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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.