本篇文章谢绝转载，也禁止用于任何商业目的。

代码下载地址：https://github.com/twinklingstar20/BumpMapping

1. 凹凸纹理简介

凹凸纹理（Bump Mapping）是计算机图形学中，使物体表面仿真出褶皱效果的技术，该技术通过（1）扰动对象平面的法向量，（2）利用扰动的法向量进行光照计算，来实现。物体表面呈现出来的褶皱效果，不是由于物体几何结构的变换，而是光照计算的结果，凹凸纹理技术的基本思想最早由James Blinn在1978年提出的^[1]。凹凸纹理的效果如图1所示^[2]，左图是一个光滑的小球，中间图是一张扰动纹理贴图，通过它来扰动小球平面的法向量，再经过光照计算，就能产生右图所示的带有褶皱表面的小球。

图1. 凹凸纹理效果图

2. Blinn技术

2.1. 数学原理

任何一个三维几何面片，可以用三个带有两个变量的参数形式的表示：

$X=X(u,v),Y=Y(u,v),Z=Z(u,v) \tag{1}$

其中，$u,v$的取值在区间[0,1]之间，几何面片的局部点的偏微分表示该点的两个方向向量：

$\overrightarrow{{{P}_{u}}}=\left( {{x}_{u}},{{y}_{u}},{{z}_{u}} \right),\overrightarrow{{{P}_{v}}}=\left( {{x}_{v}},{{y}_{v}},{{z}_{v}} \right) \tag{2}$

单位向量$\overrightarrow{{{P}_{u}}},\overrightarrow{{{P}_{v}}}$表示与该局部点相切的两个方向的切线，那么该点的法向量可以通过这两个切线向量的叉积得到，如图2所示：

$\overrightarrow{N}=\overrightarrow{{{P}_{u}}}\times \overrightarrow{{{P}_{v}}} \tag{3}$

图2. 三维几何面片的法向量

实时渲染中的光照计算，特别是漫反射光和镜面反射光（Phong光照模型），依赖平面的法向量，Blinn技术的原理就是，使用一个扰动函数，对平面的法向量进行扰动，再将扰动后的法向量用于光照的计算。图3（a）~（d）描述了Blinn技术对法向量扰动的基本过程。

图3. 法向量的扰动过程

设扰动函数表示为$F(u,v)$，原始平面上的一个点 $P$，经过扰动后的位置为：

$P'=P+F\cdot \overrightarrow{N}/\left| \overrightarrow{N} \right| \tag{4}$

而扰动的法向量$\overrightarrow{N'}$仍然可以由两个方向向量的偏微分得到：

$\overrightarrow{N'}=\overrightarrow{{{P}_{u}}'}\times \overrightarrow{{{P}_{v}}'} \tag{5}$

计算新的位置点$P'$在$u,v$方向上的偏微分，得到：

$$\left\{ \begin{array}{l}\overrightarrow {{P_u}'} = \overrightarrow {{P_u}} + {F_u}\frac{{\vec N}}{{\left| {\vec N} \right|}} + F \cdot \frac{{d(\vec N/\left| {\vec N} \right|)}}{{du}}\\\overrightarrow {{P_v}'} = \overrightarrow {{P_v}} + {F_v}\frac{{\vec N}}{{\left| {\vec N} \right|}} + F \cdot \frac{{d(\vec N/\left| {\vec N} \right|)}}{{dv}}\end{array} \right. \tag{6}$$

 

为了简化问题，可以将上式简化为：

$$\begin{array}{l}\overrightarrow {{P_u}'} = \overrightarrow {{P_u}} + {F_u}\frac{{\vec N}}{{\left| {\vec N} \right|}}\\\overrightarrow {{P_v}'} = \overrightarrow {{P_v}} + {F_v}\frac{{\vec N}}{{\left| {\vec N} \right|}}\end{array} \tag{7}$$

把等式（7）代入等式（5），可以得到：

$$\overrightarrow {N'} = ({P_u} + {F_u}\frac{{\vec N}}{{\left| {\vec N} \right|}}) \times ({P_v} + {F_v}\frac{{\vec N}}{{\left| {\vec N} \right|}}) = \vec N + D \tag{8}$$

其中，$D=\frac{{{F}_{u}}(\vec{N}\times \overrightarrow{{{P}_{v}}})-{{F}_{v}}(\vec{N}\times \overrightarrow{{{P}_{u}}})}{\left| {\vec{N}} \right|}$。

2.2. 算法描述

Blinn技术的算法描述如下所示：

在光照计算之前，对于物体表面上的每个顶点（像素）：

1.  查询表面上的每个点$p$在高度贴图（Heightmap）上对应的点${{p}_{r}}$；
2. 采用有限差分法，计算点${{p}_{r}}$处的法向量$\overrightarrow{{{n}_{r}}}$；
3. 设物体表面上的点$p$的法向量为$\overrightarrow{n}$，利用法向量$\overrightarrow{{{n}_{r}}}$来扰动“真正”的法向量$\overrightarrow{n}$，得到扰动的法向量$\overrightarrow{n'}$；
4. 采用扰动的法向量$\overrightarrow{n'}$，实现点$p$的光照计算。

显然，凹凸纹理的深度感，依赖光照的位置等信息，当光照发生了变化，深度感也随之发生变化。

图4. 浮雕纹理

一种Blinn技术的变种称为浮雕（凹凸）纹理（Emboss Bump Mapping），浮雕纹理技术使用两张纹理贴图来显示凹凸效果，一张纹理贴图用于产生浮雕效果，再与另一张纹理贴图混合，产生凹凸效果。如图4所示，通过对一张黑白纹理贴图的两次渲染，得到浮雕效果，再通过一次渲染，使浮雕效果与另一张彩色纹理贴图纹理混合，产生凹凸效果。

这里介绍下NEHE^[4]实现的浮雕纹理的原理，该技术只使用到漫反射分量，没有镜面反射的效果，会由于下采样造成锯齿。漫反射光的计算公式可以表示为：

$C=\left( \overrightarrow{l}\cdot \overrightarrow{n} \right)\cdot {{D}_{l}}\cdot {{D}_{m}} \tag{9}$

其中，$\overrightarrow{l}$表示指向光源的方向向量，${{D}_{l}}$表示漫反射光的颜色，${{D}_{m}}$表示材质的漫反射颜色。

凹凸纹理的目的就是扰动法向量$\overrightarrow{n}$，而浮雕纹理通过新的等式^[5]来估计$\overrightarrow{l}\cdot \overrightarrow{n}$，达到类似扰动的目的，即：

${{F}_{d}}+m\approx \overrightarrow{l}\cdot \overrightarrow{n} \tag{10}$

其中，${{F}_{d}}$表示初始的漫反射分量值，$m$表示函数的一阶偏导，即表示凹凸函数的坡度。如图4所示，${{F}_{d}}$即右上角的彩色图表示的颜色值，$m$是通过对灰度图的

可以用一个二维的灰度图来表示凹凸函数，偏导$m$可以通过下列方法来估计，具体的示例如图5所示。

1. 查询某个像素点$P$的高度值为${{H}_{0}}$，该点对应二维灰度图的纹理坐标为$(s,t)$；

2. 根据像素点$P$与光源的位置$L$，对纹理坐标进行一定的扰动，得到新的纹理坐标$(s+\Delta s,t+\Delta t)$处的高度值${{H}_{1}}$；

3. 那么，$m={{H}_{1}}-{{H}_{0}}$。

图5. 在一维空间上，凹凸函数的一阶偏导$m$的示意图

现在考虑偏移量$(\Delta s,\Delta t)$的计算，设点$P$的法向量（Normal）为$\overrightarrow{n}$，它在切线（Tangent）和二重切线（Bitangent）分别为$\overrightarrow{s},\overrightarrow{t}$，由点$P$指向光源$L$的方向向量表示为$\overrightarrow{l}=\left( L-P \right)/\left\| L-P \right\|$，那么：

