Camera: optical system

d2 1

thin lens

small angles:

Y

Z

11

Y

21

curvature radius

22

Y

Y

Z

incident light beam

deviated beam

deviation angle ? ’’

lens refraction index: n

)11

()1(21

Yn

n

)'sin(

)sin(

' 1

1

1

1

n

)'sin(

)''sin(

'

''

2

2

2

2

Thin lens rules

a) Y=0 = 0

)11

)(1(

1

21

n

f

f

Y

parallel rays converge onto a focal plane

b) f = Y

beams through lens center: undeviated

independent of y

r

f

Y

h

Where do all rays starting from a scene point P converge ?

Z

r

Y

h

Z

f

fr

Y

h

frZ

111Fresnel law

P

Obs. For Z ∞, r f

O

p?

d

f

a

Z

if d ≠ r …

focussed image:blurring circle) <image resolution

depth of field: range [Z1, Z2] where image is focussed

image plane P

p

O

r (blurring circle)=a (d-r)/r

image of a point = blurring circle

the image of a point P belongs to the line (P,O)

p

P

O

p = image of P = image plane ∩ line(O,P)

interpretation line of p: line(O,p) =locus of the scene points projecting onto image point p

image plane

r fHp: Z >> a

Until here: where goes light ?

But: how much light does reach an image point?

Image Formation: Reflectance Map

Simplified model:• light originates at

a source• light is reflected

by an object• light collected by

camera lens and focused to image

[Hemant D. Tagare, CV course notes, adapted from B.Horn]

The Reflectance Map

dA at P receives flux of d watt:

Irradiance at P:

(watt/meter2,

spatial density of flux at P)

The Reflectance Map ctd.

• d infinitesimal solid angle, centered along incident direction

• infinitesimal flux d2 passes through it, incident on dA

,: zenith,azimuth• dA cos : fore-

shortened area dAf

Radiance of incident flux:

Units: watt per m2 per steradian

The Reflectance Map ctd.

• Object is a point (no area)

• flux d incident on it from solid angle d

Radiant intensity of flux:

units: watt per steradian

Computation of Solid Angle

Solid angle d: • solid angle centered

around direction ,

• spherical geometry: dA = r2 sin dd

Computation of Flux

Flux : • irradiance E and

radiance L are derivatives of flux flux comp. as integral

• L(P,,): radiance along , at any point P of surface

• E(P): irradiance at P: net flux received

by object

and,

Generalization: Reflected Light

• so far: radiance and radiant intensity defined in terms of incident light

• definitions also apply to reflected / emitted light flux d is assumed to have reverse direction (leaves surface)

• radiance of reflected light (dA through d) :

image intensity: proportional to irradiance E

42

2

cos4

),(f

ZLE rrr

where is the radiance reflected towards the lens ),( rrrL

Zf

p

P

O Z

Y

X

c

y

x

Z

Xfx

Z

Yfy

perspective projection

f

-nonlinear-not shape-preserving-not length-ratio preserving

•Point [x,y]T expanded to [u,v,w]T

•Any two sets of points [u1,v1,w1]T and [u2,v2,w2]T

represent the same point if one is multiple of the other

•[u,v,w]T [x,y] with x=u/w, and y=v/w

•[u,v,0]T is the point at the infinite along direction (u,v)

• In 2D: add a third coordinate, w

Homogeneous coordinates

Transformations

translation by vector [dx,dy]T

scaling (by different factors in x and y)

rotation by angle

Homogeneous coordinates

• In 3D: add a fourth coordinate, t

•Point [X,Y,Z]T expanded to [x,y,z,t]T

•Any two sets of points [x1,y1,z1,t1]T and [x2,y2,z2,t2]T

represent the same point if one is multiple of the other

•[x,y,z,t]T [X,Y,Z] with X=x/t, Y=y/t, and Z=z/t

•[x,y,z,0]T is the point at the infinite along direction (x,y,z)

Transformations

scaling

translation

rotation

Obs: rotation matrix is an orthogonal matrix

i.e.: R-1 = RT

10100

000

000

Z

Y

X

f

f

w

v

u

with

w

ux

w

vy

Scene->Image mapping: perspective transformation

With “ad hoc” reference frames, for both image and scene

Let us recall them

O Z

Y

X

c

y

x

f

scene reference- centered on lens center- Z-axis orthogonal to image plane- X- and Y-axes opposite to image x- and y-axes

image reference- centered on principal point- x- and y-axes parallel to the sensor rows and columns- Euclidean reference