JPH10320379A - Method for processing two-dimensional data - Google Patents

Abstract

PROBLEM TO BE SOLVED: To perform data processing such as differentiation without making deterioration of data by sequentially extracting a one-dimensional data stream from two-dimensional processing data and calculating an approximation expression by approximating the one-dimensional data stream with a polynomial that uses a Chebyshev function. SOLUTION: Two-dimensional image data D(I, J) consists of pixel position data z in the directions X and Y of each pixel and pixel data y(z) which shows gradation. Dot column image data D(J) of a J-th column are extracted among the data D(I, J) after resetting a variable J that shows the position of a pixel column to be extracted there. An approximate expression is calculated by approximating the extracted one-dimensional data stream D(J) with a polynomial that uses a Chebyshev function. A differential expression of the approximate expression is calculated by differentiating the approximate expression and the differential value of processing data is acquired by calculating the differential value of each data of the data stream D(J) by using the differential expression. In such cases, because an approximate expression that uses the Chebyshev function is an aperiodic function, it approximates even to processing data in which values arbitrarily fluctuate with high accuracy.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a processing method for differentiating, thinning out, inflating and converting resolution of two-dimensional data such as images.

[0002]

2. Description of the Related Art In two-dimensional digital image processing, processing of differentiating image data and extracting a contour portion in an image using the differential value is performed. Such differentiation processing includes the Prewitt (Prewitt) type and Laplacia
An n (Laplacian) type two-dimensional digital filter is generally used. By applying these digital filters to two-dimensional digital image data, a differential value of each pixel can be obtained.

[0003]

However, in the method of differentiating an image using a digital filter, for example, when an image is a multi-tone monochromatic image, every time the digital filter is applied, the gradation between adjacent pixels is reduced. The difference becomes gentle, and the deterioration of the image becomes severe. Then, when the differentiation processing is further repeated, an image having no gradation difference is finally created.

Further, as one application field of image processing,
There is an object strength evaluation process using image processing. In this process, measured values of the physical quantity of the target substance, for example, time-series data of the displacement of the substance surface are displayed using the image data. The amount of displacement is displayed as gradation data of an image. And
The image data is differentiated, the amount of distortion of the target substance is estimated from the obtained differential value, and the intensity of the target substance is evaluated based on the amount of distortion. In this case, the differential value of the image data represents the amount of distortion that serves as a reference for intensity evaluation. Therefore, the differential value of the image data needs to be obtained accurately.

[0005] However, in the conventional differentiation method using a digital filter, the differential value is degraded, and there is an inconvenience in applying to a use requiring an accurate differential value.

SUMMARY OF THE INVENTION An object of the present invention is to provide a method for processing two-dimensional data capable of performing data processing such as differentiation without causing data deterioration.

[0007]

Means for Solving the Problems and Effects of the Invention

(1) First invention A method of processing two-dimensional data according to the first invention sequentially extracts one-dimensional data strings from two-dimensional processed data, and uses the extracted one-dimensional data strings by using a Chebyshev function. Calculates the approximate expression by approximating the approximated polynomial, differentiates the obtained approximate expression, calculates the differential expression of the approximate expression, and calculates the differential value of each data of the one-dimensional data sequence using the calculated differential expression. To obtain the differential value of the processing data.

[0008] Since the approximation formula using the Chebyshev function is a non-periodic function, the approximation can be performed with high precision even on processing data whose value fluctuates arbitrarily. Also, the calculation of the differential equation is easy. Therefore, in the differentiation processing of the two-dimensional data, a high-precision differential value can be easily obtained by calculating a differential value from a differential expression of a high-precision approximate expression.

(2) Second invention The two-dimensional data processing method according to the second invention is the two-dimensional data processing method according to the first invention, wherein a high-order differential expression is used as a differential expression of the approximate expression. The high-order differential value of each data of the one-dimensional data sequence is calculated using the calculated high-order differential expression.

With the polynomial of the Chebyshev function, a higher-order differential equation can be easily calculated. For this reason, when obtaining a higher-order differential value of the processed data, a higher-order differential form of the approximate expression of the processed data is calculated, and the higher-order differential expression is calculated in a single calculation using the calculated higher-order differential expression. The value can be obtained, and the higher-order differential value calculation processing can be performed efficiently.

(3) Third invention A two-dimensional data processing method according to a third invention is the two-dimensional data processing method according to the first invention, wherein the differential value of the calculated processing data is newly processed. One-dimensional data strings are sequentially extracted from the new processed data as data, and the extracted one-dimensional data strings are approximated by a polynomial using a Chebyshev function to obtain an approximate expression, and the obtained approximate expression is differentiated. The differential expression of the approximate expression is calculated, and the differential value of each data of the one-dimensional data string is calculated by using the calculated differential expression to obtain the differential value of new processing data.

In this case, a series of processes of calculating the approximate expression for the processing data, calculating the differential expression of the approximate expression, and calculating the differential value using the differential expression can be repeated to calculate a higher-order differential value.

(4) Fourth Invention The two-dimensional data processing method according to the fourth invention is the two-dimensional data processing method according to the first invention, wherein the one-dimensional one-dimensional data sequentially extracted in one direction from the processed data is provided. Approximating the data sequence with a polynomial using Chebyshev function to obtain an approximate expression, differentiating the obtained approximate expression to calculate a differential expression of the approximate expression, and calculating a differential value using the calculated differential expression Calculates the one-way differential value of the processed data, uses the data consisting of the one-way differential value of the processed data as new processed data, and uses a Chebyshev function for a one-dimensional data sequence that is sequentially extracted in the other direction from the new processed data. Approximate by the polynomial that was found to obtain an approximate expression, differentiate the obtained approximate expression, calculate the differential expression of the approximate expression, calculate the differential value using the calculated differential expression, and calculate the other direction of the new processing data By obtaining the derivative of Physical data 2
This is for calculating a differential value differentiated in the direction.

