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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