JPH0855241A - Operation generating method of computer graphics - Google Patents

Abstract

PURPOSE:To provide the operation generating method of computer graphics which can generate the operation of an object body of computer graphics extremely naturally in real time with time-varying homotropy based upon physical definitions. CONSTITUTION:In a step 1, position vectors of time t=0 and time t=1 at respective points constituting the object of computer graphics are inputted and in a step 2, a blending function is defined on the basis of the physical definitions. In a step 3, the position vectors at respective points constituting the object body at time t=t are found, the operation of the respective points constituting the object body between the time t=0 and time t=1 is generated, and operations of all the points are generated to generate the natural operation of the object body conforming with a natural rule.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a motion generation method in computer graphics, and more particularly to generating motions of points constituting a target object in order to generate motions of the target object based on time-varying homotopy. And a motion generation method in computer graphics.

[0002]

2. Description of the Related Art FIG. 10 is a diagram showing a state of a realistic communication conference.

A conference for communicating as if people existing in different places meet together is called a "realistic communication conference". In this realistic communication conference, the image of the participant 3 is set in the virtual space 1, and the participants 7a and 7b existing in the real space 5 can hold a conference with the participant 3 in the virtual space 1.

The purpose of this immersive communication conference is to set an image of each of the conference participants located in different places in the generated virtual space, and to have the feeling that these participants sympathize with the same virtual space. And to provide an environment in which meetings or collaborative work can be conducted.

FIG. 11 is a diagram for explaining a realistic communication conference system. The realistic communication conference system is roughly divided into a transmitting side for transmitting information in the real space and a receiving side for receiving the transmitted information. The transmitting side is composed of a detection unit 11 that detects the temporal change information of a person in the real space 9. The receiving side is composed of a synthesizing unit 13 that generates and displays a person's motion or facial expression change in real time.

[0006] It is desired that the human image detected in the real space 9 and generated in the virtual space 15 looks natural from all directions in the presence communication conference. Therefore, the synthesizing unit 13 for generating and displaying a natural three-dimensional face image having a posture change or facial expression change in real time is extremely important.

FIG. 12 is a diagram showing facial expression muscles,
FIG. 13 is a diagram showing a three-dimensional wire frame model.

In the synthesizing section 13 of FIG. 11, a three-dimensional wire frame model 19 as shown in FIG. 13 based on the facial expression muscles 17a to 17v shown in FIG. 12 is used. Such a three-dimensional wireframe model 19 is used to generate a change in the facial expression of a human face image in real time. Then, in order to generate and display such a facial expression change of a human face image in real time, a time-varying homotopy (T
Ime-VaryingHomotopy) was published by the inventors, "Position examination of three-dimensional facial image expression generation based on homotopy", Technical Report of the Television Society, Vol.
17, No. 58, has been proposed.

The proposed time-varying homotopy is based on homotopy in topologies. Therefore, the process from the homotopy in topology to the proposal of the conventional time-varying homotopy will be described below.

[Homotopy in Topology] FIG. 14
FIG. 4 is a diagram for explaining homotopy in topological geometry.

Homotopy in topology is defined as follows. f: I → X and g: I → X are two different functions connecting P and Q. At this time, the area I
XI = {(x, y) | 0 ≦ x ≦ 1, 0 ≦ y ≦ 1} to X
In the connection map f: I × I → X, F (x, 0) =
f (x), F (x, 1) = g (x) and F (0, y) =
When there is a map such that P, F (1, y) = Q, F (x,
It is assumed that y) is the homotopy of the function f and the function g.

The homotopy in this topology is not directly applied to computer graphics. The reason is that the time taken to transit from the function f to the function g in this homotopy is not defined. Therefore, in order to apply this concept of homotopy to the shape processing of a target object in computer graphics and the generation of motion and the motion generation such as real-time facial expression change of a 3D human face image, the following homotopy is proposed. Was done. This proposed homotopy is a time-varying homotopy.
py). Hereinafter, the time-varying homotopy will be described.

[Conventional Time-Varying Homotopy in Topology] FIG. 15 is a diagram for explaining the conventional time-varying homotopy. In particular, FIG. 15 (a) shows a contour line at time t = 0. 15B is a diagram showing the state of FIG.
FIG. 15 is a diagram showing a state of the contour line at time t = t, and FIG. 15 (c) is a diagram showing a state of the contour line at time t = 1.