$\Delta s=\lambda \left( \overrightarrow{l}\cdot \overrightarrow{s} \right),\Delta t=\lambda \left( \overrightarrow{l}\cdot \overrightarrow{t} \right) \tag{11}$

其中，$\lambda$是一个缩放常量。

对NEHE第22节^[4]的代码进行简化，新的代码如下所示：

/************************************************************************       
\link   www.twinklingstar.cn
\author TwinklingStar
\date   2015/09/27
\file   main.cpp
****************************************************************************/
#pragma comment(lib, "glaux.lib")
#pragma comment(lib, "opengl32.lib")
#pragma comment(lib, "glu32.lib")
#pragma comment(lib, "glut32.lib")

#include <gl\glut.h>
#include <gl\glaux.h>
#include <stdio.h>
#include <string.h>
#include <math.h>

#define MAX_EMBOSS ((GLfloat)0.008f)		// Maximum Emboss-Translate. Increase To Get Higher Immersion

bool	gEmboss = false;												// Emboss Only, No Basetexture?
bool    gBumps = true;													// Do Bumpmapping?

GLfloat	gXrot;
GLfloat	gYrot;
GLfloat gXspeed;
GLfloat gYspeed;
GLfloat	gZ = -5.0f;

GLuint	gFilter = 1;
GLuint	gTexture[3];
GLuint  gBump[3];
GLuint  gInvbump[3];

GLfloat gLightAmbient[]	= { 0.2f, 0.2f, 0.2f};
GLfloat gLightDiffuse[]	= { 1.0f, 1.0f, 1.0f};
GLfloat gLightPosition[] = { 0.0f, 0.0f, 2.0f};

GLfloat gGray[]= {0.5f,0.5f,0.5f,1.0f};

// Data Contains The Faces For The Cube In Format 2xTexCoord, 3xVertex;
// Note That The Tesselation Of The Cube Is Only Absolute Minimum.
GLfloat gData[]= {
	// FRONT FACE
	0.0f, 0.0f,		-1.0f, -1.0f, +1.0f,
	1.0f, 0.0f,		+1.0f, -1.0f, +1.0f,
	1.0f, 1.0f,		+1.0f, +1.0f, +1.0f,
	0.0f, 1.0f,		-1.0f, +1.0f, +1.0f,
	// BACK FACE
	1.0f, 0.0f,		-1.0f, -1.0f, -1.0f,
	1.0f, 1.0f,		-1.0f, +1.0f, -1.0f,
	0.0f, 1.0f,		+1.0f, +1.0f, -1.0f,
	0.0f, 0.0f,		+1.0f, -1.0f, -1.0f,
	// Top Face
	0.0f, 1.0f,		-1.0f, +1.0f, -1.0f,
	0.0f, 0.0f,		-1.0f, +1.0f, +1.0f,
	1.0f, 0.0f,		+1.0f, +1.0f, +1.0f,
	1.0f, 1.0f,		+1.0f, +1.0f, -1.0f,
	// Bottom Face
	1.0f, 1.0f,		-1.0f, -1.0f, -1.0f,
	0.0f, 1.0f,		+1.0f, -1.0f, -1.0f,
	0.0f, 0.0f,		+1.0f, -1.0f, +1.0f,
	1.0f, 0.0f,		-1.0f, -1.0f, +1.0f,
	// Right Face
	1.0f, 0.0f,		+1.0f, -1.0f, -1.0f,
	1.0f, 1.0f,		+1.0f, +1.0f, -1.0f,
	0.0f, 1.0f,		+1.0f, +1.0f, +1.0f,
	0.0f, 0.0f,		+1.0f, -1.0f, +1.0f,
	// Left Face
	0.0f, 0.0f,		-1.0f, -1.0f, -1.0f,
	1.0f, 0.0f,		-1.0f, -1.0f,  1.0f,
	1.0f, 1.0f,		-1.0f,  1.0f,  1.0f,
	0.0f, 1.0f,		-1.0f,  1.0f, -1.0f
};


int loadTextures()
{
	bool status = true;
	AUX_RGBImageRec *Image=NULL;
	char *alpha = NULL;

	// Load The Tile-Bitmap For Base-Texture
	if (Image = auxDIBImageLoad("Data/Base.bmp")) 
	{											
		glGenTextures(3, gTexture);

		// Create Nearest Filtered Texture
		glBindTexture(GL_TEXTURE_2D, gTexture[0]);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_NEAREST);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTER,GL_NEAREST);
		glTexImage2D(GL_TEXTURE_2D, 0, GL_RGB8, Image->sizeX, Image->sizeY, 0, GL_RGB, GL_UNSIGNED_BYTE, Image->data);
		//                             ========
		// Use GL_RGB8 Instead Of "3" In glTexImage2D. Also Defined By GL: GL_RGBA8 Etc.
		// NEW: Now Creating GL_RGBA8 Textures, Alpha Is 1.0f Where Not Specified By Format.

		// Create Linear Filtered Texture
		glBindTexture(GL_TEXTURE_2D, gTexture[1]);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_LINEAR);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTER,GL_LINEAR);
		glTexImage2D(GL_TEXTURE_2D, 0, GL_RGB8, Image->sizeX, Image->sizeY, 0, GL_RGB, GL_UNSIGNED_BYTE, Image->data);

		// Create MipMapped Texture
		glBindTexture(GL_TEXTURE_2D, gTexture[2]);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_LINEAR);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTER,GL_LINEAR_MIPMAP_NEAREST);
		gluBuild2DMipmaps(GL_TEXTURE_2D, GL_RGB8, Image->sizeX, Image->sizeY, GL_RGB, GL_UNSIGNED_BYTE, Image->data);
	}
	else 
	{
		status = false;
	}
	if (Image) 
	{
		if (Image->data) 
			delete Image->data;
		delete Image;
		Image = NULL;
	}

	// Load The Bumpmaps
	if (Image = auxDIBImageLoad("Data/Bump.bmp")) 
	{			
		glPixelTransferf(GL_RED_SCALE,0.5f);		
		glPixelTransferf(GL_GREEN_SCALE,0.5f);
		glPixelTransferf(GL_BLUE_SCALE,0.5f);

		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_WRAP_S,GL_CLAMP);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_WRAP_T,GL_CLAMP);
		glTexParameterfv(GL_TEXTURE_2D,GL_TEXTURE_BORDER_COLOR, gGray);

		glGenTextures(3, gBump);

		// Create Nearest Filtered Texture
		glBindTexture(GL_TEXTURE_2D, gBump[0]);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_NEAREST);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTER,GL_NEAREST);
		glTexImage2D(GL_TEXTURE_2D, 0, GL_RGB8, Image->sizeX, Image->sizeY, 0, GL_RGB, GL_UNSIGNED_BYTE, Image->data);

		// Create Linear Filtered Texture
		glBindTexture(GL_TEXTURE_2D, gBump[1]);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_LINEAR);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTER,GL_LINEAR);
		glTexImage2D(GL_TEXTURE_2D, 0, GL_RGB8, Image->sizeX, Image->sizeY, 0, GL_RGB, GL_UNSIGNED_BYTE, Image->data);

