gradient vector flow a new external force for snakes

8/8/2019 Gradient Vector Flow a New External Force for Snakes

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Gradient Vector Flow: A New

External Force for Snakes

Joseph 2006/09/13


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Outline

Introduction

Background (Active ContourModel)

Gradient VectorFlow Field GVF Fields and GVF Snakes

Conclusion


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Introduction

Snakes, oractive contours, are used extensively in

computervisionand image processing applications,

particularly to locate object boundaries.

There are two general types ofactive contourmodels inthe literature today:parametric active contours and

geometric active contours .

Inthis paper, we focus on parametric active contours,which synthesize parametric curves withinanimage

domainand allow them to move toward desired features,

usually edges.


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Introduction

The internal and external forces are defined so thatthesnake will conform to an object boundary orotherdesired features withinanimage.

Typically, the curves are drawntoward the edges bypotential forces.

Additional forces, such as pressure forces, togetherwiththe potential forces comprise the external forces.

There are also internal forces designed to hold the curvetogether (elasticity forces) and to keep it from bendingtoo much (bending forces).


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Introduction

There are two key difficulties with active contouralgorithms. First, the initial contourmust, in general, be close to the true

boundary orelse it will likely converge to the wrong result.

Second, the active contours have difficulties progressing intoconcave boundary regions .

This paperdevelops anew external force foractivecontours, largely solving both problems.

This external force, which we call gradientvector flow(GVF), is computed as a diffusion ofthe gradientvectorsofa gray-level orbinary edge map derived from theimage.


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Parametric Snake Model


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A traditional snake is a curve ,

that moves through the spatial domain ofanimage to

minimize the energy functional.

where and are weighting parameters that control the

snake's tensionand rigidity

]1,0[)],(),([)( ! ssysxsX

E F


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The external energy function is derived from the

image so thatittakes onits smallervalues atthe

features ofinterest, such as boundaries.

Giv

en

a

gra

y-level

im

age ,

isa

two-d

ime

nsiona

lGaussian function with standard deviation and is the

gradient operator.

Ifthe image is a line drawing (black on white)

),( yxI

extE

),( yxGW W


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A snake that minimizes Emust satisfy the Eulerequation

This can be viewed as a force balance equation

where and

The internal force discourages stretching and bending while the

external potential force pulls the snake towards the desired

image contour.


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To find a solutionto (6), the snake is made dynamic bytreating x as function oftime tas well as s i.e. .

A solutionto (7) can be found by discretizing the equationandsolving the discrete system iteratively.

),( ts


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Internal Energy

Ifthe curve is represented by n points

,

niyxviii

......1),( !!

1} ii vvds

dv112

2

2 } iii vvvds

vd

!

!

n

i

ernal iiiiiE vvvvv

1

22

int 111 2FE


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External Energy

!

!n

i

external iiyiixE yxyx

1

22

),(),(


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Parametric Snake ModelAn example of the movement of a point, v

i, in an active contour.

The point, vi , is the location of minimum energy due to a large

gradient at that point.


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The snake solutions shownin Figs. 1(a) and 1(b) both

satisfy the Eulerequations (6) fortheirrespective energy

model.

In Fig. 1(a) shows a 64 64-pixel line-drawing ofa U-shaped object (shownin gray) having a boundary

concavity atthe top.

The potential force field where pixel

is shownin Fig. 1(b).

0.1!W

v


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Generalized Force Balance Equations

The choice of can have a profound impact on both

the implementationand the behaviorofa snake.

Broadly speaking, the external forces can be divided into

two classes: static and dynamic. Static forces are those thatare computed from the image

data, and do not change as the snake progresses.

The most general static vector field can be decomposed

into two components: anirrotational (curl-free)

componentand a solenoidal (divergence-free)

component.

Fg

ext

)(


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Gradient Vector Flow Field

We define below anew static external force

field , which we call the gradient vector flow

(GVF) field.

To obtainthe corresponding dynamic snake equation, wereplace the potential force with , yielding

This equationis solved in similar fashionto the traditionalsnakei.e., by discretizationand iterative solution.


