Blinn Phong reflection model

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Preface

In this chapter , Realize the interaction between light and object surface . The purpose is to add shading function in the rendering pipeline , Therefore, only the most basic local illumination model is discussed here . Different from global illumination , In the local illumination model , The color value of the shaded point depends only on the material properties of the shaded point surface 、 The local geometric properties of the surface and the location and properties of the light source , It has nothing to do with other surfaces in the scene .

Rendering process and scene definition

Because global illumination is not considered , Only consider the light emitted from the light source , Specifically, only one single interaction between the light source and the surface is considered . So this problem can be divided into two independent parts ：

• Define the lights in the scene （ Only point light sources are introduced here ）
• Define a reflection model that describes the interaction between materials and light （phong And Blinn-Phong Reflection model ）.

First, from the light emitted by a point light source , Because the observer sees only the light that starts from the light source and finally reaches his eyes . In other words, it can be divided into two situations ：

• Or this light will go directly into the observer's eyes after starting from the light source , What you see at this time is the color of the light source .
• Either this ray passes through a complex route and interacts with the objects in the scene many times （ Only consider it once ）, Then enter the observer's eyes , What you see at this time is the interaction between the light source and the surface material .

stay webgl in , The projection plane is often used instead of the observer . conceptually , The part of the projection plane located in the clipping window is mapped to the display , Therefore, the projection plane can be divided into many small rectangles with straight lines , Each small rectangle corresponds to a pixel on the screen . In other words, the color of the light source and the object surface determines the color of one or more pixels .

Reflected light type

After studying physics in junior high school, I know , Basically divided into 3 There are different types of reflected light .

• The ambient light ： Its characteristic is to make the scene get uniform lighting , It is the result of multiple interactions between light and scene objects . Here, simply set it as a constant to simulate .
• Diffuse light ： It is characterized by scattering incident light in all directions , And the intensity of light scattered in all directions is equal , Therefore, the reflected light seen by observers at different positions is the same .
• The mirror reflects light ： Its characteristic is that it looks shiny , Because the direction of most of the reflected light is very close to the direction of the reflection angle . The direction of the reflected light follows the law that the incident angle is equal to the reflection angle .

For the sake of simplicity , The illumination mentioned below is to establish a point , This point is the coloring point . And this point should be on the surface of the object , So the illumination of this point is the shading effect of the light source on this point .

As shown in the figure , First, let's define some unit vectors ：

• The normal of the shaded point n
• The direction of the light l
• Camera observation direction v
• Some parameters of the object surface

Light propagation and energy conservation

Light will decay in the process of propagation .

For example, in the beginning, the light is concentrated on a unit ball , That is, the innermost ball , At this point, the energy received by each point on the ball is the energy of light divided by the area of the ball .

When light travels to a radius of r On the ball , That is, when the outermost ball . According to the law of conservation of energy , Each point on the outermost ball receives less energy than , According to the formula, the energy received by the object surface is inversely proportional to the square of the distance of light propagation .

That is to say, the energy of the light received by the coloring point should be calculated , To calculate the distance from the light to the shading point .

Phong Reflection model

The reflection model is composed of Phong First of all . Not only the simulation efficiency is very high, but also the practical effect is very good , So that it can get a good rendering effect for a variety of lighting conditions and material properties .

Phong The reflection model takes into account the three interactions described above ： The ambient light 、 Diffuse and specular reflection . For each color component , All have independent ambient light components 、 Diffuse light component and specular light component . Finally, add these three components to form the final color .

Use 4 A vector to calculate the shading points p The color value of . The unknown is the reflection vector r, and r Can be n and l affirmatory

Ambient light reflection

Ambient light intensity at all points on the surface LaL_{a}La​ It's all the same . Part of the ambient light is absorbed by the surface , Another part is reflected by the surface , The intensity of the reflected part is determined by the ambient light reflection system kak_{a}ka​, So at this time RaR_{a}Ra​=kak_{a}ka​. among 0 <= kak_{a}ka​ <= 1.

therefore IaI_{a}Ia​ = kak_{a}ka​ * LaL_{a}La​

Code implementation

In the code , Adding ambient light is very simple , Just set a small constant and multiply it by the color of the object

void main () {    vec3 ambient = 0.05 * color;} Copy code 

Diffuse reflection

When a beam of parallel incident light hits a rough surface , The surface reflects light in all directions .

It can be seen that there is a certain angle between the plane normal and the illumination direction , And according to the different included angles, the shading obtained by coloring points is different .

If the energy of each light is equal . Analyze the figure on the left , It can be seen that when the object surface is perpendicular to the direction of light , Will receive all 6 Root light . In the middle picture, the object is rotated to a certain angle , Only... Can be received 3 Root light , Logically, the surface of this object should be darker than the left .

