Segmentation Using Active Contour Model and Tomlab By: Dalei Wang 29/04/2003

Content Introduction Mathematical definition of active contour, and its energies Program design and operation Performance and issues What to be done next

The Project Started out as an investigation of how the interior point optimization algorithm applies to segmentation Popular segmentation algorithms: Euler-Lagrange, Dynamic Program- ming, Greedy Algorithm

The Project Objectives: Design or find an interior point optimization algorithm. Design a self contained segmentation program based on interior point method. Later realized that interior point algorithms mostly apply to linear constrained problems, unsuitable for energy minimization

The Project Interior point method for non-linear programming exists but resources are scarce and beyond my current level Finally decided to use an “off the shelf” optimization program: Tomlab toolbox Tomlab will include an interior point solver soon

Mathematical Definition of Snake Energies Active Contour Model, a.k.a. Snake, developed by Kass et al. in 1987 An ordered set of snaxels{p 1, p 2,… p n } defined on a bounded two dimensional space [1] A snake is defined so as to possess certain energy Energy conveniently defined so to reach minimum at boundary of objects [1,2]

Mathematical Definition of Snake Energies: Internal The internal energy controls the elasticity and stiffness of the snake Grants resistance to pulling and bending Smoothness constraints: If no external force present the snake will be circular

Mathematical definition of snake energies: Internal Mathematically, internal energy is defined as  dv/ds  2 +  d 2 v/ds 2   v(s) is snake curve parametrically defined in terms of s, the curve length Must be discretized for the snake is not continuous, but defined as a set of points Discretized, the internal energy of i th snaxel is: E int (v i )=  ||v i -v i-1 || 2 +  ||v i-1 -2v i -v i+1 || 2

Mathematical definition of snake energies: Internal Total internal energy of a snake is the sum of all internal energies of individual snaxels. Can be expressed as: [x y]A[x t y t ] t, where x=[x 1 x 2 x 3..x n ] and y=[y 1 y 2 y 3..y n ] and A is and n by n matrix whose elements are expressed in terms of  and .

Mathematical definition of snake energies: External The external energy depends only on the property of the image Let z=I(x, y) be the image function, x, y are axes, z is 8 bit grey scale. Alternate definition depending on the type of image and the object one wants to extract

Mathematical definition of snake energies: External For prostate ultra-sound images (noisy, non-homogeneous background,boundary not clearly defined) the external energy is:

Mathematical definition of snake energies: External The gradient magnitude of prostate image This is the plot of the x components of image gradient The magnitude of gradient is strong near edge of prostate… …But also at false minima(noises)

Mathematical definition of snake energies: External This is the plot of Y gradient component

Mathematical definition of snake energies Minimizing the total snake energy is therefore an unconstrained (but bounded) non-linear (due to the non- linearity of image function) programming problem There is an unconstrained non-linear routine in Tomlab: ucSolve

Program Design and Operation Snake initialization: A function allows the user to select points on the prostate image For optimal results, one should select about 10 points The program automatically interpolates along the curve and samples about 50 points

Program Design and Operation Problem definition: In order to define the problem in Tomlab format, the following data is stored in global variable: image, gradient matrices of the image, matrix used in computing internal energy It is important for them to be calculated only once. They are needed for energy calculation, which is evaluated about 3000~5000 times per run

Program Design and Operation Another design issue: It is possible for some images to cause most or all snaxels to cluster to a region of the prostate edge, leaving a large gaps between some neighboring snaxels

Program Design and Operation Solution: Interpolating along the snake curve at each iteration and sample snaxels at equal intervals along the curve length

A Typical Output

The blue curve is initial snake, the red is the final Solution found in seconds and 10 iterations (tested on Pentium III 1GHZ 384 MB Ram) This result does reflect typical time cost

Issues Snakes as defined by Kass et al. has some intrinsic short comings: Extreme sensitivity to parameters Extreme sensitivity to initialization Inability to displace far from original position due to weak external force fields away from edges Will NOT converge into concavities

Example of Poor Initialization

Sensitivity to Parameters 

What’s Still to Be Done Performance analysis: Compute how close the final snake is to real contour. Compare with other algorithms Solve the problem again with interior point solver when it comes out

Bibliography [1] K. F. Lai, “Deformable Contours: Modeling, Extraction, Detection and Classiication”, University of Wisconsin- Madison, 1994 [2] A. Amini, T.Weymouth, and R. Jain, “Using Dynamic Programming for Solving Variational Problems in Vision”, in IEEE Transaction On Pattern Analysis And Machine Intelligence, Vol.12 No.9, September 1990

Bibliography [3] L. Zhang, “Active Contour Model, Snake”, Department of Computer Science, University of evade, Reno [4] K.Holmstrom.”TOMLAB -An Environment for Solving Optimization Problems in Matlab.” In M.Olsson,editor,Proceedings for the Nordic Matlab Conference ’97,October 27-28,Stock- holm, Sweden,1997.Computer Solutions Europe AB.

Thanks Prof. Salama Prof. Freeman Prof. Vanelli Jessie Shen Bernard Chiu