In this case, in one direction of the two-dimensional data, a series of processes of calculating an approximate expression using a Chebyshev function, calculating a differential expression of the approximate expression, and calculating a differential value using the differential expression is performed. Obtain the differential value of the two-dimensional data differentiated in one direction, calculate the approximate expression using the Chebyshev function for the other direction of the processing data composed of the obtained differential value, calculate the differential expression of the approximate expression, and A series of processes for calculating a differential value using a differential equation is performed. Thereby, the two-dimensional processing data is calculated using the approximate expression of the one-dimensional Chebyshev function.
Differentiation processing in the direction can be performed.

(5) Fifth Invention The two-dimensional data processing method according to the fifth invention is the two-dimensional data processing method according to the fourth invention, wherein the one-dimensional data string extracted in one direction is used. The differential value is a higher-order differential value calculated by using a higher-order differential expression of an approximate expression obtained for a one-dimensional data string extracted in one direction. The differential value of the one-dimensional data string extracted in the other direction is made up of the high-order differential value of the one-dimensional data string extracted in the other direction. It is a higher-order differential value calculated using the following differential equation.

With a Chebyshev function polynomial, a higher-order differential equation can be easily calculated. Therefore, when obtaining a higher-order differential value of the processing data, a higher-order differential expression of the approximation formula of the processing data is calculated, and the higher-order differential value is calculated by one time using the higher-order differential expression. Can be obtained. Therefore, by performing the high-order differential value calculation processing once in each of the two directions of the two-dimensional processing data, the high-order differential value calculation processing in the two directions of the two-dimensional processing data can be efficiently performed. .

(6) Sixth invention A method for processing two-dimensional data according to a sixth invention is directed to the method for processing two-dimensional data according to the fourth invention, wherein the differential value of the processed data differentiated in two directions is obtained. As new processing data,
A one-dimensional data sequence sequentially extracted in one direction from new processing data is approximated by a polynomial using a Chebyshev function to obtain an approximate expression, and the obtained approximate expression is differentiated to calculate a differential expression of the approximate expression. A differential value is calculated using the calculated differential expression to obtain a differential value in one direction of the processed data, and a one-dimensional data sequence sequentially extracted in the other direction from data consisting of the differential value of the processed data in one direction is Chebyshev Calculate the approximate expression by approximating with a polynomial using a function, differentiate the obtained approximate expression, calculate the differential expression of the approximate expression, calculate the differential value using the calculated differential expression, and obtain new processing data To obtain the differential value of.

In this case, a series of processes of calculating the approximate expression, calculating the differential expression of the approximate expression, and calculating the differential value using the differential expression for the processed data is performed in one direction and the other direction of the two-dimensional processed data. , The higher-order differential value of the two-dimensional processing data in two directions can be calculated.

(7) Seventh invention A method of processing two-dimensional data according to a seventh invention sequentially extracts one-dimensional data strings from two-dimensional processed data, and extracts each data of the extracted one-dimensional data strings. Is approximated by a polynomial using a Chebyshev function to obtain an approximate expression, and a one-dimensional data sequence different from the one-dimensional data sequence is calculated using the obtained approximate expression.

Since the approximation formula using the Chebyshev function is a non-periodic function, the approximation can be performed with high accuracy even on processing data whose value fluctuates arbitrarily. Furthermore, the Gibbs phenomenon described below does not occur at an arbitrary position in the data area, and the approximation accuracy is high. Therefore, a data string different from the original data string can be obtained with high accuracy using this approximation formula.

(8) Eighth Invention The two-dimensional data processing method according to the eighth invention is the two-dimensional data processing method according to the seventh invention, wherein the one-dimensional data processing method has a smaller number of data than the one-dimensional data sequence. Is calculated.

By this processing, it is possible to perform so-called data thinning processing in which the number of data of the original processing data is reduced and processing for reducing the resolution of an image with high accuracy.

(9) Ninth Invention The method for processing two-dimensional data according to the ninth invention is the method for processing two-dimensional data according to the seventh invention, wherein the one-dimensional data having a larger number of data than the one-dimensional data sequence Is calculated.

By this processing, so-called data padding processing in which the number of data of the original processing data is increased and processing for improving image resolution can be performed with high accuracy.

(10) Tenth invention A method for processing two-dimensional data according to a tenth invention sequentially extracts one-dimensional data strings from two-dimensional processed data and converts the extracted one-dimensional data strings into Chebyshev functions. Is approximated by a polynomial using, an approximate expression is obtained, a differentiated expression of the approximate expression is calculated by differentiating the obtained approximate expression, and a one-dimensional data sequence different from a one-dimensional data sequence is calculated using the calculated differential expression. This is to calculate the differential value of the data sequence.

Since the approximation formula using the Chebyshev function is a non-periodic function, the approximation can be performed with high accuracy even on processed data whose values fluctuate arbitrarily. Furthermore, the Gibbs phenomenon does not occur at any position in the data area,
High approximation accuracy. Therefore, differential value data of a data sequence different from the original data sequence can be obtained with high accuracy using this approximate expression.

(11) Eleventh Invention The two-dimensional data processing method according to the eleventh invention
In the two-dimensional data processing method according to the invention, the differential value of the one-dimensional data string having a smaller number of data than the one-dimensional data string is calculated.

By this processing, the so-called differential value thinning processing for reducing the number of original differential value data can be performed with high accuracy.

(12) Twelfth Invention A two-dimensional data processing method according to a twelfth invention
In the two-dimensional data processing method according to the invention, the differential value of the one-dimensional data string having more data than the one-dimensional data string is calculated.

By this processing, a so-called differential value inflating process in which the number of original differential value data is increased can be performed with high accuracy.

