3-D Computer Vision CSc 83029

3-D Computer Vision CSc83029 / Ioannis Stamos3-D Computer Vision CSc83029 / Ioannis Stamos

3-D Computer Vision3-D Computer VisionCSc 83029CSc 83029

Radiometry and ReflectanceRadiometry and Reflectance


From 2D to 3DFrom 2D to 3D

DEPTH from TWO or MORE IMAGESDEPTH from TWO or MORE IMAGES StereoStereo Optical Flow -> Factorization Method.Optical Flow -> Factorization Method.

SHAPE from SINGLE IMAGE CUESSHAPE from SINGLE IMAGE CUES SHAPE from SHADINGSHAPE from SHADING SHAPE from TEXTURESHAPE from TEXTURE ……


Shading Encodes ShapeShading Encodes Shape



n

I

Image Intensity I = f ( orientation n,surface reflectance,illumination, imaging system )

L

Surface Element

Note: Image Intensity Understanding is an under-constrained problem!


Solid AngleSolid Angle

δω=(δA cosθ)/R (steradian)

δ

δω

2



δ

δω

2

Foreshortened Area

δA’

δω= δA’/R (steradian)2



δ

δω

2

Foreshortened Area

UnitSphere

δA’

δω= δA’/R (steradian)2

Solid Angle Sustainedby a hemisphere = 2π

n

Source

Flux

Radiant Intensity of Source: Light flux (power)emitted per unit solid angle:

J=dΦ/dω (watts/steradian)

dω

dA

Surface Irradiance: Flux incident per unit surface area:

E=dΦ/dA (watts/m )Does not depend on where the light is coming from!

2

Flux=dΦ

Surface Radiance(Brightness): Flux emitted per unit foreshortened area,

per unit solid angle:L= d Φ/(dA cosθr)dω (watts/m . steradian)

Note: L depends on direction θr Surface can radiate into whole hemisphere L is proportional to irradiance E Depends on reflectance properties of surface

dω

dA

2

2

θr

2


Surface Radiance & Image IrradianceSurface Radiance & Image Irradiance

E=δP/δI:Irradiance at point p (Watts/m )δP= (δO * cosθ)* L * ΔΩ , L scene radiance at P.

2

L (Watts/m * steradian)2Trucco & Verri


F=f/d: F-number:How much lightis captured by the camera

Trucco & Verri



n

I

Image Intensity I = f ( orientation n,surface reflectance,illumination, imaging system )

)(cos4

42

fdLE

Scene radiance

L

Image irradiance

1 / F-number of lens

E

Optics

Brightness falloff

LE Assume Image irradiance is proportional to scene radiance


Bi-Directional Reflectance Distribution Function Bi-Directional Reflectance Distribution Function (BRDF)(BRDF)

n ii , rr ,

yx

z

φ

θ



n ii , rr ,

yx

z

φ

θ

iiE , : Irradiance due to source in direction ii ,

rrL , : Radiance of surface in direction rr ,

ii

rrrrii E

Lf,,,,, BRDF:



n ii , rr ,

yx

z

φ

θ

iiE , : Irradiance due to source in direction ii ,

rrL , : Radiance of surface in direction rr ,

ii

rrrrii E

Lf,,,,, BRDF:

irrif ,,For Rotationally Symmetric Reflectance Properties:BDRF: ISOTROPIC SURFACES

*Bird Feathers are often Non-Isotropic.


Reflectance ModelsReflectance Models*Reflection: An Electromagnetic Phenomenonλ

τ

σh

Two Approaches to deriving Reflectance Models:Physical Optics (Wave Optics)Geometrical Optics (Ray Optics)

Geometrical Models are approximation to Physical Models.But easier to use.


Surface Reflection

Body Reflection

Internal scattering

Body Reflection: *Diffuse Reflection *Matte Appearance *Non-Homogeneous Medium

Surface Reflection: *Specular Reflection *Glossy Appearance *Highlights. *Dominant for Metals

Image Intensity: Diffuse Comp + Specular Comp

Lambertian Reflectance ModelLambertian Reflectance Model

f Albedo: intrinsic brightness or color of surface

n

ikE cos

Theta is the angle between source direction and surface normal

s

i

A Lambertian (diffuse) surface scatters light equally in all directions



i

iirr

kL

EL

EfL

cos

, ,

Surface appears equally bright from all viewing directions

A Lambertian (diffuse) surface scatters light equally in all directions

Albedo: intrinsic brightness or color of surface

Illumination strength

f

ni

s

ikE cos



sn

L

kL icos

A Lambertian sphere

ns

Surface normal nDirection of illumination s

Commonly used in Computer Vision and Graphics

Effective albedo


What Information Does Shading What Information Does Shading EncodeEncode

In regions of constant albedo, changesof intensity correspond to changes in thesurface normal of the scene

sn LI


Ideal Specular Model (Mirrors)Ideal Specular Model (Mirrors)

n

))(()( eieikL

Perfect reflector

sr

*Very SMOOTH surface*All incident energy reflected in a single direction

ii , rr ,

ee ,v

Viewer received light only when v=r


Phong Reflectance ModelPhong Reflectance Model

b=0.3, c=0.7, n=2 b=0.7, c=0.3, n=0.5

ni cbL coscos

diffuse specular

v: viewingdirection

αθi

n

s

r


Torrance-Sparrow ModelTorrance-Sparrow ModelSpecular Reflection from Rough Surfaces.Surface Micro-Structure Model

Facet orientationMean orientation α

Micro-facet Orientation Model: (example)

2

2

21

21)(

a

eap

(Gaussian Model) Isotropic

σ: roughness parameter


Torrance-Sparrow ModelTorrance-Sparrow ModelMasking and Shadowing Effects

shadow shadowMasked Light

n sr

v

Specular Direction

),,()( vnsGpvn

PL spec

Radiance

Geometric Factor G(Masking-Shadowing)

n’β


Gradient Space (p,q)Gradient Space (p,q)

Surface normal can be represented by the intersection of the normal with a plane.

qp,

1

1,,22

qpqpn

np

q

x

y

Sourcez

1.0 ss qp ,

3-D Computer Vision CSc 83029

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