Stefano Soatto (c) UCLA Vision Lab 1 Homogeneous representation Points Vectors Transformation representation

Stefano Soatto (c) UCLA Vision Lab 1

Homogeneous representation

Points

Vectors

Transformation

representation


Lecture 4: Image formation


Image Formation

Vision infers world properties form images. How do images depend on these properties? Two key elements

Geometry Radiometry We consider only simple models of these


Image formation (Chapter 3)


Representation of images


Similar triangles <P’F’S’>,<ROF’> and <PSF><QOF>

(

fzz

11

'

1

2' ))(( ffzfz


Pinhole model


Forward pinhole


Distant objects are smaller

(Forsyth & Ponce)


Parallel lines meet

Common to draw image plane in front of the focal point. Moving the image plane merely scales the image.

(Forsyth & Ponce)


Vanishing points

• Each set of parallel lines meets at a different point– The vanishing point for this direction

• Sets of parallel lines on the same plane lead to collinear vanishing points. – The line is called the horizon for that plane


Properties of Projection

Points project to points Lines project to lines Planes project to the whole image or a half image Angles are not preserved Degenerate cases

Line through focal point projects to a point. Plane through focal point projects to line Plane perpendicular to image plane projects to part of

the image (with horizon).


Orthographic projection

yy

xx

'

'



Cameras with Lenses

(Forsyth & Ponce)



Assumptions for thin lens equation Lens surfaces are spherical Incoming light rays make a small angle with

the optical axis The lens thickness is small compared to the

radii of curvature The refractive index is the same for the

media on both sides of the lens



Blur circle

A point at distance z is imaged at point 'z from the lens

fzz

11

'

1

and so

)()()(

'' zzfz

f

fz

fzz

|''|'

zzz

db

Points a t distance are brought into focus at distance z 'z

Thus points at distance will give rise to a blur circle of diameterz

with d the diameter of the lens

-zz’

-zz’

P

P’

Q

Q’db


Interaction of light with matter Absorption Scattering Refraction Reflection Other effects:

Diffraction: deviation of straight propagation in the presence of obstacles

Fluorescence:absorbtion of light of a given wavelength by a fluorescent molecule causes reemission at another wavelength


Refraction

n1, n2: indexes of refraction


Solid Angle

2

0

2d

radian

2

]cos[2

sin2

sin

0

2

0

2

0

2

0

2

0

d

ddd

steradian (sr)

ddsind

hemisphere

Sphere: 4


Radiometric Terms


Irradiance and Radiance

Irradiance Definition: power per unit area incident on a surface

W/m2 = luxdA

dE

ddA

dL

cos

2

iL

i

A

Radiance Definition: power per unit area and projected solid angle

W/m2sr

iii dLdA

ddE cos

2

2H

iii dLE cos


Radiant Intensity

Radiant flux W

Definition: flux per unit solid angle

W/sr = cd (candela)

d

dI

2S

dI

I

[ ]


Isotropic Point Source



Radiant flux WAll directions: solid angle 4

Radiant flux per

unit solid angle W/sr

d

dI

Radiant intensity

4

I

r

drdA 2

22 4

1

44 rdA

dA

rdA

d

dA

Id

dA

dE

• Note inverse square law fall off.



Radiant flux WAll directions: solid angle 4

Radiant flux per

unit solid angle W/sr

d

dI

Radiant intensity

4

I

r

drdAcos

12

2

3

2 444 hdA

dA

rdA

d

dA

Id

dA

dE

coscos

• Note cosine dependency.

h

cosrh



Point source at a finite distance

r

2

3

4 hE

cos

• Note cosine dependency.

h

• Note inverse square law fall off.


Irradiance from Area Sources


Hemispherical Source

L

dL

dL

ddLE

i

ii

iiii

2

0

2

020

2

2

0

2

0

2

sin2

sincosL


Reflectance

Reflectance: ratio of radiance to irradiance

dLr=fr dEi

Ei

i

Ei

Lr(x,)

Li(x,i)

irr dEfLi

rr dE

dLf

The surface becomes a light source


BRDF


BRDF

i

rr

iii

irriirrriir

d

dd

dE

EdLf

/

,

;,;,,;,

2

iiriririr dEfEdL ;;;

i

iirirrr dEfL

;

ii

irirrir dE

EdLf

;;

;

iiiiiiiii dLdLdE cos

iiiiriri

dLf cos;


Reflection Equation


Perfectly Diffuse Reflectioni

Perfectly Diffuse Surface •Appears equally bright from all viewing directions (r, r)•Reflects all incident light, i.e.,

iiiiBi

dLL cos1

BrrrrBrrr LddLdLEr rr

sincos

LB(r, r) is constant for all directions (r, r)

1

s

BB E

Lf


Common Diffuse Reflectioni

Normal Diffuse Surface •Appears almost equally bright from most viewing directions (r, r), r << 90°•Reflects only a fraction of incident light, i.e.,

iiiiBBi

dLL cos

Bs

BB E

Lf

Reflectance : Albedo


Perfectly Diffuse Reflection

i

Lambertian cosine Law

s

sisiii sin

)()(E),(E

s

iiisisi

iiiiiii

iiiiiBr

E

ddE

ddL

dLfL

i i

i i

i

cos1

cos)()(1

sincos),(1

cos),(

Distant point light source


Law of Reflection

),(),( iiirrr LL

),(L rrr ),(L iii


Perfectly Specular Reflection

),(),( iiirrr LL

ii

ririiirrsf

cossin

)()(),;,(

i i

iiiiiiiiirrsrrr ddLfL

sincos),(),;,(),(

From the definition of BRDF, the surface radiance is:

To satisfy:

i i

iiiiiririrri ddLL

),()()(),(


Lambertian Examples

Scene

(Oren and Nayar)

Lambertian sphere as the light moves.

(Steve Seitz)


Lambertian + Specular Model

)(cos),,()(

cos),,()(),,(

0

00

sn

sss

iiid

PLP

dPLPPL


Lambertian + specular

• Two parameters: how shiny, what kind of shiny.• Advantages

– easy to manipulate– very often quite close true

• Disadvantages– some surfaces are not

• e.g. underside of CD’s, feathers of many birds, blue spots on many marine crustaceans and fish, most rough surfaces, oil films (skin!), wet surfaces

– Generally, very little advantage in modelling behaviour of light at a surface in more detail -- it is quite difficult to understand behaviour of L+S surfaces (but in graphics???)


Lambertian+Specular+Ambient

(http://graphics.cs.ucdavis.edu/GraphicsNotes/Shading/Shading.html)

Documents

Stefano Soatto (c) UCLA Vision Lab 1 Homogeneous representation Points Vectors Transformation representation