(13) Thirteenth Invention The method for processing two-dimensional data according to the thirteenth invention includes the following:
In the two-dimensional data processing method according to the twelfth aspect, a one-dimensional data sequence extracted from the two-dimensional processed data is approximated by a polynomial using a Chebyshev function, and the zero-order and first-order terms of the polynomial are obtained. Is omitted to obtain an approximate expression.

The zero-order term of the polynomial defines a bias component of the processing data, and the first-order term defines a straight line component passing through the coordinate origin. Therefore, by omitting the zero-order and first-order terms, it is possible to eliminate the first-order gradient component of the original processing data.

[0033]

DESCRIPTION OF THE PREFERRED EMBODIMENTS In a method for processing two-dimensional data according to the present invention, a large number of discrete processing data (digital data) are approximated by a polynomial, and another approximate expression is calculated by using an approximate coefficient of the approximate expression. For example, the processing of differentiation, thinning, padding, and resolution conversion is performed on the two-dimensional data by generating a differential function formula.

The following polynomial (1) is used as an approximate expression of the processing data. Hereinafter, this approximate expression is referred to as Chebyshev's approximate expression.

[0035]

(Equation 1)

Here, Ti (χ) is a Chebyshev polynomial and is defined by the following equation.

[0037]

(Equation 2)

The coefficient Ai of Chebyshev's approximate expression (1) is

[0039]

(Equation 3)

Is defined as Here, N indicates the number of processing data.

When the processing data D is composed of the data d and its coordinate value z, in the above equations (1) to (3), the variable 座標 on the abscissa corresponds to the coordinate value z of the processing data D, and the variable y
(Χ) corresponds to data d. In the Chebyshev polynomial (2), the variable χ is defined between [−1, 1]. That is, the variable χ takes a value between [−1, 1]. Therefore,
When the processing data whose coordinate value z is within the area [a, b] is approximated by the Chebyshev approximation equation (1), the coordinate value z of the processing data is approximated by the Chebyshev approximation equation (1) using the following equation (4). ) To variable χ.

Χ = {2z− (a + b)} / (ba) (4) Further, Chebyshev's approximate expression (1) can be easily differentiated, and the m-th order differential form is represented by the following expression (5) ).

[0043]

(Equation 4)

Here, Ti ^(m) (χ) is the m-th derivative form of the Chebyshev polynomial (2), and is obtained by differentiating the Chebyshev polynomial (2) m times. Note that m =
When 1, that is, the first-order differential form of the Chebyshev polynomial (2) is represented by the following equation (6).

[0045]

(Equation 5)

Using the above relational expressions (1) to (6), 2
Each process of the dimensional data is performed as described later.

In the method for processing two-dimensional data according to the present invention, the following advantages can be obtained by using an approximate expression using a Chebyshev polynomial.

(I) The Chebyshev polynomial (2) is a non-periodic function, and the approximation accuracy is high even when the approximation is performed with a small order, and the Gibbs phenomenon does not occur. Also, high approximation accuracy can be obtained even at the end of the area of the processing data.

The Gibbs phenomenon is that the original data used for calculating the approximate expression and the approximate data are in good agreement with each other, but the approximate accuracy deviates greatly from the expected data at positions other than the original data, and the approximate accuracy is low. A phenomenon that decreases.

Since this Gibbs phenomenon does not occur,
Inflate, thin out, and scale data using Chebyshev polynomials.
Each process of resolution conversion can be performed with high accuracy.

(Ii) The Chebyshev polynomial can easily obtain a higher-order differential form. Therefore, high-order differentiation processing can be performed efficiently.

(Iii) The zero-order term T ₀ (χ) of the Chebyshev polynomial indicates a bias component, and the first-order term T ₁ (χ)
Indicates a straight line component passing through the coordinate origin. Therefore,
Even when the data to be approximated has a uniform increment or decrement over the entire data area, it can be approximated with high accuracy.

Hereinafter, each processing method of two-dimensional data according to the present invention will be described. (A) Differential processing of two-dimensional data in one direction (1) Differentiation of Chebyshev polynomial using higher-order differential form
Processing Regarding processing for differentiating two-dimensional image data in one direction FIG.
This will be described with reference to FIG. 1A and 1B are explanatory diagrams showing a processing state of two-dimensional data, in which FIG. 1A shows two-dimensional image data to be processed, and FIG. 1B shows a state of differentiation processing in the X direction. FIGS. 2 and 3 are flowcharts of the differentiation process.

In the following, N _× pixel × N shown in FIG.
_A process of differentiating the two-dimensional image data D (I, J) composed of _y pixels in the X direction to obtain a m-th (1 or more integer) -order differential value will be described.

First, the two-dimensional image data D to be processed
(I, J) and various input data are read and stored in the storage device. The two-dimensional image data D (I, J) is composed of pixel position data z in the X and Y directions of each pixel and pixel data y (z) indicating gradation. The input data includes the number of pixels N _x in the X direction, the number of differentiations m, and instruction data indicating whether or not to remove the primary gradient component of the image data. The contents of the primary gradient component will be described later (step S2).

Next, after resetting the variable J indicating the position of the pixel row to be extracted (step S4), the counting of the variable J is started (step S6), and the two-dimensional image data D
The point sequence image data D (J) of the J column shown in FIG. 1B is extracted from (I, J). Then, the point sequence image data D
The pixel position data z in (J) is converted into the variable の of the Chebyshev polynomial (2) using the equation (4) (step S).
8).

Further, the extracted point sequence image data D (J)
Substituting each pixel data y (χ) in the expression (3) into the expression (3), calculates the coefficient Ai (i = 0 to n) of the Chebyshev approximation expression (1). The number n of terms in Chebyshev's approximate expression (1) is
To increase the approximation accuracy equal to set the number N _x of the pixel data of the series of points the image data D (J) to (step S1
0). The calculated coefficient Ai is stored in the storage device (Step S12).

