computer and robot vision ii chapter 12 illumination presented by: 傅楸善 & 張庭瑄 0963...

Computer and Robot Vision II

Chapter 12Illumination

Presented by: 傅楸善 & 張庭瑄 0963 331 533

[email protected]指導教授 : 傅楸善博士

mailto:[email protected]

DC & CV Lab.DC & CV Lab.CSIE NTU

12.1 Introduction

two key questions in understanding 3D image formation

What determines where some point on object will

appear on image?

Answer: geometric perspective projection model What determines how bright the image of some

surface on object will be?

Answer: radiometry, general illumination models, diffuse and specular

refraction of light bouncing off a surface patch: basic reflection phenomenon


12.1 Introduction

image intensity : proportional to scene radiance

scene radiance depends onthe amount of light that falls on a surface

the fraction of the incident light that is reflected

the geometry of light reflection,

i.e. viewing direction and illumination directions

I


12.1 Introduction

image intensity

: incident radiance : bidirectional reflectance function : lens collection : sensor responsivity : sensor gain : sensor offset

bCSfgJI ri

iJrfCSg

b


Joke


12.2 Radiometry

is the measurement of the flow and transfer of radiant energy in terms of both the power emitted from or incident upon an area and the power radiated within a small solid angle about a given direction.

is the measurement of optical radiation.


12.2 Radiometry

irradiance:the amount of light falling on a surfacepower per unit area of radiant energy falling on a surfacemeasured in units of watts per square meter.

radiance: the amount of light emitted from a surfacepower per unit foreshortened area emitted into a unit solid anglemeasured in units of watts per square meter per steradian

radiant intensity: of a point illumination source power per steradianmeasured in units of watts per steradianmay be a function of polar and azimuth angles


12.2 Radiometry

z-axis: along the normal to the surface element

at 0

polar angle: measured from the z-axis

(pointing north)

azimuth angle: measured from x-axis

(pointing east)

dA


12.2 Radiometry

The solid angle subtended by a surface patch is defined by the cone whose vertex is at the point of radiation and whose axis is the line segment going from the point of radiation to the center of the surface patch.


12.2 Radiometry

size of solid angle: area intercepted by the cone on a unit radius sphere centered at the point of radiation

solid angle: measured in steradians

total solid angle about a point in space:

steradians

4


12.2 Radiometry

: surface area

: distance from surface area to point of radiation

: angle the surface normal makes w.r.t. the cone axis

)( 2 Ad

2

cos

d

A

d

A


12.2 Radiometry

surface irradiance :

: area of surface patch

: constant radiant intensity of point

illumination source

)( 2mw

200

200 coscos

d

I

A

dAI

A

)(0 srwI

law of inverse squares:

irradiance varies inversely as square of

distance from the illuminated surface to source

infinitesimal slice on annulus on sphere of

radius r , polar angle , azimuth angle slice subtends solid angle ,

since

d

ddd sin)(*)sin(,,10cos rddrArd


12.2.1 Bidirectional Reflectance Function

The bidirectional reflectance distribution function

is the fraction of incident light emitted in one direction when the surface is illuminated from another direction.

ratio of the scene radiance to the scene irradiance

rf



differential reflectance model:

: polar angle between surface normal and lens center : azimuth angle of the sensor : emitting from : incident to : irradiance of the incident light at the illuminated surface : radiance of the reflected light : ratio of the scene radiance to the scene irradiance

),,,(),(),,,( eeiiriii

iieer fdJdJ

eiiJrJrf



The differential emitted radiance in the direction due to the incident differential irradiance in the direction is equal to the incident differential irradiance times the bidirectional reflectance distribution function .)1)(,,,( srf eeiir

)/( 2 srmw ),( ee

),( ii )/)(,( 2mwdJ ii

i



For many surfaces the dependence of on the azimuth angles and is only a dependence on their difference

except surfaces with oriented microstructure

e.g. mineral called tiger’s eye, iridescent feathers of

some birds

rf

);,(),,,( ieeireeiir ff

ei



An ideal Lambertian surface is one that appears equally bright from all viewing directions and reflects all incident light absorbing none

Lambertian surface:

perfectly diffusing surface with

matte appearance



reflectivity r:

unitless fraction called reflectance factor

white writing paper: r = 0.68 white ceilings or yellow paper: r = 0.6 dark brown paper: r = 0.13


white blotting paper: r = 0.8

dark velvet: r = 0.004



bidirectional reflectance distribution function for Lambertian surface

r

f eeiir ),,,(


r : irradiance, : radiance

ddd sin

EA

IL n

: polar angel, : azimuth angle, : reflectivity

LddA

Id

A

IrE n

sincos2

0

2

02



differential relationship for emitted radiance for Lambertian surface

Lambertian surface: consistent brightness no matter what viewing direction

power radiated into a fixed solid angle: same in any direction

srmwrdJ

dJi

eer 2),(


Example 12.1


Joke


12.2 Photometry

photometry: study of radiant light energy resulting in physical sensation

brightness: attribute of sensation by which observer aware of differences of observed radiant energy

radiometry radiant energy photometry luminous energy radiometry power( radiant flux ) photometry luminous flux


On Internet

Photometry: is the science of measuring visible light in units that are weighted according to the sensitivity of the human eye.


