JPS63256988A - Spline interpolation - Google Patents

Abstract

(57) [Summary] This bulletin contains application data before electronic filing, so abstract data is not recorded.

Description

DETAILED DESCRIPTION OF THE INVENTION [Field of Industrial Application] The present invention relates to a spline interpolation method, and particularly to a spline interpolation method using a graphical method in the field of image processing.

[Conventional technology]

In the field of image processing, which deals with binary figures such as two-dimensional figures and characters, these figures can be converted into contour images, or their contours can be converted into straight lines or curves (ellipses, circles) for data compression.
The image is expressed as an approximate image.

In particular, we sample some points that are characteristic on the contour,
There is a method of compressing the data of a single graphic character image by storing these points. When these images are restored and displayed on a display, basically, each sample point is connected with a straight line, but the displayed image is enlarged or reduced depending on the display means, display density, etc.

Here, in order to make the restored image as close to the original image as possible with smooth hair, it is necessary to supplement some points between the sample points, especially when the image is enlarged.

Conventionally used interpolation techniques can be roughly divided into two types: algebraic interpolation methods and geometric interpolation methods. Among these, the algebraic interpolation method uses an algebraic curve (polynomial) as the interpolation function, and the geometric spline interpolation method uses a conic section (geometric functions such as circles, ellipses, hyperbolas, and parabolas) as the interpolation function. There is.

In either case, the functions (approximation functions for interpolation) are used in the same way, and the operations to derive the functions during the interpolation process, such as the design of nodes and numerical operations (solving equations, inputting values to functions), are the same. substitution) is required.

Here, designing a node means further subdividing an interval to be interpolated in spline interpolation into appropriate small intervals.

For example, if we consider a case where the sampled points within the target interpolation interval have parts that change relatively slowly and parts that change suddenly, it is better to subdivide the parts that change suddenly to make it smoother. Can be interpolated.

However, the number of nodes is determined by the degree of polynomial used for interpolation, and the number of nodes cannot be freely selected.Currently, the optimum design method for designing nodes is not fully elucidated. −
For example, considering the B-spline interpolation method, which is a typical algebraic interpolation method, the design of internal and external nodes is important in order to improve interpolation accuracy, and depending on the design method,
Problems such as "undulation" may occur, resulting in results that are far from expectations.

Furthermore, numerical calculations to obtain approximate functions take time and effort, and the conditions are often poor, making it difficult to solve equations.

As described above, conventional interpolation methods have drawbacks in that it is difficult to optimally design nodes, calculations to determine interpolation functions take time and effort, and solutions are also difficult.

As an interpolation method for compensating for these drawbacks, the applicant of the present invention previously devised a three-point interpolation method using a drawing method (Japanese Patent Application No. 61-11E1404) as one of the geometric spline interpolation methods.

This three-point interpolation method uses both endpoints of a small interval and one sample point set in the middle of a flat curve that does not include an inflection point, and uses a line segment that connects the two endpoints for this sample point. A point symmetrical to the midpoint of is determined as a reference vertex, a line segment connecting this reference vertex and the end points is each hypotenuse, and a line segment connecting the end points is the base of the sample point. The basic condition is that is located at the midpoint of the line segment connecting the reference vertex and the midpoint of the base.r Parabola Theorem 1 Edited by Iwata, Dictionary of Geometry 6'' Maki Shoten, 198
2) When interpolating between any of the above end points and the sample point, the reference vertex is set according to the above reference conditions for the triangle formed by using those two points and the midpoint of the hypotenuse on that side as the new reference vertex. You can repeat the process of finding the midpoint of the line segment that connects the midpoint of the base and using this as the interpolation point.

This three-point interpolation method uses a drawing method to directly find interpolation points from given sample points without using functions. 3) Calculations for assigning numerical values to functions are no longer necessary, and it is possible to realize an interpolation device that is (i) fast and (ii) has a simple device configuration.

[Problem that the invention seeks to solve]

However, in the three-point interpolation method described above, since a plane curve is interpolated using only three sample points, the slope of the tangent to this curve at both end points matches the hypotenuse from the reference vertex determined as explained above, so 2 At the end points that separate two interpolation intervals, in order to maintain smooth continuity of the connection between the two curves in both intervals and increase the accuracy of approximation by interpolation, the range of one interpolation interval cannot be made too large, and sufficient data compression is required. could not be realized.

Therefore, the present invention eliminates the need for (1) design of nodes, (2) numerical calculations such as solving equations, (3) numerical substitution calculations for functions, and (i) a simple device configuration that allows for high-speed processing. The present invention aims to provide an interpolation method with a higher data compression rate than the three-point interpolation method described above.

