1. Feature-based methods
Numerical geometry of
non-rigid
objects
Feature-based methods
Alexander Bronstein
Michael Bronstein 1

2. Two perspectives Metric geometry Computer vision
Features and local descriptors
Metric geometry
Shapes as metric spaces 2

3. Invariant retrieval Illumination View Missing data Deformation
Topology
Missing data 3

4. Bags of words
Notre Dame de Paris is a Gothic cathedral in the fourth quarter of Paris, France. It was the first Gothic architecture cathedral, and its construction spanned the Gothic period.
Notre Dame de Paris is a Gothic cathedral in the fourth quarter of Paris, France. It was the first Gothic architecture cathedral, and its construction spanned the Gothic period.
construction
architecture
Italy
France
cathedral
church
basilica
Paris
Rome
Gothic
Roman
St. Peter’s basilica is the largest church in world, located in Rome, Italy. As a work of architecture, it is regarded as the best building of its age in Italy.
St. Peter’s basilica is the largest church in world, located in Rome, Italy. As a work of architecture, it is regarded as the best building of its age in Italy. 4

5. Feature detector + descriptor
Bags of features
Visual vocabulary
Feature detector + descriptor
Invariant to changes of the image
Discriminative (tells different images apart) 5

6. Images vs shapes Images Shapes Many prominent features
Few prominent features
Affine transforms, illumination,
occlusions, resolution
Non-rigid deformations, topology,
missing parts, triangulation
SIFT, SURF, MSER, DAISY, …
Curvature, conformal factor,
local distance histograms 6

7. Diffusion scale space
can be interpreted as probability of Brownian motion to remain at the same point after time (represents “stability” of the point)
Multiscale descriptor
Time (scale)
J. Sun, M. Ovsjanikov, L. Guibas, SGP 2009
M. Ovsjanikov, BB, L. Guibas, 2009 7

8. Heat kernel descriptor represented in RGB space
Heat kernel descriptors
Heat kernel descriptor represented in RGB space
J. Sun, M. Ovsjanikov, L. Guibas, SGP 2009
M. Ovsjanikov, BB, L. Guibas, 2009 8

9. Our descriptor is… Dense Defined at every point of the shape
No feature detection
Statistical weighting (reduce the influence of “dull” points)
Intrinsic
Based on diffusion geometry, hence intrinsic
Invariant to isometric deformations
Insensitive to topological noise
Consistent
Uses a geometrically consistent discretization of the
Laplace-Beltrami operator
Insensitive to different sampling and triangulations
M. Ovsjanikov, BB, L. Guibas, 2009

10. Heat kernel descriptors
Invariant to isometric deformations
Localized sensitivity to topological noise
J. Sun, M. Ovsjanikov, L. Guibas, SGP 2009
M. Ovsjanikov, BB, L. Guibas, 2009 10

11. Geometric vocabulary
M. Ovsjanikov, BB, L. Guibas, 2009 11

12. Nearest neighbor in the descriptor space
Bags of features
Given a geometric vocabulary for each point with local descriptor find representation
Hard quantization
Soft quantization
Weighted distance to
words in vocabulary
Nearest neighbor in the descriptor space
M. Ovsjanikov, BB, L. Guibas, 2009 12

13. Bags of features
Statistics of different geometric words over the entire shape
Shape distance = distance between bags of features
M. Ovsjanikov, BB, L. Guibas, 2009 13

14. Bags of features 1 Index in vocabulary 64 14
M. Ovsjanikov, BB, L. Guibas, 2009 14

15. Expressions
In math science, matrix decomposition is a factorization of a matrix into some canonical form. Each type of decomposition is used in a particular problem.
In math science, matrix decomposition is a factorization of a matrix into some canonical form. Each type of decomposition is used in a particular problem.
In math science, matrix decomposition is a factorization of a matrix into some canonical form. Each type of decomposition is used in a particular problem.
In biological science, decomposition is the process of organisms to break down into simpler form of matter. Usually, decomposition occurs after death.
In biological science, decomposition is the process of organisms to break down into simpler form of matter. Usually, decomposition occurs after death.
Matrix is a science fiction movie released in Matrix refers to a simulated reality created by machines in order to subdue the human population.
Matrix is a science fiction movie released in Matrix refers to a simulated reality created by machines in order to subdue the human population.
Matrix is a science fiction movie released in Matrix refers to a simulated reality created by machines in order to subdue the human population.
matrix
decomposition is a
the of in to by
science
form
matrix decomposition
matrix factorization
science fiction
canonical form
M. Ovsjanikov, BB, L. Guibas, 2009 15

16. Expressions
In math science, matrix decomposition is a factorization of a matrix into some canonical form. Each type of decomposition is used in a particular problem.
In particular matrix used type a some science, decomposition form a factorization of is canonical. matrix math decomposition is in a Each problem. into of
matrix
decomposition is a
the of in to by
science
form
matrix decomposition
matrix factorization
science fiction
canonical form
M. Ovsjanikov, BB, L. Guibas, 2009 16

17. Visual expressions “Inquisitor King” Inquisitor, King
Giuseppe Verdi, Don Carlo, Metropolitan Opera 17

18. Geometric expressions
Yellow
“Yellow Yellow”
No total order between points (only “far” and “near”)
Geometric expression = a pair of spatially close geometric words
M. Ovsjanikov, BB, L. Guibas, 2009 18

19. Spatially-sensitive bags of features
Distribution of pairs of geometric words
is the probability
to find word at point and
word at point
Proximity between
points and
is the statistic of geometric expressions of the form
Shape distance
M. Ovsjanikov, BB, L. Guibas, 2009 19

20. Spatially-sensitive bags of features
M. Ovsjanikov, BB, L. Guibas, 2009 20

21. Query Positives Negatives
Iso
Topo
Triang
Part
Query
Positives
Negatives

22. Shape Google Query Match 1 Match 2 Match 3 Match 25 Iso Iso 0.060
Null 0.072
Noise 0.382
Iso
Iso+Topo 0.011
Iso 0.019
Iso+Topo 0.023
Iso 0.073
M. Ovsjanikov, BB, L. Guibas, 2009 22

23. Shape Google Query Match 1 Match 2 Match 3 Match 25 Part Part 0.292
Iso+Topo 0.372
Null 0.377
Triang 0.575
Iso+Topo
Iso 0.014
Iso+Topo 0.108
Noise 0.114
Iso+Topo 0.459
M. Ovsjanikov, BB, L. Guibas, 2009 23

24. Bags of features 100% 98% 96% True Positive Rate (TPR) 90% 80% 0.1% 1%
10%
100%
M. Ovsjanikov, BB, L. Guibas, 2009
False Positive Rate (FPR) 24

25. Spatially-sensitive bags of features
100%
98%
96%
True Positive Rate (TPR)
90%
80%
0.1% 1%
10%
100%
M. Ovsjanikov, BB, L. Guibas, 2009
False Positive Rate (FPR) 25

26. Comparison SS-BoF 8% BoF Shape DNA [Reuter et al.]
Equal Error Rate (EER) 4% 0%
All
Null
Iso.
Part.
Topo.
Triang.
Iso.+Topo.
M. Reuter et al., 2006
M. Ovsjanikov, BB, L. Guibas, 2009 26