从零开始的openGL——五、光线追踪

前言

注：代码已开源在 github 中， https://github.com/leo6033/CSU_CS_Experiment/tree/master/计算机图形学

前面介绍了基本图形、模型、曲线的绘制，但是，在好像还没有感受到那种3D游戏里一些能惊艳到自己的效果，即真实感还不是很足。这篇文章中介绍的光线追踪，是实现真实感必不可少的。拿下面的两张图片来对比

对比一下是不是被下面这张图片的效果惊艳到了？可以很明显感觉到，下面的这个图片效果要好的多。这篇博客将介绍如何实现这样的效果。

光线求交

这里暂时只介绍光线与球面和三角面片的求交

光线与球面相交

射线的方程：

\[R(t) = A+tD
\]

球面的隐式方程：

\[(X-C)^2=r^2
\]

联立两式：

\[(A+tD-C)^2=r^2
\]

然后通过判别式：$$\Delta=4[(A-C) \cdot D]^2 - 4(A-C)^2+r2$$来判断是否相交。

交点法向量：

\[N=\frac{P-C}{||P-C||}
\]

bool Sphere::intersectLocal( const ray& r, isect& i ) const
{
    // YOUR CODE HERE:
	// 光线与球面相交
    // Add sphere intersection code here.
	Vec3d A = r.getPosition();
	Vec3d D = r.getDirection();
	Vec3d C= Vec3<double>();
	double _r = 1.0;
	double a = D.length2();
	double b = 2 * (A - C) * D;
	double c = (A - C).length2() - _r;
	double delta = b * b - 4 * a * c;
    // it currently ignores all spheres and just return false.
	if (delta >= 0) {
		double t1 = (-b + sqrt(delta)) / (2 * a);
		double t2 = (-b - sqrt(delta)) / (2 * a);
		if (t1 <= RAY_EPSILON)
			return false;
		else {
			double t;
			if (t2 <= RAY_EPSILON) {
				t = t1;
				i.outsideTheObject = false;
			}
			else {
				t = t2;
				i.outsideTheObject = true;
			}
			// 焦点设置
			i.obj = this;
			i.setT(t);
			Vec3d P = r.at(t);
			Vec3d Normal = P;
			if (D*Normal > 0)
				Normal = -Normal;
			Normal.normalize();
			i.setN(Normal);
			return true;
		}
	}
    return false;
}

光线与三角面片相交

射线的方程：

\[R(t) = A+tD
\]

三角面片点法式方程：

\[N(p-p_1)=0
\]

联立两式得：

\[t=\frac{N\cdot p_1 - N \cdot A}{n\cdot D}
\]

求出t后，便得到交点坐标，然后可通过同向法来判别交点是否在平面内。

// Calculates and returns the normal of the triangle too.
bool TrimeshFace::intersectLocal(const ray& r, isect& i) const
{
	// YOUR CODE HERE:
	// Add triangle intersection code here.
	// it currently ignores all triangles and just return false.
	//
	// Note that you are only intersecting a single triangle, and the vertices
	// of the triangle are supplied to you by the trimesh class.
	//
	// You should retrieve the vertices using code like this:
	//
	// const Vec3d& a = parent->vertices[ids[0]];
	// const Vec3d& b = parent->vertices[ids[1]];
	// const Vec3d& c = parent->vertices[ids[2]];
	const Vec3d& a = parent->vertices[ids[0]];
	const Vec3d& b = parent->vertices[ids[1]];
	const Vec3d& c = parent->vertices[ids[2]];

	Vec3d edge1 = b - a;
	Vec3d edge2 = c - a;
	// 计算平面法向量
	Vec3d nor = edge1 ^ edge2;
	nor.normalize();

	// 判断是否与平面平行
	float x = nor * r.getDirection();
	if (x == 0)
		return false;
	// Ax + By + Cz = d
	float d = nor * a;
	float t = (d - nor * r.getPosition()) / x;
	if (t <= RAY_EPSILON)
		return false;
	Vec3d intersection_point = r.at(t);
	Vec3d edge3 = intersection_point - a;
	// 同向法判断是否在平面内
	if (((b - a) ^ (intersection_point - a)) * nor <= 0)
		return false;
	else if (((c - b) ^ (intersection_point - b)) * nor <= 0)
		return false;
	else if (((a - c) ^ (intersection_point - c)) * nor <= 0)
		return false;
	else {
		//交点设置
		i.obj = this;
		i.setT(t);
		i.setN(nor);
		return true;
	}

