1 Minimum Ratio Contours For Meshes Andrew Clements Hao Zhang gruvi graphics + usability +...
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Transcript of 1 Minimum Ratio Contours For Meshes Andrew Clements Hao Zhang gruvi graphics + usability +...
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Minimum RatioContours For Meshes
Andrew Clements Hao Zhang
gruvi
graphics + usability + visualization
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Introduction
Problem: Feature extraction and segmentation of 3D mesh models for the purpose of object recognition Object parts are delimited by contours
Given an initial contour, search for a ‘better’ contour
What are features? Explained later
Contributions Applying minimum ratio to meshes Energy definition Efficiency
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Outline
Previous Work & Motivation Algorithm Overview Ambient Graph Construction Minimum Ratio Contour (MRC) Algorithm Results
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Snake Methods
Widely applied for the task of feature extraction and segmentation
Techniques work with images and meshes Formulated as a minimization problem
Snake is parameterized by v(s), and energy is defined as
Internal energy controls length and smoothness External energy controls feature adaptation A search for the snake with the lowest energy is
performed Gradient descent, graph minimization
ds )external(v )internal(v )(SE
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Drawbacks of Snakes
Local in nature When using gradient descent, snake cannot jump
out of local minima Global minimum does not yield a meaningful
result Trivial solution results with classical energy
definition
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Minimum Ratio Methods
Previously applied for the task of image segmentation
Energy of a contour is defined as a ratio
G(v)ds
)()(
dsvFCE
F(v) controls feature adaptation and smoothness G(v) is a general measure of length Removes bias towards short contours Trivial solutions are not minimizers
Due to normalization by length
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Minimum Ratio Methods
To find a solution, problem is discretized Goal is to find the minimum ratio cycle in a graph A global solution can be obtained in polynomial
time Requires at least O(n2) time to find minimizing
cycle in a general graph
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Method Differences
Snaking method uses a Total Energy
MRC uses a Ratio Energy
ds )(external )internal(v )( vSE
v)arclength(
ds )external(v )internal(v )( CE
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Outline
Previous Work & Motivation Algorithm Overview Ambient Graph Construction Minimum Ratio Contour (MRC) Algorithm Results
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Algorithm Overview
InitialContour
AmbientGraph
InputMesh
MinimumRatio
Contour
AmbientGraph
Construction
MRCAlgorithm
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Outline
Previous Work & Motivation Algorithm Overview Ambient Graph Construction
Refinement Energy definition
Minimum Ratio Contour (MRC) Algorithm Results
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Ambient Graph Construction
Ambient graph models the space of admissible contours
Nodes in ambient graph correspond to directed edges of mesh
Arcs in ambient graph are inserted between nodes of successive directed edges
Weights can be assigned to arcs which encode bending between nodes
Contours on mesh map to cycles in ambient graph
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Sample Ambient Graph
Mesh Ambient Graph
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Refined Ambient Graph
Problem: irregular mesh connectivity Contours may not be smooth
Refine mesh before constructing ambient graph Smoother contours are possible
Refinement scheme inserts extra chords passing through faces of mesh Subdivision is not sufficient
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Energy Motivation
Denominator weight is taken to be Euclidean length
Numerator weight controls feature adaptation serves to attract the contour to features which are
perceptually salient
ii
ii
c
cden
numCE
)(
)()(
Each arc in ambient graph is assigned a numerator and denominator weight
Energy of a contour C is defined as
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Energy Considerations
What features should be segmented? Minima Rule
A theory which describes where the humans perceive boundaries between parts
Boundaries consist of surface points at the negative minima of principal curvatures
Contour Steering: favour contours aligned with principle curvature directions
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Outline
Previous Work & Motivation Algorithm Overview Ambient Graph Construction Minimum Ratio Contour (MRC)
Algorithm Results
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MRC Algorithm Overview
Initial Contour Strip Boundaries Edge Cut Gate Segments Acyclic Edge Graph Optimization
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MRC Algorithm Overview
Initial Contour Strip Boundaries Edge Cut Gate Segments Acyclic Edge Graph Optimization
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MRC Algorithm Overview
Strip Boundaries Mimic flow of initial
contour Constructed by ‘dilating’
initial contour
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MRC Algorithm Overview
Edge Cut Disconnects search
space Used in the acyclic graph
construction
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MRC Algorithm Overview
Gate Segments Help orient flow Inserted at constrictions
between adjacent strip boundaries
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MRC Algorithm Overview
Acyclic Edge Graph select nodes from ambient
graph – orient edges in search space
Edge cut nodes are duplicated
Paths from edge cut nodes in acyclic graph correspond to contours in search space
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MRC Algorithm Overview
Optimization A series of Minimum
Ratio Path (MRP) problems are solved, one for every edge in the edge cut
The path with minimum ratio corresponds to the contour with least ratio
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Solving The MRP Problem
Reduces to a series of decisions determining whether a negative path exists in an acyclic graph Can be performed in linear time
Linear vs. Binary Search Experimentally, a constant number of iterations is
needed for linear search Affirms other researchers observations
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Outline
Previous Work & Motivation Algorithm Overview Ambient Graph Construction Minimum Ratio Contour (MRC) Algorithm Results
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Results – Regular vs. Refined
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Results – Escaping Local Minima
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Results – Iterations
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Results – Constraints
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Questions?
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Future Work
Numerator weights that incorporate area Use Stoke’s Theorem
MRP + Length Combine ratio with length Currently have algorithm to handle minimum mean
path with length Generalize to MRP + length Reduce running time from O(n2)
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MRC Algorithm Overview
Initial Contour Strip Boundaries Edge Cut Gate Segments Acyclic Edge Graph Optimization