JPH0786938B2 - Shading method and device - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION [Object of the Invention] (Industrial field of application) The present invention relates to computer graphics, and in particular, approximates the surface of an object as a collection of polygons, and makes the internal color and brightness continuous. TECHNICAL FIELD The present invention relates to a method and apparatus for generating a shaded image by dynamically changing the image.

(Prior Art) Recently, in the field of three-dimensional computer graphics, a method of obtaining a realistic image by shading the visible surface of an object is used. Specifically, the surface of the object is approximated by a collection of polygons, and the inside of each polygon is not painted with a constant color and brightness, but by continuously changing the color and brightness, a more realistic image can be obtained. It has gained.

Gouraud shading and Phong shading are typical methods for obtaining the color and brightness of a polygonal surface.

Gouraud shading is a method of approximating the color and brightness of each point in a polygon by linear interpolation using the color and brightness of each vertex. Since only linear interpolation processing is necessary as this processing, it is relatively easy to realize and high-speed processing is possible. However, the quality of the image obtained is inferior to phon shading.

On the other hand, the phon shading is to obtain the normal vector of the surface of the object at each point in the polygon from the normal vector at each vertex, and based on that, calculate the brightness of the reflected light from the following equation. .

I = IaKa + I ₁ Kd · + I ₁ Ks (·) ^m (1) where the first term is environmental reflected light, the second term is diffuse reflected light, and the third term
The term is specular reflection light, Ia is the intensity of ambient light, Ka is the reflection coefficient for ambient light, I ₁ is the intensity of point source light, Kd is the reflection coefficient of diffuse reflection light, Ks is the reflection coefficient of specular reflection light, Is a unit vector indicating the direction of the ray of the point light source, is a unit vector indicating the direction of the center of the specularly reflected ray from the point light source, is a unit vector indicating the direction of the line of sight, and m is a constant depending on the material of the surface of the object. is there. These relationships are as shown in FIG.

Here, the normal vector n of a point in a polygon as shown in FIG. 12 is a normal vector at each vertex, ₀ , ₁ , and ₂ are linearly interpolated, and then normalized to length 1. It is obtained by doing.

′ = X + y + = ′ / | ′ | (2) Linear interpolation of the normal vector at the apex ′ is represented by a linear expression of x, y coordinates on the display screen.

Calculation of this normal requires complicated calculation including square root and division, as can be seen from the equation (2). Therefore, there is a problem that high speed processing cannot be performed.

Therefore, as a method of performing this phon shading at high speed, Fast Phong shading has been developed.

This fastphone shading is a normal vector,
Alternatively, it is a method of accelerating the processing by Taylor-expanding the brightness represented by using it with respect to x and y and approximating it as a quadratic polynomial of x and y. However, in the case of this method, although the processing is sped up, there is a problem that the degree of approximation is poor and the quality of the image is significantly deteriorated as shown below.

As an example, as shown in FIG. 13, a case where the normal vector is constant in the y direction of the display screen and changes only in the x direction will be described.

It is assumed that when x = x ₀ , the normal vector is ₀ , and when x = X ₁ , the normal vector is ₁ .

This is converted into a variable t as follows, and Taylor expansion is performed at t = 1/2.

_{t = (x-x 0)} / (x 1 -x 0) expansion result to the secondary becomes the following equation, = _{((1 + 0 1)} / ^{2) -1/2} · _{{0 +} 1) _{/ 2 + (0 - 1)} (t-1/2) - (1- 0 1) /
(1 + _{0 1} ) * ( ₀ + ₁ ) (t-1 / 2) ² } (3) Although t = 1/2 gives a correct value, t = 0 (x-
x ₀ ), t = 1 (x = x ₁ ) does not have a correct value. For this reason, the luminance becomes discontinuous at the left and right boundaries of the x region, and a so-called Mach band effect occurs.

(Problems to be Solved by the Invention) As described above, foss shading is highly realistic.
Although it is a method of creating a three-dimensional computer graphics image, it has a problem that the processing is complicated and high-speed processing cannot be performed. Furthermore, in the fastphone shading, the quality of the generated image is extremely low, and there is a problem that a strong Mach band is generated.