}

当然，这里还可以使用重心坐标法来实现

光线衰减

在现实场景中，光线也是会衰减的，比如看同一场景，距离远近不同看到的清晰度也就不同，这是距离衰减。还有阴影衰减，当有物体遮挡住部分光的时候，会形成一定的阴影，这就是阴影衰减产生的效果。

距离衰减

点光源：

\[A_{j}^{d i s t}=\min \left\{1, \frac{1}{a_{j}+b_{j} r_{j}+c_{j} r_{j}^{2}}\right\}
\]

double PointLight::distanceAttenuation( const Vec3d& P ) const
{
	// You'll need to modify this method to attenuate the intensity
	// of the light based on the distance between the source and the
	// point P.  For now, we assume no attenuation and just return 1.0
	Vec3d d = P - position;
	double r = d.length(); //距离
	return min(1.0, 1.0 / (constantTerm + linearTerm * r + quadraticTerm * r*r));
//	return 1.0;
}

平行光源：

double DirectionalLight::distanceAttenuation( const Vec3d& P ) const
{
	// distance to light is infinite, so f(di) goes to 0.  Return 1.
	return 1.0;
}

阴影衰减

点光源：

首先判断光线是否被遮挡，然后再判断是否超出光强所能打到的距离

Vec3d PointLight::shadowAttenuation(const Vec3d& P) const
{
    // YOUR CODE HERE:
    // You should implement shadow-handling code here.
	Vec3d d = getDirection(P);
	isect i;
	ray shadowRay(P, d);
	if (this->getScene()->intersect(shadowRay, i)) {
		double tLight = (P - position).length();
		if (i.t < tLight)
			return Vec3d(0, 0, 0);
		else
			return Vec3d(1, 1, 1);
	}
    return Vec3d(1,1,1);
}

平行光：

只需判断是否被遮挡即可

Vec3d DirectionalLight::shadowAttenuation( const Vec3d& P ) const
{
    // YOUR CODE HERE:
	Vec3d d = getDirection(P);
	isect i;
	ray shadowRay(P, d);
	if (this->getScene()->intersect(shadowRay, i)) {
		return Vec3d(0, 0, 0);
	}
    // You should implement shadow-handling code here.
    return Vec3d(1,1,1);
}

光线追踪

先来份伪代码

光线跟踪中的四种射线：

• 视线：由视点与象素(x，y)发出的射线

• 阴影测试线：物体表面上点与光源的连线

• 反射光线

• 折射光线

光线追踪的过程

phong光照模型

由物体表面上一点P反射到视点的光强I为环境光的反射光强\(I_e\)、理想漫反射光强\(I_d\)、和镜面反射光\(I_s\)的总和，即

\[I=I_ak_a + I_lk_d(L \cdot N)+k_s\sum_{i=1}^{m}[I_{pi}(R \cdot V)^n]
\]

在washington CSE 457的课件中给出的公式为

\[l_{\text {direct }}=k_{e}+k_{e} I_{L s}+\sum_{f} A_{j}^{\text {shadow}} A_{j}^{\text {dist}} I_{L j} B_{j}\left[k_{d}\left(\mathbf{N} \cdot \mathbf{L}_{j}\right)+k_{s}\left(\mathbf{N} \cdot \mathbf{H}_{j}\right)^{n_{s}}\right]
\]

其中\(k_d\)项表示漫反射，采用Lamber模型，\(k_s\)项表示镜面反射

\[I_{d}=I_{p} K_{d} *(L \cdot N)
\]

\[I_{s}=k_{s} I_{p}(R \cdot V)^{n}
\]

即可写出下列代码

// Apply the Phong model to this point on the surface of the object, returning
// the color of that point.
Vec3d Material::shade( Scene *scene, const ray& r, const isect& i ) const
{
	// YOUR CODE HERE

	// For now, this method just returns the diffuse color of the object.
	// This gives a single matte color for every distinct surface in the
	// scene, and that's it.  Simple, but enough to get you started.
	// (It's also inconsistent with the Phong model...)