		// Create MipMapped Texture
		glBindTexture(GL_TEXTURE_2D, gBump[2]);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_LINEAR);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTER,GL_LINEAR_MIPMAP_NEAREST);
		gluBuild2DMipmaps(GL_TEXTURE_2D, GL_RGB8, Image->sizeX, Image->sizeY, GL_RGB, GL_UNSIGNED_BYTE, Image->data);

		for (int i = 0; i < 3 * Image->sizeX * Image->sizeY; i++)
			Image->data[i] = 255 - Image->data[i];

		glGenTextures(3, gInvbump);

		// Create Nearest Filtered Texture
		glBindTexture(GL_TEXTURE_2D, gInvbump[0]);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_NEAREST);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTER,GL_NEAREST);
		glTexImage2D(GL_TEXTURE_2D, 0, GL_RGB8, Image->sizeX, Image->sizeY, 0, GL_RGB, GL_UNSIGNED_BYTE, Image->data);

		// Create Linear Filtered Texture
		glBindTexture(GL_TEXTURE_2D, gInvbump[1]);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_LINEAR);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTER,GL_LINEAR);
		glTexImage2D(GL_TEXTURE_2D, 0, GL_RGB8, Image->sizeX, Image->sizeY, 0, GL_RGB, GL_UNSIGNED_BYTE, Image->data);

		// Create MipMapped Texture
		glBindTexture(GL_TEXTURE_2D, gInvbump[2]);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_LINEAR);
		glTexParameteri(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTER,GL_LINEAR_MIPMAP_NEAREST);
		gluBuild2DMipmaps(GL_TEXTURE_2D, GL_RGB8, Image->sizeX, Image->sizeY, GL_RGB, GL_UNSIGNED_BYTE, Image->data);

		glPixelTransferf(GL_RED_SCALE,1.0f);	
		glPixelTransferf(GL_GREEN_SCALE,1.0f);			
		glPixelTransferf(GL_BLUE_SCALE,1.0f);
	}
	else
	{
		status = false;
	}
	if (Image) 
	{
		if (Image->data) 
			delete Image->data;
		delete Image;
	}	

	return status;
}

void doCube (void) 
{
	int i;
	glBegin(GL_QUADS);
	// Front Face
	glNormal3f(0.0f, 0.0f, +1.0f);
	for (i = 0; i < 4; i++) 
	{
		glTexCoord2f(gData[5*i], gData[5*i+1]);
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	// Back Face
	glNormal3f( 0.0f, 0.0f,-1.0f);
	for (i = 4; i < 8; i++) 
	{
		glTexCoord2f(gData[5*i], gData[5*i+1]);
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	// Top Face
	glNormal3f(0.0f, 1.0f, 0.0f);
	for (i = 8; i < 12; i++) 
	{
		glTexCoord2f(gData[5*i], gData[5*i+1]);
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	// Bottom Face
	glNormal3f(0.0f,-1.0f, 0.0f);
	for (i = 12; i < 16; i++) 
	{
		glTexCoord2f(gData[5*i], gData[5*i+1]);
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	// Right face
	glNormal3f(1.0f, 0.0f, 0.0f);
	for (i = 16; i < 20; i++) 
	{
		glTexCoord2f(gData[5*i], gData[5*i+1]);
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	// Left Face
	glNormal3f(-1.0f, 0.0f, 0.0f);
	for (i = 20; i < 24; i++) 
	{
		glTexCoord2f(gData[5*i], gData[5*i+1]);
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	glEnd();	
}

void matMultVector(GLfloat *M, GLfloat *v) 
{// Calculates v=vM, M Is 4x4 In Column-Major, v Is 4dim. Row (i.e. "Transposed")
	GLfloat res[3];
	res[0] = M[ 0]*v[0] + M[ 1] * v[1] + M[ 2] * v[2] + M[ 3] * v[3];
	res[1] = M[ 4]*v[0] + M[ 5] * v[1] + M[ 6] * v[2] + M[ 7] * v[3];
	res[2] = M[ 8]*v[0] + M[ 9] * v[1] + M[10] * v[2] + M[11] * v[3];;	
	v[0] = res[0];
	v[1] = res[1];
	v[2] = res[2];
	v[3] = M[15];											// Homogenous Coordinate
}

void setUpBumps(GLfloat *n, GLfloat *c, GLfloat *l, GLfloat *s, GLfloat *t) 
{
	GLfloat v[3];							// Vertex From Current Position To Light	
	GLfloat lenQ;							// Used To Normalize		
	// Calculate v From Current Vector c To Lightposition And Normalize v	
	v[0] = l[0] - c[0];		
	v[1] = l[1] - c[1];		
	v[2] = l[2] - c[2];		
	lenQ=(GLfloat) sqrt(v[0] * v[0] + v[1] * v[1] + v[2] * v[2]);
	v[0] /= lenQ;		v[1] /= lenQ;		v[2] /= lenQ;
	// Project v Such That We Get Two Values Along Each Texture-Coordinat Axis.
	c[0] = (s[0] * v[0] + s[1] * v[1] + s[2] * v[2]) * MAX_EMBOSS;
	c[1] = (t[0] * v[0] + t[1] * v[1] + t[2] * v[2]) * MAX_EMBOSS;	
}

bool doMesh1TexelUnits(void) 
{
	GLfloat c[4] = {0.0f, 0.0f, 0.0f, 1.0f};
	GLfloat n[4] = {0.0f, 0.0f, 0.0f, 1.0f};					// Normalized Normal Of Current Surface		
	GLfloat s[4] = {0.0f, 0.0f, 0.0f, 1.0f};					// s-Texture Coordinate Direction, Normalized
	GLfloat t[4] = {0.0f, 0.0f, 0.0f, 1.0f};					// t-Texture Coordinate Direction, Normalized
	GLfloat l[4];										// Holds Our Lightposition To Be Transformed Into Object Space
	GLfloat Minv[16];									// Holds The Inverted Modelview Matrix To Do So.
	int i;								

	glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);

	// Build Inverse Modelview Matrix First. This Substitutes One Push/Pop With One glLoadIdentity();
	// Simply Build It By Doing All Transformations Negated And In Reverse Order.
	glLoadIdentity();								
	glRotatef(-gYrot, 0.0f, 1.0f, 0.0f);
	glRotatef(-gXrot, 1.0f, 0.0f, 0.0f);
	glTranslatef(0.0f, 0.0f, -gZ);
	glGetFloatv(GL_MODELVIEW_MATRIX,Minv);
	glLoadIdentity();
	glTranslatef(0.0f, 0.0f, gZ);

	glRotatef(gXrot, 1.0f, 0.0f, 0.0f);
	glRotatef(gYrot, 0.0f, 1.0f, 0.0f);	

	// Transform The Lightposition Into Object Coordinates:
	l[0] = gLightPosition[0];
	l[1] = gLightPosition[1];
	l[2] = gLightPosition[2];
	l[3] = 1.0f;

	matMultVector(Minv, l);

	/*	PASS#1: Use Texture "Bump"
	No Blend
	No Lighting
	No Offset Texture-Coordinates */
	glBindTexture(GL_TEXTURE_2D, gBump[gFilter]);
	glDisable(GL_BLEND);
	glDisable(GL_LIGHTING);
	doCube();

