Joshua I. CohenBrown University

September 2001 – May 2002

“Computational Procedures For Extracting Landmarks In Order To Represent The

Geometry Of A Sherd”

Introduction

Project Process

- Feature extraction

- Reconstruction of the original 3D object using the extracted features Motivation

Data Collection

Sherd

- ShapeGrabber

- Polyworks

- .mat files

Data Collection Cont’d Breakcurve Breakcurve Algorithm (Xavior Orriols) 1. Subdivides sherd into smaller planes recursively starting from centroid

2. Singular value decomposition 3 orthogonal vectors

3. Project points into 2D plane

4. Find edge points.

Breakcurve Algorithm Pitfall

Data Collection Cont’d .iv files (Dongjin Han)

Polynomial Approximation of Curves

Containing High Curvature Points Design Decision f2D Software Two Cases To Consider

1. Polynomial Approximation of a High Curvature Segment

2. Polynomial Approximation of a Breakcurve

Polynomial Approximation of a High Curvature Segment

Gradient-1 Gradient-1 R.R.

Degree 4

Degree 11

Polynomial Approximation of a Breakcurve

Gradient-1 Gradient-1 R.R.

Degree 4

Degree 11

Advantages of Corners

Good Landmarks Segmentation of Breakcurve Better

Representation Locally Lower Degree Polynomial Fit (3 or 4)

- Computation Time

- Stability

Landmark Extraction Algorithm

Pre-Processing1. Find Normals on Breakcurve

- Patch- Eigenvector Associated With Minimum Eigenvalue- Check Direction

2222 )()()( zczycyxcx BDBDBD

1

1

PzPyPxPz

Py

Px

M

S = M1 + M2 + M3 + … + Mk

2. Order Breakcurve Points

Landmark Extraction Algorithm Corner Detection

1. Concatenate Breakcurve

2. Polynomial Degree One Fitting To Approximate Tangent Vectors

- tR and tL

- 2 Planes: ax + by + cz + d = 0 x + y + z + = 0

- Eigenvectors associated with 2 smallest eigenvalue

- Normalize

3. Ensure Tangent Vectors have the Right Direction

Line of Intersection of 2 Planes = [a,b,c] x [,,]

)( k

c

nck c BBt

)( k

c

nck c BBt

Distance Positive Distance Negative

Landmark Extraction Algorithm

Corner Detection Cont’d4. Compute Angle

- goodness of fit

180

)()(cos

1

RcLc

RcLc

tmagtmag

tt

Landmark Extraction Algorithm Corner Detection Cont’d

5. Find Local Minimum Angles (Corners)

- smaller than angle of neighbors

- smaller than angle threshold

Angle Threshold: 145 degrees Angle Threshold: 135 degrees

Landmark Extraction Algorithm Computing Curvature Extrema

1. Segment Breakcurve at Corners

2. Project Breakcurve into 2D using Local Projection

- Global vs Local

- Bmid , Nmid

- Rotated perpendicular to [1,0,0], x components are 0

100

0

0

0

010

0

),,(2222

2222

222

22

222

222222

22

yx

x

yx

yyx

y

yx

x

zyx

yx

zyx

z

zyx

z

zyx

yx

zyxM

midmidmidcmidc N)N)B - ((B - )B - (B M

Landmark Extraction Algorithm Computing Curvature Extrema Cont’d

3. Gradient-1 2D Curve Fitting of Projected Breakcurve Segments

- ipfit_5.3.0

- gradient-1 and gradient-1 ridge regression w/specified degree

- ipfit_5.3.0 vs f2D

Degree = 3

Landmark Extraction Algorithm Computing Curvature Extrema Cont’d

4. Compute Curvature of Projected Breakcurve Segments

- Obtain points on g(x,y) in [Bxmin, Bxmax, Bymin, Bymax]

- Order according to contour

- Compute Curvature

2

322

22

),(),(

),(),(),(),(),(2),(),(),(

yxgyxg

yxgyxgyxgyxgyxgyxgyxgyxK

yx

xyyyxxyxx

Landmark Extraction Algorithm Computing Curvature Extrema Cont’d

5. Find Curvature Extrema of Projected Breakcurve Segments

- Minima: K < 0, K < Neighbors, K < Threshold

- Maxima: K > Neighbors, K > Threshold

Threshold = 0.012

Landmark Extraction Algorithm Computing Curvature Extrema Cont’d

6. Obtain Landmarks by Combining Curvature Extrema of

Projected Breakcurve Segments with the Corners

Analysis of Results Curvature Extrema Not Always Accurate Problems

1. The Polynomial Fit is Not Always Very Good

2. Points on g(x,y) are Approximate

Without Any K Threshold

Conclusion

Correct Curvature Problems- f2D software, g(x,y) = 0 and dotprod(T,K) = 0

Corners match for p6ed and p10ed Groundwork of Landmark Detector Established

References1. Linear Algebra and Its Applications, Gilbert Strang, International

Thomson Publishing, 3rd edition, 1988.

2. Numerically Invariant Signature Curves, Mireille Boutin

3.      Numerical Recipes http://www.nr.com/

4.      Numerical Recipes in C: The Art of Science, William H. Press, Cambridge University Press, 2nd edition, 1993

5.      Scientific Computing An Introduction With Parallel Computing, Gene Golub, Academic Press, 1993

6. Wolfram Research http://mathworld.wolfram.com/

 Thanks to the following people for all their help: 

Professor David CooperDr. Mireille Boutin

Andrew Willis