WO1999064988A1

WO1999064988A1 - Two-dimensional data interpolating system

Info

Publication number: WO1999064988A1
Application number: PCT/JP1999/003044
Authority: WO
Inventors: Kazuo Toraichi; Kouichi Wada
Original assignee: Fluency Research & Development Co., Ltd.
Priority date: 1998-06-10
Filing date: 1999-06-08
Publication date: 1999-12-16
Also published as: TW442739B; JPH11353306A

Abstract

A two-dimensional data interpolating system in which the amount of calculation is small and few errors are produced. A data processor includes a discrete value extracting unit (10), an X-direction sampling function calculating unit (20), an X-direction convolution unit (30), Y-direction sampling function calculating unit (40), and an X-direction convoluting unit (50). The discrete value extracting unit (10) extracts data on pixels contained in a predetermined area around a point concerned to be subjected to interpolation. The X-direction sampling function calculating unit (20) and X-direction convoluting unit (30) calculate interpolation values (X-direction interpolation values) by using a sampling function differentiatable only once over the whole region and having finite values based on the data on pixels whose Y coordinates are the same. The Y-direction sampling function calculating unit (40) and Y-direction convolution calculating unit (50) calculate the interpolation values corresponding to the point concerned by using the sampling function based on the X-direction interpolation values.

Description

Description Two-dimensional data interpolation method Technical field

The present invention relates to a two-dimensional data interpolation method 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 a “finite base” and is described. I do. Background art

Conventionally, as a data interpolation method for obtaining a value between predetermined sample values, a method of performing data interpolation using a sampling function has been known.

FIG. 8 is an explanatory diagram of a conventionally known sampling function called a sinc function. This sine function appears when the Dirac delta function is inverse Fourier-transformed, and becomes 1 only at the sample point at t = 0, and becomes 0 at all other sample points. Specifically, the sine function, when the sampling frequency is f,

(0 = ί…). According to this equation (1), interpolation by the sinc function involves shifting the function sin {7rf (t-kT)} /; rf (t-kT) by kT in the time axis direction and multiplying by the sample value. It can be seen that the addition is realized by performing a so-called convolution operation.

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.

In addition, two-dimensional data such as an image can be interpolated by using the above-described overnight interpolation method. As the conventional methods used for the interpolation processing of image data, the nearest neighbor interpolation method, bilinear interpolation method, cubic convolution interpolation method and the like are known. For example, when obtaining the value of image data to be interpolated (interpolated) by cubic convolution interpolation, discrete data of two pixels before and after each point in the X and y directions with respect to the point of interest are P, P Assuming _{1 2} etc., the value P of the interpolation data is

Is calculated by

Where f (t) is

/ ((3)

Which is an approximation of the sinc function described above with a cubic function.

By the way, when the sinc function described above is used as a sampling function, the value of each sampling function corresponding to the sampling points from -∞ to + ∞ is theoretically added by convolution to obtain an accurate interpolation value. Can be obtained. However, if the above-described interpolation calculation is actually performed by various processors or the like, the processing is terminated in a finite interval.Therefore, an error due to the truncation occurs, and the interpolation calculation is performed using a small number of sample values. In such a case, there is a problem that sufficient accuracy cannot be obtained.

For example, in the case of the cubic convolution interpolation method shown in equation (2), in order to simplify the calculation, the sinc function is approximated by a cubic function, and pixels distant by more than two pixels are forced The calculation is performed assuming that there is no effect of the error, and the error increases. Disclosure of the invention

The present invention has been made in view of the following points. An object of the present invention is to provide a two-dimensional de-interpolation method which can reduce the number of errors and has few errors.

The two-dimensional de-interpolation method of the present invention uses sampling functions that are finitely differentiable and have values of a finite order, and are arranged at equal intervals on a two-dimensional space defined by two variables. Interpolation between discrete data is performed for each of the two variables, and for each of the two variables, only the discrete data included in this finite range need to be interpolated. Since the amount is small and no truncation error occurs, 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 existing in the natural world are considered to need to be differentiable because they change smoothly, but the number of differentiable times does not have to be infinite, but rather only once. Then, it is thought that natural phenomena can be sufficiently approximated.