	/* PASS#2:	Use Texture "Invbump"
	Blend GL_ONE To GL_ONE
	No Lighting
	Offset Texture Coordinates 
	*/
	glBindTexture(GL_TEXTURE_2D, gInvbump[gFilter]);
	glBlendFunc(GL_ONE,GL_ONE);
	glDepthFunc(GL_LEQUAL);
	glEnable(GL_BLEND);	

	glBegin(GL_QUADS);	
	// Front Face	
	n[0] = 0.0f;		n[1] = 0.0f;		n[2] = 1.0f;			
	s[0] = 1.0f;		s[1] = 0.0f;		s[2] = 0.0f;
	t[0] = 0.0f;		t[1] = 1.0f;		t[2] = 0.0f;
	for (i = 0; i < 4; i++) 
	{	
		c[0] = gData[5*i+2];		
		c[1] = gData[5*i+3];
		c[2] = gData[5*i+4];
		setUpBumps(n, c, l, s, t);
		glTexCoord2f(gData[5*i] + c[0], gData[5*i+1] + c[1]); 
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	// Back Face	
	n[0] = 0.0f;		n[1] = 0.0f;		n[2] = -1.0f;	
	s[0] = -1.0f;		s[1] = 0.0f;		s[2] = 0.0f;
	t[0] = 0.0f;		t[1] = 1.0f;		t[2] = 0.0f;
	for (i = 4; i < 8; i++) 
	{	
		c[0] = gData[5*i+2];		
		c[1] = gData[5*i+3];
		c[2] = gData[5*i+4];
		setUpBumps(n, c, l, s, t);
		glTexCoord2f(gData[5*i] + c[0], gData[5*i+1] + c[1]); 
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	// Top Face	
	n[0] = 0.0f;		n[1] = 1.0f;		n[2] = 0.0f;		
	s[0] = 1.0f;		s[1] = 0.0f;		s[2] = 0.0f;
	t[0] = 0.0f;		t[1] = 0.0f;		t[2] = -1.0f;
	for (i = 8; i < 12; i++) 
	{	
		c[0] = gData[5*i+2];		
		c[1] = gData[5*i+3];
		c[2] = gData[5*i+4];
		setUpBumps(n, c, l, s, t);
		glTexCoord2f(gData[5*i] + c[0], gData[5*i+1] + c[1]); 
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	// Bottom Face
	n[0] = 0.0f;		n[1] = -1.0f;		n[2] = 0.0f;		
	s[0] = -1.0f;		s[1] = 0.0f;		s[2] = 0.0f;
	t[0] = 0.0f;		t[1] = 0.0f;		t[2] = -1.0f;
	for (i = 12; i < 16; i++) 
	{	
		c[0] = gData[5*i+2];		
		c[1] = gData[5*i+3];
		c[2] = gData[5*i+4];
		setUpBumps(n, c, l, s, t);
		glTexCoord2f(gData[5*i] + c[0], gData[5*i+1] + c[1]); 
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	// Right Face	
	n[0] = 1.0f;		n[1] = 0.0f;		n[2] = 0.0f;		
	s[0] = 0.0f;		s[1] = 0.0f;		s[2] = -1.0f;
	t[0] = 0.0f;		t[1] = 1.0f;		t[2] = 0.0f;
	for (i = 16; i < 20; i++) 
	{	
		c[0] = gData[5*i+2];		
		c[1] = gData[5*i+3];
		c[2] = gData[5*i+4];
		setUpBumps(n, c, l, s, t);
		glTexCoord2f(gData[5*i] + c[0], gData[5*i+1] + c[1]); 
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}
	// Left Face
	n[0] = -1.0f;		n[1] = 0.0f;		n[2] = 0.0f;		
	s[0] = 0.0f;		s[1] = 0.0f;		s[2] = 1.0f;
	t[0] = 0.0f;		t[1] = 1.0f;		t[2] = 0.0f;
	for (i=20; i<24; i++) 
	{	
		c[0] = gData[5*i+2];		
		c[1] = gData[5*i+3];
		c[2] = gData[5*i+4];
		setUpBumps(n, c, l, s, t);
		glTexCoord2f(gData[5*i] + c[0], gData[5*i+1] + c[1]); 
		glVertex3f(gData[5*i+2], gData[5*i+3], gData[5*i+4]);
	}		
	glEnd();

	/* PASS#3:	Use Texture "Base"
	Blend GL_DST_COLOR To GL_SRC_COLOR (Multiplies By 2)
	Lighting Enabled
	No Offset Texture-Coordinates
	*/
	if (!gEmboss) 
	{
		glTexEnvf (GL_TEXTURE_ENV, GL_TEXTURE_ENV_MODE, GL_MODULATE);
		glBindTexture(GL_TEXTURE_2D, gTexture[gFilter]);
		glBlendFunc(GL_DST_COLOR, GL_SRC_COLOR);	
		glEnable(GL_LIGHTING);
		doCube();
	}

	return true;
}

bool doMeshNoBumps(void) 
{
	glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
	glLoadIdentity();
	glTranslatef(0.0f, 0.0f, gZ);

	glRotatef(gXrot, 1.0f, 0.0f, 0.0f);
	glRotatef(gYrot, 0.0f, 1.0f, 0.0f);	
	glDisable(GL_BLEND);
	glBindTexture(GL_TEXTURE_2D,gTexture[gFilter]);	
	glBlendFunc(GL_DST_COLOR,GL_SRC_COLOR);
	glEnable(GL_LIGHTING);

	doCube();
	return true;
}

void init()
{
	if (!loadTextures()) 
		return;

	glEnable(GL_TEXTURE_2D);
	glShadeModel(GL_SMOOTH);
	glClearColor(0.0f, 0.0f, 0.0f, 0.5f);
	glClearDepth(1.0f);
	glEnable(GL_DEPTH_TEST);
	glDepthFunc(GL_LEQUAL);
	glHint(GL_PERSPECTIVE_CORRECTION_HINT, GL_NICEST);

	//Initialize the lights.
	glLightfv( GL_LIGHT1, GL_AMBIENT, gLightAmbient);
	glLightfv( GL_LIGHT1, GL_DIFFUSE, gLightDiffuse);	
	glLightfv( GL_LIGHT1, GL_POSITION, gLightPosition);

	glEnable(GL_LIGHT1);
}

void draw(void)
{
	if (gBumps) 
	{
		doMesh1TexelUnits();	
	}
	else
	{
		doMeshNoBumps();
	}

	gXrot += gXspeed;
	gYrot += gYspeed;
	if (gXrot > 360.0f) 
		gXrot -= 360.0f;
	if (gXrot < 0.0f) 
		gXrot += 360.0f;
	if (gYrot > 360.0f) 
		gYrot -= 360.0f;
	if (gYrot < 0.0f) 
		gYrot += 360.0f;

	glutPostRedisplay();
}

void reshape(GLsizei w,GLsizei h)
{
	glViewport(0, 0, w, h);

	glMatrixMode(GL_PROJECTION);
	glLoadIdentity();

	// Calculate The Aspect Ratio Of The Window
	gluPerspective(45.0f,(GLfloat)w/(GLfloat)h,0.1f,100.0f);

	glMatrixMode(GL_MODELVIEW);
	glLoadIdentity();
}

void keyboard(unsigned char key, int x, int y)
{
	//If escape is pressed, exit
	if(key == 27)
	{
		exit(0);
	}
	else if (key == 'E' || key == 'e')
	{
		gEmboss =! gEmboss;
	}						
	else if (key == 'B' || key == 'b')
	{
		gBumps =! gBumps;
	}				
	else if (key == 'F' || key == 'f')
	{
		gFilter = (gFilter + 1) % 3;
	}			
	else if (key == 'Z' || key == 'z')
	{
		gZ -= 0.02f;
	}
	else if (key == 'X' || key == 'x')
	{
		gZ += 0.02f;
	}
	else if (key == 'W' || key == 'w')
	{
		gXspeed -= 0.01f;
	}
	else if (key == 'S' || key == 's')
	{
		gXspeed += 0.01f;
	}
	else if (key == 'A' || key == 'a')
	{
		gYspeed += 0.01f;
	}
	else if (key == 'D' || key == 'd')
	{
		gYspeed -= 0.01f;
	}
}

int main(int argc,char ** argv)
{
	glutInit(&argc,argv);
	glutInitDisplayMode(GLUT_SINGLE | GLUT_RGB);
	glutInitWindowSize(400,400);

	glutCreateWindow("Emboss Bump Mapping");

	init();

	glutReshapeFunc(reshape);
	glutDisplayFunc(draw);
	glutKeyboardFunc(keyboard);

	glutMainLoop();
	return(0);
}

3. 法线贴图技术

法线贴图（Normal Mapping）也是通过对光照计算的法向量进行扰动，实现图像的凹凸效果。最早^[11]由Krishnamurthy& Levoy^[8]提出从模型中提取模型信息的想法，尔后Cohen等^[9], Cignoni等^[10]提出将模型的法向量信息存储在纹理当中的想法，就是法线贴图的基本思想。

