aula 09. deformable contours

8/13/2019 Aula 09. Deformable Contours

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Deformable Contours

Raul Queiroz FeitosaGilson A. O. P. Costa


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Content

Introduction

The Energy Functional

A Greedy Algorithm

11/14/2013 Deformable Contours 3


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Introduction

The problem consists of fitting a curve of arbitraryshape to a set of image edge points.

Thesnake (active contour)is a kind of elastic

band of arbitrary shape that is attracted towards the

target contour.

An energy functionalmodels the forces acting uponthe snake, searching the condition of minimum

energy.


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Introduction


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Content

Introduction


A Greedy Algorithm



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Elements of Energy Functional

Continuity Term

Forces the edge points to be equally spaced.

For a chain ofN adjacent image points p1,, pN

Econt= (d - | pi- pi-1|)2

where dis the average distance between pairs (pi

, pi-1

).


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Edge Attraction Term

Moves the contour toward edges, i.e. where the

magnitude of the gradient is high. Thus

Eimage= - | I|

where I is the image and Ithe magnitude of gradient.


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Smoothness Term

Avoids oscillations of the deformable contour

Penalizes high contour curvatures

Ecurv= | (pi+1- pi) - (pi- pi-1) |2

Ecurv= | pi-1-2pi+ pi+1|2


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Content

Introduction


A Greedy Algorithm



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A Greedy Algorithm

Problem Statement

Starting from p1,,pN find the deformable contour p1,,pNwhich fits the target image contour best, by minimizing the

energy functional

with i,i,i 0 andEcont,EcurvandEimagedefined as

before.

N

i

imageicurviconti EEE1


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A Greedy Algorithm

Steps1. Minimization:compare the energy functional values at each

location in a (typically small) neighborhood around each edge.

2. Corner Elimination:If a curvature maximum is found at point

pj, jis set to zero. This will allow that corner to exist.3. Normalization:the termsEcontandEcurvare divided by the

largest value in the neighborhood. ForEimagethe gradient is to benormalized as

whereM and m are the maximum and minimum of | I|

over the neighborhood, respectively.

mM

mI


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A Greedy Algorithm

Letfbe the minimum fraction of snake points that must move in eachiteration before convergence, and V(p) a small neighborhood of pointp. In the beginning pi=piandd=d(used inEcont).

While fraction greater thanfof the snake points move in an iteration:

1. For each i=1,,N, find the location of V(pi) for which the functional isminimum, and move the snake point pi to that location,

2. For each i=1,...,N estimate the curvature kof the snake at pias

k=| pi-1-2pi+ pi+1|2

and look for local maxima. Set i

=0 for all pj

at which the curvature has alocal maximum and exceeds a user-defined minimum value;

3. Update the value of the average distance, d.


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Comments

1. i,i,imay be assigned all to 1, or i=i=1 and i=1.2;

Set high if there is a deceptive Image Gradient

Set high if smooth edged Feature, low if sharp edges

Set high if contrast between Background and Feature is low

2. To remove corners formed too far away from image

edges impose a further condition:

Point pjis a corner if and only if the curvature is locally

maximum at pjand the norm of the intensity gradient at pjissufficiently large.


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Exercise

Download the Demoof the Greedy Algorithm shown in

these slides, unzip the files, read the help of function

snake_demo, and experiment with the image of a Brain CT.
http://localhost/var/www/apps/conversion/tmp/scratch_7/Snake.ziphttp://localhost/var/www/apps/conversion/tmp/scratch_7/Snake.zip


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aula 09. deformable contours

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