The present invention is to solve the problems of the above conventional phon shading.
An object of the present invention is to provide a shading method and an apparatus thereof capable of performing phon shading with high accuracy at high speed and capable of obtaining an image of good quality.

[Means for Solving the Problems] (Means for Solving the Problems) In order to solve the above problems, the present invention approximates the surface of an object as a collection of polygons, and continuously changes the color and brightness inside the object to create shadows. A computer graphics shading method for generating an image with a mark, wherein a normal vector at each vertex of the polygon is obtained, and the length of the normal vector is maintained between these normal vectors. Interpolate the curved surface created by this interpolated normal vector into a cubic Bezier.
r) It is approximated by a curved surface, the luminance is represented by a polynomial of x and y in the screen coordinates by the von reflection model, and the value of this polynomial is obtained by using the forward difference.

Further, the present invention is a computer graphics shading device for approximating the surface of an object as a collection of polygons, and continuously changing the color and brightness inside the polygon to generate a shaded image. A means for obtaining a normal vector at each vertex of a polygon and an interpolation between these normal vectors while maintaining the length of the normal vector, and a curved surface formed by the interpolated normal vector is a cubic Bezier ( Bezier) curved surface, the brightness is represented by a polynomial of x and y in the screen coordinates by the Phong reflection model, and the calculation means for obtaining the value of this polynomial using the forward difference and the value of the polynomial obtained by this calculation means And storage means for storing.

(Operation) In the reflected light model of the phone, the brightness of the reflected light from the object is modeled as shown in equation (1).

In Phong shading, the normal vector of each point of the polygon is linearly interpolated with the normal vector at the vertex, and the length is 1
Calculated as normalized to.

As an example, consider a case where the normal vector changes only in the x direction of the display screen and is constant in the y direction.

In this case, the normal vector at x = x ₀ is to be given by the normal vector _{1 0,} x = x _1. t = (x−x ₀ ) /
When the variable transformation is performed with (x ₁ −x ₀ ), the normal vector in Phong shading is obtained by the following equation.

_{'= (1-t) 0} + t 1 =' / | '| = {(1-t) 0 + t 1} / {1 -2 (1-t) t (1- 0 · 1)} 1/2 ... (4) The calculation of n requires complicated calculation including the square root.

So, in this invention, as an approximation method with this polynomial,
The following approximation is used instead of the Taylor expansion at t.

(1-2 (1-t) tA) ^l ≈1-2 ^l (1-t) tA (5) This is the expansion of A up to the first order, and accuracy is not generally guaranteed, but t = 0, At a value close to t = 1,
It is a highly accurate approximation. In particular, the value at t = 0,1 and the value of the first derivative are correct values.

Under this approximation, the normal vector is
It is represented by a cubic expression of t. This is the curve that makes the normal line 3
This is equivalent to approximating with the Bezier curve below.

_{^{= (1-t) 3 0}} +3 (1-t) 2 t A +3 (1-t) t 2 B + t 3 1 , _{where, A = 0 + (1 -} (0 · 1) 0) / 3 B = ₁ + _{(0 - (0, 1) 1)} / 3 (6) which is further converted back to x, the normal vector n can be expressed as cubic equation in x.

Is expressed by a polynomial of x, the brightness of the reflected light is also expressed by a polynomial of x when only the model environment light and diffused light of Phong are considered, and therefore the brightness can be calculated using the forward difference. .

The above is the case where the normal vector changes in only one direction, but in the general case, the following is performed.

Assuming that the polygon is a triangle, as shown in FIG.
Variable conversion from screen coordinates (x, y) to (s, t). The normal vector n is represented by a bicubic expression of s and t. This corresponds to approximating the surface formed by the normals with a cubic Bezier curved surface. Next, it is inversely transformed and expressed by a bicubic expression of x and y, and a forward difference with respect to x and y is performed.

By approximating the surface formed by the normal lines with the Bezier curved surface in this way, the operation can be speeded up, and the Mach band effect can be reduced to obtain an image of good quality.

(Embodiment) An embodiment of the present invention will be described below with reference to the drawings.

