JP2002288668A - Curve linear transformation method - Google Patents

Abstract

PROBLEM TO BE SOLVED: To produce a transformed curve, in a state of a transformed degree of the curve agreeing with an identification degree of recognition, and to combine the curves of the same transformation degree, when the transformation curves having the same transformation degree are combined. SOLUTION: A correlation function used conventionally as criterion in a identification method, is applied as the transformation degree. One reference curve is determined to realize three means for producing the transformation curve having the predetermined correlation function with the reference curve. That is, (1) a transformation method by changing an inclination, without changing the length of each intercept, (2) a transformation method by changing the length without changing the inclination of each intercept, and (3) a combination method for equally producing an aimed transformation curve by combining both, when the curve is represented by connecting divided intercepts.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method of deforming a curve, which is a basic technique of a handwritten character recognition method and a graphic recognition method, and more particularly to a method of generating data used for learning and testing.

[0002]

2. Description of the Related Art Curve deformation is one of the basic techniques in pattern recognition. It is used for matching two curves. One is an unknown input pattern and the other is a recognition dictionary pattern. Either one is deformed. Curve deformation methods are roughly classified into four.
(1) A method using a normally distributed random number. This is a method in which parameters or coordinates of a curve are changed by a normally distributed random number.
(2) A method of fusing several appropriately selected patterns. The coordinates or parameters are synthesized according to the concept of a convex set. (3) A method of moving a point on a curve to a target point by giving a deformation rule. A non-linear function or fuzzy function is used as the deformation rule. Also applies to morphing. These possible deformations are treated collectively in quantities called deformation vector fields. (4) A method using the energy of a curve. A physical equation to be satisfied by the curve is set and deformed so as to satisfy the constraint condition.

[0003] Curve deformation methods can also be used to artificially generate curve samples. Artificial samples are useful in the following cases. If only a few samples are available, increase the number of samples. When the independence between the training sample set and the test sample set is doubtful, an artificial sample set as a substitute for either is provided. Curve deformation methods that can be used for artificial sample generation are (1) and (2) above. The reason is that it is necessary to be able to generate all possible deformation curves. (3) and (4) are methods for bringing a certain pattern closer to the other pattern. Therefore, not every deformation curve can be generated from one or several standard curves. The method (2) creates a new set based on the set of deformation curves generated in (1).

In the method (1), a random number is used. There, a degree of deformation is defined, and a deformation curve is generated within the limited range. The degree of deformation was not tied to the recognition discrimination method. That is, when a sample set of deformation curves is generated by the method (1) using the conventional deformation degree, the set may be directly related to the set of curves distinguished by the recognition discrimination degree. There was a disadvantage that it could not be done.
Therefore, there is a problem that an effective learning sample cannot be prepared for learning the recognition method.

Further, the method (2) has a drawback that even if the deformation curves having the same deformation degree are combined, the deformation degrees of the combined curves have different values.

[0006]

The problem to be solved is that the degree of deformation of the curve does not coincide with the discrimination degree of recognition, and when the deformed curves having the same degree of deformation are combined, different deformation levels are obtained. The point is that the degree curves are combined.

[0007]

According to the present invention, a correlation coefficient which has been used for a long time as a criterion of a discrimination method is defined as a degree of deformation. Three means for defining one standard curve and generating a deformation curve with the determined correlation coefficient have been realized.
That is, when the curve is represented by the connection of divided intercepts,
From [1] a deformation method by changing the inclination without changing the length of each section, [2] a deformation method by changing the length without changing the inclination of each section, and [3] a synthesis of both, This is a synthesis method that can generate the desired deformation curve uniformly. The present invention is characterized in that these means are realized as one method.

[0008]

DESCRIPTION OF THE PREFERRED EMBODIMENTS When a deformation curve having a predetermined correlation coefficient is generated, the means [1] provides a range in which the slope of the intercept can be changed, and furthermore, the means [1] and [2].
Then, by giving the basic form of the deformation, and by means [3], by giving the procedure of uniquely generating the desired deformation curve from the group of curves generated by means [1] and [2],
The present invention has been realized.

[0009]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS A curve is a set of intercepts in which discrete points on the curve are connected by broken lines. The standard curve is represented by the following equation.

