data need to be smoothed for various purposes, but conventional methods have limitations such as computational efficiency and accuracy in processing high resolution data. Therefore, when the digital filtering method for processing 3D spatial data according to the present invention is applied to the smoothing of high resolution terrain data,

Can be handled by

You can also do this with For example, when the resolution of latitude and longitude is 0.1 degrees, the number of lattice data is about 5400000 pieces. However, when the present invention is applied thereto, the efficiency of calculation is improved by several tens to one hundred times than the conventional method. In addition, since the present invention uses a Fourier transform, the accuracy is maintained at a very high level regardless of the resolution.

이며, 해상도는 2048×1024 격자이다.Comparing the unsmoothened terrain data of FIG. 10 with the smoothed terrain data according to the present invention of FIG. 11, it can be seen that the whole is smoothed. In particular, it can be seen that the terrain of the Tibetan plateau region has changed considerably smoothly. In this case, the order of the filter is 3, and the filter coefficient is

The resolution is 2048 x 1024 grids.

Thus, when the present invention is applied to the smoothing of high resolution terrain data, it can be smoothed at high speed while maintaining high accuracy.

Next, a case in which the digital filtering method for processing 3D spatial data according to the present invention is applied to the smoothing of super high resolution terrain data will be described with reference to the accompanying drawings. Among the contents to be described later, overlapping descriptions of the digital filtering method for processing 3D spatial data according to the present invention described above will be omitted or briefly described.

FIG. 12 shows uneven smoothed terrain data having a lattice resolution of 21600 × 0800, and FIG. 13 shows smoothed terrain data of FIG. At this time, the enlarged region of FIGS. 12 and 13 is an enlarged Hawaiian island to see the effect of the smoothing.

이다.This embodiment applies when the resolution of the terrain data is about 1000 times larger than the above-described embodiment, and the terrain data used in this embodiment has a resolution of 21600 ㅧ 10800 (about 2Km × 2Km) and the number of grids is about 200 million or more. to be. In addition, the order of the filter for processing such terrain data is 3, and the filter coefficient is

to be.

When the digital filtering method for processing 3D spatial data according to the present invention is applied to the smoothing of ultra-high resolution terrain data, the unsmoothened terrain data of FIG. 12 can be smoothed as shown in FIG. 13. In particular, it can be seen that the islands of Hawaii have changed slowly.

As such, the present invention can accurately process terrain data of very high resolution.

Next, a case in which the digital filtering method for processing 3D spatial data according to the present invention is applied to noise filtering from a photographic image image will be described with reference to the accompanying drawings. Among the contents to be described later, overlapping descriptions of the digital filtering method for processing 3D spatial data according to the present invention described above will be omitted or briefly described.

14 is a circuit diagram including noise, and FIG. 15 is a circuit diagram of removing noise of FIG. 14.

In some cases, it is difficult to identify a picture because noise is included in image data such as a picture. Even in this case, the present invention can be used quite usefully.

It can be seen that the parts which were impossible to identify due to the noise appear very clearly.

이며, 해상도는 1024×512 격자이다.14 is a planar image data, it is preferable to attach the photographic data to a narrow area centered on the equator as much as possible in order to process it as a part of the spherical coordinate data. This embodiment exemplifies what is set to 30 degrees in the longitude and latitude directions, respectively. In this case, since it is affixed on a curved surface, distortion of a photograph is concerned, but it has little effect on a practical surface. At this time, the filter order is 8, and the filter coefficient is

The resolution is 1024 × 512 grid.

As described above, when the digital filtering method for processing 3D spatial data according to the present invention is applied to noise filtering from a photographic image image, parts which cannot be identified due to noise as shown in FIG. 14 are very clearly shown as shown in FIG. 15. It can be seen.

Although described above with reference to the drawings and embodiments, those skilled in the art can be variously modified and changed within the scope of the invention without departing from the spirit of the invention described in the claims below. I can understand.

For example, the present invention can be applied to various fields such as medicine, space science, earth science, biology, and the like.

1 is a view for explaining the grid data conversion of the three-dimensional spatial data processing digital filtering method according to the present invention, Figure 2 is a view showing the boundary conditions of the three-dimensional spatial data processing digital filter according to the present invention, Figure 3 is a schematic diagram showing the difference between the polar boundary conditions according to the east-west wave number of the three-dimensional spatial data processing digital filter according to the present invention.

FIG. 4 shows the cast phenomenon that occurs when a small component is removed by adding a vortex field of typhoon scale arbitrarily to the vortex of the global meteorological observation data, and FIG. It is a graph showing the cross section.

FIG. 6 is a non-lattice data showing a human head, FIG. 7 is a grid data obtained by converting FIG. 6 to a grid data, FIG. 8 is a grid data with noise added to the grid data of FIG. Grid material with noise removed from the grid data

FIG. 10 is uneven smoothed terrain data having a lattice resolution of 2048 × 1024, and FIG. 11 is smoothed terrain data of FIG.

FIG. 12 is uneven smoothed terrain data having a lattice resolution of 21600 × 10800, and FIG. 13 is smoothed terrain data of FIG.

14 is a circuit diagram including noise, and FIG. 15 is a circuit diagram of removing noise of FIG. 14.

Claims (5)

In the digital filtering method for three-dimensional spatial data processing by spectralizing an arbitrary function of a spherical surface by a double Fourier series, The basis function of the double Fourier series includes a function that satisfies an extreme condition, The function has a first derivative of zero using an even function when the east-west wave is even, And a single sinusoidal function when the east-west wave is odd. delete delete delete delete

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