As described above, there are many advantages to using a sampling function that is finitely differentiable and finite, but it has conventionally been thought that there is no sampling function that satisfies such a condition. However, the present inventor's research has found a function satisfying the above conditions.

Specifically, the sampling function H (t) to which the present invention is applied is represented by one F (t + 1/2) / 4 + F (F, where the third-order B-spline function is F (t). t) -F (t-1/2) / 4. This sampling function H (t) is a finite function that can be differentiated only once in the entire region and whose value converges to 0 at t = ± 2, and satisfies the above two conditions. By using such a function H (t) to perform interpolation for a discrete time interval, it is possible to perform an interpolation operation with a small amount of operation and high accuracy. 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.

Also, the above-mentioned third-order B-spline function F (t) is given by (4 t ² + 12 t + 9) / 4 for-3 / 2≤t-1- / 2 can be expressed as — 2 t ² + 3/2, and l / 2 ^ t and 3/2 can be expressed as (4 t ² — 12 t + 9) / 4. Sampling described above by piecewise polynomial Since the operation of the function can be performed, the content of the operation is relatively simple and the amount of operation can be reduced.

Also, instead of using the B-spline function to represent the sampling function as described above, it can be represented by a quadratic piecewise polynomial. Specifically, for -2-3/ ^2, (1 t ^2-4 t-4) / 4 and for 1 3/2 ≤ t <-1, (3 t ² + 8 t + 5) / 4 and (5 t ² + 12 t + 7) / 4 for one l ≤ t <— 1 ^{2 and} (-7 t ² +4) / 4 for — 1/2 ≤ t <1/2 in, about 1/2 district 1 - at ^{(5 t 2 12 t + 7} ) / 4, for 1≤ t <3/2 is - in ^{(3 t 2 8 t + 5} ) / 4, 3/2 ^ for ^{t≤2 (- t 2 + 4 t} - 4) / by using a sampling function is defined by 4, Ru can be performed above interpolation process.

In addition, the two-dimensional data interpolation method of the present invention includes a discrete data extraction unit, first and second sampling function operation units, and first and second convolution operation units in order to perform the above-described interpolation operation. I have. The discrete data extracting means extracts a plurality of discrete data existing in a predetermined range around a point of interest on a two-dimensional space defined by two variables. First, the first sampling function operation means and the first convolution operation means perform a convolution operation on one of the two variables using the above-described sampling function. Then, the second sampling function operation means And the second convolution operation means performs convolution operation on the other of the two variables using the sampling function described above. In this way, it is possible to perform data interpolation between multiple discrete values simply by calculating the value of the sampling function separately for each of the two variables and performing a convolution operation on this result. The amount of processing required can be greatly reduced, and as described above, the use of a finite number of sampling functions eliminates truncation errors, thereby increasing the processing accuracy. BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a diagram showing a configuration of a data processing device of the present embodiment,

Figure 2 is a diagram showing the range of pixel data extracted around the point of interest,

Figure 3 shows the relationship between pixel data arranged at regular intervals in the X direction and the interpolation position between them. Diagram showing the relationship,

FIG. 4 is an explanatory diagram of a sampling function used in the operation in the sampling function operation unit, FIG. 5 is a diagram showing a relationship between pixel data along the X direction and an interpolated value in the X direction therebetween, and FIG. A diagram showing a specific example of calculating an X-direction interpolation value,

FIG. 7 is a diagram showing the 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 the s i n c function,

FIG. 9 is an explanatory diagram of data interpolation using the sinc function. BEST MODE FOR CARRYING OUT THE INVENTION

The data processing apparatus according to an embodiment to which the two-dimensional data interpolation method of the present invention is applied is arranged at regular intervals in a two-dimensional space by using a sampling function that is infinitely differentiable and has a finite number of values. It is characterized in that interpolation between the obtained discrete data is performed. Hereinafter, a data processing device according to an embodiment will be described in detail with reference to the drawings.