首先介绍下什么是法线纹理，法线纹理是一张存储法向量的纹理。纹理的每个像素存储着RGB信息，它可以转化为法向量。设某个像素点的颜色为$\left( r,g,b \right)$，那么该点的法向量为$\left( r*2-1,g*2-1,b*2-1 \right)$；相反，对于法向量$\left( x,y,z \right)$，其中，$0\le x,y,z\le 1$，它在法线纹理中存储的颜色为$\left( x*0.5+0.5,y*0.5+0.5,z*0.5+0.5 \right)$。举个例子，对于法向量$\left( 0.3,0.4,-0.866 \right)$，它在法线纹理中的颜色值$\left( \left( 0.3,0.4,-0.866 \right)*0.5+\left( 0.5,0.5,0.5 \right) \right)\cdot 255=\left( 166,\text{ }179,\text{ }17 \right)$。法线贴图纹理如图6所示。

图6. 法线纹理^[12]

设人眼在位置$O$(原点)，模型上存在一个点$P$，对应的纹理坐标为$\left( s,t \right)$，法线贴图的颜色值为$\left( r,g,b \right)$，光源在点$Q$(对象空间)，那么如何计算点$P$处的光照值呢？这里，引入切线空间的概念^{[13, 14]}。

如图7所示表示位置$A$处的切线空间，它是一个相对于纹理坐标系的坐标系统，三条坐标轴分别为：法向量$\overrightarrow{N}$，切线(Tangent)$\overrightarrow{T}$，双切线(Bitangent) $\overrightarrow{B}$（在世界空间下）。在切线空间下，这三条坐标轴分别为$\left( 0,0,1 \right),\left( 1,0,0 \right),\left( 0,1,0 \right)$。相对于光源，要计算出光源点$Q$在点$P$上的颜色值，即需要计算视点坐标系统下光照向量$\overrightarrow{PQ}$、视点向量$\overrightarrow{PO}$在切线空间下的坐标值。

图7. 切线空间

设三维空间上有三个点${{P}_{1}},{{P}_{2}},{{P}_{3}}$，纹理坐标分别为$\left( {{s}_{1}},{{t}_{1}} \right),\left( {{s}_{2}},{{t}_{2}} \right),\left( {{s}_{3}},{{t}_{3}} \right)$，则有：

$$\left\{ \begin{array}{l}\left( {{s_1}',{t_1}'} \right) = \left( {{s_2},{t_2}} \right) - \left( {{s_1},{t_1}} \right)\\\left( {{s_2}',{t_2}'} \right) = \left( {{s_3},{t_3}} \right) - \left( {{s_1},{t_1}} \right)\end{array} \right.,\left\{ \begin{array}{l}{D_1} = {P_2} - {P_1}\\{D_2} = {P_3} - {P_1}\end{array} \right. \tag{12}$$

可以得到等式：

$$\left\{ \begin{array}{l}{D_1} = {s_1}'\overrightarrow T + {t_1}'\overrightarrow B \\{D_2} = {s_2}'\overrightarrow T + {t_2}'\overrightarrow B \end{array} \right. \tag{13}$$

转化为矩阵表示为：

$$\left( {\begin{array}{*{20}{c}}{{D_{1,x}}}&{{D_{1,y}}}&{{D_{1,z}}}\\{{D_{2,x}}}&{{D_{2,y}}}&{{D_{2,z}}}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{{s_1}'}&{{t_1}'}\\{{s_2}'}&{{t_2}'}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{T_x}}&{{T_y}}&{{T_z}}\\{{B_x}}&{{B_y}}&{{B_z}}\end{array}} \right) \tag{14}$$

通过计算$\left( s,t \right)$矩阵的逆矩阵，可以得到等式：

$$\left( {\begin{array}{*{20}{c}}{{T_x}}&{{T_y}}&{{T_z}}\\{{B_x}}&{{B_y}}&{{B_z}}\end{array}} \right) = \frac{1}{{{s_1}{t_2} - {s_2}{t_1}}}\left( {\begin{array}{*{20}{c}}{{t_2}}&{ - {t_1}}\\{ - {s_2}}&{{s_1}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{D_{1,x}}}&{{D_{1,y}}}&{{D_{1,z}}}\\{{D_{2,x}}}&{{D_{2,y}}}&{{D_{2,z}}}\end{array}} \right) \tag{15}$$

 

通过等式（15）计算出切线向量和双切线向量，那么可以得到变换矩阵：

$${M_1} = \left( {\begin{array}{*{20}{c}}{{T_x}}&{{T_y}}&{{T_z}}\\{{B_x}}&{{B_y}}&{{B_z}}\\{{N_x}}&{{N_y}}&{{N_z}}\end{array}} \right) \tag{16}$$

 

通过变换矩阵${{M}_{1}}$，可以将点由对象空间变换到切线空间，它的逆矩阵就可以把坐标由切线空间变换为对象空间下的坐标，需要计算出矩阵${{M}_{1}}$的逆矩阵即可。采用Gram-Schmidt正交化方法，可以计算出新的切线向量和双切线向量：

$$\left\{ \begin{array}{l}\overrightarrow {T'} = \overrightarrow T - \overrightarrow N \left( {\overrightarrow N \cdot \overrightarrow T } \right)\\\overrightarrow {B'} = B - \left( {\overrightarrow N \cdot \overrightarrow B } \right)\overrightarrow N - \left( {\overrightarrow {T'} \cdot \overrightarrow B } \right)\overrightarrow {T'} \end{array} \right. \tag{17}$$

实际上，我们并不需计算双切线向量$\overrightarrow{B'}$，可以通过$\overrightarrow{N'}\times \overrightarrow{T'}$出$\overrightarrow{B'}$，所以逆矩阵可以表示为：

$${M_2} = \left( {\begin{array}{*{20}{c}}{{T_x}'}&{{T_y}'}&{{T_z}'}\\{{B_x}'}&{{B_y}'}&{{B_z}'}\\{{N_x}'}&{{N_y}'}&{{N_z}'}\end{array}} \right) \tag{18}$$

至此，我们得到了将坐标由对象空间变换到切线空间的变换矩阵${{M}_{2}}$。那么，光照方向$\overrightarrow{PQ}$和物点位置$\overrightarrow{PO}$转化为切线空间下的坐标为：${{M}_{2}}\cdot \overrightarrow{PQ}$和${{M}_{2}}\cdot \overrightarrow{PO}$。点$P$在法线纹理的纹理颜色值为$\left( r,g,b \right)$，它表示的法向量为$\left( x,y,z \right)$，就可以利用Phong光照模型计算出$P$的RGB颜色了。

