CN113674378B - Curve image construction method based on interpolation and approximation subdivision technology - Google Patents

Curve image construction method based on interpolation and approximation subdivision technology Download PDF

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CN113674378B
CN113674378B CN202110974243.5A CN202110974243A CN113674378B CN 113674378 B CN113674378 B CN 113674378B CN 202110974243 A CN202110974243 A CN 202110974243A CN 113674378 B CN113674378 B CN 113674378B
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point
subdivision
interpolation
vertex
approximation
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CN113674378A (en
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檀结庆
童广悦
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Hefei University of Technology
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T11/00—2D [Two Dimensional] image generation
    • G06T11/001—Texturing; Colouring; Generation of texture or colour
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T10/00—Road transport of goods or passengers
    • Y02T10/10—Internal combustion engine [ICE] based vehicles
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Abstract

The invention relates to a curve image construction method based on interpolation and approximation subdivision technology, which solves the defects of inflexibility and poor smoothness of a constructed curve image compared with the prior art. The invention comprises the following steps: acquiring image control vertex information; setting image side information; generating new vertexes for vertexes of the same type; generating new vertexes for the vertexes of different types; and (6) generating a curve image. The invention combines the approximation subdivision method and the interpolation subdivision method, and adjusts the interpolation points and the approximation points according to the requirement of the graph, thereby solving the problems of more flexible graph construction, better smoothness of the generated graph curve and the like.

Description

Curve image construction method based on interpolation and approximation subdivision technology
Technical Field
The invention relates to the technical field of image processing, in particular to a curve image construction method based on combination of interpolation and approximation subdivision technology.
Background
Subdivision methods have become an important piece of research in the fields of computer-aided geometry design and graphics in recent years. The subdivision method is to insert new vertexes into given initial control vertexes according to a selected subdivision rule for given control polygons, wherein the new vertexes are weighted averages of the control vertexes, and the new vertexes are connected into new control polygons. And repeating the steps continuously, and generating a limit curve finally along with the increase of the subdivision times.
The subdivision method is generally divided into: interpolation subdivision and approximation subdivision. The most classical of interpolation subdivision methods is Dyn et al and Hassan et al, which propose a four-point double interpolation subdivision method and a four-point triple interpolation subdivision method, and the generated limit curves respectively reach C 1 And C 2 Continuous. The most classical approach to subdivision is a three-point triple approach subdivision method proposed by cubic B spline curve, hassan and the like, and the generated limit curves all reach C 2 Continuous.
In recent years, the approximation subdivision method and the interpolation subdivision method have also been continuously improved, thereby achieving higher order continuity. The limit curve generated by the interpolation subdivision method passes through all the original control vertexes, and the limit curve generated by the approximation subdivision method does not pass through the original control vertexes. However, there is a case where it is desired to generate a limit curve that passes through some of the initial control vertices and approaches the rest of the points. In this case, neither the interpolation subdivision method nor the approximation subdivision method can be independently performed, resulting in inflexible construction of the limit curve.
Disclosure of Invention
The invention aims to solve the defects of inflexibility and poor smoothness of a curve image constructed in the prior art, and provides a curve image construction method based on interpolation and approximation subdivision technology.
In order to achieve the above object, the technical scheme of the present invention is as follows:
a curve image construction method based on interpolation and approximation subdivision technology comprises the following steps:
11 Acquiring image control vertex information): obtaining a given column of control vertices { p } i Control vertex { p }, control vertex i The method comprises the steps of dividing the interpolation point into an interpolation point and an approximation point, wherein the interpolation point is marked as I, the approximation point is marked as A, the position of the interpolation point I is set to be unchanged in the subdivision iteration process, and the approximation point A is continuously replaced by a new vertex generated in the subdivision iteration process;
12 Set image side information): setting p i 、p i+1 Two points define an edge, and the points on the edge are classified into the following four types: type one, p i 、p i+1 Marked as an approximation point a; type two, p i Marked as approximation point A, p i+1 Marking as interpolation point I; type three, p i Marked as interpolation point I, p i+1 Marked as an approximation point A; type four, p i ，p i+1 Marked as interpolation point I;
13 Generating new vertices for vertices of the same type: for type one p i 、p i+1 Three-point triple approximation subdivision method for generating new vertex and pair type four p i ，p i+1 Generating new vertexes by a four-point triple interpolation subdivision method, which are marked as interpolation points I, wherein the generated new vertexes are marked as approximation points;
14 Generating new vertices for non-type vertices: p of type two i Marked as approximation point A, p i+1 Marked as interpolation points I andtype three p i Marked as interpolation point I, p i+1 Generating new vertexes according to the set rules marked as the approximation point A, wherein the set rules meet the limit curve to meet C 2 The new vertexes which are continuous and generated are marked as approximation points;
15 Generation of a curve image: generating updated vertexes according to the characteristic usage rules of the new vertexes, and generating limit curves continuously and iteratively, namely generating curve images.
