ron numerical geometry of images: shape reconstruction ron kimmel geometric image processing lab...
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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
Shape reconstruction techniques
Shape from: Inputs
Shading
Stereo
Auto-stereograms
Photometric stereo
Structured/coded light
IR-pulses (3DV-Systems)
Stereo
Shape from Stereo
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
Image Formation
F. Guichard 93Mondrian world:Lambertian surface patches
lyxNyxyxI
),,(),(),(
Image formationLambetian
model
V
lN
)cos(
,),(
lNyxI
Lambertian surfaces
Images formation
.1
1),( ˆfor
,),,(),(
2z
yxIzl
lyxNyxI
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
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
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
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
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
Shape from shading
Horn: characteristic strip expansion Bruckstein: equal height contours Rouy-Turin: Minimal cost approach Kimmel-Sethian: Fast Marching
Summary: Shape reconstruction
Shape from shading Explicit methods (Horn/Bruckstein)
Open Questions
A working automatic shape from stereo algorithm does not exist (but we are getting there).
www.cs.technion.ac.il/~ron