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Image Processing:4.4. Radiometry andRadiometry and
Shading ModelsShading Models
Aleix M. [email protected]
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
3
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
4
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
5
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)
6
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
7
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.