FIG. 1 shows that all processing is performed by software using a processor, and the apparatus is composed of a processor 11, a main memory 13 connected to the processor 11 and a system bus 12, and an image memory 14.

The main memory 13 stores a program for operating the processor 11 and data to be processed.
The image memory 14 stores the computer graphics image processed by the processor 11. Image memory 14
May be common to the main memory 13.

FIG. 2 shows the software processing by the processor 11, and the operation of this embodiment will be described with reference to this.

Note that the polygon is a triangle here.

First, as shown in FIG. 3, x and y coordinates of each vertex of a triangle to be shaded and normal vectors ₀₀ , ₀₃ , ₃₀ are obtained, and ₃₃ is obtained by the following equation (step ST1). ₃₃ = ₀₃ + ₃₀ - ₀₀ ... (7) Next, as shown in FIG. 4, x that contains a polygon, obtains a rectangle y plane (step ST2), as shown in FIG. 5, x, Performs linear variable conversion from y to s, t (step ST
3).

After this, from the normal vector ₀₀ , ₀₃ , ₃₀ , ₃₃ , the next 1
Calculate two vectors (step ST4). ₀₁ = ₀₀ + ₍₀₃ - ₍₀₀ - _{03) 00)} / 3 ₀₂ = ₀₃ + ₍₀₀ - ₍₀₀ - _{03) 03)} / 3 ₃₁ = ₃₀ + ₍₃₃ - _{(30, 33) 30)} / 3 ₃₂ = ₃₃₊ ( _30- ( ₃₀・₃₃ ) ₃₃ ) / ₃₁₀ = ₀₀ + ( ₃₀₋ ( ₀₀・₃₀ ) ₀₀ ) / ₃₂₀ = ₃₀ + ( ₀₀₋ ( ₀₀・₃₀ ) ₃₀ ) / ₃₁₃ = ₀₃ + ₍₃₃ - ₍₀₃ - _{33) 03)} / 3 ₂₃ = ₃₃ + ₍₀₃ - ₍₀₃ - _{33) 33)} / 3 ₁₁ = ₁₀ + ₍₁₃ - ₍₁₀ - _{13) 10)} / 3 ₁₂ = ₁₃ + ₍₁₀ − ( ₁₀ · ₁₃ ) ₁₃ ) / ₃₂₁ = ₂₀ + ( ₂₃₋ ( ₂₀ · ₂₃ ) ₂₀ ) / ₃₂₂ = ₂₃ + ( ₂₀₋ ( ₂₀ · ₂₃ ) ₂₃ ) / 3 The normal vector is the above ₀₀ , _Using 16 vectors nij (0 ≤ i, j ≤ 3) including ₀₃ , ₃₀ , and ₃₃ and the Bernstein polynomial Bi () of the third degree, it is expressed as a bicubic expression of s, t as follows. It

= ΣBi (s) nijBi (t) (9) where B ₀ (t) = (1-t) ³ , B ₁ (t) = 3 (1-t) ² t, B ₂ (t) = 3 (1-t) t ² , B ₃ (t) = t ³ Next, from the above 16 vectors nij, the one obtained by removing the third term from the equation (1) showing the reflected light model of the phon is used. To calculate 16 luminance coefficients Iij (step ST
Five).

The brightness I is expressed as a bicubic expression of s, t as follows using Iij and a cubic Bernstein polynomial.

I = ΣBi (S) IijBj (t) (10) The s, t obtained above is inversely converted into x, y to obtain the brightness I as a bicubic expression F (x, y) of x, y ( Step ST6).

The above steps ST3, ST4, ST5, ST6 need not be performed in this order as long as they are equivalent as a result.

Hereinafter, using this F (x, y), the brightness of the point corresponding to the coordinates x, y is obtained by the forward difference method.

First, F (x, y) is transformed into the following form, and the coefficient Cij (0 ≦ i, j
≦ 3) is obtained (step ST7).