When the coefficient Ai of the Chebyshev approximation (1) is obtained, it is determined whether or not the primary gradient component of the image data is to be removed with reference to the input data (step S14).
Here, the primary gradient component refers to, for example, a uniform change component of the density when the density of the image in the image area changes uniformly in a certain direction. The process of removing the uniform change component from each pixel of the image data is referred to as the removal of the primary gradient component. Note that this process includes a process of uniformly raising or lowering the pixel data value of the entire image data.

When removing the first-order gradient component, the zero-order coefficient A out of the coefficients Ai of the Chebyshev approximation (1) is used.
₀ and the first-order coefficient A ₁ are each replaced with 0 (step S16).

Next, the m-th order differential form T ^(m) (χ) of the Chebyshev polynomial (2) obtained in advance and the above-described step S1
A variable χ is sequentially given to the m-th order differential expression (5) of the Chebyshev approximation formula defined by the coefficient Ai obtained at 0 to calculate the m-th order differential value Y ^(m) (χ) of each pixel in the J column. (Step S18). The calculated m-order differential value Y ^(m) (χ) is stored in the storage device (step S20).

Further, it is determined whether or not the J column subjected to the differentiation processing is the last column N _y . Last unless column N _y of shifts to the m-th order differential processing of the point sequence image data of the next column is performed from step S6 repeats the processing in step S20. When the differentiation processing of all the J columns is completed (step S22), the m-th order differential value Y ^(m) (χ) in the X direction of the two-dimensional image data obtained by the above differentiation processing is output (step S2).
4).

With the above processing, the two-dimensional image data D
A differential value obtained by differentiating (I, J) m times in the X direction is obtained.

As described above, the Chebyfev polynomial (2)
Can easily find a higher-order differential form. Therefore, even in the process of obtaining a higher-order (m ≧ second-order) differential value, a higher-order differential form can be obtained in advance, and a higher-order differential value of image data can be immediately obtained by one calculation.

FIG. 4 is a data diagram showing an example of execution of the above-described differentiation process, and FIG. 5 is a diagram showing the processing result of FIG.
In the present execution example, a process of obtaining a secondary differential value for one-dimensional data whose number of data extracted from the two-dimensional image data is 21 is performed.

In FIG. 4, the first column shows the pixel position (z) of the one-dimensional data, and the second column shows the pixel value y (z) of the one-dimensional data. Also, the third
In the column, an approximate value (hereinafter referred to as an interpolated value) Y (算出) calculated from Chebyshev's approximation formula (1) according to the differentiation processing flow shown in FIGS. 2 and 3 is shown, and in the fourth column, The second derivative calculated from the differential equation (5) of the Chebyshev approximation is shown. In the fifth column, a one-dimensional [1, -2, 1] Laplac
The figure shows a second-order differential value obtained by applying an ian type digital filter to one-dimensional data.

In FIG. 5, the pixel value y (z) in the second column of FIG. 4 is indicated by a black circle as original data. Also, the interpolation value Y (χ) in the third column almost coincides with the original data, and is not shown. The white triangle points indicate the interpolation data obtained by the padding process of the data described later, and the description is omitted here. Further, the solid line indicates the function that was the basis for artificially creating the pixel value y (z) of the one-dimensional data.

As is clear from FIGS. 4 and 5, the pixel value y (z) at each pixel position (z) and the interpolation value Y (χ) obtained from the Chebyshev approximation (1) are in good agreement. I have. As a result, it was found that the approximation accuracy of Chebyshev's approximation formula (1) was high.

FIG. 6 shows the second derivative (the fourth column) based on the Chebyshev approximation formula in FIG.
It is a figure which shows the 2nd derivative value (5th column) by the ian type digital filter. As is clear from FIG.
In the differential processing using Chebyshev's approximation formula, the second derivative is approximated by a smooth function, and Gib as seen in the second derivative by a Laplacian type digital filter is used.
bs phenomenon has not occurred. It was also found that the position of the inflection point was determined with high accuracy.

(2) Calculation of differential form of Chebyshev polynomial
Higher-order Differential Processing to be Repeated In the above-described differential processing of (1), a higher-order differential form of the Chebyshev polynomial is calculated in advance, and a higher-order differential value is calculated by one calculation. A method of repeatedly calculating a higher-order differential value will be described.

The processing object is the same as in the case (1). FIGS. 7 and 8 are flowcharts of the differential processing. In the following, N _x pixels × N _y shown in FIG.
A process for sequentially differentiating the two-dimensional image data D (I, J) composed of pixels in the X direction to obtain an m-th (1 or more integer) -order differential value will be described.

First, the two-dimensional image data D to be processed
(I, J) and various input data are read and stored in the storage device. The image data D (I, J) is composed of pixel position data z and pixel data y (z) indicating the gradation of each pixel in the X and Y directions. Further, the input data includes the number of pixels N _x in the X direction, the number of differentiations m, and instruction data on whether or not to remove the primary gradient component of the image data (step S32).

Next, after resetting the variable J indicating the position of the pixel row to be extracted and the variable M indicating the count value of the number of times of differentiation (steps S33 and S34), the counting of the variables J and M is started (steps S33 and S34). Step S36, S3
8) From among the two-dimensional image data D (I, J), FIG.
The point sequence image data D (J) of the J column shown in (b) is extracted. Then, the pixel position data z in the point sequence image data D (J) is converted into the variable の of the Chebyshev polynomial (2) using Expression (4) (Step S40).

Further, the extracted point sequence image data D (J)
The coefficient Ai (i = 0 to n) of the Chebyshev approximation formula (1) is calculated by substituting the pixel data y (χ) in the formula into the formula (3). The number of terms of the Chebyshev polynomial (2) is set to be equal to the number N of pixel data of the point sequence image data D (J) in order to increase the approximation accuracy (step S42). The calculated coefficient Ai is stored in the storage device (step S4).
4).

