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Image Processing:4.4. Radiometry andRadiometry and

Shading ModelsShading Models

Aleix M. Martinezaleix@ece.osu.edu

Radiometry

•By understanding how light travels from asource to a surfaces and how this creates abrightness pattern, we will be able toestimate 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

ns

a

I(u,v)

A 3D objectilluminatedwith a lightsource.

Intensity atimage point(u,v)T.

Reconstruction

•Since an image depends on its illumination,it should be possible to recover its 3Dinformation.

Images underdifferentbut knownillumination.

Convex Objects

y

45o x’

y’

S5

S3S2

S4

S1

45o

45o

60o

Objectx

zCamera

I1

I2

I3I4I5

Hemisphere of Direction

2

•Questions:–how “bright” will a surfaces be? –what is “brightness”?•measuring light,•interactions between light and surfaces.

•Core idea: think the light arriving at asurface around any point defines ahemisphere of directions.•Simplest problems can be dealt with by

reasoning about this hemisphere.

Simulating the hemisphere

Construct a 3D sphere tosimulate a large number ofillumination directions.

Eliminate the shadows causedby the known illumination3D reconstruction of the face.

Generate a new viewwith a new illumination

•By analogy with angle(in radians), the solidangle subtended by aregion at a point is thearea projected on aunit sphere centered atthat point.•The solid angle

subtended by a patcharea dA is given by:

.sin

,cos

2

dddr

dAd

Solid Angle Radiance

•The power (amount of energy per unit time)traveling at some point in a specifieddirection, per unit area perpendicular to thedirection of travel, per unit solid angle.

•Units: watts per square meter per steradian(wm-2sr-1).

,,PL

Irradiance

•Incident power per unit area notforeshortened.•A surface experiencing radiance L(x)

coming in from dexperiences irradiance:

.cos,, dL P

BRDF•The BRDF (Bidirectional Reflectance

Distribution Function) is the ratio betweenthe incoming radiance and the outgoingirradiance at a point P:

•This is given by the properties of the objectmaterial.

.

cos,,,,

,,,dwL

L

iiii

oooiioobd

PP

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Original image

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

•Obtaining the BRDF of objects like a face isstill a challenging task. At MERL,researchers have developed a special systemto obtain this with high accuracy.

BRDF on a 3D surface Smoothness

In many instances, it is convenient to smooth the resultsof the BRDF to create a more realistic image.

Eliminateunwantedareas

Interpolate for smoothness Radiosity

•The total power leaving a point on a surfaceper unit area on the surface (Wm-2).

•Note that this is independent of thedirection.

.cos,,

dLB o PP

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Radiosity and Constant Radiance

•Radiosity of a surface whose radiance is independent ofangle (e.g. that cotton cloth):

.

sincos

cos

cos,,

2

0

2

0

xL

ddxL

dxL

dxLxB

o

o

o

o

Albedo•A common, reasonable assumption is that

the light leaving a surface is independent ofthe exit angle.•Directional Hemisphere Reflectance: the

fraction of incident irradiance in a givendirection that is reflected back, whatever thedirection of reflection.

.cos,,,

cos,,

cos,,,

00 ooiidb

iiii

ooooo

iiph

dw

dwL

dwL

P

P

•The second most common assumption isthat this directional hemisphere reflectancefunction does not depend on the direction ofthe illumination (i.e., most directionsproduce the same illumination effect).•This is reasonable if the object is convex.

.cos

cos,,, 00

oo

ooiibdd

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 areaon the radiating surface.

• similar to radiosity: asource can have both– radiosity, because it reflects,–exitance, because it emits.

•General idea:

•But what aspects of theincoming radiance will wemodel?

B(x)E (x)radiosity due toincoming radiance

d

Radiosity due to a point source

d

2

Radiosity due to a point source

•Radiosity is

B xLo xd xLi x, cosid

d xLi x, cosidD

d xsolid angle Exitance term cosi

d xcosi

r x2 Exitance term and some constants

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Standard nearby point source model

•N is the surface normal•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

2xr

xSxNxd

Standard distant point source model

•Issue: nearby point source gets bigger if one getscloser–the sun doesn’t for any reasonable binding of closer

•Assume that all points in the model are close toeach other with respect to the distance to thesource. 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:

xSxNx dd

Shadows cast by a point source Photometric Stereo

i

i

i

yx

yxyxk

yxkByxI

Vg

SN

),(

),(),(

),(),(

Dealing with shadows

I12(x, y)

I22(x, y)

..In

2(x, y)

I1(x, y) 0 .. 00 I2(x, y) .. .... .. .. 00 .. 0 In(x, y)

V1T

V2

T

..Vn

T

g(x, y)

Known Known Known Unknown

Recovering normal andreflectance

•Given sufficient sources, we can solve the previousequation (most likely need a least squares solution) for

g(x, y)•Recall that N(x, y) is the unit normal.•This means that x,y) is the magnitude of g(x, y).•This yields a check–If the magnitude of g(x, y) is greater than 1, there’s a

problem•And N(x, y) = g(x, y) / x,y)

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Example Recovered reflectance

Recovered normal fieldSurface recovered by integration

•Consider an n-pixel image to be a point in an n-dimensionalspace, I Rn.

•Each pixel value is a coordinate of x.•This is the same we saw in stereo vision, when we compute

point correspondences. We now want to find a representationthat is invariant to the illumination.

x1

x2

xnI n

I2

I1Illumination Sphere

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Illumination Cone

N-dimensionalImage Space

Single light source imageslie on cone boundary

2-light sourceimage

x1

x2

xn

Illumination Cone

Original (Training) Images

x,y fx(x,y) fy(x,y)albedo (surface normals)

Surface. f(x,y)(albedo textured

mapped on surface).

3D linear subspace

[Georghiades,Belhumeur,

Kriegman 01]Single Light Source Face Movie

Eyes for Relighting

From an image of an eyewe can extract theinformation of thedirection and type ofambient illumination.