最后给出法线贴图技术的着色器代码如下所示：

顶点着色器代码NormalMapping.vert：

#version 330 core
layout(location = 0) in vec3 Vertex;
layout(location = 1) in vec2 VertexUV;
layout(location = 2) in vec3 NormalModelSpace;
layout(location = 3) in vec3 TangentModelSpace;
layout(location = 4) in vec3 BitangentModelspace;
out vec2 TexVertUV;
out vec3 PosWorldSpace;
out vec3 LightDirTangSpace;
out vec3 EyeDirTangSpace;
uniform mat4 ModelViewProj;
uniform mat4 View;
uniform mat4 Model;
uniform mat3 MV3x3;
uniform vec3 LightPos;
void main(){
	// Output position of the vertex, in clip space : MVP * position
	gl_Position =  ModelViewProj * vec4(Vertex, 1);
	// Position of the vertex, in worldspace : Model * position
	PosWorldSpace = (Model * vec4(Vertex,1)).xyz;
	// Vector that goes from the vertex to the camera, in camera space.
	// Assume the camera is at (0,0,0) in world space.
	vec3 eyeDirCamSpace = (View * (vec4(0,0,0, 1) - vec4(PosWorldSpace,1))).xyz;
	// Vector that goes from the vertex to the light, in camera space.
	vec3 lightDirCamSpace = (View * (vec4(LightPos,1) - vec4(PosWorldSpace, 1))).xyz;
	// UV of the vertex. No special space for this one.
	TexVertUV = VertexUV;
	// model to camera = ModelView
	vec3 TangentCamSpace 	= MV3x3 * TangentModelSpace;
	vec3 BitangentCamSpace 	= MV3x3 * BitangentModelspace;
	vec3 NormalCamSpace 	= MV3x3 * NormalModelSpace;
	// You can use dot products instead of building this matrix and transposing it. See References for details.
	mat3 TBN = transpose(mat3(TangentCamSpace,BitangentCamSpace,NormalCamSpace)); 
	LightDirTangSpace 	= TBN * lightDirCamSpace;
	EyeDirTangSpace 	= TBN * eyeDirCamSpace;
}

片断着色器代码NormalMapping.frag：

#version 330 core
in vec2 TexVertUV;
in vec3 LightDirTangSpace;
in vec3 EyeDirTangSpace;
out vec3 color;
uniform sampler2D DiffTexSampler;
uniform sampler2D NormTexSampler;
uniform vec3 LightPos;
void main(){
	// Light emission properties
	// You probably want to put them as uniforms
	vec3 lightColor = vec3(1,1,1);
	float LightPower = 3.0;
	// Material properties
	vec3 d = texture2D(DiffTexSampler, TexVertUV).rgb;
	// Normal of the computed fragment, in camera space
	vec3 n = normalize(texture2D(NormTexSampler, vec2(TexVertUV.x, TexVertUV.y)).rgb * 2.0 - 1.0);
	// Direction of the light (from the fragment to the light)
	vec3 l = normalize(LightDirTangSpace);
	vec3 v = normalize(EyeDirTangSpace);
	float power = 0.3;
	// ambient lighting
	float iamb = 0.1;
	// diffuse lighting
	float idiff = clamp(dot(n, l), 0, 1);
	float ispec = clamp(dot(v + l, n), 0, 1) * power;
	color = d * (iamb + idiff + ispec);
}

4. 视差贴图技术

4.1. 综述

视差贴图技术（ParallaxMapping）可以看成是法线贴图技术的加强版，并没有改变模型的结构，而只是通过改变光照的计算达到3D视感。对于法线贴图技术来说，它存在一个问题，例如一个墙壁上的砖块，当你以一定的倾斜角观察它的时候，依旧可以看到砖块之间的缝隙，砖块间没有互相遮挡的效果，视差贴图技术就是为了解决这个问题，如图8所示，图(a)和图(b)分别是法线贴图技术和视差贴图技术达到的效果，显然图(b)的石头立体感更加显著。

图8. 法线贴图技术和视差贴图技术的效果图

视差贴图技术最早由Kaneko等^[15]在2001年提出的，后来被持续改进，主要有下面几个变种：1）Parallax Mapping with Offset Limiting，2）Steep Parallax Mapping，3）Relief Parallax Mapping，4）Parallax Occlusion Mapping（嗯，不知道怎么翻译这几个名词）。Dujgta^[16]对这几个技术有非常清晰的阐述，这里分别对这几个技术进行介绍。

特别注意：视差贴图技术解决的凹凸纹理中无法解决互相遮挡的问题。

为了实现视差贴图技术，至少需要有三种纹理贴图：高度贴图纹理，法线贴图纹理和漫反射贴图纹理，分别如图9(a)、(b)、(c)所示。深度贴图纹理的R通道、G通道、B通道和A通道的值是相同的，黑色的表示深度值为0，白色的表示深度值为1；法线贴图纹理，有RGB三个颜色通道，因此，可以把深度贴图纹理和法线贴图纹理合并为一张纹理，RGB通道表示法线，A通道表示深度；漫反射贴图纹理用于模型表示的颜色渲染。

图9. 三种贴图纹理，（a）深度贴图纹理，（b）法线贴图纹理，（c）漫反射贴图纹理

视差贴图技术跟法线贴图技术的区别在于，它需根据眼睛位置和深度贴图信息，对法线进行偏移。原理如图10所示，法线贴图技术计算视点向量与纹理平面的交点在${{t}_{0}}$位置，但实际上，由于纹理表现存在一定的高度值，采用${{t}_{0}}$处的纹理坐标计算出的颜色值存在一定的误差。视差贴图技术的目的就是：基于视点向量$\vec{v}$，计算出真实的位置${{t}_{1}}$处的纹理坐标。

图10. 视差贴图技术原理示意图

这里给出视差贴图技术的着色器的代码结构，顶点着色器与法线贴图技术的相同，但是片断着色器添加了对得到的纹理坐标进一步进行扰动，从而采用新的纹理坐标处的法向量进行光照的计算。

顶点差色器代码ParallaxMapping.vert：

#version 330 core
layout(location = 0) in vec3 Vertex;
layout(location = 1) in vec2 VertexUV;
layout(location = 2) in vec3 NormalModelSpace;
layout(location = 3) in vec3 TangentModelSpace;
layout(location = 4) in vec3 BitangentModelspace;
out vec2 TexVertUV;
out vec3 EyeTangSpace;
out vec3 LightTangSpace;
uniform mat4 ModelViewProj;
uniform mat4 View;
uniform mat4 Model;
uniform mat3 MV3x3;
uniform vec3 LightPos;
uniform int gSwitch;
void main(){
	// Output position of the vertex, in clip space : MVP * position
	gl_Position =  ModelViewProj * vec4(Vertex, 1);
	// the View matrix is identity here.
	// Position of the vertex, in worldspace : Model * position
	vec3 posWorldSpace = (Model * vec4(Vertex,1)).xyz;
	// Vector that goes from the vertex to the light.
	vec3 lightPosWorldSpace = (View * vec4(LightPos,1)).xyz;
	vec3 lightDirWorldSpace = normalize(lightPosWorldSpace - posWorldSpace);
	vec3 camPosWorldSpace = vec3(0,0,0);
	vec3 camDirCamSpace = normalize(camPosWorldSpace - posWorldSpace);
	// UV of the vertex. No special space for this one.
	TexVertUV = VertexUV;
	// model to camera = ModelView
	vec3 TangentCamSpace = MV3x3 * TangentModelSpace;
	vec3 BitangentCamSpace = MV3x3 * BitangentModelspace;
	vec3 NormalCamSpace = MV3x3 * NormalModelSpace;
	// You can use dot products instead of building this matrix and transposing it. See References for details.
	mat3 TBN = transpose(mat3(TangentCamSpace,BitangentCamSpace,NormalCamSpace)); 
	LightTangSpace = TBN * lightDirWorldSpace;
	EyeTangSpace =  TBN * camDirCamSpace;
}

