Lecture 1 -Introduction and Overview

ECE 600 – Summer 2009Introduction to Shape AnalysisCourse Supplements ….

Aly A. Farag

University of LouisvilleAcknowledgements:

Help with these slides were provided by Shireen Elhabian

Farag, Summer 09

What is it all about ?!!!

• The field of shape analysis involves methods forquantifying the shape components of visual data andderiving information from it.

• Bottom-line, we want to describe shape informationnumerically.

• Techniques from statistics, geometry and more generalmathematics are applied to shape data objects to obtainanalytic and summary conclusions.

Farag, Summer 09

Eeeew !!!

• We don’t really need torepresent/describe all the detailsof an object

• We need just to approximateits shape

• Enough …

– That it looks right when displayed.

– That we can derivemeaningful/useful information tothe application at hand/

Farag, Summer 09

Why Shape?

• To describe any real-life object on the computer,we must tart with shape (2D/3D).

Farag, Summer 09

Why study Shape?

• Most real-world objects have acharacteristic shape relative toother objects.

• Across the object’s population,instances vary in shape, whileretaining the “key features” ofthe shape, i.e. shape variesstatistically

• In medical applications,abnormal shape variations oftencharacterize disease

Farag, Summer 09

Where does the shape come from?• Modeling (by hand)

• Acquired real-world objects –shape acquisition throughsampling of real world objects(2D via image acquisitiondevices, 3D via laser scanners).

Farag, Summer 09

What is Shape?• The most commonly cited definition is given by D. G. Kendall [*] as

follows:

– ”Shape is all the geometrical information that remains when location, scale,and rotation effects are filtered out from an object”.

• In other words, a shape is invariant to Euclidean similaritytransformations of scaling, translation, and rotation. Two objects havethe same shape if they can be mapped onto each other by atranslation, rotation, and scaling.

[*] D. G. Kendall. The diffusion of shape. Advances in Applied Probability, 9:428 – 430, 1977.

Farag, Summer 09

Shape Models• What constitutes a shape model?

• How do we represent variation in such a model?

• How do we measure variation of an instancefrom the model?

• How can we display/represent the variation?

• How can we use a shape model to find instancesof the shape, for example in images that arehard to segment?

Farag, Summer 09

Shape Models• There are plenty of models in literature, each

accompanies with various approach ofimplementation, we will be interested in:

– Active Shape Models (ASMs)

– Active Appearance Models (AAMs)

– Morphable Models (MMs)

Farag, Summer 09

Main Pipeline

• Which model to use mainly depends on theapplication and the shapes you have.

• However, all models share a common theme,which is divided into two stages.

Model Training Model Fitting

Farag, Summer 09

Model Training

• We start with training data containing shapes of differentinstances of the object to be modeled.

• Models such as ASM keep track only with shape, howevermodels such as AAM and MM make us of shape andappearance, where AAM is defined in 2D space while MM isdefined in the 3D space.

Farag, Summer 09

Model Training

• The training stage often involves registration ofdifferent shapes (rigidly) such that points of individualshapes will be in correspondence.

• Usually Generalized Procrustas Analysis is employed,however we can also use any other registrationalgorithm.

Farag, Summer 09

Model Training

• To capture shape variation, statistical data modelingtechniques are employed, such as principle componentanalysis (PCA), linear discriminant analysis (LDA) andindependent component analysis (ICA).

Farag, Summer 09

Model Fitting• In this stage, it basically involves minimization of objective

function, in an iterative manner, in order to generate an instancewhich resemble the real-world object. This is the area of imagewarping, transformations and non-linear optimizationtechniques.

Farag, Summer 09

What for ?!!! Applications

• Traditional biological and medical applicationsstudy how shape– changes during growth

– changes during evolution

– is related to size

– is affected by disease

– related to other covariates

• Automatic object recognition - differentiation

Farag, Summer 09

Ingredients

• This course utilize different conceptsand algorithms from :

– Topology of surfaces

– Computational geometry

– Graph theory

– Numerical linear algebra

– Computer graphics

– Differential geometry

– Others …

Farag, Summer 09

Organization

• Prof. Aly A. Farag

– Lutz , room 412

– Website: www.cvip.lousville.edu

– Office hours: …

• Communication:

– Course materials, reading materials, projects and homeworkswill be posted on blackboard every Thursday and collectedthe following Thurday.