[0010]

(Equation 1)

In the above equation, wk (k = 1,
2,. . . , N) is a quantity obtained by dividing the given standard curve into n intercepts and expressing each intercept by a complex number of x and y coordinates. wk is represented by the following equation.

[0012]

(Equation 2)

The intercept of the curve to be deformed is expressed by the following equation.

[0014]

(Equation 3)

In a modification in which the slope of the intercept is changed without changing the length of the intercept, the correlation coefficient between the standard curve and the deformed curve is given by the following equation.

[0016]

(Equation 4)

In the above equation, cos (Δθ) is the value of the specified correlation coefficient. Δθk is the angle formed by the k-th intercept of the deformation curve with the k-th intercept of the standard curve. ak is represented by the following equation.

[0018]

(Equation 5)

FIG. 1 is a diagram of an embodiment of the method of the present invention, and is a diagram for explaining a range in which the inclination of the intercept can be changed by means [1]. 1 indicates a line segment 0A, and the length between the coordinate origin 0 and the point A is 1. The line segment 0A has a length a
k (k = 1, 2,..., n). The angle between the line segment 0A in FIG. 1 and the horizontal axis xe is Δθ. In the line segment 0A, all the sections have inclinations of the same angle Δθ with respect to the horizontal axis. The position of the perpendicular from point A to the horizontal axis is C
It is. The length of 0C is cos (Δθ), which is the value of the specified correlation coefficient. An arbitrary point B on the vertical line AC is taken. The connected section sequence from the origin 0 to the point B while maintaining the length ak of the section k is one of the modifications. This is indicated by 2 in FIG. There are many ways to determine the slope Δθk of each section k with respect to the horizontal axis when the point B is specified. As if moving the A end of the straight chain 0A to the B end, the form of the slack chain could be various. The range where the chain is placed is within an ellipse whose focal point is point 0 and point B and whose major axis is twice as long. This is indicated by 3 in FIG. When one end of any section k rides on the circumference of the ellipse, that point and point 0,
The point B is placed in a straight line. Consider a case where the slope Δθ1 of the section 1 having one end on the point 0 is determined, then the Δθ2 of the section 2 is determined, and the process proceeds in the order of. Section 1
Is determined, the chain (2, 3,..., N) is placed with the end of section 1 (not point 0) and point B as the focal point and twice the major axis is 1-a1. Within the ellipse. This procedure is sequentially recursive. When the last n-1 and n sections are left to be determined, their common endpoints must be located on the circumference of the ellipse. The upside down of FIG. 1 also gives possible variations.

FIG. 2 shows an example of a standard curve, and FIG. 3 shows an example of a modification in which the slope is changed without changing the length of the intercept.

FIG. 4 is another embodiment of the arrangement of the chains of FIG. Point 0 and point B are 3 of Tchebycheff
They are connected by a curve of degree polynomial. By using a low-order polynomial to some extent, it is possible to prepare a basic form of deformation that does not cause jerky deformation.

Next, when changing the length without changing the inclination of the intercept, the correlation coefficient between the standard curve and the deformation curve is as follows.

[0023]

(Equation 6)

Here, ak〜 is given by the following equation.

[0025]

(Equation 7)

The intercept wk ~ of the deformation curve shown in Equation 3 is expressed by Equation 2
Is changed according to the following equation from wk of the standard curve shown in FIG. This gives the basic form of the deformation.

[0027]

(Equation 8)

Here, Tm () is the m-th order Tcheby.
This is a polynomial of cheff. Since the term related to wk on the right side of the above equation is a real number, wk〜 changes only by length without changing the slope from wk. Unknown real constants c and d contained in this term
Satisfies the quadratic algebraic equation by substituting equation 8 into equations 7 and 6. Given c, d can be determined. The magnitude of c only changes the size of the scale of the deformation curve, but does not change the shape.

The embodiment of FIG. 5 is an example in which the standard curve of FIG. 2 is modified in this way.

