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 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.