片段着色器代码ParallaxMapping.frag：

#version 330 core
in vec2 TexVertUV;
in vec3 EyeTangSpace;
in vec3 LightTangSpace;
out vec3 color;
uniform sampler2D DiffTexSampler;
uniform sampler2D NormTexSampler;
uniform vec3 LightPos;
uniform int gSwitch;
float gHeightScale = 0.1;

// Calculates lighting by Blinn-Phong model and Normal Mapping
// Returns color of the fragment
vec3 normalMappingLighting(in vec2 t, in vec3 l, in vec3 v){
	// restore normal from normal map
	vec3 n = normalize(texture(NormTexSampler, t).rgb * 2.0 - 1.0);
	vec3 d = texture(DiffTexSampler, t).rgb;
	float power = 0.3;
	// ambient lighting
	float iamb = 0.1;
	// diffuse lighting
	float idiff = clamp(dot(n, l), 0, 1);
	float ispec = clamp(dot(v + l, n), 0, 1) * power;
	return d * (iamb + idiff + ispec);
}
vec2 pallaxMapping(in vec3 v, in vec2 t){
    ……………………
}
void main(){
	// Direction of the light (from the fragment to the light)
	vec3 l = normalize(LightTangSpace);
	// Direction of the eye (from the fragment to the eye)
	vec3 v = normalize(EyeTangSpace);
	vec2 t = pallaxMapping2(v, TexVertUV);
	color = normalMappingLighting(t, l, v);
}

4.2. Parallax Mapping with Offset Limiting

最简单的视差贴图技术，通过单步扰动纹理坐标来实现，称之为Parallax Mapping with Offset Limiting，虽然它的表现却让人很失望，但它是后续几种视差技术的基础，这里先对该技术进行介绍。

图11. Parallax Mapping with Offset Limiting原理示意图

如图11所示，设视点向量$\vec{v}$是在切线空间下的数值，那么它的z轴与纹理平面互相垂直，我们的目的是根据纹理坐标${{t}_{s}}$和它的深度值$H\left( {{t}_{s}} \right)$估计出一个纹理坐标${{t}_{p}}$使它尽可能的靠近${{t}_{d}}$。估计方程式可以表示为：

$${t_p} = {t_s} + {\vec v_{xy}}/{\vec v_z}*H\left( {{t_s}} \right)*scale \tag{19}$$

图12. Parallax Mapping with Offset Limiting效果图对比，（a）法线贴图效果图，(b)视差贴图效果图

其中，通常$scale$的取值介于$\left[ 0,0.5 \right]$之间^[16]，该算法得到的效果如图12所示，相应的片断着色器代码如下所示：

vec2 pallaxWithOffsetLimit(in vec3 v, in vec2 t){
	float height =  texture(NormTexSampler, t).a;    
	vec2 offset = v.xy / v.z * height * gHeightScale;
	return t - offset;
}

4.3. Steep Parallax Mapping

图13. SPM原理示意图

与Parallax Mapping with Offset Limiting技术不同的是，Steep Parallax Mapping(SPM)有对计算出的新的纹理坐标进行校验，判断它是否尽可能的接近真实的纹理坐标值。它把深度图的范围$\left[ 0,1 \right]$，平均分成$n$份，每次移动一点，直到找到一个点在层表面的下方（即当前的层的深度比采样点的深度值大），停止查询，选择当前点作为目标点。设$n=8$为例，如图13所示，第一次判断点${{t}_{0}}$，该点在第1层上方（即从上往下数第一条蓝绿色线），搜索下一个点；判断点${{t}_{1}}$，它也在第2层上方，搜索下一个点；直到找到${{t}_{4}}$，即是我们要找的目标点。

如果$n$值越大，则计算出来的纹理坐标越接近实际值，但会降低性能；如果$n$的取值较小，就会产生明显的锯齿效果，如图14所示。

图14. SPM效果图对比，（a）法线贴图效果图，(b)视差贴图效果图

SPM技术的着色器代码如下所示：

vec2 steepPallaxMapping(in vec3 v, in vec2 t){
	// determine number of layers from angle between V and N
	const float minLayers = 5;
	const float maxLayers = 15;
	float numLayers = mix(maxLayers, minLayers, abs(dot(vec3(0, 0, 1), v)));
	// height of each layer
	float layerHeight = 1.0 / numLayers;
	// depth of current layer
	float currentLayerHeight = 0;
	// shift of texture coordinates for each iteration
	vec2 dtex = gHeightScale * v.xy / v.z / numLayers;
 	// current texture coordinates
	vec2 currentTextureCoords = t;
	// get first depth from heightmap
 	float heightFromTexture = texture(NormTexSampler, currentTextureCoords).a;
	// while point is above surface
	while(heightFromTexture > currentLayerHeight) {
		// to the next layer
		currentLayerHeight += layerHeight;
		// shift texture coordinates along vector V
		currentTextureCoords -= dtex;
		// get new depth from heightmap
		heightFromTexture = texture(NormTexSampler, currentTextureCoords).a;
   }
   return currentTextureCoords;
}

4.4. Relief Parallax Mapping and Parallax Occlusion Mapping

Relief Parallax Mapping（RPM）与SPM技术相比，多了一个后处理。如图13所示，即找到点${{t}_{4}}$后，再在${{t}_{3}}$和${{t}_{4}}$进行一定步骤的二分查找，但是产了较为明显的效果改进，如图15所示。

图15.效果图，（a）SPM效果图，(b) RPM效果图

RPM技术的着色器代码如下所示：

vec2 reliefPallaxMapping(in vec3 v, in vec2 t){
	// determine required number of layers
	const float minLayers = 10;
	const float maxLayers = 15;
	float numLayers = mix(maxLayers, minLayers, abs(dot(vec3(0, 0, 1), v)));
	// height of each layer
	float layerHeight = 1.0 / numLayers;
	// depth of current layer
	float currentLayerHeight = 0;
	// shift of texture coordinates for each iteration
	vec2 dtex = gHeightScale * v.xy / v.z / numLayers;
 	// current texture coordinates
	vec2 currentTextureCoords = t;
	// depth from heightmap
	float heightFromTexture = texture(NormTexSampler, currentTextureCoords).a;
	// while point is above surface
	while(heightFromTexture > currentLayerHeight) {
		// go to the next layer
		currentLayerHeight += layerHeight; 
		// shift texture coordinates along V
		currentTextureCoords -= dtex;
		// new depth from heightmap
		heightFromTexture = texture(NormTexSampler, currentTextureCoords).a;
   }
	// Start of Relief Parallax Mapping
	// decrease shift and height of layer by half
	vec2 deltaTexCoord = dtex / 2;
	float deltaHeight = layerHeight / 2;
	// return to the mid point of previous layer
	currentTextureCoords += deltaTexCoord;
	currentLayerHeight -= deltaHeight;
	// binary search to increase precision of Steep Paralax Mapping
	const int numSearches = 5;
	for(int i = 0; i < numSearches; i++){
		// decrease shift and height of layer by half
		deltaTexCoord /= 2;
		deltaHeight /= 2;
		// new depth from heightmap
		heightFromTexture = texture(NormTexSampler, currentTextureCoords).a;
		// shift along or agains vector V
		if(heightFromTexture > currentLayerHeight) // below the surface{
			currentTextureCoords -= deltaTexCoord;
			currentLayerHeight += deltaHeight;
		}
		else // above the surface{
			currentTextureCoords += deltaTexCoord;
			currentLayerHeight -= deltaHeight;
 		}
	}
	return currentTextureCoords;
}