F = (x, y) = Σgi (y) Cijfj (x) (11) f ₀ (x) = 1 f ₁ (x) = x−X ₀ f ₂ (x) = (x−X ₀ ) ( X−X ₀ −1) / 2 f ₃ (f) = (x−X ₀ ) · (x−X ₀ −1) · (x−X ₀ −2) / 6 g ₀ (y) = 1 g ₁ (Y) = y−Y ₀ g ₂ (y) = (y−Y ₀ ) (y−Y ₀ −1) / 2 g ₃ (y) = (y−Y ₀ ) · (y−Y ₀ −1) ) · (Y−Y ₀ −2) / 6 Next, the brightness of one line is obtained from the forward difference in the x and y directions and written in the image memory 14 (step ST8).

6 and 7 show a flowchart for obtaining the forward difference in the x and y directions, FIG. 6 shows the processing in the y direction, and FIG. 7 shows the processing in the x direction in FIG. Shows.

That is, in FIG. 6, from y = Y ₀ to y = Y ₁ shown in FIG.
Up to one line, C _0i , C _1i , and C _2i are changed as shown in step STST15, the equation (11) is executed, and the forward difference in the y direction is obtained (steps ST11 to 15).

In the middle of this, in step ST12, the forward difference in the x direction is obtained according to the flowchart shown in FIG.

That is, the processing in the x direction is performed from x = X ₀ to x shown in FIG.
= 4 work registers W ₀ , W ₁ , up to X ₁
The forward difference in the x direction is obtained while changing the contents of W ₀ , W ₁ , and W ₂ of W ₂ and W ₃ (steps ST21 to ST26).

In addition, during this process, it is determined whether or not the pixel position to be processed is inside the triangle (step ST22). If the pixel position is inside the triangle, the content (luminance I) of the work register W ₀ is (x, y). Is written in the address in the image memory 14 corresponding to (step ST23).

The work registers W ₀ , W ₁ , W ₂ , W ₃ may be the resists in the processor 11 or the variable areas stored in the main memory 13.

According to the above-described embodiment, the normal vectors are interpolated with each other by the Pezier curved surface, and the luminance is represented by the polynomial of x and y by the Phong reflection model. Therefore, since the calculation of the normal vector does not include the square root, the calculation can be performed at a higher speed than in the conventional phon shading.

Moreover, since the continuity of the luminance at the boundary of the polygonal area can be maintained, the Mach band effect is small,
The room can be improved compared to fast phone shading.

Next, a second embodiment of the present invention will be described with reference to FIG.

FIG. 8 shows the hardware of the part of the one-line processing in the x direction in the first embodiment.

A processor 22, a main memory 23, a forward difference processing unit 24, a region determination unit 25, a memory control unit 26, and an image memory 27 are connected to the system bus 21.

In this embodiment, the processor 22 performs the processing from step ST1 to step ST7 of the first embodiment. Further, the updating of the coefficient Cij in the y direction in step ST8 is also performed by the processor.
22 does. However, this part can also be made into hardware to further reduce the processing time.

FIG. 9 shows the forward difference processing unit 24. The forward difference processing unit 24 sequentially obtains the value of the third-order polynomial, and has four registers W ₀ , W ₁ , W ₂ , W ₃ and three adders A.
_It is composed of ₀ , A ₁ and A ₂ .

In such a configuration, when the function F (x) of x is given by the following equation, F (x) = Ax ³ + Bx ² + Cx + D (12) Registers W ₀ , W ₁ , W ₂ , W ₃ Set the following initial values.

W ₀ = D W ₁ = A + B + C W ₂ = 6A + 2B W ₃ = 6A After setting these initial values, the following operations are simultaneously performed using the adders A ₀ , A ₁ and A ₂ to update each register.

W ₀ ← W ₀ ＋ W ₁ W ₁ ← W ₁ ＋ W ₂ W ₂ ← W ₂ ＋ W ₃ As a result, the register C ₀ can be set to F
Values of (0), F (1), F (2) ... Are obtained.

In this embodiment, the processor sets the coefficients C ₀₀ , C ₀₁ , C ₀₂ , C ₀₃ in the registers W ₀ , W ₁ , W ₂ , W ₃ before processing one line.

FIG. 10 shows the structure of the area determination unit 25.