When the coefficient Ai of the Chebyshev approximation equation (1) is obtained, it is determined whether or not to remove the primary gradient component of the image data with reference to the input data (step S46).

When removing the first-order gradient component, the 0th-order coefficient A of the coefficients Ai of the Chebyshev approximation (1) is used.
₀ and the first-order coefficient A ₁ are each replaced with 0 (step S48).

Next, a first-order differential equation of the Chebyshev polynomial (2) is calculated, and the first differential equation of the Chebyshev equation defined by the obtained differential equation and the coefficient Ai obtained in step S42.
(M = 1) The variable χ is sequentially given to the differential equation (5) to calculate the primary differential value Y ′ (χ) of each pixel in the J column (step S50). The calculated primary differential value Y ′ (χ) is stored in the storage device (step S52).

With the above processing, one differentiation processing is completed, and a first-order differential value Y ′ (χ) is obtained.

The current number of times of differentiation M is compared with the inputted number of times of differentiation m. If the number of times of differentiation M is smaller than the number of times of differentiation m, the differentiation is repeated again (step S5).
4).

First, the calculated primary differential value Y ′ (χ)
Is replaced with the pixel data y (χ) in the J column (step S5).
6). Then, returning to step S38, step S52
The process up to is repeated. By this processing, the Chebyshev approximation formula (1) is again obtained for the differential value Y ′ (χ) obtained in the first differentiation processing, and its differential form is calculated to obtain a second-order differential value Y ^{( 2)} (χ) is required.

Such a differentiation process is repeated until the number of times of differentiation m is reached. As a result, the m-th order differential value Y ^(m) (の) of the point sequence image data in the first column is calculated.

Further, the processing shifts to the m-th order differentiation processing of the point sequence image data of the next row (step S58), and the processing from step S34 to step S56 is repeated. When the differentiation processing for all the columns is completed (step S58), the m-th order differential value in the X direction of the obtained two-dimensional image data is output (step S60).

With the above processing, the two-dimensional image data D
A differential value obtained by differentiating (I, J) m times in the X direction is obtained.

In the above-described differentiation processing, it is preferable that the differential value of the pixel data y (示 す) indicating the gradation is calculated as a real number during the processing, and is converted into an integer at the time of output. By doing so, the accuracy of the calculation can be increased.

(B) Differentiation Processing of Two-Dimensional Data in Two Directions (1) Differentiation of Chebyshev Polynomial Using Higher Differential Form
The approximate expression (1) of the processing Chebyshev is a one-dimensional function. Therefore, when differentiating two-dimensional data in the two directions of the X direction and the Y direction, a differentiating process is performed in a first direction, for example, the X direction, and a second processing is performed on the obtained differential value. Differentiation processing is performed in the direction, that is, the Y direction.

In the following, N _× pixel × N shown in FIG.
_The two-dimensional image data D (I, J) composed of _y pixels is differentiated in the X direction to obtain an m (one or more integer) derivative value, and further differentiated in the Y direction to an s (one or more integer) order. The process for obtaining the differential value of is described.

In FIG. 1, (a) shows the two-dimensional image data to be processed, (b) shows the state of the differential processing in the X direction, and (c) shows the state of the differential processing in the Y direction. I have. 9 to 11 are flowcharts of the differentiation processing.

First, the two-dimensional image data D to be processed
(I, J) and various input data are read and stored in the storage device. The image data D (I, J) is composed of pixel position data z and pixel data y (z) indicating the gradation of each pixel in the X and Y directions. The input data includes the number of pixels N _x and N _y in the X direction and the Y direction, the number of differentiations m and s, and instruction data as to whether or not to remove the primary gradient component of the image data (step S52). .

Next, after the variables I and J indicating the position of the pixel row to be extracted are reset (step S54), the direction of extracting the pixel data is determined. Here, the X direction is selected (step S56). The counting of the variable J is started (step S58), and point sequence image data D (J) of column J shown in FIG. 1B is extracted from the two-dimensional image data D (I, J). Then, the pixel position data z in the point sequence image data D (J) is converted into a Chebyshev polynomial (2) using the equation (4).
(Step S60).

Further, the extracted point sequence image data D (J)
Substituting each pixel data y (中) in the expression (3) into the expression (3), the coefficient Ai (i = 0 to n) of the Chebyshev approximation expression (1) is calculated (step S62). The calculated coefficient Ai is stored in the storage device (Step S64).

When the coefficient Ai of the Chebyshev approximation equation (1) is obtained, it is determined whether or not to remove the primary gradient component of the image data with reference to the input data (step S66).

When removing the first-order gradient component, the 0th-order coefficient A of the coefficients Ai of the Chebyshev approximation (1) is used.
₀ and the first-order coefficient A ₁ are each replaced with 0 (step S68).

Next, the m-th order differential expression T ^(m) (χ) of the Chebyshev polynomial (2) obtained in advance and the above-described step S
A variable χ is sequentially given to the m-th order differential expression (5) of the Chebyshev approximation formula defined by the coefficient Ai obtained in step 62 to calculate the m-th order differential value Y ^(m) (χ) of each pixel in the J column. (Step S70). The calculated m-th order differential value Y ^(m) (χ) is stored in the storage device (step S72).

Further, the processing shifts to the m-th order differentiation processing of the pixel data of the next point in the series of points, and the processing from step S58 to step S72 is performed.
Is repeated. When the differentiation processing for all the J columns is completed (step S74), the two-dimensional image data D (I,
A differential value obtained by performing the m-order differentiation in the X direction of J) is obtained.