RPM技术通过在${{t}_{3}}$和${{t}_{4}}$范围内做二分查找，得到最终的纹理坐标；但是POM技术只基于当前的层信息，对${{t}_{3}}$、${{t}_{4}}$进行加权计算得到最终的结果。所以POM的性能比RPM技术更佳，但是效果比它略差，由于忽略了更多的细节，有可能会产生错误的结果。RPM和POM的效果对比如图16所示，并不会看出明显的差异。

图15.效果图，（a）RPM效果图，(b) POM效果图

POM技术的着色器代码如下所示：

vec2 occlusionPallaxMapping1(in vec3 v, in vec2 t){
	// determine optimal number of layers
	const float minLayers = 10;
	const float maxLayers = 15;
	float numLayers = mix(maxLayers, minLayers, abs(dot(vec3(0, 0, 1), v)));
	// height of each layer
	float layerHeight = 1.0 / numLayers;
	// current depth of the layer
	float curLayerHeight = 0;
	// shift of texture coordinates for each layer
	vec2 dtex = gHeightScale * v.xy / v.z / numLayers;
	// current texture coordinates
	vec2 currentTextureCoords = t;
	// depth from heightmap
	float heightFromTexture = texture(NormTexSampler, currentTextureCoords).a;
	// while point is above the surface
	while(heightFromTexture > curLayerHeight) {
		// to the next layer
		curLayerHeight += layerHeight; 
		// shift of texture coordinates
		currentTextureCoords -= dtex;
		// new depth from heightmap
		heightFromTexture = texture(NormTexSampler, currentTextureCoords).a;
	}
	// previous texture coordinates
	vec2 prevTCoords = currentTextureCoords + dtex;
	// heights for linear interpolation
	float nextH	= heightFromTexture - curLayerHeight;
	float prevH	= texture(NormTexSampler, prevTCoords).a - curLayerHeight + layerHeight;
	// proportions for linear interpolation
	float weight = nextH / (nextH - prevH);
	// interpolation of texture coordinates
	vec2 finalTexCoords = prevTCoords * weight + currentTextureCoords * (1.0-weight);
	// return result
	return finalTexCoords;
}

此外，提供另外一个版本的POM的实现，它是由Zink^[17]采用HLSL改为GLSL的实现版本，与前面的实现相比，边缘更加光滑，如图16所示，着色器代码如下所示：

图15. POM效果图

vec2 occlusionPallaxMapping2(in vec3 v, in vec2 t){
	int		nMaxSamples			= 15;
	int		nMinSamples			= 10;
	float	fHeightMapScale		= 0.1;
	int nNumSamples = int(mix(nMaxSamples, nMinSamples, abs(dot(vec3(0, 0, 1), v))));
	// height of each layer
	float fStepSize = 1.0 / float(nNumSamples);
	// Calculate the parallax offset vector max length.
	// This is equivalent to the tangent of the angle between the
	// viewer position and the fragment location.
	float fParallaxLimit = length(v.xy ) / v.z;
	// Scale the parallax limit according to heightmap scale.
	fParallaxLimit *= fHeightMapScale;						
	// Calculate the parallax offset vector direction and maximum offset.
	vec2 vOffsetDir = normalize(v.xy);
	vec2 vMaxOffset = vOffsetDir * fParallaxLimit;
	// Initialize the starting view ray height and the texture offsets.
	float fCurrRayHeight = 1.0;	
	vec2 vCurrOffset = vec2(0, 0);
	vec2 vLastOffset = vec2(0, 0);
	vec2 dx = dFdx(t);
	vec2 dy = dFdy(t);
	float fLastSampledHeight = 1;
	float fCurrSampledHeight = 1;
	int nCurrSample = 0;
	while ( nCurrSample < nNumSamples ){
		// Sample the heightmap at the current texcoord offset.  The heightmap 
		// is stored in the alpha channel of the height/normal map.
		//fCurrSampledHeight = tex2Dgrad( NH_Sampler, IN.texcoord + vCurrOffset, dx, dy ).a;
		fCurrSampledHeight = textureGrad(NormTexSampler, TexVertUV + vCurrOffset, dx, dy).a;
		// Test if the view ray has intersected the surface.
		if (fCurrSampledHeight > fCurrRayHeight){
			// Find the relative height delta before and after the intersection.
			// This provides a measure of how close the intersection is to 
			// the final sample location.
			float delta1 = fCurrSampledHeight - fCurrRayHeight;
			float delta2 = (fCurrRayHeight + fStepSize) - fLastSampledHeight;
			float ratio = delta1 / (delta1 + delta2);

			// Interpolate between the final two segments to 
			// find the true intersection point offset.
			vCurrOffset = ratio * vLastOffset + (1.0 - ratio) * vCurrOffset;
			
			// Force the exit of the while loop
			nCurrSample = nNumSamples + 1;	
		}
		else{
			// The intersection was not found.  Now set up the loop for the next
			// iteration by incrementing the sample count,
			nCurrSample ++;
			// take the next view ray height step,
			fCurrRayHeight -= fStepSize;
			// save the current texture coordinate offset and increment
			// to the next sample location, 
			vLastOffset = vCurrOffset;
			vCurrOffset += fStepSize * vMaxOffset;
			// and finally save the current heightmap height.
			fLastSampledHeight = fCurrSampledHeight;
		}
	}
	// Calculate the final texture coordinate at the intersection point.
	return TexVertUV + vCurrOffset;
}

参考

1. James F. Blinn. "Simulation of wrinkled surfaces." ACM SIGGRAPH Computer Graphics, vol.12, no.3, pp.286-292, 1978.
2. “Bump Mapping.” website <https://en.wikipedia.org/wiki/Bump_mapping>.
3. Tomas Akenine-Möller, Eric Haines, and Naty Hoffman. Real-time rendering. CRC Press, 2008.
4. NEHE Production. "22. Bump-Mapping, Multi-Texturing & Extensions."
5. Michael I. Gold. "Emboss Bump Mapping." NVIDIA Corporation. (ppt)
6. John Schlag. "Fast embossing effects on raster image data." Graphics Gems IV. Academic Press Professional, Inc. 1994.
7. Brian Lingard. “Bump Mapping.” website <http://web.cs.wpi.edu/~matt/courses/cs563/talks/bump/bumpmap.html>, 1995.
8. Venkat Krishnamurthy and Marc Levoy. "Fitting smooth surfaces to dense polygon meshes." Proceedings of the 23rd annual conference on Computer graphics and interactive techniques. ACM, 1996.
9. Jonathan Cohen, Marc Olano, and Dinesh Manocha. "Appearance-preserving simplification." Proceedings of the 25th annual conference on Computer graphics and interactive techniques. ACM, 1998.
10. Paolo Cignoni, et al. "A general method for preserving attribute values on simplified meshes." Visualization'98. Proceedings. IEEE, 1998.
11. “Normal Mapping.” website < https://en.wikipedia.org/wiki/Normal_mapping>.
12. Christian Petry. “Normal Mapping.” website <http://cpetry.github.io/NormalMap-Online/>.
13. Eric Lengyel. “Mathematics for 3D game programming and computer graphics.” Cengage Learning, 2012.
14. Siddharth Hegde. “Messing with Tangent Space.” website <http://www.gamasutra.com/view/feature/129939/messing_with_tangent_space.php>.
15. Tomomichi Kaneko, et al. "Detailed shape representation with parallax mapping." Proceedings of ICAT. vol.2001, 2001.
16. “Parallax Occlusion Mapping in GLSL.” website< http://sunandblackcat.com/tipFullView.php?topicid=28>.
17. Jason Zink. “A Closer Look At Parallax Occlusion Mapping.” website<http://www.gamedev.net/page/resources/_/technical/graphics-programming-and-theory/a-closer-look-at-parallax-occlusion-mapping-r3262>.