# geodesic minimal paths

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Geodesic Minimal Paths. Vida Movahedi Elder Lab, January 2010. Contents. What is the goal? Minimal Path Algorithm Challenges How can Elderlab help? Results. Goal. Finding boundary of salient objects in images of natural scenes. Minimal Path. Inputs: Two key points - PowerPoint PPT PresentationTRANSCRIPT

Geodesic Minimal PathsVida Movahedi

Elder Lab, January 2010

ContentsWhat is the goal?Minimal Path AlgorithmChallengesHow can Elderlab help?Results

GoalFinding boundary of salient objects in images of natural scenes

Minimal PathInputs: Two key points A potential function to be minimized along the path

Output:The minimal path

Minimal Path- problem formulationGlobal minimum of the active contour energy:

C(s): curve, s: arclength, L: length of curve

Surface of minimal action U: minimal energy integrated along a path between p0 and p

Ap0,p : set of all paths between p0 and p

Fast Marching AlgorithmComputing U by frontpropagation: evolving a front starting from an infinitesimal circle around p0 until each point in image is reached

ChallengesCan the minimal path algorithm solve the boundary detection problem?Key points?Potential Function?

Idea: Use Yorks multi-scale algorithm (MS)

MS AlgorithmWe have a set of contour hypotheses at each scaleThese contours can be used to find good candidates for key pointsThese contours (and some other cues) can also be used to build potential functions.Multi-scale model (coarse to fine) can also help

Key PointsSimplest approach: 3 key points, equally spaced on the MS contour (prior)Maximize product of probabilities (MS unary cue)

Rotating Key PointsConsider multiple hypothesis for key points Obtain multiple contours

Next step: Find which contour is the bestDistribution model for contour lengthsDistribution model for average Pb valueImprove method to find simple contours only

Rotating Key Points

Potential FunctionIdeas:The Sobel edge map Distance transform of MS contour (prior)Distance transform of several overlapped MS contoursBerkeleys Pb mapLikelihood based on Pb and distance to prior contour

Sobel Edge Map

Sobel Edge MapCan use the MS prior to emphasize or de-emphasize map

Distance Transform

Distance transformToo much emphasis on MS prior

Distance transform of 10 overlapped MS contours

Challenge: If MS contours are not good

Challenge: If MS contours are not good

Berkeleys Pb map

Combining Pb and DistanceNext step: learning models

SummaryThe MP algorithm provides global minimal pathsThe MS algorithm provides contour hypothesis The MS contours can be used to obtain key points and potential functions for MP algorithmNext steps:Learning models for better potential functionsObtaining simple contoursRanking contoursEvaluate multi-scale model

ReferencesLaurent D. Cohen (2001), Multiple Contour Finding and Perceptual Grouping using Minimal Paths, Journal of Mathematical Imaging and Vision, vol. 14, pp. 225-236.

Estrada, F.J. and Elder, J.H. (2006) Multi-scale contour extraction based on natural image statistics, Proc. IEEE Workshop on Perceptual Organization in Computer Vision, pp. 134-141. J. H. Elder, A. Krupnik and L. A. Johnston (2003), "Contour grouping with prior models," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, pp. 661-674.

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