Next, the process proceeds to the s-th order differentiation process in the Y direction (step S76). First, counting of the variable I indicating the position of the pixel column to be extracted is started (step S78), and FIG.
As shown in (c), the J-point sequence image data D _x (I) is selected from the image data D _X (I, J) composed of the two-dimensional m-th order differential value differentiated in the X direction. Extract. Then, the pixel position data z in the point sequence image data D _x (I) is converted into the variable の of the Chebyshev polynomial (2) using Expression (4) (Step S80).

Further, the extracted point sequence image data D
The coefficient Ai (i = 0 to n) of the Chebyshev approximation formula (1) is calculated by substituting the differential value y _x (χ) in _x (I) into the formula (3). Incidentally, the number of terms of the approximate expression of the Chebyshev is equal sets the number N _y of the differential value of the series of points the image data D _x (I) in order to improve the approximation accuracy (Step S82). The calculated coefficient Ai is stored in the storage device (step S8).
4).

When the coefficient Ai of the Chebyshev approximation equation (1) is obtained, one-dimensional two-dimensional image data is referred to with reference to the input data.
It is determined whether to remove the next gradient component (Step S8)
6). When removing the first-order gradient component, the 0th-order coefficients A ₀ and 1 of the coefficients Ai of the Chebyshev approximation (1) are used.
Respectively replaced with 0 next coefficient A ₁ and the (step S8
8).

Next, the s-order differential equation T ^(s) (χ) of the Chebyshev polynomial (2) obtained in advance and the above-described step S
The variable χ is sequentially given to the s-order differential equation (5) of the Chebyshev approximation defined by the coefficient Ai obtained by
An s-order differential value Y ^(s) (χ) of each pixel in the column is calculated (step S90). The calculated s-order differential value Y ^(s) (χ) is stored in the storage device (step S92).

Further, the processing shifts to the s-order differentiation processing of the next point-sequence pixel data in the I-th row, and the processing from step S78 to step S92 is repeated. When the differentiation processing of all I columns is completed (step S94), the differential values of the two-dimensional image data obtained in the above-described differentiation processing in the X and Y directions are output (step S96).

(2) Calculation of differential form of Chebyshev polynomial
A description will be given of another method of high-order differential processing to be repeated and another method of differential processing in two directions. In the method described below, a higher-order differential value of two-dimensional data in the X direction and the Y direction is calculated by alternately performing differential processing in the X direction and the Y direction.

The processing target is the same as in the case (1). 12 to 14 are flowcharts of the main differentiation process. In the following, N _x pixels × N _y shown in FIG.
A process of alternately differentiating the two-dimensional image data D (I, J) composed of pixels in the X direction and the Y direction to obtain an m-th (1 or more integer) -order differential value will be described.

First, the two-dimensional image data D to be processed
(I, J) and various input data are read and stored in the storage device. The image data D (I, J) is composed of pixel position data z and pixel data y (z) indicating the gradation of each pixel in the X and Y directions. The input data includes the number of pixels N _x and N _y in the X direction and the Y direction, the number of differentiations m, and instruction data indicating whether to remove the primary gradient component of the image data (step S102).

Next, after the variable M indicating the count value of the number of times of differentiation processing and the variables I and J indicating the position of the pixel row to be extracted are reset (steps S104 and S106), the differentiation direction of the pixel data is determined. Here, the X direction is selected (step S108). The counting of the variable M and the counting of the variable J are started (steps S110, S11
2) FIG. 1 out of two-dimensional image data D (I, J)
The point sequence image data D (J) of the J column shown in (b) is extracted. Then, the pixel position data z in the point sequence image data D (J) is converted into a variable の of the Chebyshev polynomial (2) using Expression (4) (Step S114).

Further, the extracted point sequence image data D (J)
The coefficient Ai (i = 0 to n) of the Chebyshev approximation formula (1) is calculated by substituting the pixel data y (χ) in the formula into the formula (3). Incidentally, the number of terms of the approximate expression of the Chebyshev is set equal to the number N _x of the pixel data of the series of points the image data D (J) in order to improve the approximation accuracy (step S116). The calculated coefficient Ai is stored in the storage device (step S11).
8).

When the coefficient Ai of the Chebyshev approximation equation (1) is obtained, it is determined whether or not to remove the primary gradient component of the image data with reference to the input data (step S12).
0).

To remove the first-order gradient component, the zero-order coefficient A out of the coefficients Ai of the Chebyshev approximation (1) is used.
₀ and the first-order coefficient A ₁ are each replaced with 0 (step S122).

Next, a first-order differential equation T ′ (χ) of the Chebyshev polynomial (2) is calculated, and is defined by the obtained differential equation T ′ (χ) and the coefficient Ai obtained in step S116. The variable に is added to the first derivative (5) of the Chebyshev approximation.
Are sequentially given to calculate a primary differential value Y ′ (χ) of each pixel in the J column (step S124). The calculated primary differential value Y ′ (χ) is stored in the storage device (step S12).
6).

By the above-described processing, one differentiation processing in the X direction is completed, and a first-order differential value Y ′ (χ) is obtained.

Further, it is determined whether or not the point sequence image data subjected to the above-described differential processing is the last N _y column point sequence image data. If it is not the last point sequence image data _Ny , a differentiation process is performed on the next J column point sequence image data. That is, the processing from step S106 to step S126 is repeatedly performed. When the differentiation processing for all the J columns is completed (step S128), the two-dimensional image data D (I,
A differential value D _x (I, J) obtained by differentiating J) once in the X direction is obtained.

Next, the process proceeds to the differentiation process in the Y direction (step S130). In the differential processing in the Y direction, as shown in FIG. 1C, image data D _x (I, J) composed of two-dimensional primary differential values differentiated in the X direction is converted to point sequence image data in the Y direction. Are sequentially extracted and differential processing is performed.