The area determination unit 25 includes three straight line determination units 31, 32, 33 having the same configuration.
And one AND gate 34. The straight line determination units 31 to 33 are hardware that determines which side a certain point is with respect to a straight line, and includes an adder Add, a constant register Cst, and an accumulator Acc.

When the equation of the three-sided straight line forming the triangle is expressed by the following equation (13), it is determined whether the point (x, y) is on either side of the straight line by the sign of f (x, y). You can judge. However, A, B,
C is selected so that the value of f (x, y) becomes negative inside the rectangle.

f (x, y) = Ax + By + C = 0 (13) The value of f (x, y) at a certain x, y value is set in the accumulator Acc, and the value of A above is set in the constant register Cst. Set. The straight line determination units 31 to 33 sequentially add the contents of the constant register Cst to the accumulator Acc and output the value of the most significant sign bit of the accumulator Acc.

The AND gate 34 takes the logical product of the outputs of the straight line determination units 31 to 33 and outputs the logical product as the output of the area determination unit 25. When the output of the AND gate 34 is "1", it means the inside of the triangular area.

The memory control unit 26 sequentially generates addresses in the x and y directions,
When the output of the area determination unit 25 is 1, the output data of the forward difference processing unit 24 is written in the image memory 27.

Next, a third embodiment of the present invention will be described.

In this embodiment, the normal vector is approximated by a quadratic equation instead of a cubic equation of x and y coordinates. The curve formed by the normal line is not approximated by a cubic Bezier curve for t obtained by coordinate conversion of x as in the equation (6), but is approximated by a quadratic Bezier curve as follows.

_{^{= (1-t) 2 0}} +2 (1-t) t · (3 - 0 · 1) / 2 · (0 + 1) / 2 + t 2 1 ... (14) By using this approximation, the first embodiment In the example, the brightness is x,
The luminance is represented by the biquadratic equation of x and y in the same manner as the expression of the bicubic equation of y. Shading is performed by the same processing as that of the first embodiment except for this difference.

Further, the method using this biquadratic equation can be implemented as hardware and can be realized with the same hardware configuration as that of the second embodiment. In this case, the forward difference hardware shown in FIG. 9 is composed of two adders and three resists.

Next, a fourth embodiment of the present invention will be described.

This embodiment is a shading method for removing the Mach band, and a case where the normal vector changes only in the x direction as shown in FIG. 13 will be described.

The first region of x (x ₀ ≤x≤ ₁ ) and the second region (x ₁ ≤x≤
In each of x ₂ ), variables are converted from x to s, t as follows.

s = (x−x ₀ ) / (x ₁ −x ₀ ) t = (x−x ₁ ) / (x ₂ −x ₁ ) ... (15) The normal vector n in the arbitrary first region is _{^{, = (1-t) 3}} 0 +3 (1-t) 2 t 0A +3 (1-t) t 2 1B + t 3 1 in the second area adjacent to the first region, the normal vector n is , = (1-s) ³ ₁ +3 (1-s) ² s _1A +3 (1-s) s ² _2B + s ³ ₂ .

Here, unlike the equation (6), the normal vectors _1B and _1A are determined so as to satisfy the following conditions.

_{(1A - 1) / (x} 2 -x 1) = (1 - 1B) / (x 1 -x 0) ... (16) 0A, 2B also determined so as to satisfy the same conditions with the left and right regions.

By doing so, the value of the first derivative of the normal vector at x = x ₁ can be made continuous, and the Mach band can be removed.

The present invention is not limited to the above-described embodiments, and it goes without saying that various modifications can be made without departing from the spirit of the invention.

[Effect of the Invention] As described in detail above, according to the present invention, the surface formed by the normals is approximated by the Bezier curved surface, and thus the square root is not included in the calculation of the normal vector. Pixel shading can be realized at a very high speed and high accuracy of pixels per clock in wear processing, and a shading method and apparatus capable of generating a high-quality and highly realistic computer graphics image by reducing the McBand effect Can be provided.

