实时全局照明技术：球谐函数光照详解

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"Spherical Harmonic Lighting技术是一种用于计算3D模型受区域光源影响的照明的技术，它能够在实时环境中捕捉、重新照明并展示全局光照风格的图像。该技术由Sloan, Kautz和Snyder在2002年的Siggraph会议上首次提出，主要用于实现超现实的模型照明效果。Spherical Harmonic Lighting实际上是一系列相互关联的技术工具箱，游戏开发社区可以充分利用这些技术来提升光照效果。" Spherical Harmonic Lighting（SH Lighting）的细节解析： Spherical Harmonic Lighting的核心在于利用球谐函数（Spherical Harmonics）来近似表示复杂的光照环境。球谐函数是一种在三维空间中描述光线分布的数学工具，它们是正交的、归一化的函数，可以在球面上形成一系列的模式，这些模式可以有效地捕获光照的复杂性，同时保持计算效率。 1. 光照计算：在3D图形学中，计算每个像素的光照通常涉及到表面法线、光源位置、颜色和材质属性等因素。传统的逐像素光照方法对于大型复杂场景和动态光源来说过于昂贵。SH Lighting通过预计算和存储光源的球谐系数，将全局光照简化为对模型表面的快速近似计算。 2. 球谐函数的性质：球谐函数可以分为多个阶数，每阶都有固定数量的函数，如第一阶有3个，第二阶有5个，以此类推。这些函数对应于不同方向和强度的光照信息。高阶函数可以捕捉更复杂的光照变化，但计算成本也随之增加。在游戏应用中，通常选择较低的阶数以平衡性能和质量。 3. 实时性与重光照： SH Lighting的优势在于能够实现实时的全局光照效果。当光源位置或属性改变时，只需更新相应的球谐系数，而非重新计算整个场景的光照。这使得在保持流畅的游戏体验的同时，还能呈现出更加丰富的光照效果。 4. 应用于游戏：对于游戏开发者而言，理解SH Lighting背后的原理和如何将其整合进游戏引擎至关重要。这包括如何采样和计算光源的球谐系数，如何将这些系数应用于3D模型的着色，以及如何优化内存占用和计算效率。 5. 扩展与改进：尽管SH Lighting提供了实时全局光照的良好基础，但它也有一些局限，例如难以处理镜面反射和透明度等复杂光学效果。因此，研究者们不断探索如混合SH Lighting、基于机器学习的方法等技术来进一步增强光照的真实感和性能。 Spherical Harmonic Lighting是游戏开发中实现逼真光照效果的重要工具，通过理解和应用这些技术，开发者能够提升游戏的视觉品质，同时保持良好的运行性能。本文旨在提供必要的背景知识，解答“为什么”的问题，并指导读者如何将SH Lighting集成到自己的游戏中。

SPHERICAL HARMONIC LIGHTING 9

In the above example we have used a set of linear basis functions,

giving us a piecewise linear approximation to the input function.

There are many basis functions we can use, but some of the most

interesting are grouped into a family of functions mathematicians

call the orthogonal polynomials.

Orthogonal polynomials are sets of polynomials that have an

intriguing property – when you integrate the product of any two

of them, if they are the same you get a constant value and if they

are different you get zero.

() ()







≠

∫

−

mnc

dxxFxF

for

for 0

We can also specify the more rigorous rule that integrating the

product of two of these polynomials must return either 0 or 1, and

this sub-family of functions are known as the orthonormal basis

functions. Intuitively, it’s a like the functions do not “overlap” each

other’s influence while still occupying the same space, the same

effect that allows the Fourier transform to break a signal into it’s

component sine waves.

These families of polynomials are often named after the

mathematicians who studied them, names like Chebyshev, Jacobi

and Hermite. The one family we are most interested in are called

the Legendre polynomials, specifically the Associated Legendre

Polynomials. Traditionally represented by the symbol P, the

associated Legendre polynomials have two arguments l and m, are

defined over the range [–1,1] and return real numbers (as opposed

to the ordinary Legendre Polynomials which return complex

values – be careful not to confuse the two).

(x)

–1

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8

The two arguments l and m break the family of polynomials into

bands of functions where the argument l is the band index and takes

any positive integer value starting from 0, and the argument m

takes any integer value in the range [0,l]. Inside a band the

polynomials are orthogonal w.r.t. a constant term and between

Figure 4. The first six associated

Legendre polynomials.

10 SPHERICAL HARMONIC LIGHTING

bands they are orthogonal with a different constant. We can

diagram this as a triangular grid of functions per band, giving us a

total of n(n+1) coefficients for an n band approximation:

()

() ()

() () ()

xPxPxP

xPxP

The process for evaluating Legendre polynomials turns out to be

quite involved, which is why they’re rarely used for

approximating 1D functions. The usual mathematical definition of

the series is defined in terms of derivatives of imaginary numbers

and requires a series of nasty cancellations of values that alternate

in sign and this is not a floating point friendly process. Instead we

turn to a set of recurrence relations (i.e. a recursive definition) that

generate the current polynomial from earlier results in the series.

There are only three rules we need:

()

(

)

(

)

PmlPlxPml

112

−−

−+−−=−

The main term of the recurrence takes the two previous bands l–1

and l–2 and generates a new higher band l from them.

()( )

(

)

1!!121

xmP −−−=

The expression is the best place to start from as it is the only rule

that needs no previous values. Note that x!! is the double factorial

function which, as (2m–1) is always odd, returns the product of all

odd integers less than or equal to x. We can use P

(x) = 1 as the

initial state for an iterative loop that hoists us up from 0 to m.

(

)

PmxP 12

This expression allows us to lift a term to a higher band.

The method for evaluating the function is first to try to generate

the highest P

possible using rule 2, which if l=m is the final

answer. Since m<l in all remaining cases, all that is left is to raise

the band until we meet the required l. We do this by calculating

m+1

using rule 3 only once (stopping if l=m+1) and finally

iterating rule 1 until the correct answer is found (noting that using

rule 1 has less floating point roundoff error than iterating rule 3).

double P(int l,int m,double x)

{

// evaluate an Associated Legendre Polynomial P(l,m,x) at x

double pmm = 1.0;

if(m>0) {

double somx2 = sqrt((1.0-x)*(1.0+x));

For more, see “Numerical Methods

in C: The Art of Scientific

Computing”, Cambridge

University Press, 1992, pp 252-

254

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