Image Processing 11. Radiometry and Shading Models Aleix M. - - PDF document

1 Image Processing

• 11. Radiometry and

Shading Models

Aleix M. Martinez aleix@ece.osu.edu

Radiometry

• By understanding how light travels from a

source to a surfaces and how this creates a brightness pattern, we will be able to estimate additional data from an image.

• Our goal is to understand the principals and

how these can be used.

• We will used them to build what’s call

shading models.

Illumination

^ n ^ s a

I(u,v) A 3D object illuminated with a light source. Intensity at image point (u,v)T.

2 Hemisphere of Direction Hemisphere of Direction

• Questions:

– how “bright” will a surfaces be (luminance)? – what is “brightness”?

• measuring light,
• interactions between light and surfaces.
• Core idea: think the light arriving at a

surface around any point defines a hemisphere of directions.

• Simplest problems can be dealt with by

reasoning about this hemisphere.

3

• By analogy with angle

(in radians), the solid angle subtended by a region at a point is the area projected on a unit sphere centered at that point.

• The solid angle

subtended by a patch area dA is given by:

( )( )

ï î ï í ì = = . sin , cos

2

f J J w J w d d d r dA d

Solid Angle Radiance

• The power (amount of energy per unit time)

traveling at some point in a specified direction, per unit area perpendicular to the direction of travel, per unit solid angle.

• Units: watts per square meter per steradian

(wm-2sr-1).

( )

f q, , P L

Irradiance

• Incident power per unit area not

foreshortened.

• A surface experiencing radiance L(x,q,f)

coming in from dw experiences irradiance:

( )

. cos , , w q f q d L P

4 BRDF

• The BRDF (Bidirectional Reflectance

Distribution Function) is the ratio between the incoming radiance and the outgoing irradiance at a point P:

• This is given by the properties of the object

material.

( ) ( ) ( )

. cos , , , , , , , dw L L

i i i i

• i

i

• bd

q f q f q f q f q r P P =

BRDF

Original image Change the BRDF to make the skin look like tanned, with added facial hair, or darker.

5 Radiosity

• The total power leaving a point on a surface

per unit area on the surface (Wm-2).

• Note that this is independent of the

direction.

( ) ( )

. cos , ,

ò

W

= w q f q d L B

• P

P

Radiosity and Constant Radiance

• Radiosity of a surface whose radiance is independent of

angle (e.g. that cotton cloth):

( ) ( ) ( ) ( ) ( ).

sin cos cos cos , ,

2 2

x L d d x L d x L d x L x B

• p

J j J J w J w J j J

p p

= = = =

ò ò ò ò

W W

Albedo

• A common, reasonable assumption is that

the light leaving a surface is independent of the exit angle.

• Directional Hemisphere Reflectance: the

fraction of incident irradiance in a given direction that is reflected back, whatever the direction of reflection.

( ) ( ) ( ) ( )

. cos , , , cos , , cos , , ,

• i

i db i i i i

• i

i ph

dw dw L dw L q f q f q r q f q q f q f q r

ò ò

W W

= = P P

6

• The second most common assumption is

that this directional hemisphere reflectance function does not depend on the direction of the illumination (i.e., most directions produce the same illumination effect).

• This is reasonable if the object is convex.

( )

. cos cos , , , rp q r q f q f q r r = = =

ò ò

W W

• i

i bd d

dw dw

ALBEDO

Sources and shading

• How bright (or what color)

are objects?

• One more definition:

Exitance of a source is

– the internally generated power radiated per unit area

• n the radiating surface.
• similar to radiosity: a

source can have both

– radiosity, because it reflects, – exitance, because it emits.

• General idea:
• But what aspects of the

incoming radiance will we model?

B(x) = E(x) + radiosity due to incoming radiance ì í î ü ý þ dw

W

ò

Radiosity due to a point source

p e d æ è ö ø

2

7 Radiosity due to a point source

• Radiosity is

B x

( )= pLo x ( )

= rd x

( )

Li x,w

( )cosqidw

W

ò

= rd x

( )

Li x,w

( )cosqidw

D

ò

» rd x

( ) solid angle ( ) Exitance term ( )cosqi

= rd x

( )cosqi

r x

( )

2

Exitance term and some constants

( )

Standard nearby point source model

• N is the surface normal
• r is diffuse albedo
• S is source vector - a vector from x to the

source, whose length is the intensity term

– works because a dot-product is basically a cosine

( ) ( ) ( ) ( )

÷ ÷ ø ö ç ç è æ ×

2

x r x S x N x

d

r

Standard distant point source model

• Issue: nearby point source gets bigger if one gets

closer

– the sun doesn’t for any reasonable binding of closer

• Assume that all points in the model are close to

each other with respect to the distance to the

• source. Then the source vector doesn’t vary much,

and the distance doesn’t vary much either, and we can roll the constants together to get:

( ) ( ) ( ) ( )

x S x N x

d d

× r

8 Shadows cast by a point source