photometric stereo · 2018. 4. 7. · photometric stereo -the math •how do we recover g if the...
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Photometric stereo
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Radiance
• Pixels measure radiance
This pixelMeasures radiance
along this ray
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Where do the rays come from?
• Rays from the light source “reflect” off a surface and reach camera• Reflection:
Surface absorbs light energy and radiates it back
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Light rays interacting with a surface• Light of radiance !" comes from light source at an
incoming direction #"• It sends out a ray of radiance !$ in the outgoing
direction #$• How does !$ relate to !" ?
#" #$
• N is surface normal• L is direction of light, making #"
with normal• V is viewing direction, making #$
with normal
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Light rays interacting with a surface
!" !#
• N is surface normal• L is direction of light, making !"
with normal• V is viewing direction, making !#
with normal
$# = & !", !# $" cos !"Output radiance along V
Bi-directional reflectance function (BRDF)
Incoming irradiance along L
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Light rays interacting with a surface
• Special case 1: Perfect mirror• ! "#, "% = 0 unless "# = "%
• Special case 2: Matte surface• ! "#, "% = !' (constant)
"# "%(% = ! "#, "% (# cos "#
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Special case 1: Perfect mirror
• ! "#, "% = 0 unless "# = "%• Also called “Specular surfaces”• Reflects light in a single, particular direction
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Special case 2: Matte surface
• ! "#, "% = !&• Also called “Lambertian surfaces”• Reflected light is independent of viewing direction
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Lambertian surfaces
• For a lambertian surface:
• ! is called albedo• Think of this as paint• High albedo: white colored surface• Low albedo: black surface• Varies from point to point
Lr = ⇢Li cos ✓i
) Lr = ⇢LiL ·N"# "$
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Lambertian surfaces
• Assume the light is directional: all rays from light source are parallel• Equivalent to a light source
infinitely far away
• All pixels get light from the same direction L and of the same intensity Li
!" !#
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Lambertian surfaces
I(x, y) = ⇢(x, y)LiL ·N(x, y)
Reflectance image
Shading image
Intrinsic Image Decomposition
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Reconstructing Lambertiansurfaces
• Equation is a constraint on albedo and normals• Can we solve for albedo and normals?
I(x, y) = ⇢(x, y)LiL ·N(x, y)
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Solution 1: Recovery from a single image• Step 1: Intrinsic image
decomposition• Reflectance image ! ", $• Shading image• Decomposition relies on priors on
reflectance image• What kind of priors?• Reflectance image captures the
“paint” on an object surface• Surfaces tend to be of uniform color
with sharp edges when color changes
LiL ·N(x, y)
Images from Barron et al, TPAMI 13
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Solution 1: Recovery from a single image
• Step 2: Decompose shading image into illumination and normals
• Called Shape-From-Shading• Relies on priors on shape: shapes are smooth
LiL ·N(x, y)
Far
Near
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Solution 2: Recovery from multiple images
• Represents an equation in the albedo and normals• Multiple images give constraints on albedo and normals• Called Photometric Stereo
I(x, y) = ⇢(x, y)LiL ·N(x, y)
Image credit: Wikipedia
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Multiple Images: Photometric Stereo
N
L1L2
V
L3
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Photometric stereo - the math
• Consider single pixel• Assume !" = 1
• Write• Gis a 3-vector• Norm of G = '• Direction of G = N
I(x, y) = ⇢(x, y)LiL ·N(x, y)
I = ⇢L ·N
I = ⇢NTL
G = ⇢N
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Photometric stereo - the math
• Consider single pixel• Assume !" = 1
• Write• Gis a 3-vector• Norm of G = '• Direction of G = N
I = ⇢NTLG = ⇢N
I = GTL = LTG
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Photometric stereo - the math
• Multiple images with different light sources but same viewing direction?
I = LTG
I1 = LT1 G
I2 = LT2 G
...
Ik = LTkG
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Photometric stereo - the math
• Assume lighting directions are known• Each is a linear equation in G• Stack everything up into a massive linear system of
equations!
