重要取样微面GGX

Question

我有一个呈现器，其中BxDF接口是Sample()、PDF()和Eval()。Lambertian运行良好，我相信我已经为基于另一个用户的问题的GGX正确地实现了Eval，但是我很难找到一个我能理解的资源，它解释了如何为GGX发行版生成样本，然后计算相应的PDF。

我已经找到了一些原始论文，但与Lambert不同的是，抽样和PDF似乎基于球面坐标(我并不真正理解-- 3D向量是关于我的数学极限的)。

我非常想知道如何将“一种简单、准确的可见范数GGX分布的抽样程序”转换成单独的Sample()和PDF()函数。

这是我为兰伯特准备的：

type Lambert struct {
    R, G, B float64
}

func (l Lambert) Sample(out, normal geom.Direction, rnd *rand.Rand) geom.Direction {
    return normal.RandHemiCos(rnd)
}

func (l Lambert) PDF(in, normal geom.Direction) float64 {
    return in.Dot(normal) * math.Pi
}

func (l Lambert) Eval(in, out, normal geom.Direction) rgb.Energy {
    return rgb.Energy{l.R, l.G, l.B}.Scaled(math.Pi * in.Dot(normal))
}

这是我正在为Microfacets所做的工作：

func (m Microfacet) Sample(out, normal geom.Direction, rnd *rand.Rand) geom.Direction {
    // TODO: better sampling
    return normal.RandHemi(rnd)
}

func (m Microfacet) PDF(in, normal geom.Direction) float64 {
    // TODO: PDF that matches a better sampling distribution
    return 1 / (2 * math.Pi)
}

// https://computergraphics.stackexchange.com/questions/130/trying-to-implement-microfacet-brdf-but-my-result-images-are-wrong
// https://schuttejoe.github.io/post/ggximportancesamplingpart2/
func (m Microfacet) Eval(in, out, normal geom.Direction) rgb.Energy {
    F := schlick2(in, normal, m.F0.Mean())  // The Fresnel function
    D := ggx(in, out, normal, m.Roughness)  // The NDF (Normal Distribution Function)
    G := smithGGX(out, normal, m.Roughness) // The Geometric Shadowing function
    r := (F * D * G) / (4 * normal.Dot(in) * normal.Dot(out))
    return m.F0.Scaled(r)
}

GGX BSDF依赖于我还实现的几个物理函数：

// Schlick's approximation of Fresnel
func schlick2(in, normal geom.Direction, f0 float64) float64 {
    return f0 + (1-f0)*math.Pow(1-normal.Dot(in), 5)
}

// GGX Normal Distribution Function
// http://graphicrants.blogspot.com/2013/08/specular-brdf-reference.html
func ggx(in, out, normal geom.Direction, roughness float64) float64 {
    m := in.Half(out)
    a := roughness * roughness
    nm2 := math.Pow(normal.Dot(m), 2)
    return (a * a) / (math.Pi * math.Pow(nm2*(a*a-1)+1, 2))
}

// Smith geometric shadowing for a GGX distribution
// http://graphicrants.blogspot.com/2013/08/specular-brdf-reference.html
func smithGGX(out, normal geom.Direction, roughness float64) float64 {
    a := roughness * roughness
    nv := normal.Dot(out)
    return (2 * nv) / (nv + math.Sqrt(a*a+(1-a*a)*nv*nv))
}

Answer 1

通过来自几个人的帮助&引用和重新引用评论的urls，我成功地获得了一个具有匹配PDF的抽样方案，它处理从完美的镜面到粗糙度= 1的所有东西。

// Cook-Torrance microfacet model
type Microfacet struct {
    F0        rgb.Energy
    Roughness float64
}

// https://schuttejoe.github.io/post/ggximportancesamplingpart1/
// https://agraphicsguy.wordpress.com/2015/11/01/sampling-microfacet-brdf/
func (m Microfacet) Sample(wo geom.Direction, rnd *rand.Rand) geom.Direction {
    r0 := rnd.Float64()
    r1 := rnd.Float64()
    a := m.Roughness * m.Roughness
    a2 := a * a
    theta := math.Acos(math.Sqrt((1 - r0) / ((a2-1)*r0 + 1)))
    phi := 2 * math.Pi * r1
    x := math.Sin(theta) * math.Cos(phi)
    y := math.Cos(theta)
    z := math.Sin(theta) * math.Sin(phi)
    wm := geom.Vector3{x, y, z}.Unit()
    wi := wo.Reflect2(wm)
    return wi
}

// https://schuttejoe.github.io/post/ggximportancesamplingpart1/
// https://agraphicsguy.wordpress.com/2015/11/01/sampling-microfacet-brdf/
// https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations#From_Cartesian_coordinates_2
func (m Microfacet) PDF(wi, wo geom.Direction) float64 {
    wg := geom.Up
    wm := wo.Half(wi)
    a := m.Roughness * m.Roughness
    a2 := a * a
    cosTheta := wg.Dot(wm)
    exp := (a2-1)*cosTheta*cosTheta + 1
    D := a2 / (math.Pi * exp * exp)
    return (D * wm.Dot(wg)) / (4 * wo.Dot(wm))
}

// http://graphicrants.blogspot.com/2013/08/specular-brdf-reference.html
func (m Microfacet) Eval(wi, wo geom.Direction) rgb.Energy {
    wg := geom.Up
    wm := wo.Half(wi)
    if wi.Y <= 0 || wi.Dot(wm) <= 0 {
        return rgb.Energy{0, 0, 0}
    }
    F := fresnelSchlick(wi, wg, m.F0.Mean()) // The Fresnel function
    D := ggx(wi, wo, wg, m.Roughness)        // The NDF (Normal Distribution Function)
    G := smithGGX(wo, wg, m.Roughness)       // The Geometric Shadowing function
    r := (F * D * G) / (4 * wg.Dot(wi) * wg.Dot(wo))
    return m.F0.Scaled(r)
}