12.2 Photometry

lumen: unit of luminous flux luminous intensity: ( w.r.t. radiance intensity )

luminous flux leaving point source per unit solid anglehas units of lumens per steradian

candela: one lumen per steradian illuminance: ( w.r.t. irradiance )

luminous flux per unit area incident upon a surfacein units of lumens per square meter

one lux: one lumen per square meter foot-candle: one lumen per square foot


12.2 Photometry

one foot =0.3048 meter

1 lux = foot - candles = 10.76 foot - candles

luminance: ( w.r.t. radiance )luminous flux per unit solid angle per unit of projected area

in units of lumens per square meter per steradian

2)3048.0(

1


12.2.3 Torrance-Sparrow Model

: diffuse reflection from Lambertian surface facets

: specular reflection from mirrorlike surface facets

dependent on the view point whereas is not

: reflected light from roughened surface consider surfaces:

rsJ

rdJ

rJ

rdJ

);,(),,,( ieeireeiir ff


Torrance-Sparrow model:

: proportion of specular reflection depending on surface

s=0: diffuse Lambertian surface s=1: specular surface

: wavelength of light

);()1();,;();,;( irdei

rsei

r JssJJ

)10( ss


: unit surface normal : unit positional vector of the light source : unit positional vector of the sensorLN

V

VN

LN

e

i

cos

cos


12.2.4 Lens Collection

lens collection: portion of reflected light coming through lens to film

: distance between the image plane and the lens : distance between the object and the lens : distance between the lens and the image of the object : diameter of the lens : angle between the ray from the object patch to the lens center

f

1r

2r

a

a1

a2



irradiance incident on differential area coming from differential area , having radiance ,

and passing through a lens having

aperture area

: foreshortened area of aperture stop

seen by

: distance from to the aperture

1da idJ2da

42aA

1dacosA

1dacos11 sr



solid angle subtended by aperture stop as seen from :

differential radiant power passing through aperture due to

21

3

21

coscos

s

A

r

A

1da

cos1dadJd i

d1da



radiant power passing through aperture from

irradiance incident to :

(radiant power reaching is )

221

14

2

cos

das

daAdJ

da

ddJ

ir

1da

21

14cos

s

daAdJd

i

2da

2da d



assume , then , thus lens magnification is

hence , therefore

21 ss 21 / ss

fs 2

221

22121 /)/(/ fsssdada

2

4

2

21

221

14 coscos

f

AdJ

f

s

das

daAdJdJ

iir



since

then the lens collection C is given by

4/2aA

42 cos)(4 f

a

dJ

dJC

i

r

2

24

4

cos

f

adJdJ

ir


12.2.5 Image Intensity

The image intensity gray level I associated with some small area of the image plane can then be represented as the integral of all light collected at the given pixel position coming from the observed surface patch, modified by sensor gain g and bias b


12.2.5 Image Intensity

: light wavelength : sensor responsivity to light at wavelength : radiance of observed surface patch : solid angle subtended by the viewing cone of camera for the pixel : distance to the observed patch : power received for the pixel position

bdrJCSgI r

22);,()(),(

)(S

);,( rJr

)();,( 2rJ r

)/( 2 srmwatt


12.3 Photometric Stereo

In photometric stereo there is one camera but K light sources having known intensities and incident vectors to a given surface patch.

In photometric stereo the camera sees the surface patch K times, one time when each light source is activated and the remaining ones are deactivated.

Kii ,...,1

Kvv ,...,1



: observed gray levels produced by the model of Lambertian reflectance

Kff ,...,1

Kkbnvgrif kkk ,...,1, n: surface normal vector of the surface patch having

Lambertian reflectance r: reflectivity of the Lambertian surface reflectance g: sensor gain b: sensor offset



if camera has been photometrically calibrated, g, b known

let and

in matrix form

nrvgi

bff k

k

kk

*

'

'1

*

*1

*

KK v

v

V

f

f

f

rVnf *



if surface normal n known then least-squares solution for reflectivity r:

if K = 3 a solution for unit surface normal n:

)()( '

*'

VnVn

Vnfr

*1

*1

fV

fVn

rVnf *13:

1:

3:*

n

Kf

KV

rVnf *



if K > 3, a least-squares solution:

*'1'

*'1'

)(

)(

fVVV

fVVVn



if g, b unknown camera must be calibrated as follows:

geometric setup with known incident angle of light source to surface normal surfaces of known reflectivities illuminated by known intensity light source



