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## PowerPoint Slideshow about 'Active Contours - SNAKES' - HarrisCezar

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Not so much for others

- Too many spurious edges
- Linking edges belonging to the same object is difficult

Active Contours or Snakes

- Results from the work of Kass et.al. in 1987
- Snakes are shapes or curves that transform themselves to take the shape of an object in an image.
- Snakes are energy minimizing splines

Snake Behavior

- An initial curve/snake is defined near the object
- The curve is iteratively refined to form the least energy contour
- Snakes operate locally, thus one must provide an initial position for them

Snake Representation

- A snake is a contour in the image plane, defined by a set of ncontrol points.

vi = (xi, yi), i = 0, .... n-1

Snake Energy

The snake - v(s) = [x(s), y(s)], is defined as a function of a real variable s ϵ [0,1]

The energy of the snake is given by -

Esnake = 0∫1 (Eint + Eext)ds

Internal Energy

- Internal energy describes the manner in which the snake moves
- It can be written as -

Eint = (α (v’)2 + β (v”)2)

- α, β specify the amount of elasticity and stiffness of the snake, respectively

Internal Energy

- First Term : α (v’)2
- If you want the snake to shrink, then make Eint directly proportional to snake length
- Instead,

Eint = Σα (distance b/w control pt. and left neighbour)2

- This makes the control points to spread out evenly

Internal Energy

- Second Term : β (v”)2
- If you want the snake to be smooth, then make Eint directly proportional to the curvature
- Instead, Eint = Σβ (curvature at control point)2
- This makes the control points to move as a smooth curve

Internal Energy

- A large positive value of α and/or β means that stretching(elasticity) and/or bending(smoothness) are tightly constrained as we search for an energy minimum
- Making β = 0 i.e. second order discontinuous, makes the snake develop a corner (least stiff)

External Energy

- External energy pulls the snake towards desirable features such as edges
- Tells the snake where to stop
- It is derived from the image
- External energy – negative gradient

Eext = -|∇I(x,y)|2

For a gray-scale image I(x,y),

Eext = -|∇ I(x,y)|2

Eext = -|∇ (Gσ(x,y) * I(x,y))|2

For a binary image

Eext = I(x,y)

Eext = Gσ(x,y) * I(x,y)

External EnergyEuler Equation

- A snake that minimizes E must satisfy the Euler equation –

α v”(s)- β v””(s) -∇Eext = 0

Force Balance Equation

Minimizing energy equation can be interpreted as a force balance equation

Fint + Fext = 0

where

Fint = α v” - β v””

and

Fext = −∇ Eext

The internal force Fint discourages stretching and bending while the external potential force Fext pulls the snake toward the desired image edges

Problems with traditional snakes

- Low capture range
- Initial snake position should be close to the object boundary
- Reason - External force field is non-existent in uniform regions
- Unable to move into boundary concavities
- Reason – External force vectors are normal to the object boundary

Traditional Snakes

(a)

(a)

(a)

(b)

(b)

(b)

(c)

(c)

- Convergence of the snake
- External Force Field
- Close-up of the force field within the boundary concavity

Gradient Vector Flow (GVF) snakes

Both problems with traditional snakes are related to external force field.

Xu and Prince formulate a new type of static external force called Gradient Vector Flow.

Fext = v

Edge Map

- For any gray-scale or binary image, let the edge map be defined such that it has large values near edges.

Eg.Eext =-|∇ I(x,y)|2

- Properties of edge maps
- Gradient vectors point toward and are normal to the edges
- Gradients are large only in the immediate neighborhood of edges
- In homogeneous regions, gradient is almost zero

How edge map properties affect traditional snake behavior

- Gradient vectors point toward and are normal to the edges

=> a snake initialized close to the edge will converge to a stable configuration near the edge

- Gradients are large only in the immediate neighborhood of edges

=> small capture range

- In homogeneous regions, gradient is almost zero

=> homogeneous regions will have no external forces

GVF Approach

- Retain the highly desirable property of the gradients near the edges
- But extend the gradient map farther away from the edges and into homogeneous regions using a computational diffusion process.
- As an important benefit, the inherent competition of the diffusion process will also create vectors that point into boundary concavities.

GVF energy functional

- We define the gradient vector flow field to be the vector fieldv(x,y)=(u(x,y),v(x,y)) that minimizes the energy functional

Ɛ = ∫ ∫ Ч(ux2+uy2+vx2+vy2)+|∇ f |2|v-∇ f |2dxdy

- When ∇ f is low, the first term dominates yielding a slowly varying field
- When ∇ f is high, the second term dominates and is minimized by setting v =∇ f.

GVF Euler equations

- It can be shown that the GVF field can be found by solving the following Euler equations –

- In homogeneous regions, second term = 0
- u and v are determined only by the Laplace equation
- v is determined from the region’s boundary
- a kind if competition among boundary vectors

Therefore, GVF snakes can move into boundary concavities

GVF Snakes

(a)

(b)

(c)

- Convergence of the snake
- External Force Field
- Close-up of the force field within the boundary concavity

Applications

- Motion Tracking
- Speech Analysis
- Traffic monitoring
- Surveillance
- Actor-driven facial animation
- Gait Analysis

Traffic Monitoring

By automatically tracking cars, the emergency services could obtain rapid warning of an accident or traffic jam.

Surveillance

A computer vision system follows an intruder on a security camera

Actor-driven facial animation

Tracked facial motions drive input channels to a cartoon cat, programmed with some exaggeration of expression

Motion Tracking

Tracking the articulated motion of a human body is applicable both to biometrics and clinical gait analysis and for actor-driven whole body animation

References

Books -

- Computer Vision – L. Shapiro and G. Stockman
- Active Contours – Blake and Isard

Technical papers -

- Snakes: Active contour models - M. Kass, A. Witkin and D. Terzopoulos.
- Snakes, Shapes, and Gradient Vector Flow - Chenyang Xu and Jerry L. Prince

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