So the brightness of the coloring point is related to the lighting direction l And the normal of the object surface n The included angle of is related . And in reality, we can also observe , For example, the earth's four seasons have different temperatures , That is, the illumination direction is related to the angle of the surface normal .

This relationship is famous Lamber's consine law The laws of ： The light direction is proportional to the cosine of the surface normal direction ：

cosθ=l∗ncosθ = l * ncosθ=l∗n

So diffuse reflection is also called Lambertian Shading：

First define the intensity of light I, When light reaches the surface of an object, the energy is I/r2I / r^2I/r2. How much light is absorbed is calculated according to the cosine of the included angle , This cosine is calculated from the point multiplication of the illumination vector and the normal vector .

Ld=kd(I/r2)max(0,n∗l)L_d = k_d(I / r^2)max(0, n * l)Ld​=kd​(I/r2)max(0,n∗l)

as for max(0, n * l) It means , Exclude the case where the cosine is negative . Because in real life （ When the surface is opaque ）, It is impossible for light to illuminate the surface of an object from a negative angle . This happens , Just assign 0.

as for k, It's an absorption coefficient . It shows how much light is absorbed by the surface of the object , How much light is reflected . When kd=1k_d=1kd​=1 when , That is, the surface of the object does not absorb light at all , All reflected out .

Code implementation

In the code , The normal vector is known

const aNormal = gl.getAttribLocation(program, 'aNormal');gl.vertexAttribPointer(aNormal, 3, gl.FLOAT, gl.FALSE, fsize * 8, fsize * 3);gl.enableVertexAttribArray(aNormal); Copy code 

Passed from vertex shader to clip shader

//  Vertex shader attribute vec3 aNormal;varying vec3 Normal;void main () {    Normal = aNormal;    ...}//  Fragment Shader varying vec3 Normal;void main () {    //  Normalize     vec3 norm = normalize(Normal);} Copy code 

The incoming and outgoing vector is calculated from the pixel position and the light source direction , The position of the pixel is also known as the vertex , The position of the light source is uniform uniform Variables store

//  Vertex shader attribute vec3 aPos;varying vec3 FragPos;void main () {    FragPos = aPos;    ...}//  Fragment Shader varying vec3 FragPos;uniform vec3 lightPos;void main () {    vec3 lightDir = normalize(lightPos - FragPos);} Copy code 

Finally, the diffuse reflection component is calculated according to the formula

//  Fragment Shader void main () {    float diff = max(dot(norm, lightDir), 0.0);    vec3 diffuse = diff * color;} Copy code 

Specular reflection

When we look at shiny objects, we see highlights . These highlights usually show different colors from ambient and diffuse reflections . And contrary to the fact that the diffuse surface is rough , The specular surface is smooth . The smoother the surface , The closer it is to the mirror , The more the reflected light is concentrated near an angle .

Phong An approximate model is proposed , When considering specular reflection , Think of the surface as smooth . The intensity of the light the observer sees depends on the direction of the reflected light r And the direction of the observer v The angle between the two α.

Ls=ksLscospαL_s = k_sL_scos^pαLs​=ks​Ls​cospα

Where the coefficient ksk_sks​(0 <= ksk_sks​ <= 1) Indicates how much of the incident specular reflected light is reflected . Index p Is the specular coefficient . As can be seen from the figure below , When p increases , The reflected light is increasingly concentrated near the reflection angle of the ideal reflector .

When not p when （ namely p=1）, From the perspective of reflection, we are 60 Look at the color dot when you're degrees , It's reasonable to say that it's very biased , But you can still see the highlights . That's not reasonable , Because in reality, you can only see highlights when you are very close , A little farther away, you can't see . That's why we add an index to normalize it .

Phone The advantage of the reflection model is , If you've put r and v Normalized to the unit vector , Then the specular reflection component can be calculated by point product operation just like diffuse reflection ：

Ls=ksLsmax((r∗v)p,0)L_s = k_sL_smax((r * v)^p, 0)Ls​=ks​Ls​max((r∗v)p,0)

Next, just calculate the reflection vector r that will do ！

Reflection angle calculation

The normal vector is given , Using the normal vector n And the incident vector l You can calculate the reflection angle . Ideal specular reflection has a good feature ： The angle of incidence is equal to the angle of reflection . As shown in the figure ：

The angle of incidence is the angle between the normal vector and the incidence vector , The reflection angle is the angle between the normal vector and the reflected light . In the plane , Only one reflection direction can satisfy the condition that the incident angle is equal to the reflection angle . But not in three-dimensional space , Because there are countless directions where the incident angle is equal to the reflection angle . So add a condition ： A point on the surface p, Incident light and reflected light must be in the same plane . These two conditions can be determined by n and l determine r.

hypothesis l and n It's already a unit vector ：

|l| = |n| = 1.