[Brief description of drawings]

FIG. 1 is a block diagram showing a first embodiment of the present invention, FIG. 2 is a flow chart for explaining the operation of FIG. 1, and FIGS. 3 to 5 are shown for explaining the operation respectively. FIG. 3 is a diagram for explaining a normal vector of a vertex of a triangle, FIG. 4 is a diagram for explaining variable conversion from x, y to s, t, and FIG. 5 is a diagram for explaining a triangle. FIG. 6 is a flow chart for explaining part of the operation of FIG. 2, FIG. 6 is a flow chart for explaining a part of the operation of FIG. 6, and FIG. FIG. 9 is a block diagram showing a second embodiment of the present invention, FIG. 9 is a block diagram of a forward difference processing unit shown in FIG. 8, and FIG. 10 is a block diagram of an area determination unit shown in FIG. Figure is a figure for explaining the model of the reflected light of the phon, Figure 12 is a figure showing the normal vector of the vertex of the polygon,
FIG. 13 is a diagram for explaining a case where the normal vector changes only in the x direction. 11, 22 …… Processor, 13, 23 …… Main memory, 14, 27…
... Image memory, 24 ... Forward difference processing section, 25 ... Area determination section, 26 ... Memory control section.

Claims (9)

[Claims] 1. A shading method of computer graphics, wherein a surface of an object is approximated as a collection of polygons, and a color and luminance inside the polygon are continuously changed to generate a shaded image, The normal vector at each vertex of the polygon is obtained, the normal vectors are interpolated between these normal vectors while maintaining the length of the normal vector, and the curved surface formed by the interpolated normal vector is a cubic Bezier A shading method characterized by approximating with a curved surface, expressing luminance with a polynomial of x and y in screen coordinates by a Phong reflection model, and obtaining a value of this polynomial using a forward difference. 2. The shading method according to claim 1, wherein the polynomial is a bicubic expression of x and y. 3. The shading method according to claim 1, wherein the polynomial is a biquadratic expression of x and y. 4. It is determined whether or not a point having x and y coordinates in the screen coordinates is inside a polygon to be shaded, and when it is determined that it is inside the polygon, the obtained polynomial is obtained. The shading method according to claim 1, wherein the value of is stored in a corresponding address of the memory. 5. A shading device for computer graphics, which approximates the surface of an object as a collection of polygons, and continuously changes the color and brightness inside the object to generate a shaded image. A means for obtaining a normal vector at each vertex of a polygon and an interpolation between these normal vectors while maintaining the length of the normal vector, and a curved surface formed by the interpolated normal vector is a cubic Bezier ( Bezier) is approximated by a curved surface, the brightness is represented by a polynomial of x and y in the screen coordinates by the Phong reflection model, and the calculation means for obtaining the value of this polynomial using the forward difference and the value of the polynomial obtained by this calculation means A shading device, comprising: a storage unit for storing. 6. The polynomial is a bicubic expression of x, y, and the arithmetic means for obtaining the value of the polynomial has first to fourth registers to which initial values are set, first and fourth registers. The contents of the second register are added and the addition output is set in the first register, and the contents of the second and third registers are added, and the addition output is set in the second register. And a third adder for adding the contents of the third and fourth registers and setting the addition output in the third register. The shading device according to claim 5, wherein 7. The polynomial is a biquadratic expression of x and y, and the arithmetic means for obtaining the value of the polynomial has first to third registers in which initial values are set, first and third The contents of the second register are added and the addition output is set in the first register, and the contents of the second and third registers are added, and the addition output is set in the second register. The shading device according to claim 5, wherein the shading device is configured by a forward difference processing means including: 8. Judging means for judging whether or not a point having x, y coordinates in the screen coordinates is inside a polygon to be shaded, and this judging means judges that the inside of the polygon. The shading device according to claim 5, further comprising: control means for storing the value obtained by the computing means in a corresponding address of the storage means. 9. The judging means comprises a register for storing a constant, a calculating means for calculating a function f (x, y) == Ax + By + C = 0 at arbitrary x and y coordinates, and the calculating means and the register. A plurality of straight line judging means having an adder for adding the contents and setting them in the calculating means; and an AND gate for obtaining a logical product of the outputs of these straight line judging means and outputting a signal indicating the inside or the outside of the polygon. The shading device according to claim 8, wherein the shading device is provided.