As shown in FIG. 15B, two different contour lines C ₁ ↑ (u, v, t) (hereinafter,
↑ is added to the vector), and C ₂ ↑ (u, v, t) is set in the three-dimensional space. Both contour lines C ₁ ↑ (u, v,
t) and C ₂ ↑ (u, v, t) are interpolated so that each node (point) on the surface is x ↑ (u, v, t).
t). These contour lines C ₁ ↑ (u, v, t),
C ₂ ↑ (u, v, t) and x ↑ (u, v, t) are expressed by the equation (1), the equation (2), and the equation (3), respectively. However, 0 ≦ u ≦ u _max , 0 ≦ v ≦ 1, 0 ≦ t ≦ 1
And C ₁ ↑ (u, v, t) and C ₂ ↑ (u, v, t)
Is perpendicular to the Y axis, and each Y coordinate is set so as to always satisfy the equation (4).

From the equations (1), (2) and (3), the contour line C ₁ ↑ (u, v, 0) and the contour line C ₂ ↑ (u,
Each node on the surface between v, 0) x ↑ (u,
It is assumed that v, 0) is set as shown in Expression (5). Expression (5) corresponds to the relationship as shown in FIG.

Here, the contour lines C ₁ ↑ (u, v, 0) and C
₂ ↑ (u, v, 0) R _n (v)
Function is introduced. This R _n (v) is a blending of both contour lines C ₁ ↑ (u, v, t) and C ₂ ↑ (u, v, t). _It is assumed that _n (v) is called a blending function. The blending function R _n (v) is classified into two types of functions as shown in, for example, the equations (6) and (7).

The variable n in the blending function R _n (v) shown in the equations (6) and (7) is defined by contour lines C ₁ ↑ (u, v, t) and C ₂ ↑ (u, v). , T) is an important parameter that determines the blending ratio.

The blending function R _n (v) shown in the equation (6) is a function as shown in FIG. 16 when specifically shown. The horizontal axis in FIG. 16 is the value of v, and the vertical axis is the value of R _n (v). Further, in FIG. 16, n = −0.9, n = −0.8, n = −0.65,
n = -0.4, n = 0.0, n = 1.1, n = 3.8,
The case of n = 9.3 is shown.

The blending function R _n (v) shown in the equation (7) is the function shown in FIG. In FIG. 17, the horizontal axis is the value of v, and the vertical axis is the value of R _n (v). Further, in FIG. 17, n = −0.97, n =
-0.85, n = -0.6, n = 0.0, n = 3.5,
The case of n = 23.8 is shown.

Further, according to the expressions (6) and (7), these blending functions R _n (v) satisfy the expression (8). The difference between the blending functions R _n (v) defined by the equations (6) and (7) is that v = 0, 1
It is whether or not the differential coefficient of the blending function R _n (v) in Eq. That is,
The blending function R _n (v) according to the equation (6) does not necessarily satisfy the equation (9), but the blending function shown in the equation (7) always satisfies the equation (9).

Here, the contour line C ₁ ↑ when t = 1
Each node x ↑ (u, v, 1) on the curved surface formed by interpolating between (u, v, 1) and C ₂ ↑ (u, v, 1) is expressed by the equation (10). expressed. The node x ↑ (u, v, 1) and the contour C ₁ ↑ (u, v,
The relationship between 1) and C ₂ ↑ (u, v, 1) is shown in FIG. 15 (c).

Further, the position vectors x ↑ (u, v, 0) and x ↑ (u, v) of each node at t = 0,1 defined by the equations (5) and (10), respectively. , 1), the position vector x ↑ (u, v, of each node at time t
t) is defined as in Expression (11). However, in general, L in the equation (11) is set to L ≠ n.

For the sake of explanation, the position of each node when t = 0
The position vector x ↑ (u, v, 0) is shown in equation (5).
The contour C in equation (5)₁↑
(U, v, 0) and C₂↑ For mixing (u, v, 0)
Blending function R _nEven if (v) is used
Good.

From the equation (11), the blending function R
_{When n} (t) is transited according to time t (0 ≦ t ≦ 1), position vector x ↑ (u, v, 0) at time t = 0
And a position vector x ↑ (u, u at time t = 1
By interpolating with the model having v, 1) by this blending function R _n (t), the target model easily transits in real time. Then, only by changing the value of the variable L, the temporal transition of the target model is significantly changed. Furthermore, the shape of the model at t = 1 can be easily deformed only by changing the value of the variable n used in the blending function R _n (t) in the equation (10). Similarly, when the blending function is used also for the equation (5), the shape of the model at the time t = 0 is easily transformed.