[Means for solving problems]

The present invention has been made to solve the above-mentioned problems.The present invention has been made in order to solve the above-mentioned problems. When the coordinates of the second endpoints and at least three sample points are given, the calculation means first calculates (1) the first . a second endpoint and each adjacent first endpoint;
The first sample point connects the second sample point. Find a second straight line, (2) Find the intersection of the first and second straight lines as a reference vertex, (3) Find a third straight line connecting the reference point and the third sample point, (4) Find a fourth straight line connecting the first and second sample points, (5) Find the intersection of the third and fourth straight lines as the first internal division point, (6) Find the first and second sample points. The first for the line segment connecting the points
Find the first internal division ratio of the internal division point of (7) the reference vertex and the first
(8) Find the second internal division ratio of the third sample point to the line segment connecting the internal division points of , and (8) calculate the line segment connecting the first (or second) sample point and the first reference vertex as Find the point that is internally divided by the second internal division ratio as a new first (or second) reference vertex, and (9) a line connecting the first (or second) sample point and the third sample point. 00) A new first (or second) reference vertex and a new first internal division point.
An internal division point that internally divides a line segment connecting the (or second) internal division points by the first internal division ratio is determined as a first (or second) interpolation point, and this first (or second) This interpolation method is characterized in that the interpolation point is set as the interpolation point between the first (or second) sample point and the third sample point.

The principle of this interpolation method using a diagrammatic method will be explained below based on the drawings.

The graphical method is a method in which the relative position of a sample point is determined according to a certain geometric rule, and this is used as an interpolation point.

First, the algorithm for finding interpolation points will be explained from a geometric perspective.

Here, in the small section that does not include the inflection point, both end points, that is, the first and second end points and the first . Second. Consider FIG. 1, given a third sample point.

Now, let the first and second end points of the section be B+, B2, and B
+, B The first and second sample points adjacent to z are T'+, and the point between Tz is a third sample point S. Next, explanation will be given using FIG. 2.

Let W be the intersection of the straight line B + T + and the straight line B2T2, let S be the sample point sandwiched between T1 and T2, that is, the third sample point, and let M be the intersection of the straight line ws and the straight line TIT2.

The interpolation method according to the present invention is characterized by two internal division ratios. That is, it is a point for the line molecule IT2.

Next, the intersection point W of the line segment WT and a straight line passing through the point S and parallel to the straight line T IT 2 is assumed. Here, two points TI correspond to triangles W, T, and S whose vertices are point W1.

Find the interpolation point between 8. As an operation, a point M1 to be internally divided is found in order.

Find the point SI that is internally divided by .

(3) Let this point SI be the interpolation point between the two points T, , S.

The above is the algorithm for finding interpolation points.

A similar operation was also performed between the two points S and T2, and the results of connecting each point with a straight line are shown in Figure 3.

Furthermore, in order to obtain fine interpolation points, the SI or S2
As the new third sample point, for example, the two points T in Fig. 2,
Between S and S, find the intersection of a straight line passing through point S and parallel to straight line 1゛1S and line segments W and T, and set it as W2, and form a triangle Wz
The same operation as above can be repeated for T+S+.

In this way, interpolation points can be found between the two points T, , T with arbitrary fineness.

Figure 4 is a comparison diagram of the spline interpolation method using this drawing method and the linear interpolation method.The curve is represented by five sample points (0), and by interpolating between them, four interpolation points can be obtained. (◎) has been obtained.

As shown in this example, the interpolation method according to the present invention provides a smoother curve compared to linear interpolation in which sample points are connected with straight lines.

Furthermore, compared to the three-point interpolation method described above, since the interpolation interval for one time can be larger, data compression can be improved.

Note that, as described above, it is possible to interpolate each interpolation interval as narrowly as possible, but from a practical standpoint, it is only necessary to obtain the necessary approximation accuracy (smoothness). Therefore, the evaluation parameters described below may be used as a guideline for controlling the number of times interpolation is repeated.

For example, when the contour curve of a character or figure is approximated by a polygonal line or a polygon, the criteria for determining whether or not to perform interpolation to supplement sample points, especially in the curved part, and when to repeat interpolation. FIG. 5 shows the evaluation parameter F that provides the termination condition for terminating the interpolation.

When focusing on one sample point B on the curve, let A and C be the two sample points that sandwich this sample point, and let M be the foot of the perpendicular drawn from sample point B to line segment AC, then the evaluation parameter is F is 8M As an example, for each sample point on the curve,
Calculate the value of this evaluation parameter F, and use F from a certain darkness value.
When the value of is small, a method of interpolating between that sample point and two adjacent sample points can be considered.

After interpolation, the evaluation parameter F is calculated again for each sample point including the newly added sample point, and the interpolation is terminated when the value of F for all sample points exceeds the darkness value.

As described above, by using such evaluation parameter F, it is possible to automatically determine the start and end of interpolation.

〔Example〕

FIG. 6 shows an embodiment of an interpolation device to which the spline interpolation method according to the present invention is applied. Block 100 is storage means and block 200 is calculation means.

