Geodesic Minimal PathsGeodesic Minimal Paths

Vida MovahediVida Movahedi

Elder Lab, January 2010Elder Lab, January 2010

ContentsContents

• What is the goal?

• Minimal Path Algorithm

• Challenges

• How can Elderlab help?

• Results

GoalGoal

• Finding boundary of salient objects in images of natural scenes

Minimal PathMinimal Path

• Inputs: – Two key points

– A potential function to be minimized along the path

• Output:– The minimal path

Minimal Path- problem formulationMinimal Path- problem formulation

• Global 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

],0[

))((~

)(L

dssCPCE

dssCPCEpUpoppop

ΑΑ)(

~inf)(inf)(

,,

Fast Marching AlgorithmFast Marching Algorithm

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

ChallengesChallenges

• Can the minimal path algorithm solve the boundary detection problem?– Key points?

– Potential Function?

• Idea: Use York’s multi-scale algorithm (MS)

MS AlgorithmMS Algorithm

• We have a set of contour hypotheses at each scale

• These contours can be used to find good candidates for key points

• These contours (and some other cues) can also be used to build potential functions.

• Multi-scale model (coarse to fine) can also help

Key PointsKey Points

• Simplest approach: 3 key points, equally spaced on the MS contour (prior)

• Maximize product of probabilities (MS unary cue)

Rotating Key PointsRotating Key Points

• Consider multiple hypothesis for key points

• Obtain multiple contours

• Next step: Find which contour is the best– Distribution model for contour lengths

– Distribution model for average Pb value

– Improve method to find simple contours only

Rotating Key PointsRotating Key Points

Potential FunctionPotential Function

• Ideas:– The Sobel edge map

– Distance transform of MS contour (prior)

– Distance transform of several overlapped MS contours

– Berkeley’s Pb map

– Likelihood based on Pb and distance to prior contour

Sobel Edge MapSobel Edge Map

Sobel Edge MapSobel Edge Map

• Can use the MS prior to emphasize or de-emphasize map

Distance TransformDistance Transform

Distance transformDistance transform

• Too much emphasis on MS prior

Distance transform Distance transform of 10 overlapped MS contoursof 10 overlapped MS contours

Challenge: Challenge: If MS contours are not goodIf MS contours are not good

Challenge: Challenge: If MS contours are not goodIf MS contours are not good

Berkeley’s Pb mapBerkeley’s Pb map

Combining Pb and DistanceCombining Pb and Distance

)|(

)|(

)|(

)|()()(),(

CxDp

CxDp

CxPbp

CxPbpDLPbLDPbL

Next step: learning models

SummarySummary

• The MP algorithm provides global minimal paths

• The MS algorithm provides contour hypothesis

• The MS contours can be used to obtain key points and potential functions for MP algorithm

• Next steps:

– Learning models for better potential functions

– Obtaining simple contours

– Ranking contours

– Evaluate multi-scale model

ReferencesReferences

Laurent 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.