CSci 8810 Image Processing Active Contours - SNAKES Meghana Viswanath March 19, 2009
Object Recognition… How? • Edge detectors Works well for some images
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
Aorta Segmentation using snakes
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 Initialization
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
Energy Minimization vi-1 vi vi+1 snake object boundary Find the local minimum and move towards it.
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 Energy
Euler 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
Subjective Contours
Applications • Motion Tracking • Speech Analysis • Traffic monitoring • Surveillance • Actor-driven facial animation • Gait Analysis
Speech 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
Questions?