A RAM is generally used as the storage means 100, and as shown in the figure, the first and second end points, the first to third sample points, the reference vertex, the first and second internal division ratios, and the new third sample points are stored.

A CPU is used as the calculation means, and the CPU sequentially performs calculations in the following steps (1) to 00) by calculation units 11 to 21.

(1) In the calculation units 11 and 12, the first . a second endpoint and each adjacent first endpoint. Connect the second sample point and connect the first sample point. Find the second straight line.

(2) The calculation unit 13 determines the intersection of the first and second straight lines as a reference vertex.

(3) A third straight line connecting the reference vertex and the third sample point is determined by the calculation unit 14.

(4) A fourth straight line connecting the first and second sample points is determined by the calculation unit 15.

(5) The calculation unit 16 determines the intersection of the third and fourth straight lines as a first internal division point.

(6) The calculation unit 17 calculates a first internal division ratio of the first internal division point for the line segment connecting the first and second sample points.

(7) The calculation unit 18 calculates the second internal division ratio of the third sample point with respect to the line segment connecting the reference vertex and the first internal division point.

(8) The calculation unit 19 divides the line segment connecting the first (or second) sample point and the first reference vertex internally by the second internal division ratio into a new first (or second) point. ) as the reference vertex.

(9) The calculation unit 20 divides the line segment connecting the first (or second) sample point and the third sample point internally by the first internal division ratio into a new first (or second) point. ) is found as the internal division point.

0ω The calculation unit 21 determines the internal division point for internally dividing the line segment connecting the first (or second) reference vertex and the first (or second) internal division point by the first internal division ratio. This first (or second) interpolation point is determined as an interpolation point between the first (or second) sample point and the third sample point.

If even finer interpolation points are required, the calculation unit 21
The steps (8) to 0ω described above can be repeated using the interpolation point obtained in 2 as a new third sample point.

Reference numeral 22 in FIG. 6 denotes an interpolation repetition number control circuit, which sets the number of interpolation times and controls the end time based on the evaluation parameter F.

〔Effect of the invention〕

As described above, the present invention does not use an interpolation (approximation) function, but directly determines interpolation points from given sample points by a graphical method.

Therefore, since the complicated procedure for deriving functions, namely (1) design of nodes, (2) numerical calculations such as solving equations, and (3) calculation of substitution into functions, is no longer necessary, it is possible to (3) Interpolation with a high data compression rate can be realized, and a smooth curve can be obtained. −

[Brief explanation of drawings]

1 to 4 are diagrams for explaining the present invention in detail,
Figure 1 is a diagram in which a curve is approximated by a polygonal line, Figure 2 is a diagram showing the procedure of the present invention, Figure 3 is a diagram showing the results of implementing the present invention, and Figure 4 is a diagram showing the conventional linear interpolation and the present invention. A comparison diagram with the interpolation method according to the invention, and FIG. 5 is an explanatory diagram of the evaluation parameters to be removed in the implementation of the invention. FIG. 6 is a block diagram of an embodiment of the present invention. 100... Storage means 200... Arithmetic means 11-21... Arithmetic unit

Claims (1)

[Claims] Arithmetic means 20 having storage means 100 and arithmetic units 11 to 21
0, and 1 does not include an inflection point in the storage means 100.
The first,
When the coordinates of the second endpoints and at least three sample points are given, the calculation means 200 performs the following step (1): the calculation units 11 and 12 calculate the first and second endpoints and their respective coordinates. First and second sample points adjacent to each end point are connected to obtain first and second straight lines. (2) The calculation unit 13 determines the intersection of the first and second straight lines as a reference vertex. (3) A third straight line connecting the reference vertex and the third sample point is determined by the calculation unit 14. (4) A fourth straight line connecting the first and second sample points is determined by the calculation unit 15. (5) The calculation unit 16 determines the intersection of the third and fourth straight lines as a first internal division point. (6) The calculation unit 17 calculates a first internal division ratio of the first internal division point for the line segment connecting the first and second sample points. (7) The calculation unit 18 calculates the second internal division ratio of the third sample point with respect to the line segment connecting the reference vertex and the first internal division point. (8) The calculation unit 19 divides the line segment connecting the first (or second) sample point and the first reference vertex internally by the second internal division ratio into a new first (or second) point. ) as the reference vertex. (9) The calculation unit 20 divides the line segment connecting the first (or second) sample point and the third sample point internally by the first internal division ratio into a new first (or second) point. ) is found as the internal division point. (10)) An internal division point at which the calculation unit 21 internally divides a line segment connecting the first (or second) reference vertex and the first (or second) internal division point by the first internal division ratio. is determined as the first (or second) interpolation point. A spline interpolation method characterized in that the first (or second) interpolation point is set as an interpolation point between the first (or second) sample point and the third sample point by performing calculations according to the following.