It is also assumed that r It's also a unit vector ：

|r| = 1

If θIθ_IθI​=θrθ_rθr​ that

cosθicosθ_icosθi​ = cosθrcosθ_rcosθr​

Using dot product operation, we can get :

cosθicosθ_icosθi​ = l * n = cosθrcosθ_rcosθr​ = n * r

The coplanar condition means that r It's written in l and n The linear combination of :

r = αl + βn（1）

Both sides of the equal sign are equal to n Do some product to get the equation （2）：

n * r = αl * n + β = l * n（2）

because r It's the unit vector , So substitute it into the equation （1） You can get the equation （3）:

1 = r * r = α2α^2α2 + 2αβl * n + β2β^2β2（3）

Combine the equation （2） And equation （3） Available

r = 2(l * n))n - l

Code implementation

The above formula for calculating the reflection angle , stay glsl There is a built-in function in reflect Realized . So the implementation in the code is very simple , The line of sight vector is also known .

//  Fragment Shader uniform vec3 viewPos;void main () {    vec3 viewDir = normalize(viewPos - FragPos);    vec3 reflectDir = reflect(-lightDir, norm);    vec3 specular = vec3(0.3) * pow(max(dot(viewDir, reflectDir), 0.0), 8.0);} Copy code 

Phong Reflection model results

Put the above 3 The sum of the two components is Phong Reflection model

L=La+Ld+LsL = L_a + L_d + L_sL=La​+Ld​+Ls​
=kaIa+kd(I/r2)max(0,n∗l)+ksLsmax((r∗v)p,0)= k_aI_a + k_d(I/r^2)max(0, n*l) + k_sL_smax((r * v)^p, 0)=ka​Ia​+kd​(I/r2)max(0,n∗l)+ks​Ls​max((r∗v)p,0)

gl_FragColor = vec4(ambient + diffuse + specular, 1.0); Copy code 

The renderings are as follows

Phong The reflection model not only has a good approximation to the real illumination , And the performance is also very high . But its specular reflection will have problems in some cases , Especially when the reflectance of the object is very low , Can cause large areas of highlights .

The reason for this problem is that the angle between the observation vector and the reflection vector cannot be greater than 90 degree . If the dot product is negative , The specular reflection becomes 0. You might think , When the angle between light and line of sight is greater than 90 When the degree of , It should not receive any light . This idea only applies to diffuse reflection .

But in specular reflection , The measured angle is not the angle between the light source and the normal , It's the angle between the line of sight and the reflected light vector , Here's the picture .

On the right , The angle between the line of sight and the reflection direction is significantly greater than 90 degree , In this case, the mirror light component is 0. This is OK in most cases , Because the observation direction is very far from the reflection direction . However , When the reflectance of an object is very small , The specular highlight radius it produces is sufficient for these opposite rays to have a large enough impact on the brightness , In this case, their contribution to the specular light component cannot be ignored .

And the reflection angle is difficult to calculate .

So in Phong On the basis of ,Blinn Expand on this , Introduced Blinn-Phong Reflection model .

Blinn-Phong Reflection model

This model is similar to Phong The difference between the models is only in the treatment of specular light components .Blinn-Phong The reflection model no longer depends on the reflection angle , Instead, we use a half range vector , That is, the unit vector in the direction of half the angle between the light and the line of sight .

Half range vector

When the viewing direction is close to the specular reflection direction , The normal direction of the object surface is close to the half range vector .

The so-called half range vector is the sum of the illumination direction vector and the observation direction vector according to the parallelogram rule and then divided by its length .

h=(v+l)/∣v+l∣h = (v + l) / |v + l|h=(v+l)/∣v+l∣

h And n Proximity means v and r near , This is it. Blinn-Phong What's special about the model . because r and v The included angle of is difficult to calculate , With this little trick , The calculation is much simpler .

Almost the same as diffuse reflection , It's just that the angle between the illumination direction and the normal direction is replaced by the angle between the half-way vector and the normal direction

Ls=ks(I/r2)max(0,n∗h)pL_s = k_s(I / r^2)max(0, n * h)^pLs​=ks​(I/r2)max(0,n∗h)p

Code implementation

vec3 halfwayDir = normalize(lightDir + viewDir);vec3 specular = vec3(0.3) * pow(max(dot(norm, halfwayDir), 0.0), 32.0); Copy code 

The renderings are as follows

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