The pair type p i 、p i+1 The generation of a new vertex by a three-point triple approximation subdivision method, both labeled as approximation point A, includes the steps of:
21 Acquiring an initial set of control verticesIs the coordinates of the initial control vertices, n is the initial number of vertices, i represents the ith symbol, P 0 Representing an initial set of control vertices, R d Representing a real set;
22 If any)For the control vertex set after the kth subdivision, n is the initial vertex number, i represents the ith symbol, ++>Represents the ith vertex coordinate, P, of the kth iteration k Represents the k iteration control vertex set, R d Representing a real set, a new vertex is recursively defined as follows>Is the ith control vertex coordinate after the k+1th subdivision, P k+1 Representing the control top point set after the (k+1) th iteration, the subdivision generation rule of the triple-three-point interpolation subdivision method is as follows:
limit curve generated by the above subdivision schemeReach C 2 The process is continuous and the process is carried out,is the ith vertex coordinate after the kth iteration,representing the vertex coordinates at 3i after the (k+1) th iteration,/o>Representing the vertex coordinates at 3i+1 after the k+1 iteration,representing the vertex coordinates at 3i+2 after the k+1th iteration.
The pair type four p i ，p i+1 The generation of new vertices, all labeled interpolation points I, using the four-point ternary interpolation subdivision method includes the steps of:
31 Acquiring an initial set of control verticesIs the coordinates of the initial control vertices, n is the initial number of vertices, i represents the ith symbol, P 0 Representing an initial set of control vertices, R d Representing a real set;
32 If any)For the control vertex set after the kth subdivision, n is the initial vertex number, i represents the ith symbol, ++>Represents the ith vertex coordinate, P, of the kth iteration k Represents the k iteration control vertex set, R d Representing a real set, a new vertex is recursively defined as follows>Is the ith control vertex coordinate after the k+1th subdivision, P k+1 Represents the kthThe subdivision generation rule of the control top point set after +1 iteration and the four-point triple interpolation subdivision method is as follows:
wherein,
when (when)The limit curve generated by the subdivision format reaches C 2 Continuously, u represents a parameter.
The generation of the new vertex for the non-type vertex comprises the following steps:
41 P) of type II i Marked as approximation point A, p i+1 Marked as interpolation point I, ifFor the control vertex set after the kth subdivision,/->Represents the ith vertex coordinate, P, of the kth iteration k Represents the k iteration control vertex set, R d Represents a set of real numbers,
then the new vertex is recursively defined as followsIs the ith control vertex coordinate after the k+1th subdivision, P k+1 Representing the control vertex set after the k+1th iteration, the subdivision generation rule is as follows:
where delta is assigned to a pointParameters of (2); and satisfies a+b+c+d=1, a < δ 2 ，δ∈(0，1)；
42 For type three p i Marked as interpolation point I, p i+1 Marked as an approximation point A, ifFor the control vertex set after the kth subdivision, if +.>For the control vertex set after the kth subdivision,/->Represents the ith vertex coordinate, P, of the kth iteration k Represents the k iteration control vertex set, R d Representing a real set, a new vertex is recursively defined as followsIs the ith control vertex coordinate after the k+1th subdivision, P k+1 Representing the control vertex set after the k+1th iteration, the subdivision generation rule is as follows:
where delta is assigned to a pointParameters of (2); and satisfies a+b+c+d=1, a < δ 2 ，δ∈(0，1)。
The generation of the curve image comprises the following steps:
51 Aiming at the acquired new vertex, the original interpolation point I is still an interpolation point, and the generated new vertex is an approximation point A;
52 For two adjacent points, using triple four-point interpolation subdivision method to generate update vertex;
53 Generating updated vertexes by using a triple three-point interpolation subdivision method aiming at two adjacent points which are approximation points;
54 Aiming at the adjacent points, the approximation points and the interpolation points, generating updated vertexes by using a new vertex generating method by using vertexes of different types;
55 Performing iterative processing on the updated vertexes, marking all the generated new vertexes as approximate points, and then generating marking conditions for each point, and iterating continuously according to the corresponding rule until a limit curve is generated, namely generating a curve image.
Advantageous effects
The curve image construction method based on interpolation and approximation subdivision technology combines the approximation subdivision method and the interpolation subdivision method, and adjusts interpolation points and approximation points according to the graph requirement, so that the graph construction is more flexible, the smoothness of the generated graph curve is better, and the like.
Drawings
FIG. 1 is a process sequence diagram of the present invention;
FIG. 2 is a diagram showing the idea of the method of the present invention;
FIG. 3 is a graph of vertex changes after subdivision of a curved image according to the present invention;
FIG. 4a is a graph of a curve image generated by a four-point ternary interpolation subdivision method in the prior art;
FIG. 4b is a graph of a curve image generated by a three-point triple approximation subdivision method in the prior art;
FIG. 4c is a graph of a curve image generated by a secondary B-spline method in the prior art;
fig. 4d is a graph of a curvilinear image generated using the method of the present invention.
Detailed Description
For a further understanding and appreciation of the structural features and advantages achieved by the present invention, the following description is provided in connection with the accompanying drawings, which are presently preferred embodiments and are incorporated in the accompanying drawings, in which:
the interpolation-approximation subdivision method of the invention constructs the graph according to the user demand as follows: firstly, dividing initial control vertexes into two types according to the requirements of users, wherein one type is a point which needs to be fixed and becomes an interpolation point, and the other type is a point which does not need to be interpolated and needs a smooth curve to get approximation, namely an approximation point; after dividing the graph into an interpolation point 'I' and an approximation point 'A', respectively analyzing and discussing four interpolation point and approximation point conditions which can appear in the graph, and generating new vertexes by using different rules according to different conditions; after the first iteration, only three cases are left for the interpolation point and the approximation point, namely, two and three types are left, so that for two adjacent points, the interpolation subdivision method of three-three points is used, and for the interpolation point and the approximation point, the new vertex is generated by using the new rule introduced above.
Namely, inputting an initial control vertex, and marking a approaching point and an interpolation point for the initial control vertex according to the requirement of a user or the requirement of a graphic modeling. The interpolation point is fixed in the iteration process, the approximation point is continuously approximated in the iteration process, and the interpolation point cannot pass through the point. (if all initial control vertices are marked as interpolation points, the interpolation-approximation algorithm is degraded to the interpolation subdivision method, and if all initial control vertices are marked as approximation points, the interpolation-approximation algorithm is degraded to the approximation subdivision method.)
As shown in fig. 1, the method for constructing the curve image based on interpolation and approximation subdivision technology comprises the following steps:
first, obtaining image control vertex information.
Obtaining a given column of control vertices { p } i Control vertex { p }, control vertex i The method comprises the steps of dividing the interpolation point into an interpolation point and an approximation point, wherein the interpolation point is marked as I, the approximation point is marked as A, the position of the interpolation point I is set to be unchanged in the subdivision iteration process, and the approximation point A is continuously replaced by a new vertex generated in the subdivision iteration process.
If all the initial control vertexes are marked as interpolation points, the interpolation-approximation subdivision method is degenerated to be an interpolation subdivision method, if all the initial control vertexes are marked as approximation points, the interpolation-approximation subdivision method is degenerated to be an approximation subdivision method, and more applications are that users mark the interpolation points and the approximation points on the initial control vertexes according to own requirements, and then the needed images are generated. The invention provides an interpolation-approximation subdivision method, which combines the interpolation subdivision method and the approximation subdivision method, but not simply and linearly combines the interpolation subdivision method and the approximation subdivision method. As shown in fig. 2, which shows a diagram representing the main idea of the interpolation-approximation subdivision method of the present invention, for interpolation points that are stationary, the rest points are approximated continuously.
Second, setting image side information: setting p i 、p i+1 Two points define an edge, and the points on the edge are classified into the following four types: type one, p i 、p i+1 Marked as an approximation point a; type two, p i Marked as approximation point A, p i+1 Marking as interpolation point I; type three, p i Marked as interpolation point I, p i+1 Marked as an approximation point A; type four, p i ，p i+1 Marked as interpolation point I.
Thirdly, generating new vertexes for vertexes of the same type: for type one p i 、p i+1 Three-point triple approximation subdivision method for generating new vertex and pair type four p i ，p i+1 And generating new vertexes by a four-point triple interpolation subdivision method, which are marked as interpolation points I, wherein the generated new vertexes are marked as approximation points.
(1) For type one p i 、p i+1 The generation of a new vertex by a three-point triple approximation subdivision method, both labeled as approximation point A, includes the steps of:
a1 Acquiring an initial set of control vertices
A2 If any)For the control vertex set after the kth subdivision,/->Represents the ith vertex coordinate, P, of the kth iteration k Represents the k iteration control vertex set, R d Representing a real set, a new vertex is recursively defined as followsIs the ith control vertex coordinate after the k+1th subdivision, P k+1 Representing the control top point set after the (k+1) th iteration, the subdivision generation rule of the triple-three-point interpolation subdivision method is as follows:
the limit curve generated by the subdivision format reaches C 2 The process is continuous and the process is carried out,is the ith vertex coordinate after the kth iteration.
(2) For type four p i ，p i+1 The generation of new vertices, all labeled interpolation points I, using the four-point ternary interpolation subdivision method includes the steps of:
b1 Acquiring an initial set of control vertices
B2 If any)For the control vertex set after the kth subdivision, a new vertex is recursively defined as followsThe subdivision generation rule of the four-point triple interpolation subdivision method is as follows:
wherein,
when (when)The limit curve generated by the subdivision format reaches C 2 Continuously, u represents a parameter, ">Is the ith vertex coordinate after the kth iteration.
The initial control vertices are marked as interpolation points and approximation points, and then the following will occur.
The interpolation point is marked as 'I', the approximation point is marked as 'A', the position of the vertex marked as 'I' is kept unchanged in the subdivision iteration process, and the vertex marked as 'A' is continuously replaced by a new vertex generated in the subdivision iteration process. P is p i ，p i+1 Two points determine an edge, and four conditions exist on the edge:
(a)p i ，p i+1 are all labeled 'a';
(b)p i labeled 'A', p i+1 Labeled 'I';
(c)p i labeled 'I', p i+1 Labeled 'A';
(d)p i ，p i+1 are labeled 'I'.
For p i ，p i+1 The edges, all labeled 'A', are subdivided into new vertices by three-point triple approximation, for p i ，p i+1 The edges, all labeled 'I', are subdivided by four-point triple interpolation to generate new vertices.
Here, for p i ，p i+1 Marking different types of vertices, a new rule is now required to be formulated to bring the generated limit curve to C 2 Continuous.
We first consider five verticesWherein->Labeled 'I', the remaining vertices labeled 'a'. After the modification has been made,generating a new vertex->Is->As shown in fig. 3.
Now assume that a new vertex is generated as follows
α, β, a, b, c, d are linear combination coefficients, and a+b+c+d=1.
The following discussion of which conditions Cai Neng are met by α, β, a, b, C, d is that the resulting limit curve is C 2 Continuous.
First, the formulas (1), (2) and (3) are written in the form of a matrix:
where D is a local subdivision matrix.
The characteristic structure of the local subdivision matrix has an important influence on the limiting properties of the subdivision pattern. Let the eigenvalue of an n-dimensional local subdivision matrix be lambda 0 ＝1≥λ 1 ≥...≥λ n-1 If the characteristic value satisfies lambda 1 2 ＝|λ 2 |＞|λ 3 I, then the corresponding subdivision format has C 2 Continuous.
Firstly, calculating eigenvalue of local subdivision matrix D, det represents determinant
It is easy to calculate the characteristic values of the local subdivision matrix D are 1, alpha-beta, alpha+beta, a, a, and if alpha > 0 and beta < 0, alpha-beta is equal to or greater than alpha+beta, and thus lambda is present 0 ＝1，λ 1 ＝α-β，λ 2 ＝α+β，λ 3 To have a subdivision format with C 2 Continuously, the feature values of the partial subdivision must satisfy a < alpha+beta < alpha-beta < 1 and (alpha-beta) 2 To simplify the relationship of expressing α, β, we introduce the parameter β0. Let α - β=δ, then α+β=δ 2 ，a＝cδ 2 C.epsilon.0.1. The parameter delta has its own geometrical meaning. Thus meeting lambda 1 2 ＝|λ 2 |＞|λ 3 And the generation limit curve satisfies C2.
The coefficients of α, β, a, b, c, d in formulas (1) - (3) therefore need to satisfy α - β=δ, α+β=δ 2 ，a＝cδ 2 C is E (0, 1). Theorem 1 gives evidence of delta geometry.
Theorem 1: let q(s) be a quadratic polynomial,the three vertices are the interpolation points of the quadratic polynomial at s= -1,0,1, respectively, then the new vertex defined by the equation (3) (4) is ∈>Then q(s) takes on the value at s= - δ, δ.
And (3) proving: since q(s) is a quadratic polynomial, let q(s) =q 0 +q 1 s+q 2 s 2 。
Because of
So that
Thus (2)
Can be similarly obtained
The syndrome is known.
For the four cases above, the different rules are used for different cases respectively, so that the constructed curve has C 2 Continuous.
Combining the above analysis and demonstration, we define the subdivision geometry rules of the interpolation-approximation subdivision method as follows:
(1) New vertexThere are four cases, each calculated according to the following rule
(1. A) ifMarked as 'I',>marked 'A', then
Where delta is assigned to a pointIs a parameter of (a).
(1. B) ifMarked as 'A',>marked as 'I', then
Where delta is assigned to a pointIs a parameter of (a).
(2) New vertexThere are four cases, each calculated according to the following rule
(2. A) ifMarked as 'I',>marked 'A', then
Wherein a < delta 2 .
(2. B) ifMarked as 'A',>marked as 'I', then
Wherein a < delta 2 .
(3) New vertexThere are two cases, each calculated according to the following rules:
(3. A) ifMarked as 'I', then
(3.b) ifMarked 'A', then
Fourth, generating new vertexes for the vertexes of different types.
P of type two i Marked as approximation point A, p i+1 Marked as interpolation point I and type three p i Marked as interpolation point I, p i+1 The point marked as the approximation point A generates a new vertex according to a new rule method, and the point is set to satisfy the limit curve and satisfy C 2 The new vertices that are continuous and generated are all marked as approximation points. The method comprises the following specific steps:
(1) For type twoMarked as approximation point a->Marked as interpolation pointsI, the subdivision generation rule is as follows:
wherein a+b+c+d=1, a=cδ is satisfied 2 Delta is assigned to a pointC, delta e (0, 1);
(2) For type threeMarked as interpolation points I, ">Labeled as approximation point a, its subdivision generation rule is as follows:
where delta is assigned to a pointParameters of (2); and satisfies a+b+c+d=1, a=cδ 2 ，c，δ∈(0，1)。
Fifth, generating a curve image: generating updated vertexes according to the characteristic usage rules of the new vertexes, and generating limit curves continuously and iteratively.
(1) Aiming at the acquired new vertex, the original interpolation point I is still an interpolation point, and the generated new vertex is an approximation point A;
(2) Iteratively generating updated vertexes by using a triple four-point interpolation subdivision method aiming at the interpolation points of two adjacent points;
(3) Generating updated vertexes by using a triple three-point interpolation subdivision method aiming at the approximation points of two adjacent points;
(4) Generating updated vertexes by using a generation method of new vertexes by using vertexes of different types aiming at the adjacent points which are approximation points and interpolation points;
(5) And carrying out iterative processing on the updated vertexes, marking all generated new vertexes as approaching points, and then continuously iterating the generated marking conditions of each point according to the corresponding rule (two adjacent points are interpolation points and use a triple four-point interpolation subdivision method, two adjacent points are approximation points and use a triple three-point interpolation subdivision method, one of the adjacent points is an approximation point and one of the adjacent points is an interpolation point, and generating a new vertex by using non-type vertexes) until a limit curve is generated, namely generating a curve image.
The interpolation-approximation subdivision method referred to in the present invention interpolates certain given vertices, approximating the remaining points (as shown in fig. 2). Because the limit curves generated by the four-point triple interpolation subdivision method and the three-point triple approximation subdivision method reach C 2 The interpolation-approximation subdivision method therefore interpolates certain given vertices using a four-point ternary interpolation subdivision method, approximating the remaining points using a three-point ternary approximation subdivision method. A brand new subdivision rule is formulated at the joint of the four-point triple interpolation subdivision method and the three-point triple approximation subdivision method, so that the characteristic value of the subdivision matrix meets C 2 Continuous.
As shown in FIG. 4, FIG. 4a is a graph of a curve generated by a four-point ternary interpolation subdivision method, FIG. 4B is a graph of a curve generated by a three-point ternary approximation subdivision method, FIG. 4C is a graph of a curve generated by a quadratic B-spline method, FIG. 4d is a graph of a curve generated by a method according to the present invention, and the limit curves generated by the methods all reach C 2 Continuous. However, from the actual demand of constructing graphics, we can know that the interpolation-approximation subdivision method is more consistent with the design of our graphics, and is more flexible in graphic design.
The foregoing has shown and described the basic principles, principal features and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, and that the above embodiments and descriptions are merely illustrative of the principles of the present invention, and various changes and modifications may be made therein without departing from the spirit and scope of the invention, which is defined by the appended claims. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (5)

1. The curve image construction method based on interpolation and approximation subdivision technology is characterized by comprising the following steps:
11 Acquiring image control vertex information): obtaining a given column of control vertices { p } i Control vertex { p }, control vertex i The method comprises the steps of dividing the interpolation point into an interpolation point and an approximation point, wherein the interpolation point is marked as I, the approximation point is marked as A, the position of the interpolation point I is set to be unchanged in the subdivision iteration process, and the approximation point A is continuously replaced by a new vertex generated in the subdivision iteration process;
12 Set image side information): setting p i 、p i+1 Two points define an edge, and the points on the edge are classified into the following four types: type one, p i 、p i+1 Marked as an approximation point a; type two, p i Marked as approximation point A, p i+1 Marking as interpolation point I; type three, p i Marked as interpolation point I, p i+1 Marked as an approximation point A; type four, p i ,p i+1 Marked as interpolation point I;
13 Generating new vertices for vertices of the same type: for type one p i 、p i+1 Three-point triple approximation subdivision method for generating new vertex and pair type four p i ,p i+1 Generating new vertexes by a four-point triple interpolation subdivision method, which are marked as interpolation points I, wherein the generated new vertexes are marked as approximation points;
14 Generating new vertices for non-type vertices: p of type two i Marked as approximation point A, p i+1 Marked as interpolation point I and type three p i Marked as interpolation point I, p i+1 Generating new vertexes according to the set rules marked as the approximation point A, wherein the set rules meet the limit curve to meet C 2 The new vertexes which are continuous and generated are marked as approximation points;
15 Generation of a curve image: generating updated vertexes according to the characteristic usage rules of the new vertexes, and generating limit curves continuously and iteratively, namely generating curve images.
2. The method for constructing a curve image based on interpolation and approximate subdivision technique as claimed in claim 1, wherein said pair type-p i 、p i+1 The generation of a new vertex by a three-point triple approximation subdivision method, both labeled as approximation point A, includes the steps of:
21 Acquiring an initial set of control vertices Is the coordinates of the initial control vertices, n is the initial number of vertices, i represents the ith symbol, P 0 Representing an initial set of control vertices, R d Representing a real set;
22 If any)For the control vertex set after the kth subdivision, n is the initial vertex number, i represents the ith symbol, ++>Represents the ith vertex coordinate, P, of the kth iteration k Represents the k iteration control vertex set, R d Representing a real set, a new vertex is recursively defined as follows> Is the ith control vertex coordinate after the k+1th subdivision, P k+1 Representing the control top point set after the (k+1) th iteration, the subdivision generation rule of the triple-three-point interpolation subdivision method is as follows:
the limit curve generated by the subdivision format reaches C 2 The process is continuous and the process is carried out,is the ith vertex coordinate after the kth iteration,/->Representing the vertex coordinates at 3i after the (k+1) th iteration,/o>Representing the vertex coordinates at 3i+1 after the (k+1) th iteration, ++1->Representing the vertex coordinates at 3i+2 after the k+1th iteration.
3.A method of constructing a curvilinear image based on interpolation and approximation subdivision technique as claimed in claim 1, wherein said pair type four p i ,p i+1 The generation of new vertices, all labeled interpolation points I, using the four-point ternary interpolation subdivision method includes the steps of:
31 Acquiring an initial set of control vertices Is the coordinates of the initial control vertices, n is the initial number of vertices, i represents the ith symbol, P 0 Representing an initial set of control vertices, R d Representing a real set;
32 If any)For the control vertex set after the kth subdivision, n is the initial vertex number, i represents the ith symbol, ++>Represents the ith vertex coordinate, P, of the kth iteration k Represents the k iteration control vertex set, R d Representing a real set, a new vertex is recursively defined as follows> Is the ith control vertex coordinate after the k+1th subdivision, P k+1 The control top point set after the (k+1) th iteration is represented, and the subdivision generation rule of the four-point triple interpolation subdivision method is as follows:
wherein,
when (when)The limit curve generated by the subdivision format reaches C 2 Continuously, u represents a parameter.
4. A method of constructing a curvilinear image based on interpolation and approximation subdivision technique as claimed in claim 1, wherein said generating new vertices for non-type vertices comprises the steps of:
41 P) of type II i Marked as approximation point A, p i+1 Marked as interpolation point I, ifFor the control vertex set after the kth subdivision,/->Represents the ith vertex coordinate, P, of the kth iteration k Represents the k iteration control vertex set, R d Represents a set of real numbers,
then the new vertex is recursively defined as follows Is the ith control vertex coordinate after the k+1th subdivision, P k+1 Representing the control vertex set after the k+1th iteration, the subdivision generation rule is as follows:
where delta is assigned to a pointParameters of (2); and satisfies a+b+c+d=1, a<δ 2 ，δ∈(0,1)；
42 For type three p i Marked as interpolation point I, p i+1 Marked as an approximation point A, ifFor the control vertex set after the kth subdivision, if +.>For the control vertex set after the kth subdivision,/->Represents the ith vertex coordinate, P, of the kth iteration k Represents the k iteration control vertex set, R d Representing a real set, a new vertex is recursively defined as follows Is the ith control vertex coordinate after the k+1th subdivision, P k+1 Representing the control vertex set after the k+1th iteration, the subdivision generation rule is as follows:
where delta is assigned to a pointParameters of (2); and satisfies a+b+c+d=1, a<δ 2 ，δ∈(0,1)。
5. The method for constructing a curve image based on interpolation and approximate subdivision technique as claimed in claim 1, wherein the generating of the curve image comprises the steps of:
51 Aiming at the acquired new vertex, the original interpolation point I is still an interpolation point, and the generated new vertex is an approximation point A;
52 For two adjacent points, using triple four-point interpolation subdivision method to generate update vertex;
53 Generating updated vertexes by using a triple three-point interpolation subdivision method aiming at two adjacent points which are approximation points;
54 Aiming at the adjacent points, the approximation points and the interpolation points, generating updated vertexes by using a new vertex generating method by using vertexes of different types;
55 Performing iterative processing on the updated vertexes, marking all the generated new vertexes as approximate points, and then generating marking conditions for each point, and iterating continuously according to the corresponding rule until a limit curve is generated, namely generating a curve image.
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