: known intensity of light source for kth trial : known incident direction of light source for kth trial : known unit length surface normal vector : known reflectivity of surface illuminated for kth trial : observed value from the camera

ki

ky

n

kr

kv

Kknvirxbgxbnvigry kkkkkkkkk ,...,1,,



let then unknown gain g and offset b satisfy

nvrix kkkk

1

1

1

2

1

Kx

x

x

Ky

y

y

2

1

b

g

Kknvirxbgxy kkkkkk ,...,1,,



this leads to the least-squares solution for bg,

K

kk

K

kkk

K

kk

K

kk

K

kk

y

yx

Kx

xx

b

g

1

1

1

1

11

2


Joke


12.4 Shape from Shading

nonplanar Lambertian surfaces of constant reflectance factor: appear shaded

this shading: secondary clue to shape of the observed surface

shape from shading: recovers shape of Lambertian surface from image shading



: unit vector of distant point light source direction

assume surface viewed by distant camera so perspective projection approximated by orthographic projection

surface point position : projected to image position

: surface expression

zyx ,, yx,

cba ,,

yxgz ,



unit vector normal to the surface at : yx,

11)()(

1

22 y

gx

g

yg

xg



gray level at , within multiplicative constant yx,

1),(),(

),(),(),(

22

yxqyxp

cyxbqyxapyxI

Where and xgp / ygq /

: reflectance map

1

,22

qp

cbqapqpR

qpR ,



: penalty constant relaxation method: minimizing original error and a sm

oothness term criterion function to be minimized by choice of p, q

22

22

22

),()1,(),(),1(

),()1,(),(),1(

),(),,(),(

crqcrqcrqcrq

crpcrpcrpcrp

crqcrpRcrIcr


Horn Robot Vision Fig 10.19

two orthographic shaded view of the same surface caption


Horn Robot Vision Fig 10.18

a block diagram of Dent de Morcles region in southwestern Switzerland



uniform brightness if planar surfaces since ,

constant surfaces with curvature: surfaces with

, provide information about surface height

first-order Taylor expression for g:

yxg ,

yxq ,

yxp ,

yxp , yxq ,

),()1,(),(),()1,(

),(),1(),(),(),1(

yxqyxgyxgy

gyxgyxg

yxpyxgyxgx

gyxgyxg

1),(),(

),(),(),(

22

yxqyxp

cyxbqyxapyxI



with boundary conditions on , we can solve unknown surface height and partial derivatives

,

yxg ,

yxq , yxp ,


12.4.1 Shape from Focus

possible to recover shape from the shading profile of object edges

basic idea: cameras do not have infinite depth of field

The degree to which edges may be defocused is related to how far the 3D edge is away from the depths at which the edges are sharply in focus.


Joke


12.5 Polarization

illumination source characterized by four factorsdirectionality: relative to surface normal in bidirectional reflectance

intensity: energy coming out from source

spectral distribution: function of wavelength

polarization: time-varying vibration of light energy in certain direction


Examples


12.5 Polarization

polarization: time-varying vibration of the light energy in certain direction

linearly polarized: changes direction by every period

circularly polarized: phase angle difference of ,thus

elliptically polarized phase angle difference of and different amplitude

180

90 wtiwt sincos 90

wtibwta sincos


Mathematical Meaning of Polarization

polarization of light mathematically described by using wave theory


Linearly Polarized


Circularly Polarized


Usefulness of Polarization in Machine Vision

At Brewster’s angle, the parallel polarized light is totally transmitted and the perpendicularly polarized light is partially transmitted and partially reflected.


Usefulness of Polarization in Machine Vision

This effect can be used to remove the specular reflections from the window or metal surfaces by looking through them at Brewster’s angle.


No Filter With Polarizer With Warm Polarizer

http://www.tiffen.com/polarizer_pics.htm


12.5.1 Representation of Light Using the Coherency Matrix

natural light: completely unpolarized

Coherency Matrix: Representation method of polarization


12.5.2 Representation of Light Intensity

The intensity of any light can be represented as a sum of two intensities of two orthogonal polarization components.

S-pol: component polarized perpendicularly to the incidence plane

P-pol: component polarized parallel to the incidence plane


12.6 Fresnel Equation


12.7 Reflection of Polarized Light

ergodic light: time average of the light equivalent to its ensemble average


12.8 A New Bidirectional Reflectance Function


12.9 Image Intensity

image intensity can be written in terms ofillumination parameters

sensor parameters

bidirectional reflectance function


12.10 Related Work

reflectance models: have been used in computer graphics and image analysis


課程網站

http://140.112.31.93 Account: CV2 Password: DCCV Ps. 注意都是大寫

http://140.112.31.93/


Project due Mar. 7

use correlation to do image matching find to minimize dydx,

Ryx

dyydxximbPIXyximaPIX),(

|),,(),,(|


P.S. 1

)()( '

*'

VnVn

Vnfr

)()()()(

)(

)()()(

'

'*

'

*'

'*'*

VnVn

Vnf

VnVn

fVnr

VnVnrfVnrVnf

13:

1:

3:*

n

Kf

KV


P.S. 2

*1

*1

*1

*1

*1*

fV

fVn

fV

fVnr

rnfVrVnfnormalize

13:

3,1:

3:*

n

KKf

KV

computer and robot vision ii chapter 12 illumination presented by: 傅楸善 & 張庭瑄 0963...

Documents

surface area

polar angle

area of surface patch

illuminated surface

point of radiationsolid

surface element

northazimuth angle

steradianstotal solid