I1 = LT1 G
I2 = LT2 G
...
Ik = LTkG
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Photometric stereo - the math
I1 = LT1 G
I2 = LT2 G
...
Ik = LTkG
I = LTG
k x 1 vector of intensities
k x 3 matrix of lighting directions
3x1 vector of unknowns
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Photometric stereo - the math
• What is the minimum value of k to allow recovery of G?• How do we recover G if the problem is
overconstrained?
I = LTGk x 1 k x 3 3 x 1
G = L�T I
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Photometric stereo - the math
• How do we recover G if the problem is overconstrained? • More than 3 lights: more than 3 images
• Least squares
• Solved using normal equations
minG
kI� LTGk2
G = (LLT )�1LI
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Normal equations
• Take derivative with respect to G and set to 0
kI� LTGk2 = IT I+GTLLTG� 2GTLI
2LLTG� 2LI = 0
) G = (LLT )�1LI
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Estimating normals and albedo from G• Recall that G = ⇢N
kGk = ⇢
G
kGk = N
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Multiple pixels
• We’ve looked at a single pixel till now• How do we handle multiple pixels?• Essentially independent equations!
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Multiple pixels: matrix form
• Note that all pixels share the same set of lights
I(1) = LTG(1)
I(2) = LTG(2)
...
I(n) = LTG(n)
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Multiple pixels: matrix form
• Can stack these into columns of a matrix
⇥I(1) I(2) · · · I(n)
⇤= LT
⇥G(1) G(2) · · · G(n)
⇤
I(1) = LTG(1)
I(2) = LTG(2)
...
I(n) = LTG(n)
I = LTG
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Multiple pixels: matrix form
I = LTG
I LTG
=#lights
#pixels #pixels
#lights
3
3
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Estimating depth from normals
• So we got surface normals, can we get depth?• Yes, given boundary conditions• Normals provide information about the derivative
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Brief detour: Orthographic projection• Perspective projection• ! = #
$ , & ='$
• If all points have similar depth• ( ≈ (*• ! ≈ #
$+, & ≈ '
$+• ! ≈ ,-, & ≈ ,.
• A scaled version of orthographic projection• ! = -, & = .
Perspective
Scaled orthographic
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Depth Map from Normal Map
• We now have a surface normal, but how do we get depth?
32
V1V2
N
Assume a smooth surface
Get a similar equation for V2• Each normal gives us two linear constraints on z• compute z values by solving a matrix equation
(cx, cy, Zx,y)
(c(x+ 1), cy, Zx+1,y)
(cx, c(y + 1), Zx,y+1)
V1 = (c(x+ 1), cy, Zx+1,y)� (cx, cy, Zx,y)
= (c, 0, Zx+1,y � Zx,y)
0 = N · V1
= (nx, ny, nz) · (c, 0, Zx+1,y � Zx,y)
= cnx + nz(Zx+1,y � Zx,y)
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Determining Light Directions
33
• Trick: Place a mirror ball in the scene.
• The location of the highlight is determined by the light source direction.
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• For a perfect mirror, the light is reflected across N:Determining Light Directions
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= - 2
So the light source directionis given by:
Determining Light Directions
35
L
N
R||||= " ⋅ $ "= $ − " ⋅ $ "= $ − " ⋅ $ "
= $ − 2 $ − " ⋅ $ "= 2 " ⋅ $ " − $
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Determining Light Directions• Assume orthographic projection• Viewing direction R = [0,0,-1]• Normal?
Z=1
("#, %#)
("', %')
("#, %#, (#)
("', %', (')
(# and (' are unknown, but:
"# − "' * + %# − %' *+ (# − (' * = -*
(# − (' can be computed
"# − "', %# − %', (# − (' is the normal
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Photometric Stereo
Input(1 of 12)
Normals (RGB colormap)
Normals (vectors) Shaded 3Drendering
Textured 3Drendering
What results can you get?
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Results
38
from Athos Georghiades
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Results
Input(1 of 12)
Normals (RGB colormap)
Normals (vectors) Shaded 3Drendering
Textured 3Drendering
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Questions?
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