	// Your mission is to fill in this method with the rest of the phong
	// shading model, including the contributions of all the light sources.
    // You will need to call both distanceAttenuation() and shadowAttenuation()
    // somewhere in your code in order to compute shadows and light falloff.
	if( debugMode )
		std::cout << "Debugging the Phong code (or lack thereof...)" << std::endl;

	Vec3d pos = r.at(i.t);
	Vec3d N = i.N;
	N.normalize();
	Vec3d Ip, L, H, Atten;
	Vec3d shadow = ke(i) + prod(scene->ambient(), ka(i));
	for (vector<Light*>::const_iterator litr = scene->beginLights();
		litr != scene->endLights(); ++litr) {
		Light* pLight = *litr;
		Ip = pLight->getColor(pos);
		L = pLight->getDirection(pos);
		H = -r.getDirection() + L;	H.normalize();
		Atten = pLight->distanceAttenuation(pos)*pLight->shadowAttenuation(pos);
		shadow += prod(Atten, prod(Ip, kd(i)*(L*N) + ks(i)*pow(H*N, 256)));
	}
	return shadow;
}

反射方向

这里的反射指的是镜面反射

计算公式：

\[R=2(V\cdot N)N-V
\]

为什么是这样呢？首先来看\(V\cdot N\)，这里N是交点处的法向量，并且是单位向量，那个即视线在法向量上的投影，再乘法向量的两倍，得到的是平行四边形的对角线，减去V便是反射后的光线的方向。

折射方向

跟反射方向一样都是公式推导

\[\begin{array}{l}{\eta=\frac{\eta_{i}}{\eta_{t}}} \\ \eta_{i} \sin \theta_{i}=\eta_{t} \sin \theta_{t} \\ {\cos \theta_{i}=\mathbf{N} \cdot \mathbf{V}} \\ {\cos \theta_{t}=\sqrt{1-\eta^{2}\left(1-\cos ^{2} \theta_{i}\right)}} \\ {\mathbf{T}=\left(\eta \cos \theta_{i}-\cos \theta_{t}\right) \mathbf{N}-\eta \mathbf{V}}\end{array}
\]

终止条件

经过上述的介绍，很容易可以想到，什么时候终止光线追踪

• 该光线未碰到任何物体

• 该光线碰到了背景

• 光线在经过许多次反射和折射以后，就会产生衰减，光线对于视点的光强贡献很小（小于某个设定值）。

• 光线反射或折射次数即跟踪深度大于一定值

因此，光线追踪的代码实现如下

// Do recursive ray tracing!  You'll want to insert a lot of code here
// (or places called from here) to handle reflection, refraction, etc etc.
Vec3d RayTracer::traceRay( const ray& r,
	const Vec3d& thresh, int depth )
{
	isect i;

	if( scene->intersect( r, i ) && depth >= 0) {
		const Material& m = i.getMaterial();

		//计算光源直射
		Vec3d I = m.shade(scene, r, i);

		//计算反射递归
		Vec3d Q = r.at(i.t);
		Vec3d R = r.getDirection() - 2 * (r.getDirection()*i.N)*i.N;
		R.normalize();
		I += prod(m.kr(i), traceRay(ray(Q, R), thresh, depth - 1));

		//计算折射递归
		double cosThetaI = -i.N*r.getDirection();
		double eta = (i.outsideTheObject) ? 1.0003 / m.index(i) : m.index(i) / 1.0003;
		if (eta*eta*(1 - cosThetaI * cosThetaI) < 1) {
			double cosThetaT = sqrt(1 - eta * eta*(1 - cosThetaI * cosThetaI));
			Vec3d T = (eta*cosThetaI - cosThetaT)*i.N - eta * r.getDirection();
			T.normalize();
			I += prod(m.kt(i), traceRay(ray(Q, -T), thresh, depth - 1));
		}
		return I;
		// An intersection occured!  We've got work to do.  For now,
		// this code gets the material for the surface that was intersected,
		// and asks that material to provide a color for the ray.  

		// This is a great place to insert code for recursive ray tracing.
		// Instead of just returning the result of shade(), add some
		// more steps: add in the contributions from reflected and refracted
		// rays.

		//const Material& m = i.getMaterial();
		//return m.shade(scene, r, i);

	} else {
		// No intersection.  This ray travels to infinity, so we color
		// it according to the background color, which in this (simple) case
		// is just black.

		return Vec3d( 0.0, 0.0, 0.0 );
	}
}

小节

到这里，光线追踪也就差不多介绍完了，这一系列博客也算是收尾了。那天在课上听其他同学展示的的时候，说是我的世界有部分的开源源码，里面有一个可以实现光追的接口，有兴趣的小伙伴可以去康康，似乎那个仅仅实现光追还无法达到很好的效果，还要加上路线追踪，emmmmm。。。。期末考完有空了我再去康康，明早图形学考试祝我好运 orz