FIG. 1 is a diagram showing a configuration of a data processing device of the present embodiment. The data processing device shown in FIG. 1 performs interpolation processing 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, X It is configured to include a 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 a two-dimensional space, pixel data including, for example, image density data and color data are considered.

The discrete value extraction unit 10 extracts and holds, from sequentially input pixel data, those corresponding to a plurality of pixels included in a predetermined range around a point of interest to be interpolated. It is a figure showing the range of pixel data extracted around a point. As shown in the figure, assuming that the point of interest to be interpolated is p and its coordinates are (x, y), two pixels before and after the point of interest P in the X and Y directions Assuming that the rectangular area is an 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.

When the coordinates (x, y) of the point of interest p are specified, the X-direction sampling function operation unit 20 calculates the distance along the X direction between the pixel corresponding to each extracted pixel data and the point of interest p. Is calculated, and the value of the sampling function corresponding to each pixel is calculated based on the calculated distance. In this way, the value of the sampling function is calculated for each of the 16 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 sequence. The value obtained by this convolution operation is an interpolated value for each X direction, and as shown by “*” in FIG. 3, based on the pixel data of each pixel along the X direction, the same value as the point of interest p is obtained. Interpolated values corresponding to those of four pixels A, B, C; and D having Y coordinates (hereinafter referred to as “X-direction interpolated values”) are obtained.

Further, when the coordinates (X, y) of the point of interest p are specified, the Y-direction sampling function operation unit 40 calculates the position of the pixel corresponding to each X-direction interpolation value and the point of interest p along the Y direction. The distance is calculated, and the value of the sampling function corresponding to each X-direction interpolation value is calculated based on the calculated distance. In this way, the value of the sampling function is calculated for each of the four X-direction interpolated values calculated by the X-direction convolution operation unit 30.

The Y-direction convolution operation unit 50 multiplies the values of the four sampling functions calculated by the Y-direction sampling function operation unit 40 by the respective X-direction interpolation values, and adds the results to obtain a value of 4. Perform the convolution operation corresponding to the X-direction interpolation values. 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 extraction unit 10 is used for the discrete data extraction unit, the X-direction sampling function operation unit 20 is used for the first sampling function operation unit, and the X-direction convolution operation unit 30 is used for the first convolution operation. As means, the Y-direction sampling function calculator 40 corresponds to the second sampling function calculator, and the Y-direction convolution calculator 50 corresponds to the second convolution calculator. Further, the X-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, 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 operations in the X-direction sampling function operation unit 20 and the Y-direction sampling function operation unit 40. The sampling function H shown in Fig. 4 (t) is a finite function that focuses on differentiability.For example, the function is differentiable only once in the entire region, and is different from 0 when the sample position t along the horizontal axis is −2 to +2. It is a finite function with finite values. Also, since H (t) is a sampling function, it has the characteristic that it becomes 1 only at the sample point at t = 0, and becomes 0 at the sample points at t = ± l, ± 2.

It has been confirmed by the inventor's research that there exists a function H (t) that satisfies the above-described various conditions (sampling function, one-time differentiable, finite table). Specifically, such a sampling function H (t) can be expressed as follows, assuming that the third-order B-spline function is F (t).

H (t) = — F (t + 1/2) / 4 + F (t) one F (t-1/2)

Can be defined as

Where the third-order B-spline function F (t) is

(4 t ² + 1 2 t + 9) / 4 ≤ t <--2 t ² + 3/2-l / 2≤t <l / 2

^{(4 t 2 - 1 2 t} + 9) / 4 l / 2≤t <3/2

It is represented by

The sampling function H (t) described above is a quadratic piecewise polynomial and uses the third-order B-spline function F (t). Function. In addition, it becomes 0 at t = ± l, ± 2.

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

When the interpolation processing using the sampling function described above is extended to interpolation processing based on pixel data discretely present in a two-dimensional space (X_Y plane) as shown in FIG. As shown in (1), the interpolation process is first performed along the X direction, and finally the interpolation value for each 有する coordinate having the same X coordinate as the target point ρ to be obtained is calculated. The interpolation process is performed again along the Υ direction to obtain the final interpolation value Ρ. You should get it.

