光度学基础和反射
理论——Photometry principle & Reflection theory
简介
光度学是1760年由朗伯建立的，且定义了光通量、发光强度、照度、亮度等主要光学光度学参量，并用数学阐明了它们之间的关系和光度学几个重要定律，如照度的叠加性定律、距离平方比定律、照度的余弦定律等，这些定律一直沿用至今，实践已证明是正确的。
在可见光波段内，考虑到人眼的主观因素后的相应计量学科称为光度学。
▪
参量定义
光度学（Photometry ）的核心定义是光通量Φ，单位是流明，可以理解为光能的功率。
其他定义与其的关系如下：
光通量Φ 单位
lm
发光强度I 单位lm/sr
I=d Φ/d ω= r 2*d Φ/(cos θ*d A)辅照度E 单位lm/m 2
E= d Φ/*d A
其中d ω=cos θ*d A/r 2辅射度L
单位lm/（sr*m 2）
L= d 2Φ/（cos θ*d A*d ω）
单位方向角内的光通量=单位圆上
单位透视缩小面积上的光通量
单位面积接受的光通量
单位透视缩小面积单位辐射角接受
/发出的光通量
BRDF=L/E
dI=r 2Ld ω
▪双向反射分布函数
BRDF被定义为物体表面处辐射度和辐照度的比值（单位sr-1）：
f(x,ωi,ωo)=L0(x,ωo)
E i(x,ωi)
=
L0(x,ωo)
d∅
dA
=
L0(x,ωo)
I i(x,ωi)
dω
dA
=
L0(x,ωo)
I i(x,ωi)cosθi/r2
整理后得到点光源情况下辐射度的公式：
L0(x,ωo)=I i(x,ωi)cosθi f(x,ωi,ωo)
r2
因：dI i(x,ωi)=r2L i(x,ωi)dω带入上式，得到光源不是点光源情况下，如光源区域在Ω内，辐射度的公式：
L0(x,ωo)=∫L i(x,ωi)cosθi f(x,ωi,ωo)dω
Ω
▪典型反射模型
Lambert：f(x,ωi,ωo)=const
I0=kI i〈l,n〉——l:light norm,n:surface norm 描述：理想漫反射模型
Lommel-Seelinger：f(x,ωi,ωo)=const
一般p(α)=1
——μ0=cosϖi,μ=cosϖo,ϖ:single scattering albedo单次照射散射率描述：月球天体模型
Lunar-Lambert：
描述：月球天体模型
Phong：I0=k a I a+k d I i〈l,n〉+k s I i〈r,v〉s
——r:perfect ref norm,v:view norm,
——I a:envirument light,s:surface smoothness 描述：漫反射与镜面反射综合模型
Oren–Nayar
The surface roughness model used in the derivation of the Oren-Nayar model is the microfacet model, proposed by Torrance and Sparrow,[2] which assumes the surface to be composed of long symmetric V-cavities. Each cavity consists of two planar facets. The roughness of the surface is specified using a probability function for the distribution of facet slopes. In particular, the Gaussian distribution is often
used, and thus the variance of the Gaussian distribution, , is a measure of the roughness of the surfaces. The standard deviation of the facet slopes (gradient of the surface elevation), ranges in .
In the Oren–Nayar reflectance model, each facet is assumed to be Lambertian in reflectance. As shown in the image at right, given the radiance of the incoming light
, the radiance of the reflected light , according to the Oren-Nayar model, is
Where
and is the albedo of the surface, and is the roughness of the surface. In the case of (i.e., all facets in the same plane), we have , and , and thus the Oren-Nayar model simplifies to the Lambertian model:
Minnaert function
From Wikipedia, the free encyclopedia
The Minnaert function is a photometric function used to interpret astronomical observations [1][2] and remote sensing data for the Earth.[3] This function expresses the radiance factor (RADF) as a function the phase angle () and the photometric latitude () and the photometric longitude ().
where is the Minnaert albedo, is an empirical parameter, is the
scattered radiance in the direction , ,is the incident radiance, and
The phase angle is the angle between the light source and the observer with the object as the center.
The assumptions made are:
the surface is illuminated by a distant point source.
the surface is isotropic and flat.
Minnaert's contribution is the introduction of the parameter , having a value between 0 and 1,[4] originally for a better interpretation of observations of the Moon. In remote sensing the use of this function is referred to as Minnaert topographic correction, a necessity when interpreting images of rough terrain.
参考文献：
[1]. 维基百科：/wiki/光度学
[2]. 维基百科：/wiki/Oren–Nayar_reflectance_model
[3]. 维基百科：/wiki/Phong_reflection_model
[4]. 维基百科：
/wiki/Bidirectional_reflectance_distribution_function
[5]. 维基百科：/wiki/Minnaert_function
[6]. http://space.unibe.ch/staff/thomas/07_Photometry_Rev.pdf
[7]. http://www.astro.uvic.ca/~tatum/plphot/plphot03.pdf，Chapter 3. A Brief History of the Lommel-Seeliger Law.
[8].郑作勇, 基于图像的带高光物体的形状和反射属性建模技术研究, 2009, 上海交通大学.
[9].杨磊, 韩九强, 王国珲，基于Shape - from - Shading 的月球表面三维形状
恢复算法研究，宇航学报，2008.11,29(6),1995~2000
[10].Maxwell B.Fairbairn，Planetary Photometry:The Lommel-Seeliger Law，The Royal Astronomical Society of Canada,NASA Astrophysics Data System。