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Edge Map

We begin by defining anedge map derived fromthe image having the property thatitis largernear

the image edges.

where 1, 2, 3, or4. First, the gradient ofan edge map has vectors

pointing toward the edges, which are normal to the edgesatthe edges.

Second, these vectors generally have large magnitudes

only inthe immediate vicinity ofthe edges. Third, in homogeneous regions, where is nearly

constant, is nearly zero.

!i

),( yxf),( yxI

f

f

),( yxI


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Gradient Vector Flow

We define the gradientvector flow field to be the vectorfield that minimizes the energyfunctional

When is small, the energy is dominated by sum ofthe squares ofthe partial derivatives ofthe vector field,

yielding a slowly varying field. Onthe otherhand, when is large, the second term

dominates the integrand, and is minimized by setting

.

f

f

fV !


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Using the calculus of variations , it can be shownthatthe

GVF field can be found by solving the following Euler

equations

where is the Laplacian operator

We note thatina homogeneous region [where is

constant], the second term in each equationis zerobecause the gradient of is zero.

),( yxI

2

),( yxf


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Equations (13a) and (13b) can be solved by treating u

and vas functions oftime and solving

The equations are knownas generalized diffusion

equations, and are knownto arise in such diverse fieldsas heat conduction, reactorphysics, and fluid flow.

]),(),([ 22 yxfyxf yx

)],(),,([),,(),,( 2 yxftyxutyxutyxu xt ! Q

)],(),,([),,(),,( 2 yxftyxvtyxvtyxv yt ! Q

]),(),([22

yxfyxf yx


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Where

Any digital image gradient operatorcan be used to calculate

and .

The coefficients , , and , canthen be

computed and fixed forthe entire iterative process.

yf xf

),( yxb ),(2 yxc),(1 yxc


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To set up the iterative solution, letthe indices , , and

correspond to , , and , respectively, and letthe spacing

between pixels be and the time step foreach iteration be .

jx

niy tx( y( t(


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Substituting these approximations into (15) gives ouriterativesolutionto GVF as follows:

where ,

Convergence can be made to be fasteron coarserimagesi.e.,when and are larger.

When is large and the GVF is expected to be a smootherfield,the convergence rate will be slower(since must be kept small).

x( y(Q

t(


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GVF Fields and GVF Snakes

In our first experiment, we computed the GVF field

forthe line drawing of figure using .

First, the GVF field has a much largercapture

range. A second observationis thatthe GVF vectors are

pointing somewhat downward into the top ofthe U-

shape, which should cause anactive contourto

move fartherinto this concave region.


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Convergence to a Concave Region

Fig. 3. (a) Convergence ofa snake using (b) GVF external forces,

and (c) shown close-up withinthe boundary concavity.


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GVF external forcestraditional potential forces


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Convergence ofa snake using

traditional potential forcesConvergence ofa snake using

GVF external forces


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Streamlines of external force fields

Stream lines of particles in (a) a potential force field and (b) a GVF field.


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Snake Initialization


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Snake Initialization


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Gray-level Images

To compute GVF forgray-level images, the edge-map

function must first be calculated.

Two possible choices forthe edge-map are

or .

),(),(1 yxIyxf !

),( yxf

)),(),((),(

2

yxIyxGyxf

! W


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Gray-level Images

(a) A magnetic resonance

image of the left ventrical

of a human heart (short-

axis section).

(b) The edge mapwith .

(c) The computed GVF.

(d) Initial and intermediate

contours (gray curves)

and the final contour

(white curve) of the GVFsnake.


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Conclusion

We have introduced anew external force model forsnakes called gradientvector flow (GVF).

The field is calculated as a diffusion ofthe gradientvectors ofa gray-level orbinary edge map.

We have shownthatitallows for flexible initializationofthe snake and encourages convergence toboundary concavities.

Finally, the GVF framework might be useful in

defining new connections between parametric andgeometric snakes, and might form the basis foranew geometric snake.

gradient vector flow a new external force for snakes

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