summer project presentation presented by:mehmet eser advisors : dr. bahram parvin associate prof....
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Summer Project Presentation
Presented by:Mehmet Eser
Advisors :
Dr. Bahram Parvin
Associate Prof. George Bebis
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Introduction
What is morphing ? In what areas is morphing used ?
What methods are used for morphing for solid shapes?
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
What are Solid Shapes?
A slice from a brain MRI scan
Extracted & Rendered Isosurface
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Problem Definition
Interpolation of solid shapes
Let S be a deformable closed surface such that a family of evolved surfaces with initial conditions at
Construct intermediate solid shapes satisfying smoothness and continuity in time
]1,0[),( tS
10 )1(&)0( SSSS
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Approach to The Problem
Defining the intermediate interpolated shapes implicitly:
such that The givens of the problem
tSf t )( }),,(|),,{( tzyxfzyxSt
0)(1)1(
0)(0)0(
1
0
SfS
SfS
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Regularization Method
A numerical solution method Applied to the ill-posed problems The original problem is converted into a well-
posed problem by satisfying some smoothness constraint.
A smoothing parameter which controls the trade-off between an error term and the the amount of smoothing (regularization)
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Gradients can be helpful?
t=0.2
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Approach to The Problem
Gradients can be used for finding a unique solution to the problem
Disadvantages of this approach
Global average may be small But locally gradient of f may change sharply (not
good for a smooth interpolation of curves)
2||min f
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Purposed Method
Minimization of the supremum of the For minimization of the supremum of the
gradients of the functions sup can be written as follows (in series):
1,2,3.... N , ) dx |f Ñ| ( (f)H 1/2N2NN
|f|
|f| R
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Purposed Method
The minimization of this function can be achieved by using the Euler equation
The result of the min of is the following
(f)HN
)N (if
0)ffffffff2(f
ffffff
zxxzzyyzxyyx
zz2
zyy2
yxx2
x
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Implementation
Distance Field Transforms Finding an approximation to the problem
with Distance Field Transform.
Employing the regularization term Generation of the Morphing
S(1)&S(0)
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Distance Transformation
Distance Transformations
Obtained in time for 3D
D(x,y,z)
)2( NO
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
An example to Distance Transform
Original Image Distance Image
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
DT’s of a Cube and a Sphere
A slice of a distance transformed cube
A slice of a distance transformed sphere
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Signed Distance Transform
Calculation of signed distance transform Take negative of the distance value if the pixel is
inside the object Take positive of the distance value if the pixel is
outside the object Morphing region is defined as
}0Distance SignedDistance Signed | )z,y,{(x, R finalinitial
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Interpolating Surfaces
R
V1
V0
C1 Vi
V0 A B
P Q S
)anceSignedDiststance/(SignedDianceSignedDist
),,(
|)||/(||
and
101
i
izyxT
BSAS|AS)f(V
|BS| |QS||AS| |PS|
i
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Why Distance Field ?
A smooth and natural interpolation of surfaces
Can be carried out at any desired resolution A good initial seed for the iteration with ILE
PDE ‘s can be calculated finite difference formulas
)ffffffff(f
ffffffFτ
zxxzzyyzxyyx
zzzyyyxxxv
2
)( 222
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Numerical Solution to ILE
Get the interpolated surfaces Iterate using regularization term-ILE
v iteration number step size F interpolated volume
(1) 1 )τ(FFF vvv
Lawrence Berkeley National Laboratory
& UNR Computer Vision Laboratory
Iteration
1.Initialize F with boundary conditions
2.Initialize R with the approximated morphing 3.Update all points inside R with equation (1)4.Compute 5.Repeat 3 & 4 till the local minimum of sup|F| is
reached. 6.Obtain morphed volumes
S(t) = {(x,y,z,) | F(x,y,z) = t }
1)(&0)( 10 SFSF
||sup F
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