That is, according to the concept of the time-varying homotopy as shown in the equations (1) to (11), the first-order differential of the static shape of the target object in computer graphics as shown in FIG. It was possible to easily generate a surface having a discontinuity or a surface having a continuous first derivative as shown in FIG. The surface with the discontinuity of the first derivative shown in FIG.
A surface realized by a blending function R _n (v) as shown in FIG. 6 and having continuous first differentials as shown in FIG.
It was easily generated by the blending function R _n (v) shown in FIG. Further, the real-time motion of the target object could be easily generated as shown in FIG. Figure 2
20 (a) to 20 (i) show the state of the target object at each time.
In particular, FIG. 20 (a) is a diagram showing a state at time t = 0, and FIG. 20 (i) is a diagram showing a case at time t = 1.

[0026]

[Equation 1]

[0027]

However, although the conventional time-varying homotopy is based on the mathematical definition in topological geometry, it is originally necessary when considering the motion of the target object in computer graphics. It was not necessarily based on the essential physical definition.

Therefore, for example, even when an external force is applied to a target object by computer graphics,
The time-varying homotopy in which the external force was taken into consideration was not defined. Furthermore, the transfer function that should be defined by the digital control theory cannot be defined because it is not always based on the physical definition in the conventional time-varying homotopy.

Therefore, an object of the present invention is to extend a time-varying homotopy based on a mathematical definition to a time-varying homotopy based on a physical definition so that the motion of a target object in computer graphics can be performed more naturally and in real time. An object of the present invention is to provide a method for generating an action in computer graphics that can be generated.

[0030]

According to a first aspect of the present invention, there is provided a method for generating motions in computer graphics. In order to generate motions of a target object based on time-varying homotopy, motions of points constituting the target object are calculated. A method for generating motion in computer graphics, wherein a time-varying homotopy is physically defined, a position vector of a point at a first time is given, and a position vector of a point at a second time is given. A second step defining a first step and a mixing function indicative of a mixing ratio of a position vector of the point at the first time and a position vector of the point at the second time given in the first step; And a third step of obtaining a position vector at a predetermined time between the first time and the second time of the point based on the mixing function defined in the second step. .

In claim 2, the time-varying homotopy of claim 1 defines a transfer function for defining a mixing function.

In the third aspect, the time-varying homotopy of the first aspect defines an external force applied to the target object.

[0033]

In the motion generating method in computer graphics according to the first aspect of the invention, the time-varying homotopy is physically defined, and the position vector at the first time point of the point forming the target object is given, Given a position vector of a point at a second time, a mixing function is defined that indicates the ratio of the mixture of the position vector of the point at the first time and the position vector of the point at the second time, and the mixing function A position vector at a predetermined time between the first time and the second time is obtained based on the above, and a motion of a point from the first time to the second time is generated to further configure a target object. The motion of the target object can be generated by generating the motions of all points.

In the motion generating method in computer graphics according to the second aspect of the invention, the transfer function for the time-varying homotopy to define the mixing function can be defined to generate the motion of the target object based on, for example, digital control theory. .

In the motion generation method in computer graphics according to the third aspect of the present invention, the time-varying homotopy defines the external force applied to the target object, and the motion of the target object to which the external force is applied can be generated.

[0036]

EXAMPLE Since the motion of the object of interest in the natural world follows the physical law, the motion rule of the object of interest is also in computer graphics animation, especially in the synthesizing unit 13 of FIG. 11 in the immersive communication conference system. , Should follow the physical definition.

For example, a person's natural facial expression and subtle movements in real time are very often based on facial muscles existing on the surface of the face. In consideration of this, it is also necessary for the synthesizing unit 13 in the immersive teleconferencing system to define the muscle model of the facial expression muscle and its motion rule as the operation method. Therefore, it is an important task to faithfully define the equation of motion or transfer function unique to each muscle based on the motion characteristics of each muscle.

On the other hand, when simulating with a digital computer so that various data measured by continuous quantity (analog quantity) such as position, displacement, velocity, acceleration, etc. of a target object in nature are processed, These data are discretized (digitized). Therefore, we propose a new time-varying homotopy based on the new digital control theory by complementing and combining the mathematical definition based on the time-varying homotopy in the conventional topological geometry with the physical definition in the natural world. . The definition of the new time-varying homotopy will be described below.

