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Contour detection using Hough Transform

Contour detection using Active Snakes Contours

Contour detection using Geometric Active Contours (GAC)

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Hough TransformDefinition:

Extracts geometric primitives from raster images.

Identifies lines in an image, positions of arbitrary shapes like circle, ellipse

Importance of Hough Transform 3

Basic ideaEach white point if considered in isolation could lie on an infinite number of straight lines

In the Hough transform each point votes for every line it could be on

The lines with the most votes win

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Representation of linesAny line can be represented by two numbers

Here we will represent the yellow line by (w

In other words we define it using- a line from an agreed origin- of length w- at angle to the horizontal

sin(cos( yxw

Algorithm:Create and w for all possible linesCreate an array A indexed by and w for each point (x,y)for each angle

w = x*cos()+ y*sin()A[,w] = A[,w]+1end

end

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7 Two points representation

Single point representation

Representation of multiple lines

Active Snake ContoursDefinition: an energy-minimizing spline

that represents an object boundary or some other salient image feature as a parametric curve.

It was first introduced in 1987 by Kass et al

The contour is defined in the (x, y) plane of an image as a parametric curve

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ModelingContour is said to possess an energy (Esnake) which is defined as

the sum of the three energy terms.

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Elastic energy Bending energy

21 ( ) | |2elastic sE s v ds

s

( )sdv svds

21 ( ) | |2bending sss

E s v ds

2 2int

1 | | | | )2elastic bending s ss

s

E E E v v ds

External energy

It is derived from the image.

Define a function Eimage(x , y) so that it takes on its smaller values at the features of interest, such as boundaries.

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( ( ))ext images

E E v s ds

Image forces• The image forces E image are derived from the image data

over which the snake lies• Three important features the snake can be attracted to are

line, edge and termination functions. The total image energy can be expressed as a weighted combination of the three.

The simplest useful image functional is the image intensity Eline =I (x,y)

E edge = square of intensity variation E term =curvature of lines

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termtermedgeedgelinelineimage EEEE

Example

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MOTION TRACKING

Geometric Active Contours

Level set method is used instead of parameters to trace multiple objects more specifically .

The level set method is a numerical technique for tracking interfaces and shapes.

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Level set methodAdvantage: perform numerical computations Make it very easy to follow shapes that change topology.

the level set method represents a  closed curve, Γ using an auxiliary function ф, called the level set function.

 Γ is represented as the zero level set of ф by Γ={(x , y)|ф(x , y)=0}

If the curve Γ moves in the normal direction with a speed v, then the level set function  satisfies the 

level set equation: ∂ф /∂t=v| ф|

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Approach: Curve deformation velocity, v regularity of the curveshrinks or expands the curve towards the boundary.

Given a 2-D scalar function, ф(x , t) geometric active contour formula is:

∂ф /∂t=c(к+Ѵ0) ф

the use of curvature к has the effect of smoothing the contour, while

Ѵ0 has the effect of shrinking or expanding contour at a constant speed.

c(к+Ѵ0) determines the overall evolution speed of level sets of ф(x , t)

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Level set operator constant

where, c=c(x)= 1 / [1+| |G σ (x) * I(x)] here, x is any location in the image, I(x) and

G σ (x) is a 2-D Gaussian function with standard deviation σ - filtering noise

is the gradient operator- measure of intensity variation

* is the 2-D image convolution operator

Advantage:Several objects can be detected simultaneously without previous

knowledge of their exact number in the scene

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level set function determining the shape of the object

x − y plane

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Initial contour

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Detecting multiple objects using geometric contours

amoeba

SummaryHough transform – cannot detect shapes other

than lines and circles

Active Snakes Contours – selection of parameters - multiple objects

Geometric Active Contours – overcomes above disadvantages using level

sets - detecting multiple objects

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Thank You…

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