First, the counting of the variable I is started (step S132), and the point sequence image of column I is selected from the two-dimensional image data D _x (I, J) composed of the differential value in the X direction stored in step S126. The data D _x (I) is extracted. Then, the pixel position data z of the point sequence pixel data D _x (I) is converted into a variable χ using Expression (4) (Step S13)
4).

Further, the extracted point sequence image data D
Substituting the pixel data y _x (χ) in _x (I) into equation (3), the coefficient Ai (i = 0 to Chebyshev approximation equation (1))
n) is calculated. In addition, the number of terms of the Chebyshev polynomial is determined by the point sequence image data D _x (I) in order to improve the approximation accuracy.
Equal to set the number N _y of the pixel data (step S
136). The calculated coefficient Ai is stored in the storage device (Step S138).

When the coefficient Ai of the Chebyshev approximation equation (1) is obtained, it is determined whether or not to remove the primary gradient component of the image data with reference to the input data (step S14).
0). When removing the first-order gradient component, the 0th-order coefficients A ₀ and 1 of the coefficients Ai of the Chebyshev approximation (1) are used.
Respectively replaced with 0 next coefficient A ₁ and a (step S1
42).

Next, the first order differential equation T ′ (χ) of the Chebyshev polynomial (2) is calculated, and the Chebyshev approximation defined by the obtained differential equation T ′ (χ) and the coefficient Ai obtained in step S136. The variable χ is sequentially given to the primary differential equation (5) of the equation to calculate the primary differential value Y ′ (χ) of each pixel in the J-th column (step S144). The calculated primary differential value Y '
(Χ) is stored in the storage device (step S146).

With the above-described processing, one-time differentiation processing in the Y direction with respect to the point-sequence image data of the I-th row is completed, and a first-order differential value Y ′ (χ) is obtained. Further, the process proceeds to the differentiation processing of the point sequence image data of the next column (step S148), and the processing from step S132 to step S146 is repeated.
When the differentiation processing of all J columns is completed (step S14)
8) A first-order differential value obtained by differentiating the two-dimensional image data once in each of the X direction and the Y direction is obtained.

Further, when a higher-order differential value is obtained (step S150), the two-dimensional image data D (I, J) is replaced with the differential value calculated by the above processing (step S152), and the step is repeated. Step S1 from S106
Step 50 is repeated.

(C) Inflating, thinning, and dividing two-dimensional data
Resolution conversion processing Here, each processing of padding, thinning-out, and resolution conversion of two-dimensional image data using the Chebyshev approximation formula will be described. The padding of two-dimensional image data refers to a process of increasing the number of pixels by interpolating new data between original pixel data, and the thinning process reduces the number of pixels by omitting the original pixel data at predetermined intervals. Refers to processing. Furthermore, resolution conversion refers to a process of changing the number of pixels of original pixel data and the size of pixels.

The Chebyshev polynomial has the advantage that the approximation accuracy for one-dimensional data is high and the Gibbs phenomenon does not occur. Therefore, by using the Chebyshev approximation formula (1) using the Chebyshev polynomial, it is possible to perform each of the inflating, thinning-out, and resolution conversion processes with high accuracy.

For example, FIG. 15 shows data of a process for obtaining a second-order differential value for one-dimensional data consisting of 21 pieces of pixel data and for inflating the number of pixels twice. The one-dimensional data to be inflated is the same as the one-dimensional data shown in FIG. Further, the processing for obtaining the secondary differential value is the same as the procedure shown in the flowcharts of FIGS. Further, the result of the padding process shown in FIG. 15 is shown in FIG.

In FIG. 15, the first column shows the pixel position (z) of each pixel of the one-dimensional data, and the second column shows the pixel value y (z) of each pixel. Here, as the pixel position (z), from 0.0 to 20.0.
The value given in 0 increments is the position of the original one-dimensional data, and 0.5 to 19.5 from 0.5 to 19.5 as the pixel position (z).
The value given in increments of 0 is the position of the inflated data.

The third column shows the pixel value (interpolated value) Y (χ) calculated from the Chebyshev approximation (1) obtained in steps S8 and S10 in FIG. Here, the interpolation value Y (χ) corresponding to the pixel position (z) in which the value of the pixel value y (z) is not entered is the value of the inflated pixel obtained by the inflating process. This interpolation value Y
(Χ) is indicated by a white triangle in FIG. The solid line in FIG. 5 indicates a function serving as a basis for calculating the pixel value y (z) of the original one-dimensional data. The positional relationship between the white triangle and the solid line indicates that the inflated data is accurately interpolated.

The first column in FIG. 15 shows the second derivative of the pixel corresponding to all the inflated pixel positions (z). Note that the fifth column shows the secondary differential value obtained by applying a conventional one-dimensional Laplacian type digital filter for comparison. In the flowcharts of FIGS. 2 and 3, in step S18, the m-th order differential equation (5) of the Chebyshev approximation equation (1) for the one-dimensional data to be processed is obtained. Therefore, by giving a variable (χ) corresponding to the inflated or decimated pixel position (z) to this differential equation (5), inflated or decimated data of the original one-dimensional data or the data composed of the original differential value is obtained. be able to.

Further, in the case of the resolution changing process, an arbitrary pixel position (z) is corresponded using Chebyshev's approximate expression (1) or its differential expression (5) obtained in the same manner as the padding or thinning-out process. The resolution can be changed by obtaining the pixel value and returning the pixel value to a pixel position corresponding to the resolution.

[Brief description of the drawings]

FIG. 1 is an explanatory diagram showing two-dimensional data used in a two-dimensional data processing method of the present invention.

FIG. 2 is a flowchart of a process of differentiating two-dimensional image data in one direction.

FIG. 3 is a flowchart of a process of differentiating two-dimensional image data in one direction.

FIG. 4 is a data diagram showing an execution example of the differentiation processing of FIGS. 2 and 3;

FIG. 5 is a diagram showing a processing result of FIG. 4;

6 is a diagram showing a second-order differential value based on the Chebyshev approximation formula shown in FIG. 4 and a comparative example thereof.