Finally, the above-mentioned [1] Deformation method by changing the inclination without changing the length of each section, and [2] Deformation method by changing the length without changing the inclination of each section And are synthesized. Among these, the deformation method using the Tchebycheff polynomial gives a basic pattern of the deformation. In order to generate deformation curves uniformly, it is necessary to combine the deformation curves of the basic patterns. This is shown by the following equation.

[0031]

(Equation 9)

Here, v indicates a composite curve, a subscript i indicates a term relating to a deformation curve formed by changing the inclination, and a subscript j indicates a term relating to a deformation curve formed by changing the length. There are M1 and M2 respectively. λi and λj are positive signs, and their sum is 1. λ0 takes a negative value and serves to subtract the standard curve w from the combination of the deformation curves. To make the correlation coefficient ρ (w, v) equal to cos (Δθ), it is necessary to solve for λ0. This is determined by solving a quadratic algebraic equation.

Λ0 and λi, λ thus obtained
The sum of j is not 1. Therefore, by dividing λ0, λi, and λj using the absolute value of these sums, the sum of the results becomes 1. In this way, it is possible to obtain a composite curve of the values of the specified correlation coefficient.

As described above, the correlation coefficient cos (Δ
θ) are generated and multiplied by a positive constant coefficient to obtain a sum. The sum of the constant coefficients is set to 1. The set of curves formed from such sums has a correlation coefficient of cos
All deformation curves equal to or greater than (Δθ) are included.

[0035]

As described above, according to the curve deformation method of the present invention, the correlation coefficient which has been used for a long time as a criterion for the discrimination method is used as the degree of deformation, and the phase determined for one standard curve is used. We provide three means for generating deformation curves with relational numbers. That is, when a curve is represented by a series of divided sections, [1] a deformation method by changing the slope without changing the length of each section, and [2] changing the length without changing the slope of each section. And [3] a method of transforming from a combination of both. The present invention realizes a synthesizing method capable of uniformly generating a desired deformation curve by realizing these means as one method. According to the present invention, there is an advantage that a sample set can be generated which can be directly related to a set of curves distinguished by recognition discrimination.

[Brief description of the drawings]

FIG. 1 is a view for explaining a deformation method of changing a slope without changing the length of a segment of a curve. Correlation coefficient value = 0.95.
(Example 1)

FIG. 2 is a diagram showing an example of a standard curve. (Example 1)

FIG. 3 is a view showing an example in which a standard curve is modified by the method of FIG. 1; The left and right curves were deformed separately. Correlation coefficient value = 0.95. (Example 1)

FIG. 4 is an example in which the chains of FIG. 1 are arranged by a curve of a third-order polynomial of Tchebycheff. (Example 2)

FIG. 5 is an example in which the standard curve of FIG. 2 is deformed by a deformation method of changing the length without changing the inclination of the segment of the curve. The left and right curves were deformed separately. Correlation coefficient value =
0.95. Use the third-order polynomial of Tchebycheff. c = 0.7. D = 0.95 for the left curve. D = 0.86 for the right curve. (Example 3)

[Explanation of symbols]

1 A line segment indicating that all sections have the same angle of inclination Δθ with respect to the horizontal axis. 2 A chain indicating that the connected section sequence is one of the transformations that change the slope. 3 Ellipse showing the area where the chain is placed. xe The horizontal axis coordinate on the plane on which the deformation method for changing the inclination is performed without changing the length of the divided section of the curve. ye The vertical axis on the plane on which the deformation method for changing the inclination is changed without changing the length of the divided section of the curve. Axial coordinates A (xe, ye) = (0,0), end point of a line segment having a length of 1 from the (0,0) and forming an angle of [Delta] [theta] with respect to the horizontal axis. Foot of a perpendicular drawn from A to the horizontal axis x Horizontal axis coordinate of plane representing curve y Vertical axis coordinate of plane representing curve

Claims (1)

[Claims] 1. A given standard curve is represented by a polygonal line connecting a series of discrete points on the standard curve, and the curve is deformed by changing the inclination of each polygonal line, that is, the intercept, without changing the length of the intercept. It consists of a method, a method of deforming the curve by changing the length without changing the slope of each intercept, and a method of deforming a standard curve by synthesizing a group of deformation curves by both methods into one curve. Generating a deformation curve having a standard curve and a specified correlation coefficient.