joshua i. cohen brown university september 2001 – may 2002 “computational procedures for...
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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 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
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
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