FIG. 7 is a flowchart illustrating another example of the differential processing of two-dimensional image data in one direction.

FIG. 8 is a flowchart illustrating another example of the differential processing of two-dimensional image data in one direction.

FIG. 9 is a flowchart of a process of differentiating two-dimensional image data in two directions.

FIG. 10 is a flowchart of a process of differentiating two-dimensional image data in two directions.

FIG. 11 is a flowchart of a process of differentiating two-dimensional image data in two directions.

FIG. 12 is a flowchart of another example of the differential processing of two-dimensional image data in two directions.

FIG. 13 is a flowchart of another example of the differential processing of two-dimensional image data in two directions.

FIG. 14 is a flowchart of another example of a differentiation process of two-dimensional image data in two directions.

FIG. 15 is a data diagram of an inflating process of one-dimensional data.

[Explanation of symbols]

 Ai Chebyshev approximation coefficient D (I), D (J) Point sequence image data

Claims (13)

[Claims] 1. A one-dimensional data sequence is sequentially extracted from two-dimensional processing data, and the extracted one-dimensional data sequence is approximated by a polynomial using a Chebyshev function to obtain an approximate expression. Differentiating an approximate expression to calculate a differential expression of the approximate expression, calculating a differential value of each data of the one-dimensional data sequence using the calculated differential expression, and obtaining a differential value of the processing data. A two-dimensional data processing method. 2. A higher-order differential equation is calculated as a differential equation of the approximation equation, and a higher-order differential value of each data of the one-dimensional data sequence is calculated using the calculated higher-order differential equation. 2. The method for processing two-dimensional data according to claim 1, wherein: 3. The calculated differential value of the processing data is used as new processing data, a one-dimensional data sequence is sequentially extracted from the new processing data, and the extracted one-dimensional data sequence is used by a Chebyshev function. An approximate expression is obtained by approximating the obtained approximate expression, a differential expression of the approximate expression is calculated by differentiating the obtained approximate expression, and the first expression is calculated using the calculated differential expression.
2. A differential value of each data of a dimensional data sequence is calculated to obtain a new differential value of the processed data.
A method for processing the described two-dimensional data. 4. An approximate expression is obtained by approximating a one-dimensional data sequence sequentially extracted in one direction from the processing data by a polynomial using a Chebyshev function, and the obtained approximate expression is differentiated to obtain the approximate expression. A differential expression is calculated, a differential value is calculated using the calculated differential expression to obtain a one-way differential value of the processing data, and data comprising the one-way differential value of the processing data is converted into a new processing data. And a one-dimensional data sequence sequentially extracted in the other direction from the new processing data is approximated by a polynomial using a Chebyshev function to obtain an approximate expression. The obtained approximate expression is differentiated to differentiate the approximate expression. A differential value of the processed data differentiated in two directions is calculated by calculating an expression, calculating a differential value using the calculated differential expression, and obtaining a differential value of the new processed data in the other direction. Characterized by The method for processing two-dimensional data according to claim 1. 5. The differential value of the one-dimensional data string extracted in one direction is determined by using a higher-order differential equation of an approximate expression obtained for the one-dimensional data string extracted in the one direction. The new processing data is composed of the higher-order differential value differentiated in one direction, and the differential value of the one-dimensional data string extracted in the other direction is ,
5. The high-order differential value calculated by using a high-order differential expression of an approximate expression obtained for the one-dimensional data string extracted in the other direction. How to process dimensional data. 6. A differential value of the processed data differentiated in two directions is set as new processed data, and a one-dimensional data sequence sequentially extracted in one direction from the new processed data is approximated by a polynomial using a Chebyshev function. Calculating an approximate expression, differentiating the obtained approximate expression to calculate a differential expression of the approximate expression, calculating a differential value using the calculated differential expression, and differentiating the processed data in one direction. Value, and a one-dimensional data sequence sequentially extracted in the other direction from the data consisting of the differential value of the processed data in one direction is approximated by a polynomial using a Chebyshev function to obtain an approximate expression. Is differentiated to calculate a differential expression of the approximate expression, and a differential value is calculated using the calculated differential expression to obtain a differential value differentiated in two directions of the new processed data. The two-dimensional data according to claim 4. Data processing method. 7. A one-dimensional data sequence is sequentially extracted from two-dimensional processing data, and each data of the extracted one-dimensional data sequence is approximated by a polynomial using a Chebyshev function to obtain an approximate expression. And calculating a one-dimensional data sequence different from the one-dimensional data sequence using the approximation formula. 8. The method for processing two-dimensional data according to claim 7, wherein a one-dimensional data string having a smaller number of data than the one-dimensional data string is calculated. 9. The two-dimensional data processing method according to claim 7, wherein a one-dimensional data string having a larger number of data than the one-dimensional data string is calculated. 10. A one-dimensional data sequence is sequentially extracted from two-dimensional processing data, and the extracted one-dimensional data sequence is approximated by a polynomial using a Chebyshev function to obtain an approximate expression. Calculating a differential value of the one-dimensional data sequence different from the one-dimensional data sequence by using the calculated differential expression to differentiate the expression. How to process the data. 11. The two-dimensional data processing method according to claim 10, wherein a differential value of the one-dimensional data string having a smaller number of data than the one-dimensional data string is calculated. 12. The two-dimensional data processing method according to claim 10, wherein a differential value of a one-dimensional data sequence having a larger number of data than the one-dimensional data sequence is calculated. 13. A method of approximating the one-dimensional data string extracted from two-dimensional processing data with a polynomial using a Chebyshev function, and omitting zero-order and first-order terms of the polynomial to obtain an approximate expression. The method for processing two-dimensional data according to claim 1, wherein: