JP2739436B2 - Motion display generation method in computer graphics - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a motion display generation method in computer graphics, and in particular, to display motion of a point constituting a target object in order to display and generate a motion of the target object based on a time-varying homotopy. The present invention relates to a method for generating an operation display in computer graphics that can be generated.

[0002]

2. Description of the Related Art FIG. 18 is a diagram showing a state of a realistic communication conference.

[0003] A conference in which people existing in different places communicate with each other as if they are gathered together is called a real-life communication conference. In the presence communication conference, an image of the participant 3 is set in the virtual space 1, and the participants 7 a and 7 b existing in the real space 5 can hold a conference with the participant 3 in the virtual space 1.

[0004] The purpose of this real-life communication conference is to set images of conference participants located in different places in the generated virtual space, and to have a feeling that these participants sympathize with the same virtual space. And provide an environment in which meetings or collaborative work can be performed.

[0005] FIG. 19 is a diagram for explaining a realistic communication conference system. The presence communication conference system is roughly divided into a transmitting side for transmitting information in a real space and a receiving side for receiving the transmitted information. The transmitting side includes a detection unit 11 that detects temporal change information of a person in the real space 9. The receiving side is composed of a synthesizing unit 13 for generating and displaying a motion or a facial expression change of a person in real time.

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

FIG. 20 is a diagram showing facial expression muscles.
FIG. 21 is a diagram showing a three-dimensional wire frame model.

In the synthesizing unit 13 shown in FIG. 19, a three-dimensional wire frame model 19 as shown in FIG. 21 based on the facial expression muscles 17a to 17v shown in FIG. 20 is used. Using such a three-dimensional wireframe model 19, a change in the expression of a human face image in real time is generated. In order to generate and display the facial expression change of the human face image in real time, a time-varying homotopy (T
im-VaryingHomotopy), "Position study of three-dimensional facial image expression generation based on homotopy" by the inventors, Technical Report of the Institute of Television Engineers of Japan, Vol.
17, No. 58 ,.

[0009] The proposed time-varying homotopy is based on homotopy in topology. Therefore, the process from the homotopy in the topology to the proposal of the conventional time-varying homotopy will be described below.

[Homotopy in Topology] FIG. 22
FIG. 3 is a diagram for explaining homotopy in topology.

[0011] The homotopy in the topology is defined as follows. f: I → X and g: I → X are two different functions connecting P and Q. At this time, the region I
× I = {(x, y) | 0 ≦ x ≦ 1,0 ≦ y ≦ 1} to X
To the connected map f: I × I → X, F (x, 0) =
f (x), F (x, 1) = g (x) and F (0, y) =
When there is a mapping such that P, F (1, y) = Q, F (x,
y) is assumed to be a homotopy between the function f and the function g.

The homotopy in the topology is not directly applied to computer graphics. The reason is that in this homotopy, the time taken to transition from the function f to the function g is not defined. Therefore, the following homotopy is proposed so that the concept of homotopy can be applied to shape processing and motion generation of target objects in computer graphics, as well as motion generation such as facial expression changes in real time of 3D human face images. Was done. The proposed homotopy is a time-varying homotopy.
py). Hereinafter, the time-varying homotopy will be described.

[Conventional Time-Varying Homotopy in Topology] FIG. 23 is a diagram for explaining a conventional time-varying homotopy. In particular, FIG. 23A shows a contour line at time t = 0. FIG. 23B shows the state of FIG.
FIG. 23 is a diagram showing the state of the contour line at time t = t, and FIG. 23C is a diagram showing the state of the contour line at time t = 1.

As shown in FIG. 23B, two different contours C ₁ ↑ (u, v, t) (hereinafter, referred to as “t”) at time t.
ベ ク ト ル is assigned to the vector), and C ₂ ↑ (u, v, t) is set in the three-dimensional space. Both contour lines C ₁ ↑ (u, v,
t), C ₂ ↑ (u, v, t) are interpolated so that each node (point) on the surface is x ↑ (u, v, t).
t). These contours C ₁ ↑ (u, v, t),
C ₂ ↑ (u, v, t) and x ↑ (u, v, t) are expressed by equations (1), (2) and (3), respectively. Where 0 ≦ u ≦ u _max , 0 ≦ v ≦ 1, 0 ≦ t ≦ 1
C ₁ ↑ (u, v, t) and C ₂ ↑ (u, v, t)
Are set to be perpendicular to the Y axis, and each Y coordinate always satisfies Expression (4).

According to the equations (1), (2) and (3), the contour C ₁ ↑ (u, v, 0) and the contour C ₂ ↑ (u,
v, 0) and each node x ↑ (u,
(v, 0) is set as in equation (5). Equation (5) corresponds to the relationship as shown in FIG.

Here, the contour lines C ₁ ↑ (u, v, 0) and C
₂ R _n (v) to be interpolated between ↑ (u, v, 0)
Is introduced. This R _n (v) is obtained by blending the contour lines C ₁ ↑ (u, v, t) and C ₂ ↑ (u, v, t). _{Let n} (v) be referred to as a blending function. The blending function R _n (v) is classified into, for example, two types of functions as shown in Expressions (6) and (7).

The variables n in the blending function R _n (v) shown in the equations (6) and (7) are the contours C ₁ ↑ (u, v, t) and C ₂ ↑ (u, v , T) are important parameters that determine the blending ratio.

The blending function R _n (v) shown in the equation (6) is a function as shown in FIG. 24 when specifically illustrated. The horizontal axis in FIG. 24 is the value of v, and the vertical axis is the value of R _n (v). Further, in FIG. 24, n = −0.9, n = −0.8, n = −0.65,
n = −0.4, n = 0.0, n = 1.1, n = 3.8,
The case where n = 9.3 is shown.

The blending function R _n (v) shown in the equation (7) is a function shown in FIG. In FIG. 25, the horizontal axis is the value of v, and the vertical axis is the value of R _n (v). Further, in FIG. 25, n = −0.97, n =
−0.85, n = −0.6, n = 0.0, n = 3.5,
The case where n = 23.8 is shown.

According to equations (6) and (7), these blending functions R _n (v) satisfy equation (8). The difference between the blending functions R _n (v) defined in the equations (6) and (7) is that v = 0,1
Whether the differential coefficient of the blending function R _n (v) in Equation (6) satisfies the condition of Expression (9). That is,
The blending function R _n (v) according to the expression (6) does not always satisfy the expression (9), but the blending function shown in the expression (7) always satisfies the expression (9).

Here, the contour line C ₁ ↑ at t = 1
(U, v, 1) and each node x ↑ (u, v, 1) on a curved surface formed by interpolating between C ₂ ↑ (u, v, 1) as shown in Expression (10) expressed. The node x ↑ (u, v, 1) and the contour C ₁ ↑ (u, v,
1) and C ₂ ↑ (u, v, 1) are shown in FIG.

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,
t) is defined as in equation (11). However, in general, L in Expression (11) is set to L ≠ n.

For the sake of explanation, the order of each node when t = 0
The replacement vector x ↑ (u, v, 0) is given by equation (5).
The contour line C in equation (5) was used.₁↑
(U, v, 0) and C_Twoに For mixing (u, v, 0)
Even the blending function R _nEven if (v) is used
Good.

From equation (11), the blending function R
_{When L} (t) is shifted according to time t (0 ≦ t ≦ 1), the position vector x ベ ク ト ル (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) using this blending function R _L (t), the target model easily transitions 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 Expression (10). Similarly, when the blending function is used for Expression (5), the shape of the model at the time t = 0 is also easily deformed.

That is, by 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. Could easily generate a discontinuous surface or a surface where the first derivative is continuous as shown in FIG. The surface where the first derivative is discontinuous shown in FIG.
The surface realized by the blending function R _n (v) as shown in FIG. 4 and in which the first derivative is continuous as shown in FIG.
It was easily generated by the blending function R _n (v) shown in FIG. In addition, the motion of the target object in real time could be easily generated as shown in FIG. FIG.
8, (a) to (i) of FIG. 28 are diagrams showing the state of the target object at each time,
In particular, FIG. 28A is a diagram showing a state at time t = 0, and FIG. 28I 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 the topology, it is originally required when considering the operation of the object to be processed in computer graphics. It was not always based on essential physical definitions.

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

Accordingly, 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, and to more naturally realize the motion of a target object in computer graphics in real time. An object of the present invention is to provide a method for generating an operation display in computer graphics that can generate a display.

[0030]

According to a first aspect of the present invention, there is provided a method for generating a motion display in computer graphics, comprising the steps of: generating a motion of a target object based on a time-varying homotopy; A motion display generation method in computer graphics for displaying and generating a motion, wherein a time-varying homotopy is physically defined, a displacement vector of a point at a first time is obtained, and a displacement of the point at a second time is calculated. First steps in which vectors are determined and data representing their displacement vectors is transmitted, and the position at a first time of a point corresponding to the displacement vector of the point represented by the transmitted data at the first time A vector is determined and based on a mixing function indicating the ratio of mixing of the displacement vector of the point represented by the transmitted data at the first time and the displacement vector of the point at the second time. A second step of obtaining a displacement vector at a predetermined time between the first time and the second time of the point; and a position vector at a first time of the point obtained in the second step; And a third step of obtaining and displaying a position vector of the point at a predetermined time based on a displacement vector of the point at a predetermined time.

According to the second aspect, the first time and the second time of the first aspect are times between the third time and the fourth time, and the first step is the third time of the point. A fourth step in which a position vector at the time point is given and a position vector of the point at the fourth time point is provided, and the position vector of the point given in the fourth step at the third time point and the point A fifth step of defining a mixing function indicating a mixing ratio with the position vector at the fourth time, and a first time and a second time based on the mixing function defined in the fifth step, respectively. The position vector at the third time of the point given in the fourth step, and the first time and the second time of the point obtained in the sixth step. Based on each position vector,
Determining a displacement vector at each of the first time and the second time of the point.

In the third aspect, the time-varying homotopy of the second aspect defines a transfer function for defining the mixing function defined in the fifth step.

According to the fourth aspect, the time-varying homotopy of the first or second aspect defines an external force applied to the target object.

[0034]

According to the first aspect of the present invention, a time-varying homotopy is physically defined, and a displacement vector of a point constituting a target object at a first time is obtained. , A displacement vector of the point at a second time is determined, data representing those displacement vectors is transmitted, and a position vector corresponding to the displacement vector of the point represented by the transmitted data at the first time is determined. A predetermined time between the first time and the second time based on a mixing function indicating a mixing ratio of a displacement vector of the point represented by the transmitted data at the first time and a displacement vector at the second time. Is obtained, and a position vector at a predetermined time is obtained based on the position vector of the point at the first time and the displacement vector at the predetermined time. Is displayed together with the operation of the point toward et second time is generated, can be displayed generating an operation of the object by the operation of all the points are generated and displayed to further configure the target object.

According to a second aspect of the present invention, there is provided a motion display generating method in computer graphics, wherein a position vector of a point at a third time is provided, a position vector of the point at a fourth time is provided, and A mixing function indicating a mixing ratio of the position vector at the time 3 and the position vector of the point at the fourth time is defined, and the position vector at each of the first time and the second time is defined based on the mixing function. Is determined based on the position vector of the point at the third time and the position vector of the point at the first and second times, respectively. Vector can be obtained.

According to a third aspect of the present invention, there is provided a motion display generation method in computer graphics, wherein a time-varying homotopy defines a transfer function for defining a mixing function, and displays the motion of a target object based on, for example, digital control theory. Can be generated.

According to the motion display generation method for computer graphics according to the fourth aspect of the present invention, the time-varying homotopy defines an external force applied to the target object, and the motion of the target object to which the external force is applied can be displayed and generated.

[0038]

Since the motion of an object in the natural world follows physical laws in the first place, the motion rules of the object in computer graphics animation, especially in the synthesizing unit 13 in FIG. , Should follow the physical definition.

For example, a person's natural facial expressions and subtle movements in real time are extremely based on facial muscles present on the surface layer of the face. Taking this into consideration, the synthesizing unit 13 in the immersive communication conference system also needs to define the muscle model of the facial muscle and its motion rule as an operation method. Therefore, it is important to faithfully define a motion equation or a transfer function unique to each muscle based on the motion characteristics of each muscle.

On the other hand, when simulating with a digital computer to process various data measured in continuous quantities (analog quantities) such as the position, displacement, velocity, and acceleration of an object in nature, These data are discretized (digitized). Therefore, we propose a time-varying homotopy based on a new digital control theory by combining the mathematical definition based on the time-varying homotopy in the conventional topological geometry described above with the physical definition in the natural world. . Hereinafter, the definition of the new time-varying homotopy will be described.

[Linear time-invariant system in continuous time]
Blending function R _n (v) used in time-varying homotopy
A condition that satisfies the condition based on the expression (9) is a response output of a DC servomotor which is a linear time-invariant system in a digital control system.

FIG. 1 is a diagram for explaining a DC servomotor. In the DC servo motor 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
Is the voltage amplification factor of the power amplifier 27, and M is the motor constant of the motor 29. Further, the inductance of the armature is negligible, and the field is a low excitation control composed of a permanent magnet.

According to the principle of the DC servo, since the transmitted electromotive force of the armature is M (dθ / dt), the equation (12) is satisfied. Further, since the generated torque is Mi, Expression (13) is satisfied. Then, after the current i is erased from the equations (12) and (13), x ₁ =
If θ, x ₂ = dθ / dt, equation (14) is satisfied. Therefore, the output y is expressed as in equation (15). Each of the first (15) of x ₁ and x _2, represents the rotation angle of the rotor, the rotational angular velocity.

Next, equations (14) and (15) are
For simplicity, the constant p
_c, Vector x_c↑, c_c↑, b_c↑, and matrix A_c
Is determined. Equation (16) is replaced with equation (14) and (1)
Substituting into equation 5) gives the continuous time system of the DC servo motor.
The equation of state in this case is expressed as in equation (17).
And the transfer function G_c(S) is expressed by the following equation (18).
And the poles are 0, -p_cAnd there is no zero.
Here, in equation (18), Y_c(S) and U _c(S)
Is obtained by Laplace transform of y and u in equation (17).
It is.

On the other hand, the transfer function G _c (s) of the continuous time system as shown in the equation (18) can be derived from the following equation of motion. For example, when generating facial expressions in computer graphics, Terzo
Based on the anatomical point, based on the anatomical viewpoint, the equation of motion of each node constituting the wire frame based on the facial muscles is defined as shown in Expression (19). Here, for simplicity, a one-dimensional equation of motion at each node is described, _{where mn} is the mass of each node, g _n ,
q _n , h _n , 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 of each node.

In the equation (19), x _nc = x _nc1 , d
_{Assuming that} x _nc / dt = x _nc2 , the equation of motion of the continuous time system with respect to the equation of motion of each node constituting the wireframe of the human face image is expressed by the following equation (20). When the output y _nc is regarded as the displacement, the y _nc is the second (2
It is expressed as in equation (1). Here, a constant u _nc , a vector x _nc ↑, b _nc ↑, c _nc ↑, and a matrix _Anc are represented by (22)
Determined as in the formula.

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

If Equation (25) can be assumed from Equations (17) and (23), which are the state equations of a linear time-invariant system in a continuous time system, the transfer function is also expressed by Equation (18). Equation (26) is established by equation (24). That is, when the assumption of Expression (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 forming the wireframe of the human face image are regarded as equivalent. Therefore, the following description is based on this assumption.

[0049]

(Equation 2)

[0050]

(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, the continuous-time signals u [t], x］ [t], y
Assuming that [t] is sampled at a sampling period T and A / D-converted to discrete-time signals u (i), x ↑ (i), y (i), these signals are 27). Then, as shown in equation (28), the vector b
{, C} and a matrix A are defined.

When the equations (27) and (28) are substituted into the equation (17), the state equation of the linear time-invariant system in the discrete time system is expressed by the equation (29).

Here, the following definition is specifically made so that the response output of the DC motor which is a linear time-invariant system is treated as a blending function defined in the time-varying homotopy. That is, in the DC servo motor described above, if M, R, K, J, and D satisfy Expression (30) with respect to Expression (16), _Ac and b _c ↑ in Expression (16) are satisfied. Is as shown in Expression (31). Then, in order to convert this continuous-time system into a discrete-time system, _Ac and b _c } in Expression (31) become Expression (32) according to Expression (28).

Further, in this discrete time system, the input u (i) is determined as in the following 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, it is easily apparent that the speed (dy (i) / di) satisfies the expression (34). Further, when the function Bld (i) is determined as in the equation (35) so that the response output y (i) is normalized, the function Bld (i) satisfies the equation (36). . This is clear from FIG.

Since the expression (34) corresponds to the expression (9) and the expression (36) corresponds to the expression (8), the response output y (i) of the linear time-invariant system in the discrete time system is obtained. )
Is defined as a function Bld (i)
Can be regarded as a blending function satisfying the conditional expression (9) among the blending functions defined in the time-varying homotopy described above.

Thus, a new blending function Bld derived from a linear time-invariant system in a discrete time system
Using (i) to control the operation of a target object in computer graphics is described in (2).
As long as Expression 5) holds, it is the same as operating each node of the wireframe model based on Expression (19), which is the equation of motion. Therefore, the time-varying homotopy proposed here is not a time-varying homotopy based on the mathematical definition of topological geometry, but rather a physical definition in the natural world that is different from a time-varying homotopy based on that mathematical definition. Is based on a digital control theory that takes into account inevitable data discretization in a digital computer.

Therefore, by the time-varying homotopy proposed here, the static shape of the surface in which the first-order differentiation of the object in computer graphics is discontinuous as shown in FIG. The first derivative as shown can easily generate a static shape of a continuous surface, and can generate not only the real-time motion of the target object as shown in FIG. 28 but also the natural motion of the target object more naturally. Can be generated.

An application example of computer graphics animation using time-varying homotopy based on digital control theory will be described below. As an application example,
It shows cylinder motion, generation of wrinkles when a forehead is moved in a person face image, and opening and closing of a mouth. The generation of moving images of these target objects is considered to be effective in the synthesizing unit 13 in the presence communication conference system of FIG. 19 as described above.

[0060]

(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), output y (i) and blending function B for the real-time operating cylinder of FIG.
It is the graph which showed ld (i). FIG. 3A shows that t =
FIG. 5 is a diagram showing a state of the cylinder when the value is 0, which corresponds to i = 0 in FIG. FIG. 3 (j) shows the state of the cylinder when t = 1, and corresponds to i = 20 in FIG. 3 (b) to FIG. 3 (i)
FIG. 4 is a diagram showing the state of each cylinder when i = 0 to i = 20 in FIG. 4 are divided into ten.

A cylinder 31 composed of a set of points in FIG.
It is assumed that an external force f as shown by an input u (i) in FIG. 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 the operation in real time as shown in FIG.

In the operation generation of the cylinder shown in FIG.
There is little difference in appearance from the cylinder motion generation shown in FIG. However, unlike the operation generation of the cylinder shown in FIG. 28, the operation generation of the cylinder shown in FIG. 3 is given an input u (i) (external force) as shown in FIG. Generation is in progress.

[Generation of Motion of Face Wrinkle] FIG. 5 is a diagram showing a time-series continuous image of wrinkles of the face in real time. FIG. 6 shows the input in generating the motion of the face wrinkles shown in FIG. u (i), output y (i) and blending function Bl
4 is a graph showing d (i). FIG. 5A shows the time t
FIG. 7 is a diagram illustrating a wrinkle state of the face when = 0, and is a diagram corresponding to i = 0 in FIG. 6. FIG. 5 (j) shows the time t = 1.
FIG. 7 is a diagram showing a state of wrinkles of a face in the case of FIG.
It is a figure corresponding to No. 20. 5 (b) to 5 (i)
FIG. 7 is a diagram showing wrinkle states of respective faces when i = 0 to i = 20 in FIG. 6 are divided into ten.

In the case of raising the forehead 35 in a three-dimensional human face image by computer graphics,
It is assumed that an external force f indicated by an input u (i) in FIG. 6 is applied. The displacement is calculated by a time-varying homotopy based on a blending function derived from the above-described discrete-time nonlinear time-invariant system. Based on this, the amount of displacement of each node located on the face portion of the wire frame constituting the human face image is determined, and the real-time motion is generated as shown in FIG. Was.

[Generation of mouth opening / closing operation] FIG. 7 is a diagram showing a time-series continuous image of mouth opening / closing operation generation.
7A is a diagram showing a wire frame model corresponding to each of FIGS. 7A to 7J. FIG. FIG. 7A is a diagram showing a human face image at time t = 0, and FIG.
(J) is a diagram showing a human face image at time t = 1.

Sixteen feature points are defined around the mouth 39 in FIG. For each feature point, a transfer function and a blending function that constitute a linear time-invariant system in an external force and a discrete time system are defined. The external force acts radially from the center of the opening of the mouth, in short. As a result, the wire frame model opened the mouth as shown in FIG. 8, and the person in the person face image opened the mouth as shown in FIG.

FIG. 9 is a diagram showing the internal configuration of a synthesizing unit for executing the operation display generating method in computer graphics according to one embodiment of the present invention. FIG. 10 corresponds to the synthesizing unit shown in FIG. FIG. 11 is a diagram showing internal processing of the displacement vector calculation unit and the display unit.
FIG. 12 is a flowchart showing a method for generating an operation display in computer graphics according to an embodiment of the present invention. FIG. 12 is a detailed flowchart of step (represented by S in the drawing) 1 of FIG.

As shown in FIG. 9 and FIG.
Includes a displacement vector calculating unit 41 for calculating a displacement vector.
And a display unit 43 for displaying an operation of the target object on a display or the like. The displacement vector calculator 41 performs an operation corresponding to step 1 in FIG.
The operation corresponding to the step is performed. That is, the displacement vector calculator 41 calculates the displacement vector Δx (i) of a point constituting the target object based on the time-varying homotopy physically defined in step 1 (for example, it can represent an external force applied to the target object). And the displacement vector Δx
The data representing (i) is transmitted to the display unit 43. Then, in step 2, the display unit 43 obtains a position vector x (i) corresponding to the displacement vector Δx (i) and displays the position vector x (i) on a display or the like.

Next, step 1 will be described in detail with reference to FIG. In step 11, time t =
A position vector at each of 0 and time t = 1 is input. At times t = 0 and t = 1, i = 0 in the function shown in FIG. 4 as described above, and at time t = 1, i = 20 in FIG. Similarly, in the function shown in FIG. 6, the time t = 0 is i = 0, and the time t = 1 is i
= 20.

The position vector in FIG. 3 is a position vector of each point because the cylinder 31 itself is a set of points. Thereby, the cylinder 31 is motion-generated by generating the motion of the position vector of each point from the time t = 0 to the time t = 1.

The position vector in FIG. 7 is the position vector of each node of the wire frame model shown in FIG.
By generating the motion during the process 1, the motion of the wireframe model is generated. As a result, as shown in FIG. 7, an operation is generated for a human face image. Similarly, the position vector shown in FIG. 5 is a position vector of each node of the wire frame model, not shown.

Next, in step 12, a blending function is defined based on the physical definition. The physical definition includes, for example, an external force acting on a target object operating in computer graphics, and a transfer function as described above.

In step 13, a position vector at time t = t is obtained. At time t = t in FIG. 3, each of FIGS. 3B to 3I is shown, and the position vector is a position vector of each point constituting the cylinder 31. In the case of FIG. 7, t = t is each of FIGS. 7B to 7I, and the position vector is a position vector of each node constituting the wire frame model of FIG. Similarly, t = t in FIG. 5 corresponds to FIG.
5 (i), and the position vector is a position vector of each node of the wire frame model not shown. Since the position vectors of the points constituting the target object (cylinder, wireframe model) are obtained in step 13, the motion of the target object is generated by obtaining all the position vectors for the points.

Next, at step 14, at time t = t
Is obtained, and in step 15, data representing the displacement vector at time t is transmitted to the display unit 43.

Thus, the displacement vector calculating section 41
In, a displacement vector is obtained, and data representing the displacement vector is transmitted to the display unit 43.

FIG. 13 is a diagram showing data transmitted from the displacement vector calculation unit 41 to the display unit 43. FIG.
, The horizontal axis is the value of i, and the vertical axis is the value of the transmitted data.

Each of i = A, i = B and i = C is t
Between the value of i corresponding to = 0 and the value of i corresponding to t = 1. I = A calculated by the displacement vector calculator 41,
The displacement vectors of i = B and i = C can be used to determine the extremely good and natural real-time operation of the target object by a new time-varying homotopy that takes into account not only the mathematical definition but also the physical definition as described above. It is obtained as a displacement vector to generate. The obtained value of the displacement vector is transmitted as a data value to the display unit 43 as a data value, and an extremely good and natural real-time operation of the target object is displayed.

More specifically, the new time-varying homotopy is defined by combining the conventional mathematical definition and the physical definition based on the equation of motion based on dynamics while 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 control theory, and a blending function is derived by normalizing the output. However, more stable behavior of the target object in computer graphics is generated, taking into account 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. It would appear to provide better motion display generation if displayed. Therefore, in the following, an embodiment that takes that into consideration will be described.

A moving target object that is a target in the natural world is originally a control target in a continuous time system. Therefore, attention should be paid to the response between sample points in the discrete time system described above. Therefore, x ↑ (i,
m) and y (i, m) are determined. At this time, the vector b {(m), c} and the matrix A (m) as shown in Expression (38)
Is defined. Then, the equation of state between the sample points is determined as in equation (39) using equation (38). The state equation between the sample points defined by the equation (39) is different from the state equation shown by the equation (29). The embodiment shown in FIGS. 10 to 12 has been realized based on the state equation shown in Expression (29).

Next, a new blending function Bld
(I, m) is determined as in equation (40). This fourth (4
0) The blending function Bld (i, i,
m) is also different from the blending function Bld (i) defined in Expression (35). Blending function Bl
Using d (i, m), time t = iT and time t = (i +
1) Between T and Δx ↑ (i, m) is the (4)
It is expressed by the expression 1).

[0082]

(Equation 5)

FIG. 14 is a diagram showing the internal processing of the synthesizing unit for executing the operation display generating method in computer graphics according to another embodiment of the present invention, and FIG. FIG. 21 is a flowchart showing a method for generating an operation display in computer graphics based on an extended Z transform as shown in Expression (41). FIG.
4 and FIG. 15, in step 101,
The displacement vectors Δx ↑ (i) and Δx ↑ (i + 1) are obtained based on the time-varying homotopy (for example, an external force can be represented) physically defined by the displacement vector calculation unit 41. The detailed processing of this step 101 is described in FIG.
Is realized by the processing of the flowchart shown in FIG. That is, ΔX ↑ (i) and Δx ↑ (i + 1) in step 101 are obtained by the processing in steps 11 to 15 and transmitted to the display unit 43.

Next, in step 211 of step 201, the position vector x ↑ (i) corresponding to the displacement vector Δx ↑ (i) is obtained, and in step 212, Δx ↑ is determined using the blending function Bld (i, m).
(I, m) is required.

Then, in step 301, the position vector x ↑ (i) and the displacement vector Δx ↑ (i, m)
, A position vector x ↑ (i, m) is obtained and displayed.

FIG. 16 is a diagram showing data transmitted from the displacement vector calculation unit 41 to the display unit 43 by the operation display generation method in the computer graphics shown in FIGS. 14 and 15.

As shown in FIG. 16, 1 / m data is obtained between i = A and i = B and between i = B and i = C, respectively. To display unit 43
Is transmitted to That is, as shown in FIG. 10, the amount of data transmitted from the displacement vector calculation unit 41 to the display unit 43 is i as shown in FIG. 10 in the operation display generation method in computer graphics corresponding to the flowchart shown in FIG.
In the operation display generation method in the computer graphics according to the flow chart shown in FIG. 15, there are mi data as shown in FIG. Accordingly, in the motion display generation method in the computer graphics shown in FIG.
When transmitting (i), all data of the displacement vector Δx (i) was transmitted without performing interpolation along the time axis, but an extended Z-transform as shown in FIG. 15 was used. By using the operation display generation method in computer graphics, the transmission amount of the displacement vector Δx (t) at each point can be reduced to 1 / (the number of divisions at the time of interpolation) corresponding to m. Therefore, the motion of the target object can be generated and displayed at higher speed by computer graphics, and the load on the computer can be reduced.

FIG. 17 shows a time-series continuous image of cylinders operating in real time when the extended Z-transform is considered as a kind of interpolation with respect to the time axis. FIG. 17A shows the state of the cylinder when t = 0, and FIG.
1 shows the state of the cylinder.

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

1. With the conventional time-varying homotopy, it was not possible to define the external force acting on the target object operating with computer graphics, but with the new time-varying homotopy, the external force can be defined.

2. In a conventional time-varying homotopy, a transfer function cannot be defined when defining a blending function. However, a transfer function can be defined in a new time-varying homotopy.

From 3.1 and 2, the physical definition is added to the time-varying homotopy, which has conventionally been limited to the mathematical definition, and is combined while complementing it. It will be possible to display real-time behavior, and much more applications are expected than when time-varying homotopy remained in mathematical definition.

4. Furthermore, when defining the time-varying homotopy based on the above-mentioned digital control theory, interpolation along the time axis of displacement can be performed by using a state equation that also takes into account the response between sample points, In computer graphics, faster operation of a target object and reduction of the load on a computer can be realized.

[0094]

As described above, according to the present invention, a displacement vector of a point constituting a target object in computer graphics at a first time is obtained, and a displacement of the point at a second time is obtained. Vectors are determined, data representing their displacement vectors is transmitted, and a position vector at a first time of a point corresponding to the displacement vector of the point represented by the transmitted data at the first time is determined. First and second points in time of a point based on a mixing function indicating the ratio of mixing of the displacement vector of the point at the first time and the displacement vector of the point at the second time represented by the extracted data. A displacement vector at a predetermined time is obtained, and a position vector of the point at a predetermined time is obtained based on the obtained position vector of the point at a first time and the displacement vector of the point at a predetermined time. Asked Because it is displayed is, it is possible to produce as much as possible natural and real-time operation of the object of interest in computer graphics,
The operation can be displayed and generated as fast as possible. For example, the load on the computer can be reduced as much as possible.

[Brief description of the drawings]

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

FIG. 2 shows an input u (i) and an input u in a discrete time system
5 is a graph showing the output y (i), the speed dy (i) / di, and the value of the 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 a time-varying homotopy based on a physical definition.

FIG. 4 is a graph showing an input u (i), an output y (i), and a blending function Bld (i) for the operation generation of the cylinder shown in FIG. 3;

FIG. 5 is a diagram showing a time-series continuous image of face wrinkles operating in real time by a 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) with respect to the generation of the motion of the face wrinkle shown in FIG.

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

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

FIG. 9 is a diagram showing an internal configuration of a synthesizing unit that processes an operation display generation method in computer graphics according to an embodiment of the present invention.

FIG. 10 is a diagram corresponding to the synthesizing unit shown in FIG. 9,
FIG. 5 is a diagram illustrating internal processing of a displacement vector calculation unit and a display unit.

FIG. 11 is a flowchart showing a method for generating an operation display in computer graphics according to an embodiment of the present invention.

FIG. 12 is a flowchart showing details of step 1 in FIG. 11;

FIG. 13 is a graph showing data transmitted in step 1 of FIG.

FIG. 14 is a diagram showing internal processing of a synthesizing unit for executing a method for generating an operation display in computer graphics according to another embodiment of the present invention.

FIG. 15 is a flowchart showing a method for generating an operation display in computer graphics according to another embodiment of the present invention.

FIG. 16 is a graph showing data transmitted in step 101 of FIG.

17 is a diagram showing a time-series continuous image of a cylinder operating in real time by a time-varying homotopy based on a physical definition, obtained by the operation display generation method in computer graphics shown in FIG.

FIG. 18 is a diagram illustrating a state of the presence communication conference.

FIG. 19 is a schematic block diagram of a device of the communication conference system with a sense of presence.

FIG. 20 is a diagram showing the structure of facial expression muscles.

FIG. 21 is a diagram showing a three-dimensional wire frame model.

FIG. 22 is a diagram for explaining homotopy in topology.

FIG. 23 is a diagram for explaining a time-varying homotopy based on a conventional mathematical definition.

FIG. 24 is a graph showing an example of a blending function R _n (v).

FIG. 25 is a graph showing another example of the blending function R _n (v).

FIG. 26 shows a blending function R shown in FIG.
It is the figure which showed the surface where the 1st-order differentiation of the static shape of the object obtained by _n (v) was discontinuous.

FIG. 27 shows 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) was continuous.

FIG. 28 is a diagram showing a real-time motion generation of a cylinder obtained by a time-varying homotopy based on a conventional mathematical definition.

[Explanation of symbols]

 13 Combining unit 41 Displacement vector calculating unit 43 Display unit

Claims (4)

(57) [Claims] 1. A motion display generation method in computer graphics for displaying and generating a motion of a point constituting a target object in order to display and generate a motion of a target object based on a time-varying homotopy. The homotopy is physically defined, a displacement vector of the point at a first time is determined, a displacement vector of the point at a second time is determined, and data representing the displacement vectors is transmitted. Determining a position vector of the point at a first time corresponding to a displacement vector of the point at a first time represented by the transmitted data; and determining a position vector of the point represented by the transmitted data. A predetermined value between the first time and the second time of the point is determined based on a mixing function indicating a mixing ratio of the displacement vector at the first time and the displacement vector of the point at a second time. A second step of obtaining a displacement vector at the time of, based on a position vector of the point at a first time and a displacement vector of the point at a predetermined time obtained in the second step, And a third step of obtaining and displaying a position vector of the point at a predetermined time. 2. The method according to claim 1, wherein the first time and the second time are times between a third time and a fourth time, and the first step is a third time at the point. The position vector of the point at a fourth time is given, and the position vector of the point given at the fourth step at a third time and the point A fifth step of defining a mixing function indicating the proportion of mixing with the position vector at the fourth time of the following: based on the mixing function defined in the fifth step,
A sixth step of obtaining a position vector at each of the first time and the second time; a position vector at a third time of the point given in the fourth step; The first time and the second time of the point are obtained based on the position vectors of the point at the first time and the second time, respectively, obtained in the step.
And calculating a displacement vector at each of the times. 3. The method according to claim 2, wherein the time-varying homotopy defines a transfer function for defining the mixing function defined in the fifth step. 4. The method according to claim 1, wherein the time-varying homotopy defines an external force applied to the target object.