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. 3 and the same Y coordinate as this pixel are shown. Is shown with respect to the surrounding pixel data having. The pixel coordinates of Y coordinate _{Yj + 1} and X coordinate Xi _{+ 1} , Xi ₊ 2, Xi ₊ 3, Xi ₊ 4 are represented by Pi + ₁ , j ₊ K Pi + 2, _{j + 1.} , P i ₊ 3, _{j + 1} , P i ₊ 4 _{+ 1,} and an interpolation value P _j corresponding to a predetermined position X a (distance a from X _{i + 2} ) between X coordinates Xi ₊₂ and Xi + 3 _Suppose you want ₊₁ .

Generally, to obtain interpolated values P _{j + 1} by using the sampling function, for it its surrounding pixels de Isseki determined the value of the sampling function at the position of the interpolation value P _{j +1,} and have use this By performing the convolution operation, the interpolation value Ρ ” ₊₁ can be obtained. Since the si nc function is a function that converges to 0 at the sample point of t = ± ∞, if you try to find the interpolation value Ρ ” _{+ 1} accurately, each pixel data of each X coordinate up to X = ± ∞ Correspondingly, it was necessary to calculate the value of the sinc function at the position of the interpolation value P _{j +1} and use this to perform the convolution operation.

However, since the sampling function H (t) used in the present embodiment converges to 0 at the sampling points of t = ± 2, the pixel data up to t = ± 2, that is, the interpolation value Ρ " ₊₁ After that, it is sufficient to take into account a total of four pixel data, two each. Therefore, in order to obtain the X direction interpolation value Ρ ” ₊₁ shown in FIG. 5, four pixel data P _{i +1} , _{j with} X coordinates X _{i + 1} , Xi ₊₂ , Xi ₊₃ , Xi ₊₄ ₊₁ , Pi ₊ 2 _{+ 1}ヽ Pi ₊ 3, j _{+ 1} and Pi ₊ 4 _{+ 1} only need to be considered, and the amount of computation can be greatly reduced. Moreover, other pixel data should be considered originally, but they are not neglected in consideration of the amount of calculation and accuracy, etc., and need not be considered theoretically. do not do.

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. The procedure of the interpolation process, and FIG. 6 (A;) as shown in ~ (D), 4 single pixel data _{_{Pi + 1, j + 1,}} P i + 2, j + 1, P i + 3, j + _The peak heights at t = 0 (center position) of the sampling function H (t) shown in Fig. 4 are matched for each of ₁ , P i ₊ 4 and j ₊₁ . Find the value of the sampling function at the interpolation position Xa of. For example, the pixel data P _{i +1 +1 at} X _{i +1} shown in FIG. _6A will be specifically described. The distance between the interpolation position Xa and the pixel position Xi _{+ 1} is 1 + a when the distance between each pixel position is normalized to be 1. Therefore, the value of the sampling function at the interpolation position when the center position of the sampling function H (t) is adjusted to the pixel position Xi _{+ 1} is H (1 + a). Actually, in order to adjust the peak height at the center position of the sampling function H (t) so that it coincides with the pixel data P _{i + 1} , _j _{+ 1} , the above-mentioned H (1 + a) is converted to P, _{j The} value H (1 + a) · P _i , _j ₊₁ multiplied by ₊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 operation unit 20, and the operation of multiplying H (1 + a) by P _{i +1} and _{j +} _{1 is performed} by the X-direction convolution operation unit 30. Will be Similarly, as shown in FIGS. 6 (B) to 6 (D), corresponding to the other three pixel data, each operation result H (a) · P _{i +2} , _{j + 1} , H (1—a) · PH (2-a) · P _{i +} , _{j +1} are obtained.