[Linear Invariant System in Continuous Time] Assuming that the blending function R _n (v) used in the time-varying homotopy satisfies the condition based on the equation (9),
The response output of a DC servomotor, which is a linear time-invariant system in a digital control system, can be mentioned.

FIG. 1 is a diagram for explaining a DC servo motor. In the DC servomotor 21, J is the moment of inertia of the rotor 23 including the load, R is the resistance of the armature, D is the viscous friction coefficient of the bearing, u is the input voltage, K is the voltage amplification factor of the power amplifier 27, and M is The motor constant of the motor 29 is used. Further, the inductance of the armature is negligible, and the field is low excitation control composed of permanent magnets.

According to the principle of DC servo, since the electromotive force of the armature is M (dθ / dt), the equation (12) is established. Further, since the generated torque is Mi, the equation (13) is established. Therefore, after the current i is erased from the equations (12) and (13), x ₁ =
When θ and x ₂ = dθ / dt are set, the equation (14) is established. Therefore, the output y is expressed by the equation (15). Each of x ₁ and x _{2 in} the equation (15) represents the rotation angle of the rotor and the rotation angular velocity.

Next, the equations (14) and (15) are
For simplicity, as shown in the equation (16), the constant p
_c, Vector x_c↑, c_c↑, b_c↑, and matrix A_c
Determine. Expression (16) is replaced by Expression (14) and Expression (1
Substituting into equation (5), the continuous time system of the DC servo motor
The equation of state in the equation is expressed by the equation (17).
And the transfer function G_c(S) is expressed by the equation (18).
And its pole is 0, -p_cAnd there is no zero.
Here, in the formula (18), Y_c(S) and U _c(S)
Is obtained by the Laplace transform of y and u in equation (17).
Be done.

On the other hand, the transfer function G _c (s) of the continuous time system as shown in the expression (18) can be derived from the following equation of motion. For example, when generating facial expressions in computer graphics, Terzo
Poulous was defined based on the anatomical viewpoint, and the equation of motion of each node constituting the wire frame based on the facial muscles was defined as shown in the equation (19). However, for simplicity, a one-dimensional equation of motion at each node is described here, where m _n is the mass of each node, g _n ,
q _n , h _n , and f _n are various external forces applied to each node,
γ _n is a damping constant, c _n is a spring constant, and x _nc is a displacement amount of each node.

In the equation (19), x _nc = x _nc1 , d
When x _nc / dt = x _nc2 , the motion equation of the continuous-time system with respect to the motion equation of each node forming the wire frame of the human face image is expressed as in Expression (20). Further, when the output y _nc is regarded as the displacement amount, the y _nc becomes the second (2
It is expressed as in equation (1). Here, the constant u _nc , the vector x _nc ↑, b _nc ↑, c _nc ↑, and the matrix A _nc are set to (22)
Set like the formula.

The expression (22) is replaced by the expression (20) and the expression (2)
Substituting into equation (1), equations (20) and (21), which are state equations, become equation (23). Transfer function G of continuous time system for equation of motion of each node
_nc (s) is expressed by the equation (24).

If the equation (25) can be assumed by the equations (17) and (23) which are the state equations of the linear time-invariant system in the continuous time system, the transfer function is also the equation (18). The expression (26) is established by the expression (24). That is, when the assumption of the equation (25) holds,
The linear time invariant system in the continuous time system of the DC servo motor and the linear time invariant system in the continuous time system of each node constituting the wireframe of the human face image are considered to be equivalent. Therefore, the description will be given below based on this assumption.

[0047]

[Equation 2]

[0048]

(Equation 3)

[Linear Time Invariant System in Discrete Time System] Generating a moving image by computer graphics is a kind of simulation in a digital computer. In computer graphics, various data of the target object is discretized (digitized). Therefore, all physical definitions such as dynamics and control in a continuous time system need to be processed discretely. Therefore, continuous time signals u [t], x ↑ [t], y
If [t] is sampled at a sampling period T and is A / D converted into signals u (i), x ↑ (i), y (i) in the discrete time system, these signals are It is expressed by the equation (27). Then, as in equation (28), the vector b
↑, c ↑ and matrix A are defined.

By substituting the expressions (27) and (28) into the expression (17), the state equation of the linear time-invariant system in the discrete time system is expressed by the expression (29).

Here, in order to handle the response output of the DC motor, which is a linear time-invariant system, as a blending function defined in the time-varying homotopy, the following definition is specifically made. That is, in the above-mentioned DC servo motor, assuming that M, R, K, J, and D satisfy the expression (30) with respect to the expression (16), A _c and b _c ↑ of the expression (16). Becomes as shown in Expression (31). Then, in order to convert this continuous time system into a discrete time system, according to the equation (28), A _c and b _c ↑ in the equation (31) become the equation (32).

Further, in this discrete time system, the input u (i) is determined as in the equation (33), and the response output y
When (i) is obtained, the relationship is obtained as shown in FIG. In FIG. 2, in addition to the input u (i) and the output y (i),
The speed dy (i) / di and the function Bld (i) are shown.

From the output y (i) in FIG. 2 and, more clearly, the velocity dy (i) / di, it is easily seen that the equation (34) is satisfied. Furthermore, this response output y
The function Bld (i) is the third (3) so that (i) is normalized.
When defined by the equation (5), the function Bld (i) satisfies the equation (36). This is also clear from FIG.

Since the equation (34) corresponds to the equation (9) and the equation (36) corresponds to the equation (8), the response output y (i) of the linear time invariant system in this discrete time system is shown. )
The function Bld (i) defined by normalizing
Can be regarded as a blending function that satisfies the conditional expression (9) among the blending functions defined in the time-varying homotopy described above.

In this way, the new blending function Bld derived from the linear time-invariant system in the discrete time system
Using (i) to control the motion of a target object in computer graphics is described in (2)
As long as expression (5) is satisfied, it is the same as operating each node of the wire frame model based on expression (19), which is a motion equation. Therefore, the time-varying homotopy proposed here is not a time-varying homotopy based on the conventional mathematical definition of topological geometry, but a physical definition in nature to a time-varying homotopy based on the mathematical definition. It is based on a digital control theory that takes into account the inevitable discretization of data in a digital computer.

Therefore, according to the time-varying homotopy proposed here, the static shape of the surface in which the first derivative is discontinuous as shown in FIG. 18 and the surface of the target object in computer graphics which has been possible in the past and in FIG. Not only can the static shape of the surface having the first-order differential as shown in the figure be easily generated, and the motion of the target object in real time as shown in FIG. Will be able to generate.

Therefore, an application example of computer graphics animation using time-varying homotopy based on the digital control theory will be shown below. As an application example,
Fig. 7 shows cylinder motion, generation of wrinkles when moving a forehead in a human face image, and opening and closing of the mouth. It is considered that the moving image generation of these target objects is effective in the synthesizing unit 13 in the presence communication conference system of FIG. 11 as described above.

[0058]

[Equation 4]

[Cylinder Motion] FIG. 3 is a diagram showing a time-series continuous image of a cylinder operating in real time.
FIG. 4 shows the input u (i), the output y (i) and the blending function B for the cylinder operating in real time in FIG.
3 is a graph showing ld (i). In FIG. 3A, t =
FIG. 5 is a diagram showing a state of the cylinder in the case of 0, which corresponds to i = 0 in FIG. 4. FIG. 3 (j) is a diagram showing the state of the cylinder when t = 1, and corresponds to i = 20 in FIG. 3 (b) to 3 (i),
FIG. 5 is a diagram showing the state of each cylinder when i = 0 to i = 20 in FIG. 4 is divided into 10.

A cylinder 31 composed of a set of points in FIG.
It is assumed that an external force f as represented by the input u (i) in FIG. 4 is applied in real time to the center position of the bottom surface 33 of the. At this time, the blending function Bld (i) derived from the discrete-time linear time-invariant system according to the equation (29) is shown in FIG.
, And its blending function Bld
Based on the time-varying homotopy according to (i), the cylinder 31
Changes its operation in real time as shown in FIG.

In the motion generation of the cylinder shown in FIG.
There is not much difference in appearance from the motion generation of the cylinder shown in 0. However, unlike the operation generation of the cylinder shown in FIG. 20, the operation generation of the cylinder shown in FIG. 3 is given an input u (i) (external force) as shown in FIG. 4 and operates according to the external force. It is generating.

[Facial Wrinkle Motion Generation] FIG. 5 is a diagram showing a time-series continuous image of face wrinkles, and FIG. 6 is an input in the face wrinkle motion generation shown in FIG. u (i), output y (i) and blending function Bl
6 is a graph showing d (i). FIG. 5A shows time t.
FIG. 7 is a diagram showing a wrinkled state of a face when = 0, and a diagram corresponding to i = 0 in FIG. 6. In FIG. 5A, time t = 1
7 is a diagram showing the state of wrinkles of the face in the case of
It is a figure corresponding to 20. 5 (b) to 5 (i)
FIG. 7 is a diagram showing the wrinkle state of each face of i = 0 to i = 20 of FIG. 6 divided into 10.

In the direction of pulling up the forehead 35 in the three-dimensional human face image by computer graphics,
It is assumed that the external force f indicated by the input u (i) in FIG. 6 is applied. The displacement amount is calculated by the time-varying homotopy based on the blending function derived from the non-linear time-invariant system of the discrete time system described above. Based on this, the displacement amount of each node located in the face surface portion of the wire frame forming the human face image is determined, and as shown in FIG. 5, the motion is generated in real time and the wrinkles 37 are generated. It was

[Mouth Opening / Closing Motion Generation] FIG. 7 is a diagram showing time-series continuous images of the mouth opening / closing motion generation, and FIG.
FIG. 8 is a diagram showing a wireframe model corresponding to each of FIGS. 7A to 7J. FIG. 7A is a diagram showing a human face image at time t = 0.
(J) is a diagram showing a human face image at time t = 1.

Sixteen characteristic points are defined around the mouth 39 in FIG. An external force, a transfer function that constitutes a linear time-invariant system in a discrete time system, and a blending function are defined for each feature point. Briefly, the external force acts radially from the center of the mouth opening. As a result, the wireframe model opened the mouth as shown in FIG. 8, and the person in the human face image opened the mouth as shown in FIG. 7.

FIG. 9 is a flow chart of a motion generation method in computer graphics according to an embodiment of the present invention.

In step (denoted by S in the drawing) 1, position vectors at time t = 0 and time t = 1 are input. At times t = 0 and t = 1, as described above, i = 0 in the function shown in FIG.
= 1 is i = 20 in FIG. Similarly, in the function shown in FIG. 6, time t = 0 is i = 0, and time t = 1 is i = 20.

The position vector in FIG. 3 is the position vector of each point because the cylinder 31 itself is a set of points. As a result, the position vector of each point generates motion between the time t = 0 and the time t = 1, whereby the cylinder 31 generates motion.

The position vector in FIG. 7 is the position vector of each node of the wire frame model shown in FIG. 8, and the position vector is from time t = 0 to time t =
By generating the motion while reaching 1, the motion of the wire frame model is generated. As a result, the motion is generated even for the human face image as shown in FIG. 7. Similarly, the position vector shown in FIG. 5 is a position vector of each node of the wire frame model, which is not shown.

Next, in step 2, the blending function is defined based on the physical definition. The physical definition includes, for example, an external force acting on a target object that operates in computer graphics, and the transfer function described above.

In step 3, the position vector at time t = t is obtained. Time t = t corresponds to each of FIGS. 3B to 3I in FIG. 3, and the position vector is the position vector of each point forming the cylinder 31. In the case of FIG. 7, t = t is each of FIG. 7 (b) to FIG. 7 (i), and the position vector is the position vector of each node forming the wire frame model of FIG. Similarly, t = t in FIG. 5 is each of FIG. 5 (b) to FIG. 5 (i), and the position vector is the position vector of each node of the wire frame model (not shown). By this step 3, since the position vectors of the points forming the target object (cylinder, wire frame model) are obtained, by calculating all the position vectors for the points, the motion of the target object is generated.

In this way, a new time-varying homotopy is defined that takes into account not only the mathematical definition but also the physical definition.
Very good and natural real-time motions of target objects of interest in computer graphics are generated.

More specifically, the new time-varying homotopy is defined by combining the conventional mathematical definition and the physical definition by the equation of motion based on dynamics, complementing each other. An external force is defined by a state equation that can describe a linear time-invariant system in a discrete-time system based on digital control theory, and the output is normalized to derive a blending function. However, a more stable behavior of the target object in computer graphics is generated after considering the stability conditions, controllability, observability, etc. of the linear time-invariant system in the discrete time system defined by the newly proposed time-varying homotopy. If it is done, it will be good in the future.

Therefore, it is considered that such digital control theory or digital signal processing will make a great contribution especially when constructing the realistic communication conference system, particularly in the synthesizing section 13 of FIG.

Next, the differences between the conventional time-varying homotopy and the time-varying homotopy proposed in the present invention will be summarized.

1. In the conventional time-varying homotopy, the external force acting on the target object that operates in computer graphics could not be defined, but in the new time-varying homotopy, the external force can be defined.

2. In the conventional time-varying homotopy, the transfer function could not be defined when defining the blending function, but in the new time-varying homotopy, the transfer function can be defined.

From 3.1 and 2, by adding a physical definition to the time-varying homotopy, which was conventionally limited to a mathematical definition, and combining them while complementing each other, a more natural and It is possible to display real-time behavior, and much more applications are expected than when the time-varying homotopy remained in its mathematical definition.

[0079]

As described above, according to the present invention, a position vector at a first time of a point constituting a target object which is a target in computer graphics is given, and a position of the point at a second time is given. A vector is provided, and a mixing function is defined that indicates a mixing ratio of the position vector at the first time and the position vector at the second time, and the first time and the second time are defined based on the defined mixing function. Since the position vector at a predetermined time between the times of is generated and the motion of the point is generated and the motion of the target object is generated based on the physically defined time-varying homotopy, the target in computer graphics It is possible to generate a natural and real-time motion of the target object as much as possible.

[Brief description of drawings]

FIG. 1 is a diagram for explaining the principle of a DC servo motor.

FIG. 2 is an input u (i), an input u in a discrete time system.
6 is a graph showing the values of output y (i), speed dy (i) / di, and blending function Bld (i) when the value of (i) is changed to 1, 2, 0 according to i.

FIG. 3 is a diagram showing a time-series continuous image of a cylinder operating in real time by time-varying homotopy based on a physical definition.

4 is a graph showing an input u (i), an output y (i) and a blending function Bld (i) for motion generation of the cylinder shown in FIG.

FIG. 5 is a diagram showing a time series continuous image of a wrinkle of a face operating in real time by time-varying homotopy based on a physical definition.

6 is a graph showing an input u (i), an output y (i), and a blending function Bld (i) for the facial wrinkle motion generation shown in FIG.

FIG. 7 is a diagram showing a time-series continuous image of opening and closing of a mouth operating in real time by time-varying homotopy based on physical definition.

FIG. 8 is a diagram showing a state of the three-dimensional wire frame model with respect to the mouth opening / closing motion generation shown in FIG. 7;

FIG. 9 is a flowchart showing a motion generation method in computer graphics according to an embodiment of the present invention.

FIG. 10 is a diagram showing a state of a realistic communication conference.

FIG. 11 is a schematic block diagram of a device of a realistic communication conference system.

FIG. 12 is a diagram showing the structure of facial expression muscles.

FIG. 13 is a diagram showing a three-dimensional wire frame model.

FIG. 14 is a diagram for explaining homotopy in topology (topology).

FIG. 15 is a diagram for explaining time-varying homotopy based on a conventional mathematical definition.

FIG. 16 is a graph showing an example of a blending function R _n (v).

FIG. 17 is a graph showing another example of the blending function R _n (v).

FIG. 18 is a blending function R shown in FIG.
It is the figure which showed the surface where the 1st derivative of the static shape of the object obtained by _n (v) was discontinuous.

FIG. 19 is a blending function R shown in FIG.
It is the figure which showed the shape of the surface where the 1st derivative of the static shape of the object obtained by _n (v) is continuous.

FIG. 20 is a diagram showing real-time motion generation of a cylinder obtained by time-varying homotopy based on a conventional mathematical definition.

Claims (3)

[Claims] 1. A motion generation method in computer graphics for generating a motion of a target object based on a time-varying homotopy, the motion generating method in computer graphics, wherein the time-varying homotopy is a physical property. Defined firstly, a position vector at a first time of the point is given, and a position vector at a second time of the point is given, and the step given in the first step A second step of defining a mixing function indicating a mixing ratio of the position vector of the point at the first time and the position vector of the point at the second time; and the mixing defined in the second step. Based on the function
A third step of obtaining a position vector of the point at a predetermined time between the first time and the second time, the motion generating method in computer graphics. 2. The motion generation method in computer graphics according to claim 1, wherein the time-varying homotopy defines a transfer function for defining the mixing function. 3. The motion generation method in computer graphics according to claim 1, wherein the time-varying homotopy defines an external force applied to the target object.