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

Interpolation Region

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

Lawrence Berkeley National Laboratory

& UNR Computer Vision Laboratory

Results

Special Thanks to

LBL Vision Group LBL Vision Group (Dr. Bahram Parvin lead)(Dr. Bahram Parvin lead)

UNR Computer Vision Laboratory UNR Computer Vision Laboratory (Assoc. Prof. George Bebis lead)(Assoc. Prof. George Bebis lead)

National Science Foundation (NFS)National Science Foundation (NFS)