ron numerical geometry of images: shape reconstruction ron kimmel geometric image processing lab...
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![Page 1: ron Numerical Geometry of Images: Shape Reconstruction Ron Kimmel Geometric Image Processing Lab Computer Science Department Technion-Israel](https://reader036.vdocuments.mx/reader036/viewer/2022062313/56649d3f5503460f94a184fc/html5/thumbnails/1.jpg)
www.cs.technion.ac.il/~ron
Numerical Geometry of Images:
Shape ReconstructionRon Kimmel
Geometric Image Processing Lab
Computer Science Department Technion-Israel Institute of Technology
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Shape reconstruction techniques
Shape from: Inputs
Shading
Stereo
Auto-stereograms
Photometric stereo
Structured/coded light
IR-pulses (3DV-Systems)
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Stereo
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Shape from Stereo
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Shape from autostereograms
for i = (stripesize+1):xsize+stripesize, for j = 1:ysize, stereo(j,i) = stereo(j,i-floor(stripesize*(1-z(j,i)))); end %for jend %for i
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Image Formation
F. Guichard 93Mondrian world:Lambertian surface patches
lyxNyxyxI
),,(),(),(
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Image formationLambetian
model
V
lN
)cos(
,),(
lNyxI
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Lambertian surfaces
Images formation
.1
1),( ˆfor
,),,(),(
2z
yxIzl
lyxNyxI
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Shape from photometric stereo
Given Lambertian shading model, multiple images each with different light source.
At each point we have 3 unknowns that can be computed from at least 3 images.
Next, the problem is surface integration from its gradient vector field
.1
}1,,{
,),,(),(),(
2z
zzN
lyxNyxyxI
yx
ii
,,, yx zz
},{ yx zzz
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Shape from photometric stereo
Next, the problem is surface reconstruction from
Define the integral measure
For which the Euler-Lagrange is a Poisson equation
or in more compact notations
dxdyqzpzzE yx 22 )()()(
},{ qpz
yxyyxx qpzz
),( yxfz
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Shape from photometric stereo
Efficient numerical algorithm for solving a Poisson equation depends on the boundary conditions.
For periodic b.c. Fourier transform is an option
),( yxfz
22
1
2222
),(ˆ),(
)()()(
)()()()),((
)),((),(ˆ
)),((),(ˆ
vu
vufyxz
zvuzvzu
zzzzyxz
yxfvuf
yxzvuz
yyxxyyxx
F
FFF
FFFF
F
F
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Shape from photometric stereo
Efficient numerical algorithm for solving a Poisson equation depends on the boundary conditions.
For point constraints, see Kimmel-Yavneh Algebraic Multi-Grid approach.
),( yxfz
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Multigrid Poisson solvers
Solve a Poisson equation
),( yxfz
2 2 2
solve ( , )
search for in a coarse scale 2
ˆsolving for ,
we have that our solution is given by
ˆ.
h h h
h h h h
h
h h h
z f x y
r z f
r h
z r
z r
z z z
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Shape from shading
Horn: characteristic strip expansion Bruckstein: equal height contours Rouy-Turin: Minimal cost approach Kimmel-Sethian: Fast Marching
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Summary: Shape reconstruction
Shape from shading Explicit methods (Horn/Bruckstein)
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Open Questions
A working automatic shape from stereo algorithm does not exist (but we are getting there).
www.cs.technion.ac.il/~ron