The X-direction convolution operation unit 30 calculates the four operation results H (1 + a) _{Pi + 1} , _{j + 1} , H (a) _{Pi + 2} , _{j + 1} , H (1—a) _{.Pi + 3} , _{j + 1} , H (2—a) · _{Pi + 4} , _{j + 1} performs a convolution operation to correspond to pixel A shown in FIG. Outputs the X direction interpolation value P _{j +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 ”and P are converted to the X-direction convolution calculation unit 30. Output from

Next, by using the four X-direction interpolation values output from the X-direction convolution operation unit 30 in this manner, interpolation processing along the Y direction is performed, and an interpolation value corresponding to the point of interest P is obtained. .

FIG. 7 is a diagram illustrating a relationship between four X-direction interpolation values arranged at regular intervals in the Y direction and interpolation values therebetween. As described above, since the sampling function H (t) used in the present embodiment converges to 0 at the sampling points of t = ± 2, a total of four X-direction interpolations, two each, above and below the point of interest p Just take the value into account. Thus, to obtain interpolated values P shown in FIG. 7, Y coordinate _{_{Y j + 1, Y j +}} , Yj + 3, 4 one X-direction interpolation value P of Y, P j _+, P j, P j only Should be considered.

The distance between the interpolation position Yb and the pixel position corresponding to the X-direction interpolation value P _{j +1} is If the distance between the interpolated values is normalized to 1, then 1 + b. Therefore, the value of the sampling function at the interpolation position when the center position of the sampling function H (t) is adjusted to the pixel position corresponding to the X direction interpolation value Ρ ” ₊₁ is H (1 + b). Actually, in order to match the peak height at the center position of the sampling function H (t) so as to coincide with the X-direction interpolation value _{Pj + 1} , the above-mentioned H (1 + b) is calculated as _{Pj + 1} The multiplied value H (1 + b) · P _{j +1} is the desired value. 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, corresponding to the other three X-direction interpolated values P _{j +2} , P _{j +} , and P, each operation result H (b) at the interpolation position YbP _{j +2} , H (1 -b ) · P _{J + 3} , H (2 -b) · P _{j + 4} are obtained.

The Y-direction convolution operation unit 50 calculates the four operation results この (1 + b) 'P _{j + 1} , H (b) · P _{j +} , H (1-b) · P _{J + 3} , H (2 -b) · Ρ is added to perform a convolution operation, and an interpolation value P corresponding to the point of interest (x, y) shown in FIGS. 2 and 3 is output.

As described above, since the data processing device of the present embodiment uses a finite number of functions that can be differentiated only once in the entire region as the sampling function, the amount of calculation required for the interpolation processing between pixel data is greatly reduced. Can be reduced. This makes it possible to reduce the load on the processing device and reduce the processing time when processing inflated processing data in the interpolation processing on the image.

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

In addition, since the sampling function used in the present embodiment is of finite size, conventionally, there is no truncation error that occurs when the pixel data to be processed is reduced to a finite number, and aliasing distortion is prevented. Thus, 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 embodiment, the sampling function Is a finite function that can be differentiated only once in the entire region, but the number of differentiable times may be set to two or more. As shown in FIG. 4, the sampling function of the present embodiment converges to 0 at t = ± 2, but may converge to 0 at t = ± 3 or more. In the embodiment described above, the sampling function H (t) is defined using the B-spline function F (t), but the sampling function H (t) is calculated using a quadratic piecewise polynomial.

(One t ^2-4 t-4) / 4-2≤t <-3/2

(3 t ² + 8 t + 5) / 4-3 / 2≤t <-l

(5 t ² + 1 2 t + 7) / 4-1≤ t <-1/2

(One 7 t ² + 4) / 4-1 / 2≤ t <1/2

^{(5 t 2 - 1 2 t} + 7) / 4 1 / 2≤ t <1

^{(3 t 2 - 8 t +} 5) / 4 1≤ t <3/2

(One t ² + 4 t-4) 3 / 2≤ t≤ 2

Can be equivalently expressed as

In the above-described embodiment, the interpolation processing is first performed along the X direction using the pixel data corresponding to each pixel arranged two-dimensionally, and thereafter, the interpolation processing is performed. Interpolation processing is performed along the Y direction using the X direction interpolation value, and finally the interpolation value P corresponding to the point of interest p (X, y) is obtained, but the order of performing the interpolation processing is changed. It may be. That is, first, interpolation processing is performed along the Y direction, and then interpolation processing is performed along the X direction using the Y direction interpolation values obtained by this interpolation processing, and finally the point of interest p (x, y ) May be obtained. Industrial applicability

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 value of a finite order, and is included in the section of the finite order. Since only the discrete data to be processed need 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.