Farag, Summer 09

Prerequisites

• Familiarity with basic calculus, linear algebra nd vectorcalculus is preferred.

• Coding skills: mix of Matlab and C# will be used,however help will be presented to acquire such skills.

Farag, Summer 09

Course Overview

• Mathematical Preliminaries (1 week)

– Basics of geometric topology

– Basics of linear algebra

– Quiz #1; Homework #1

Topology is a field of mathematics and is relatedto geometry. In both of these subjects one studiesthe shape of things. In geometry, onecharacterizes, for example, a can of pineapple byits height, radius, surface area and volume. Intopology, one tries to identify the more subtleproperty that makes it impossible to get thepineapple out of the tin, no matter what shape itis battered into, as long as one does not puncturethe can.

Ref: Topology of Surfaces by L. C. Kinsey (Springer,1993)

Linear algebra is the branch of mathematicsconcerned with the study of vectors, vector spaces,linear maps (transformations) and systems of linearequations. Did you hear words like projection onspace, null spaces, column and row spaces, … wewill learn such things through this nice lecture.

Ref:

Farag, Summer 09

Course Overview

• Shape representations (1.5 week)

– What is Shape?

– Data-based representation

– Model-based representation

– Quiz #2; Homework #2; Project #1To process, analyze, understand, deform, construct or even render a shape,we first need to know how to represent/describe a shape to the computer.Ref:1. Alexandre Hardy, Willi-Hans Steeb, Mathematical tools of computer

graphics with C# implementations, Wordware Publishing, Inc.

Farag, Summer 09

Course Overview• Basic tools of computational geometry: (2.5

weeks)

– Polygon triangulation

– Polygon partitioning

– Convex hulls in 2D/3D

– Voronoi diagrams

– Delaunay triangulations

– Shape operators/descriptors

– Texture mapping

– Transformations

– Quiz #3; Homework #3; Project #2

Computational geometry is the study ofalgorithms for solving geometric problemson a computer. Let’s see how we candesign such algorithms, where polygonsplay a much larger role, while geometricmodeling mainly works on continuouscurves and surfaces.

Ref:1. M. de Berg, O. Cheong, M. van

Kreveld, and M. Overmars,Computational Geometry: Algorithms andApplications, 3rd ed. Springer, Berlin.

2. J. O'Rourke, Computational Geometry inC (Cambridge Tracts in TheoreticalComputer Science). CambridgeUniversity Press, October 1998.

Farag, Summer 09

Course Overview

• Basics of non-linear programming(optimization) (1 week):

– Problem statement

– Minimization in one dimension

– Multi-dimension search

– Quiz #4; Homework #4; Project #3

Optimization is central to any probleminvolving decision making which entailschoosing between various alternatives,governed by an objective function.Optimization theory and methods dealwith selecting the best solution (alternative)in the sense of the given objectivefunciton.

Ref:1. E. K. P. Chong and S. H. ˙ Zak, An

Introduction to Optimization. New York, NY: John Wiley and Sons, Inc., 1996, ISBN 0-471-08949-4, xiii+409 pp.

Farag, Summer 09

Course Overview

• Statistical data modeling:

– Principle component analysis

– Linear discriminant analysis

– Independent component analysis

– Quiz #5; Homework #5; Project #4

Farag, Summer 09

Course Overview

• Modeling: problem statement (1 week):

– Active shape models

– Active appearance models

– Morphable models

– Quiz #6; Homework #6; Project #5

Farag, Summer 09

Course Overview

• Active Appearance Models (2 weeks):

– Image warping

– Linear appearance variation

– AAMs into action

– Quiz #7; Homework #7; Project #6

Farag, Summer 09

Farag, Summer 09