Shape recovery from medical image data using extended superquadrics
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Shape Recovery from Medical Image Data Using Extended Superquadrics. Talib Bhabhrawala Advisor : Dr. Venkat Krovi Department of Mechanical and Aerospace Engineering State University of New York at Buffalo Master of Science Thesis Defense December 14 th , 2004. Overview.

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Shape recovery from medical image data using extended superquadrics l.jpg

Shape Recovery from Medical Image Data Using Extended Superquadrics

Talib Bhabhrawala

Advisor : Dr. Venkat Krovi

Department of Mechanical and Aerospace Engineering

State University of New York at Buffalo

Master of Science Thesis Defense

December 14th, 2004


Overview l.jpg
Overview Superquadrics

  • Introduction

  • Background

  • Methodology Development

  • Case Studies

  • Interfaces

  • Conclusion & Future Work

IntroductionBackground Methodology Results Conclusion


Introduction l.jpg
Introduction Superquadrics

Ubiquitous availability of computation and communication infrastructure

Create, manipulate & distribute such data.

IntroductionSuperquadrics Methodology Results Conclusion


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Computer Vision & Animation Superquadrics

Life sciences

Engineering

Introduction

Application Areas

Model Based Reconstruction

  • Building geometric shape models from raw input data

  • Data reduction, Analysis, Manipulation, Storage

Point Cloud Data is adequate for Visualization

  • Models & Methods are defined by the final application

  • Visualization –Surface Geometry

  • Dynamic & Finite Element Analysis –Volumetric Information

Introduction Superquadrics Methodology Results Conclusion


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Desired Characteristics Superquadrics

Low Order Models

  • Computational ease.

  • Fitting, Visualizing & Analysis

  • Parametric Models

  • - Intuitive and Easy to use

    • - Meaningful and repeatable

    • - Great Success in Engineering

  • Models whose nature is

  • approximation

    • - Tractability for infinite dimensional data


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Research Issues Superquadrics

Which kind of a parametricapproximation framework would be most suitable for rapid, easy, accurate and computationally inexpensive shape modeling and conversion to volumetric solid model from a dense sampling of the surface?

How can we leverage the same framework to additionally parametrically explore multi-resolution hierarchical indexing, storage, searching, reconstruction and retrieval?

Introduction Superquadrics Methodology Results Conclusion


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Superquadrics (SQ) Superquadrics

powerful & compact shape representation

  • flexible family of parametric objects

  • using low order parameterization, variety of shapes maybe obtained

  • simple mathematical representation

  • good explicit and implicit form

Introduction Superquadrics Methodology Results Conclusion


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Spherical Product Superquadrics

A 3D surface can be obtained by the spherical product of two 2D curves.

When a half circle in a plane orthogonal to the (x, y) plane.

is crossed with the full circle in (x, y) plane

Introduction Superquadrics Methodology Results Conclusion


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Superellipses Superquadrics

A superellipse is a closed curve defined by

Introduction Superquadrics Methodology Results Conclusion


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Superquadrics Superquadrics

Superellipsoids are obtained from superellipses

a1, a2, a3 - scaling factors

ε1 , ε2 - relative roundness & squareness.

Introduction Superquadrics Methodology Results Conclusion


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Superquadrics Superquadrics

Introduction Superquadrics Methodology Results Conclusion


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Implicit Representation Superquadrics

Valuable single implicit function.

  • The object is continuous everywhere.

  • Point membership classification can be done

  • Inside–outside function.

Introduction Superquadrics Methodology Results Conclusion


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SQ Discussion Superquadrics

  • Advantages

    • – can model a diverse set of objects

    • – compact representations

    • – controllability and intuitive meaning

    • – can be recovered from 3D information robustly

Limitations

– Basic representation can only model symmetrical shapes

Introduction Superquadrics Methodology Results Conclusion


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Literature Superquadrics

  • Superquadric’s applications

    • Computer environments (Montiel, 1997; Pentland, 2000)

    • Graphics & vision (Chella, 2000; Jacklic, 2000)

  • Local and Global Deformations

    • Nonlinear deformable models (Solina & Bajcsy, 1991)

    • Simulating equations of motion (Terzopoulos, 1993)

  • Increasing the DOF

    • Segmentation (e.g. Löffelman and Gröller, 1994)

    • blending multiple models (DeCarlo & Metaxas, 1998)

    • free form deformations (Bardinet et al., 1994)

Introduction Superquadrics Methodology Results Conclusion


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Literature Superquadrics

To represent more complex shapes there is a trade off between

- degrees of freedom & expressive power

Zhou and Kambhamettu (2001) first examined

- exponents need not be fixed

- possibly be spatially varying functions

- extended superquadrics

Introduction Superquadrics Methodology Results Conclusion


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Extended Superquadrics (ESQ) Superquadrics

Analogous to a SQ it is definedby

  • & are the latitude & longitude angles

  • Exponents are now functions of these angles.

Introduction Superquadrics Methodology Results Conclusion


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Inside-Outside Function Superquadrics

where

Measure the difference between a modeled shape and the given data set

Introduction Superquadrics Methodology Results Conclusion


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Exponent Functions Superquadrics

The shape of the exponent functions have to be controllable

We introduce a spline as the exponent function.

This interpolated curve acts like a look up table for the algorithm

Introduction Superquadrics Methodology Results Conclusion


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Intuitive Example Superquadrics

Introduction Superquadrics Methodology Results Conclusion


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Problem Statement Superquadrics

  • Recovering a superquadric model from a set of 3D points

    • Superquadric model

    • Vector of superquadric parameters

    • Input Points

    • Minimize

  • Least square distance between SQ surface & data points


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Initial Model Definition Superquadrics

SQ in a localcoordinate system

Five parameters which

define the size & shape

SQ in the general positionTransform the points to the object coordinated system

Introduction Superquadrics Methodology Results Conclusion


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Initial Model Definition Superquadrics

Applying the Inverse Transform & using Euler angles

Additional six parameters which

define the position and orientation

Case of Extended Superquadrics

  • Exponent is a spline interpolating ‘p’ control points increases the total number of parameters by 2(p-1).

Number of parameters are now 9+2(p-1).

Introduction Superquadrics Methodology Results Conclusion


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Moment Based Estimation Superquadrics

Object recognition and pose estimation

Obtain the rotation matrix and eigen vectors.

Orient axes along minimal & maximal moment of inertia

Farthest range point along each coordinate axis which gives an estimate of a1, a2 & a3

Introduction Superquadrics Methodology Results Conclusion


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Error of Fit Function Superquadrics

The error-of-fitfunction is define using the inside–outside function

EOF variesquickly where the exponents are large

and slowly where exponents are small

Added to remove the bias

Introduction Superquadrics Methodology Results Conclusion


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Error of Fit Function Superquadrics

Ambiguity in Description

  • A set of exponent functions in conjunction with scaling parameters can generate the same shape as another set

To solve the ambiguity, the minimum volume constraint is added

Metric to be minimized for the “fitting”

Introduction Superquadrics Methodology Results Conclusion


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Optimization Problem Superquadrics

Minimize

where

Variables

Introduction Superquadrics Methodology Results Conclusion


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Choice of Optimization Method Superquadrics

  • The conventional method used is the Levenberg-Marquardt algorithm

  • Fastand accurate

  • Problems of local minima

  • Heavily dependent on initial estimates

Introduction Superquadrics Methodology Results Conclusion


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Genetic Algorithms Superquadrics

  • Inspired by Biological Evolution and its principles

  • The evolution of life on earth can be regarded as one long optimization process though it’s up to debate if this process has reached a optimum yet…

Introduction Superquadrics Methodology Results Conclusion


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Genetic Algorithms Superquadrics

Salient Features

  • Requires little insight into the problem

  • Ideal if a problem is non convex or has a very large multimodal solution space

  • Heuristics Based, does not require derivatives

  • Provides with “Good” Solutions

  • Ideal exploratory tool to examine new approaches


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Genetic Algorithms Superquadrics

Components of a GA

  • initialize population;

    • evaluate population;

    • while TerminationCriteriaNotSatisfied

    • {

      • select parents for reproduction;

      • perform crossover & mutation;

      • evaluate population;

    • }

- Encoding technique (double vector, binary)

- Object function (environment)

- Genetic operators (selection, mutation, crossover)

“Typical” tuning parameters


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Shape Recovery Algorithm Superquadrics

Introduction Superquadrics Methodology Results Conclusion


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2D Case Study Superquadrics

Introduction Superquadrics MethodologyResults Conclusion


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3D Case Study Superquadrics

Introduction Superquadrics MethodologyResults Conclusion


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Partitioning Line Superquadrics

d

Iterative Segmentation & Recovery

  • Difficult to fit a complex model using a single extended superquadric

  • Segment an object into primitives

Maximum Error

minimize

Two Superquadrics to approximate the data

EOF1= 0.575

EOF2= 0.326

Introduction Superquadrics MethodologyResults Conclusion


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Volume Segmentation Using ESQ Superquadrics

  • 2D contours are obtained and are stacked

  • Topological Accuracy is high

  • Loses Compact representation

  • Laborious process & model has inconsistencies

  • Requires a post-processing step

Introduction Superquadrics MethodologyResults Conclusion


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PC Based Interface Superquadrics

Introduction Superquadrics MethodologyResults Conclusion


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Web Based Interface Superquadrics

Introduction Superquadrics MethodologyResults Conclusion


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Conclusion Superquadrics

  • Flexible enough for an asymmetric object that deform smoothly on spheres

  • Variable coefficients of the continuous exponents offer a compact parameter space and broad coverage

  • The descriptive parameterization is directly incorporated into the model formulation

Introduction Superquadrics Methodology Results Conclusion


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Future Work Superquadrics

  • A more intuitive and robust segmentation scheme

  • Techniques for creating “tailored” models from such simple general purpose models

  • More intelligent precursor steps to improve convergence speed of the algorithm

  • Systematic way to extract and store characteristic signatures of shape

Introduction Superquadrics Methodology Results Conclusion


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Thank You! Superquadrics

Acknowledgments:

Dr. V. Krovi, Dr. C. Bloebaum &

Dr. A. Patra


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