Claims

The scope of the claims

1. Using a sampling function that is finitely differentiable and has a finite number of values, performs convolution operation corresponding to a plurality of discrete data arranged at equal intervals in a two-dimensional space defined by two variables. The two-dimensional data interpolation method is performed separately for each of the two variables to interpolate a value between the discrete data.

2. The two-dimensional data interpolation method according to claim 1, wherein the sampling function is a function that can be differentiated only once in the entire area.

3. The sampling function is defined as F (t), where the third-order B-spline function is

2. The method according to claim 1, wherein H (t) = — F (t + 1/2) / 4 + F (t) -F (t-1 / 2) / 4. Dimensional data interpolation method.

4. The third-order B-spline function F (t) is

— For 3 / 2≤t <—1/2, (4 t ² + 12 t + 9) / 4, for one half ^ t く 1/2, — 2 t ² +3/2,

For l / 2≤t <3/2 is ^{(4 t 2 - 12 t +} 9) / claims, characterized by being represented by 4 of the third term, wherein the two-dimensional data interpolation method.

5. The sampling function is:

— For 2 ^ t Kui 3/2, (1 t ^2-4 t-4) / 4,

— For 3 / 2≤t <—1, (3 t ² + 8 t + 5) / 4

— For l≤t <—l / 2, (5 t ² + 12 t + 7) / 4

— 1/2 ^ t く 1/2 is (1 7 t ² + 4) / 4,

For 1/2 ≤ t <1, it is (5 t ² — 12 t + 7) / 4,

For 1 t <3/2, (3 t ² — 8 t + 5) / 4,

^2. The two-dimensional data interpolation method according to claim 1, wherein 3 / 2≤t≤2 is defined by (1 t ² +4 t-4) / 4.

6. Discrete data extraction means for extracting a plurality of the discrete data present in a predetermined range around a point of interest to be subjected to an interpolation operation on the 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 that defines the two-dimensional space, between the point of interest and each discrete data The sampling function H (t 1) First sampling function calculating means for calculating

A plurality of sequences along one of the two variables is obtained by performing a convolution operation along one of the two variables using the values of the plurality of sampling functions calculated by the first sampling function operation means. A first convolution operation means for obtaining a first interpolation value for each of the plurality of first interpolation values extracted by the first convolution operation means, each of which corresponds to the other of the two variables Along the direction, a second sampling function calculating means for calculating the sampling function H (t 2) with a distance between the point of interest and the first interpolation value as t 2,

By performing a convolution operation along the other of the two variables using the values of the plurality of sampling functions calculated by the second sampling function operation means, a second interpolation value corresponding to the point of interest is obtained. A second convolution operation means for obtaining

4. The two-dimensional data interpolation method according to claim 3, comprising:

7. Discrete data extracting means for extracting a plurality of the discrete data present in a predetermined range around a point of interest to be subjected to an interpolation operation in the 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 that defines the two-dimensional space, between the point of interest and each discrete data First sampling function calculating means for calculating the sampling function H (t 1) with a distance t 1,

A plurality of sequences along one of the two variables is obtained by performing a convolution operation along one of the two variables using the values of the plurality of sampling functions calculated by the first sampling function operation means. A first convolution operation means for obtaining a first interpolation value for each of the plurality of first interpolation values extracted by the first convolution operation means, each of which corresponds to the other one of the two variables. A second sampling function calculating means for calculating the sampling function H (t 2) with the distance between the point of interest and the first interpolation value as t 2 along the direction of

6. The two-dimensional data interpolation method according to claim 5, comprising: