JPH11353306A - Two-dimensional data interpolating system - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a two-dimensional(2D) data interpolating system capable of reducing operation quantity and the occurrence of errors. SOLUTION: A data processor is provided with a discrete value extracting part 10, an X direction sampling function operating part 20, an X direction convolution operating part 30, a Y direction sampling function operating part 40, and a Y direction convolution operating part 50. The extracting part 10 extracts pixel data included in a prescribed range around a noticed point to be interpolated, the operating parts 20, 30 are used for calculating an interpolation value (X direction interpolation value) by using the limited number of sampling functions to be differentiated only once in the whole area based on pixel data of which Y coordinate is the same. The operating parts 40, 50 are used for calculating an interpolation value corresponding to the noticed point by using these sampling functions based on the X direction interpolation value.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

[0001] 1. Field of the Invention [0002] The present invention relates to a two-dimensional data interpolation system for interpolating values between discrete data arranged in a two-dimensional space. In this specification, a case where the value of a function has a finite value other than 0 in a local region and becomes 0 in other regions is referred to as “finite base” and described. .

[0002]

2. Description of the Related Art Conventionally, as a data interpolation method for obtaining a value between given sample values, a method of performing data interpolation using a sampling function has been known.

FIG. 8 is an explanatory diagram of a sampling function called a sinc function which has been conventionally known. This sinc
The function appears when the Dirac delta function is subjected to the inverse Fourier transform, and becomes 1 only at the sample point at t = 0, and becomes 0 at all other sample points. Specifically, si
When the sampling frequency is f, the nc function is

[0004]

(Equation 1)

[0005] According to the equation (1), the interpolation by the sinc function is sin sπf (tk
T)｝ / πf (t−kT)
It can be seen that this is realized by performing a so-called convolution operation that is shifted by T and multiplied by the sample value and added.

FIG. 9 is an explanatory diagram of data interpolation using the sampling function shown in FIG. As shown in the figure, values other than each sample point are interpolated using all sample values.

Further, two-dimensional data such as an image can be interpolated by using the above-described data interpolation technique. As a conventional method used for image data interpolation processing, a nearest neighbor interpolation method, a bilinear interpolation method, a tertiary convolution interpolation method, and the like are known.

For example, when obtaining the value of image data to be interpolated (interpolated) by the cubic convolution interpolation method, discrete data of two pixels before and after each pixel in the x direction and the y direction with respect to a point of interest are obtained. When P _11, P _12, etc.,
The value P of the interpolation data is

[0009]

(Equation 2)

Is calculated by

Where f (t) is

[0012]

(Equation 3)

The sinc function is approximated by a cubic function.

[0014]

By the way, the above-mentioned s
When the inc function is used as the sampling function, it is theoretically possible to obtain an accurate interpolation value by adding the values of the sampling functions corresponding to the sampling points from −∞ to + ∞ by convolution. it can. However, when the above-described interpolation calculation is actually performed by various processors, etc., the processing is terminated in a finite interval, so an error due to the termination occurs, and when the interpolation calculation is performed using a small number of sample values, There was a problem that sufficient accuracy could not be obtained.

For example, in the case of the third-order convolution interpolation method shown in the equation (2), sin is used to simplify the calculation.
The c-function is approximated by a cubic function, and the calculation is performed assuming that there is no effect of pixels separated by two pixels or more, and errors increase.

The present invention has been made in view of the above points, and an object of the present invention is to provide a two-dimensional data interpolation method which can reduce the amount of calculation and has few errors.

[0017]

In order to solve the above-mentioned problems, a two-dimensional data interpolation method according to the present invention uses a sampling function that is finitely differentiable and has a finite number of values.
Interpolation between discrete data arranged at equal intervals on a two-dimensional space defined by two variables is performed for each of the two variables, and for each of the two variables, only the discrete data included in this finite range is Since it is sufficient to perform the interpolation calculation, the amount of calculation is small, and no truncation error occurs, so that good interpolation accuracy can be obtained.

In particular, as the above-mentioned sampling function, it is preferable to use a function that can be differentiated only once over the entire area of a finite range. Various signals that exist in nature,
It is considered that differentiability is necessary because it changes smoothly, but the number of differentiable times does not necessarily have to be infinite. Conceivable.

As described above, there are many advantages by using a sample function that is finitely differentiable and finite, but
Conventionally, it has been considered that there is no sampling function that satisfies such a condition. However, the research by the present inventors has found a function that satisfies the above conditions.

Specifically, the sampling function H (t) to which the present invention is applied is -F (t + 1/2) / 4 + F (t), where F (t) is a third-order B-spline function. -F (t-
1/2) / 4. This sampling function H
(T) is differentiable only once in the entire region, and t = ± 2
Is a finite function whose value converges to 0, and satisfies the above two conditions. By performing interpolation between discrete data using such a function H (t), an interpolation operation with a small amount of calculation and high accuracy can be performed.
Therefore, for example, when image data existing in a two-dimensional space is considered as discrete data, highly accurate real-time processing can be performed.

The third-order B-spline function F
(T) is (4t ² ) for −3.2 ≦ t <− ／.
+ In 12t + 9) / 4, with -2t ² + 3/2 for -1 / 2 ≦ t <1/2, the 1/2 ≦ t <3/2 may be represented by ^{(4t 2 -12t + 9) /} 4 Since the above-described operation of the sampling function can be performed by such a piecewise polynomial using a quadratic function, the content of the operation is relatively simple and the amount of operation can be reduced.

Further, instead of using a B-spline function to represent a sampling function as described above, it is also possible to represent a sampling function using a quadratic piecewise polynomial. Specifically, −2 ≦ t <−3
/ −2 is (−t ² −4t−4) / 4, −3/2
For ≦ t <−1, (3t ² + 8t + 5) / 4, and −
For 1 ≦ t <− ／, (5t ² + 12t + 7) /
In (4), for −１ ／ ≦ t <１ ／, (−7t ² +
4) / 4, for 1/2 ≦ t <1, (5t ² −12)
(t + 7) / 4, and for 1 ≦ t <3/2, (3t ² −
8t + 5) / 4 and (−t ² ) for 3/2 ≦ t ≦ ^2.
By using the sampling function defined by + 4t-4) / 4, the above-described interpolation processing can be performed.

Further, in the two-dimensional data interpolation system of the present invention, in order to perform the above-described interpolation operation, discrete data extraction means, first and second sampling function operation means, first and second convolution operation means It has. A plurality of discrete data existing in a predetermined range around a point of interest on a two-dimensional space defined by two variables is extracted by the discrete data extracting means. Then, first, the first sampling function calculating means and the first
The convolution operation means performs a convolution operation using one of the two variables using the above-mentioned sampling function, and then performs the second sampling function operation means and the second convolution operation means on the sample described above for the other of the two variables. The convolution operation is performed using the conversion function. In this way, data interpolation between multiple discrete values can be performed simply by calculating the value of the sampling function separately for each of the two variables and performing a convolution operation on the result, which is necessary for the interpolation process. The amount of processing can be greatly reduced, and the truncation error is eliminated by using a finite number of sampling functions as described above, so that the processing accuracy can be improved.

[0024]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS A data processing apparatus according to one embodiment to which a two-dimensional data interpolation method according to the present invention is applied uses a sampling function which is finitely differentiable and has a finite number of values. Is characterized by performing interpolation between discrete data arranged at regular intervals. Hereinafter, a data processing apparatus according to an embodiment will be described in detail with reference to the drawings.

FIG. 1 is a diagram showing the configuration of the data processing apparatus of the present embodiment. The data processing device shown in FIG. 1 performs an interpolation process based on input discrete data in a two-dimensional space, and includes a discrete value extraction unit 10, an X-direction sampling function operation unit 20, an X-direction convolution operation unit. 30, a Y-direction sampling function operation unit 40, and a Y-direction convolution operation unit 50. Hereinafter, as discrete data in the two-dimensional space, pixel data including, for example, image density data and color data is considered.

The discrete value extracting unit 10 extracts and holds, from sequentially input pixel data, data corresponding to a plurality of pixels included in a predetermined range around a point of interest to be interpolated. FIG. 2 is a diagram illustrating a range of pixel data extracted around a point of interest. As shown in the figure, if a point of interest to be interpolated is p and its coordinates are (x, y), a rectangular area of two pixels before and after in the X direction and the Y direction with respect to this point of interest p Is the extraction target range, the discrete value extraction unit 10 extracts pixel data corresponding to each of a total of 16 pixels included in this range.

The X-direction sampling function calculator 20 calculates the point of interest p
When the coordinates (x, y) are designated, the distance along the X direction between the pixel corresponding to the extracted pixel data and the point of interest p is calculated, and each pixel is calculated based on the calculated distance. Compute the value of the sampling function corresponding to. In this way, the value of the sampling function is calculated for each of the 16 pieces of pixel data output from the sample value extracting unit 10.

The X-direction convolution operation unit 30 multiplies each of the 16 sampling function values calculated by the X-direction sampling function operation unit 20 by the value of each pixel data, and compares the result with the same Y coordinate. The convolution operation along the X direction is performed by adding for each series. The value obtained by this convolution operation is an interpolation value for each X direction, and as indicated by “*” in FIG. 3, based on the pixel data of each pixel along the X direction, Interpolated values (hereinafter, referred to as “X-directional interpolated values”) corresponding to the four pixels A, B, C, and D having the Y coordinate are obtained.

When the coordinates (x, y) of the point of interest p are designated, the Y-direction sampling function calculation unit 40 calculates the coordinates of the pixel corresponding to each X-direction interpolation value and the point of interest p in the Y direction. The distance along the X-direction is calculated, and the value of the sampling function corresponding to each X-direction interpolation value is calculated based on the calculated distance. Thus, the value of the sampling function is calculated for each of the four X-direction interpolation values calculated by the X-direction convolution operation unit 30.

The Y-direction convolution operation unit 50 multiplies each of the four sampling function values calculated by the Y-direction sampling function operation unit 40 by each X-direction interpolation value, and adds the results. A convolution operation corresponding to the four X-direction interpolation values is performed. The value obtained by this convolution operation is the interpolation value finally obtained corresponding to the point of interest p.

The above-described discrete value extracting unit 10 serves as discrete data extracting means, the X-direction sampling function operating unit 20 serves as first sampling function operating means, and the X-direction convolution operating unit 30 serves as first convolution operating means. , The Y-direction sampling function operation unit 40
And the Y-direction convolution operation unit 50 corresponds to the second convolution operation means. Also, X
The direction interpolation value corresponds to the first interpolation value, and the interpolation value corresponding to the point of interest p corresponds to the second interpolation value.

Next, the details of the data interpolation processing performed by the above-described data processing device will be described. FIG. 4 is an explanatory diagram of a sampling function used in the calculations in the X-direction sampling function calculation unit 20 and the Y-direction sampling function calculation unit 40. The sampling function H (t) shown in FIG. 4 is a finite function focusing on differentiability. For example, the sampling function H (t) is differentiable only once in the entire region, and the sampling position t along the horizontal axis is from -2. This is a finite function having a finite value other than 0 when +2. Also, since H (t) is a sampling function, t =
It has a feature that it becomes 1 only at a sample point of 0 and becomes 0 at a sample point of t = ± 1, ± 2.

It has been confirmed by the present inventors that a function H (t) that satisfies the above-described various conditions (sampling function, one-time differentiability, finite base) exists. Specifically, when the sampling function H (t) is a third-order B-spline function as F (t), H (t) = − F (t + ／) / 4 + F (t) − F (t
−1/2) / 4.

Here, the third-order B-spline function F (t)
It ^{is, (4t 2 + 12t + 9} ) / 4; -3 / 2 ≦ t <-1/2 -2t 2 +3/2; -1 / 2 ≦ t <1/2 (4t 2 -12t + 9) / 4; 1/2 ≦ t <3/2.

The above-mentioned sampling function H (t) is a quadratic piecewise polynomial and uses the third-order B-spline function F (t), so that only one differentiability is guaranteed in the entire region. It is a finite function. Also, it becomes 0 at t = ± 1, ± 2.

As described above, the above-described function H (t) is a sampling function, is a differentiable function only once in the entire region, and is a finite function that converges to 0 at t = ± 2. By performing superposition based on each pixel data using the sampling function H (t), it is possible to interpolate the value between discrete pixel data using a function that can be differentiated only once.

In the case where the above-described interpolation processing using the sampling function is extended to interpolation processing based on pixel data discretely present in a two-dimensional space (XY plane) as shown in FIG. As shown in FIG. 3, first, interpolation processing is performed along the X direction, and the same X as the target point p to be finally obtained is obtained.
An interpolation value for each Y coordinate having coordinates is obtained, and then this X value is calculated.
The interpolation process may be performed again along the Y direction using the direction interpolation value to obtain the final interpolation value P.

FIG. 5 is a diagram showing the relationship between pixel data arranged at regular intervals in the X direction and interpolated values therebetween. For example, the interpolated value corresponding to pixel A shown in FIG. The relationship with surrounding pixel data having coordinates is shown. Y coordinate is Y _{j + 1} and X coordinate is X _{i + 1} , X _{i + 2} ,
Pixel data of X _{i + 3} and X _{i + 4} are represented by P _{i + 1, j + 1} ,
Let P _{i + 2, j + 1} , P _{i + 3, j + 1} , P _{i + 4, j + 1} be the X coordinate X
Predetermined position Xa between _{i + 2} and Xi _{+ 3} (distance a from Xi _{+ 2} )
Let us consider a case where an interpolation value P _{j + 1} corresponding to is obtained.

[0039] Generally, in order to determine the interpolated value P _{j + 1} by using the sampling function, obtains the value of the sampling function at the position of the interpolated value P _{j + 1} for each of the surrounding pixel data, using the same By performing the convolution operation, the interpolation value P _{j + 1 is obtained.}
Can be requested. Since the sinc function is a function that converges to 0 at a sample point of t = ± 、, if the interpolation value P _{j + 1 is} to be obtained accurately, it corresponds to each pixel data of each X coordinate up to X = ± ∞. Then, it is necessary to calculate the value of the sinc function at the position of the interpolation value P _{j + 1} , and to perform the convolution operation using this.

However, the sampling function H (t) used in this embodiment converges to 0 at t = ± 2 sampling points.
= 2, that is, a total of four pixel data, two before and after the interpolated value P _{j + 1} . Therefore, in order to obtain the X-direction interpolation value P _{j + 1} shown in FIG. 5, four pixel data P _{i + 1} whose X coordinates are X _{i + 1} , X _{i + 2} , X _{i + 3} , and X _{i + 4} are obtained. _{, j + 1} , P _{i + 2, j + 1} ,
Only P _{i + 3, j + 1} and P _{i + 4, j + 1} need to be considered, and the amount of calculation can be greatly reduced. In addition, other pixel data, which should be originally considered, are not neglected in consideration of the calculation amount and the accuracy, etc., and do not need to be considered theoretically, so that a truncation error does not occur. .

FIG. 6 is a detailed explanatory diagram of the interpolation processing by the X-direction sampling function operation unit 20 and the X-direction convolution operation unit 30. As the procedure of the interpolation processing, FIG.
As shown in (D), four pieces of pixel data P _{i + 1, j + 1} , P
_{For each of i + 2, j + 1} , _{Pi + 3, j + 1} , and _{Pi + 4, j + 1} , t = 0 (center position) of the sampling function H (t) shown in FIG. And the value of the sampling function at each interpolation position Xa at this time is determined.

For example, the pixel data P _{i + 1 and j + 1 at} X _{i + 1} shown in FIG. 6A will be specifically described. The distance between the interpolation position Xa and the pixel position X _{i + 1} is 1 + a when the distance between the pixel positions is normalized to be 1. Therefore, when the center position of the sampling function H (t) is adjusted to the pixel position X _{i + 1} , the value of the sampling function at the interpolation position is H
(1 + a). Actually, in order to adjust the peak height at the center position of the sampling function H (t) so as to match the pixel data P _{i + 1, j + 1} , the above-mentioned H (1 + a) is calculated by P
_The value H (1 + a) · P _{i + 1, j + 1} multiplied by _{i + 1, j + 1} is the value to be obtained. In the configuration shown in FIG. 1, H (1 + a) is calculated by the X-direction sampling function calculation unit 20, and an operation of multiplying H (1 + a) by P _{i + 1 and j + 1} is performed by the X-direction convolution calculation unit 30.

Similarly, as shown in FIGS. 6B to 6D, the interpolation position Xa corresponding to the other three pixel data is set.
, H (a) · P _{i + 2, j + 1} , H (1-
a) · P _{i + 3, j + 1} and H (2-a) · P _{i + 4, j + 1} are obtained.

The X-direction convolution operation unit 30 calculates the four operation results H (1 + a) ·
P _{i + 1, j + 1} , H (a) · P _{i + 2, j + 1} , H (1-a) · P
_A convolution operation is performed by adding _{i + 3, j + 1} and H (2-a) · P _{i + 4, j + 1} to obtain an X-direction interpolation value P _{j +} corresponding to pixel A shown in FIG. Outputs ₁ .

The same interpolation calculation is performed for each of the other pixels B, C, and D shown in FIG. 3, and the other three X-direction interpolation values P _{j + 2} , P _{j + 3} , and P _{j + 4} is output from the X-direction convolution operation unit 30.

Next, by using the four X-direction interpolation values output from the X-direction convolution operation unit 30 as described above, interpolation processing along the Y direction is performed, and the interpolation value corresponding to the point of interest p is obtained. Is required.

FIG. 7 is a diagram showing the relationship between four X-direction interpolation values arranged at regular intervals in the Y direction and interpolation values therebetween. As described above, the sampling function H used in the present embodiment
Since (t) converges to 0 at t = ± 2 sample points, a total of four X-direction interpolation values, two upper and lower two points with the point of interest p interposed therebetween, may be taken into account. Therefore, in order to obtain the interpolated value P shown in FIG. 7, the Y coordinate is Y _{j + 1} , Y _{j + 2} , Y _{j + 3} , Y _{j + 4}
Of four X-direction interpolation values P _{j + 1} , P _{j + 2} , P _{j + 3} , P _{j + 4}
You only need to consider

The distance between the interpolation position Yb and the pixel position corresponding to the X-direction interpolation value P _{j + 1} is 1 + b when the distance between the X-direction interpolation values is normalized to be 1. Therefore, the sampling function H is stored at the pixel position corresponding to the X-direction interpolation value P _{j + 1.}
The value of the sampling function at the interpolation position when the center position of (t) is adjusted is H (1 + b). Actually, in order to adjust the peak height at the center position of the sampling function H (t) so as to match the X-direction interpolation value P _{j + 1} , the above-described H (1+
b) a P _{j + 1} times the value H (1 + b) · P j + 1 is the value to be obtained. In the configuration shown in FIG. 1, H (1 + b) is calculated by the Y-direction sampling function operation unit 40, and an operation of multiplying it by P _{j + 1} is performed by the Y-direction convolution operation unit 50.

Similarly, the other three X-direction interpolation values P
_{In accordance with j + 2} , _{Pj + 3} , and _{Pj + 4} , each operation result H (b) · _{Pj + 2} , H (1-b) · at the interpolation position Yb.
P _{j + 3} and H (2-b) · P _{j + 4} are obtained.

The Y-direction convolution operation unit 50 calculates the four operation results H (1 + b) · P _{j + 1} , H
(B) · P _{j + 2} , H (1-b) · P _{j + 3} , H (2-b)
Perform a convolution operation by adding P _{j + 4} ,
An interpolation value P corresponding to the point of interest p (x, y) shown in FIGS. 2 and 3 is output.

As described above, since the data processing apparatus of this embodiment uses a finite number of functions that can be differentiated only once in the entire area as the sampling function, the amount of calculation required for the interpolation processing between pixel data is reduced. Can be significantly reduced. This allows
In the interpolation processing on an image, it is possible to reduce the load on the processing device and to reduce the processing time when a large amount of processing data is handled.

In particular, since only a total of 16 pixel data need be considered as a processing target, the amount of calculation can be reduced, and the sampling function is represented by a simple quadratic piecewise polynomial. The value of the sampling function can be obtained by a simple product-sum operation, and the amount of operation can be further reduced from this point.

Further, since the sampling function used in the present embodiment is of a finite scale, there is no truncation error that occurs when the pixel data to be processed is reduced to a finite number, and the generation of aliasing distortion is prevented. As a result, an interpolation result with a small error can be obtained.

The present invention is not limited to the above embodiment, and various modifications can be made within the scope of the present invention. For example, in the above-described embodiment, the sampling function is a finite number of functions that can be differentiated only once in the entire region. However, the number of differentiable times may be set to two or more. Also,
As shown in FIG. 4, the sampling function of the present embodiment is t = ±
Although it converged to 0 at 2, it may be converged to 0 at t = ± 3 or more.

In the above-described embodiment, the sampling function H (t) is defined using the B-spline function F (t). However, the sampling function H (t) is calculated using a quadratic piecewise polynomial. (−t ² −4t−4) / 4; −2 ≦ t <−3/2 (3t ² + 8t + 5) / 4; −3 ≦≦ t <−1 (5t ² + 12t + 7) / 4; −1 ≦ t ^{<-1/2 (-7t 2 +4) /} 4; -1 / 2 ≦ t <1/2 (5t 2 -12t + 7) / 4; 1/2 ≦ t <1 (3t 2 -8t + 5) / 4; 1 ≦ t <3/2 (−t ² + 4t−4) / 4; 3/2 ≦ t ≦ 2.

Further, in the above-described embodiment, interpolation processing is first performed along the X direction using pixel data corresponding to each pixel arranged two-dimensionally, and then X obtained by this interpolation processing is obtained. The interpolation processing is performed along the Y direction using the direction interpolation value, and finally the interpolation value P corresponding to the point of interest p (x, y) is obtained. However, the order in which the interpolation processing is performed is changed. You may. That is, first, Y
An interpolation process is performed along the direction, and then an interpolation process is performed along the X direction using the Y-direction interpolation value obtained by the interpolation process, and finally an interpolation process corresponding to the point of interest p (x, y) is performed. The value P may be obtained.

[0057]

As described above, according to the present invention, an interpolation operation between discrete data is performed using a sampling function that is finitely differentiable and has a finite number of values. Since only the discrete data included in the section needs to be subjected to the interpolation calculation, the amount of calculation is small, and no truncation error occurs, so that an interpolation result with a small error can be obtained.

[Brief description of the drawings]

FIG. 1 is a diagram illustrating a configuration of a data processing device according to an embodiment.

FIG. 2 is a diagram showing a range of pixel data extracted around a point of interest.

FIG. 3 is a diagram illustrating a relationship between pixel data arranged at regular intervals along an X direction and an interpolation position therebetween.

FIG. 4 is an explanatory diagram of a sampling function used in an operation in a sampling function operation unit.

FIG. 5 is a diagram illustrating a relationship between pixel data along the X direction and an X direction interpolation value therebetween;

FIG. 6 is a diagram showing a specific example of calculating an X-direction interpolation value.

FIG. 7 is a diagram showing a relationship between X-direction interpolation values arranged at regular intervals along the Y direction and interpolation values at points of interest therebetween.

FIG. 8 is an explanatory diagram of a sinc function.

FIG. 9 is an explanatory diagram of data interpolation using a sinc function.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 10 Discrete value extraction part 20 X direction sampling function operation part 30 X direction convolution operation part 40 Y direction sampling function operation part 50 Y direction convolution operation part

Claims (6)

[Claims] 1. A convolution corresponding to a plurality of discrete data arranged at equal intervals in a two-dimensional space defined by two variables using a sampling function that is finitely differentiable and has a value of a finite order. A two-dimensional data interpolation method, wherein an operation is separately performed for each of the two variables to interpolate a value between the discrete data. 2. The method according to claim 1, wherein the sampling function is
A two-dimensional data interpolation method, wherein the whole area is a function that can be differentiated only once. 3. The sampling function according to claim 1, wherein the sampling function is H (t) = − F (t + ／) / 4 + F (t), where F (t) is a third-order B-spline function. −F (t
A two-dimensional data interpolation method defined by (−1/2) / 4. 4. The method according to claim 3, wherein the third-order B-spline function F (t) is (4t ² + 12t +
9) / 4, for -1 / 2≤t <1/2, -2t ² +3/2, and for 1 / 2≤t <3/2, (4t ² -12t + 9)
A two-dimensional data interpolation method represented by / 4. 5. The sampling function according to claim 1, wherein the sampling function is (−t ² −4t−4) / − 2 ≦ t <−3/2.
4, with respect to −3 ≦ t <−1, (3t ² + 8t + 5) /
4, for -1 ≦ t <− ／, (5t ² + 12t + 7)
For −1 / 2 ≦ t <１ ／, (−7t ² +4) / 4
For 1/2 ≦ t <1, (5t ² −12t + 7) / 4
For 1 ≦ t <3/2, (3t ² −8t + 5) / 4
The two-dimensional data interpolation method, wherein 3/2 ≦ t ≦ 2 is defined by (−t ² + 4t−4) / 4. 6. A discrete data extracting means according to claim 3, wherein said discrete data extracting means extracts a plurality of discrete data present in a predetermined range around a point of interest to be subjected to an interpolation operation in said two-dimensional space. For each of the plurality of discrete data extracted by the discrete data extraction means, along a direction corresponding to one of the two variables defining the two-dimensional space, between the point of interest and each discrete data A first sampling function calculating means for calculating the sampling function H (t1) with a distance being t1, and using a plurality of sampling function values calculated by the first sampling function calculating means, By performing a convolution operation along one of the two variables, a first convolution operation means for obtaining a first interpolation value for each of a plurality of streams along one of the two variables; and a first convolution operation means. The distance between the point of interest and the first interpolated value is defined as t2 along the direction corresponding to the other of the two variables for each of the plurality of first interpolated values extracted by Function H (t
2) using a second sampling function calculating means for calculating 2), and using the values of the plurality of sampling functions calculated by the second sampling function calculating means, to perform a convolution calculation along the other of the two variables And a second convolution operation means for obtaining a second interpolation value corresponding to the point of interest by performing the two-dimensional data interpolation method.