Fourier Descriptors For Shape Recognition Applied to Tree Leaf Identification By Tyler Karrels.
Shape Descriptors I
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Transcript of Shape Descriptors I
Shape Descriptors IShape Descriptors I
Thomas Funkhouser
CS597D, Fall 2003Princeton University
Thomas Funkhouser
CS597D, Fall 2003Princeton University
3D Representations
What properties are required for analysis and retrieval?
Intuitive specification Yes No No NoGuaranteed continuity Yes No No NoGuaranteed validity Yes No No NoEfficient boolean operations Yes No No NoEfficient rendering Yes Yes No NoAccurate Yes Yes ? ?Concise ? ? ? YesStructure Yes Yes Yes Yes
Edi
ting
Dis
play
Ana
lysi
s
Ret
riev
al
Property
Shape Analysis Problems
Examples:• Feature detection• Segmentation• Labeling• Registration• MatchingRetrieval• Recognition• Classification• Clustering
“How can we find 3D models best matching a query?”“How can we find 3D models best matching a query?”
1)
2)
3)
4)
Query
Ranked Matches
Shape
Definition from Merriam-Webster’s Dictionary:• a : the visible makeup characteristic of a
particular item or kind of item b : spatial form or contour
Shape
Shape is independent of similarity transformation
(rotation, scale, translation, mirror)
=
Shape Similarity
Need a shape distance function d(A,B) that:• matches our intuitive notion of shape similarity• can be computed robustly and efficiently
Perhaps, shape distance function should be a metric:• Non-negative: d(A,B) 0 for all A and B• Identity: d(A,B) = 0 if and only if
A=B• Symmetry: d(A,B) = d(B,A) for all A
and B• Triangle inequality: d(A,B) + d(B,C) d(A,C)
Example Distance Functions
Lp norm:
Hausdorff distance:
Others (Fréchet, etc.)
pp
ii baBAd1
),(
),(~
),,(~
max),(
minmax),(~
ABdBAdBAd
baBAd iiBbAa
Shape Matching
Compute shape distance function for pair of 3D models• Can matching two objects• Can find most similar object among a small set
Are these the same chair?
Shape Retrieval
Find 3D models with shape most similar to query• Searching large database must take less than O(n)
Is this blue chair in the database?
Shape Retrieval
Build searchable shape index
ShapeRetrieval
SimilarObjects
ShapeIndex
ShapeDescriptor
ShapeAnalysis
ShapeAnalysis
Databaseof
3D Models
GeometricQuery
Shape Retrieval
Find 3D models with shape similar to query
3D Query
3D Database
Best Matches
Challenge
Need shape descriptor that is:• Concise to store• Quick to compute• Efficient to match• Discriminating
3D Query ShapeDescriptor
3D Database
BestMatches
Challenge
Need shape descriptor that is:Concise to store• Quick to compute• Efficient to match• Discriminating
3D Database
3D Query ShapeDescriptor
BestMatches
Challenge
Need shape descriptor that is:• Concise to storeQuick to compute• Efficient to match• Discriminating
3D Database
3D Query ShapeDescriptor
BestMatches
Challenge
Need shape descriptor that is:• Concise to store• Quick to computeEfficient to match• Discriminating
3D Database
3D Query ShapeDescriptor
BestMatches
Challenge
Need shape descriptor that is:• Concise to store• Quick to compute• Efficient to matchDiscriminating
3D Database
3D Query ShapeDescriptor
BestMatches
Challenge
Need shape descriptor that is:• Concise to store• Quick to compute• Efficient to match• Discriminating Invariant to transformations• Insensitive to noise• Insensitive to topology• Robust to degeneracies
Different Transformations(translation, scale, rotation, mirror)
Challenge
Need shape descriptor that is:• Concise to store• Quick to compute• Efficient to match• Discriminating• Invariant to transformations Insensitive to noise• Insensitive to topology• Robust to degeneracies
Scanned Surface
Image courtesy ofRamamoorthi et al.
Challenge
Need shape descriptor that is:• Concise to store• Quick to compute• Efficient to match• Discriminating• Invariant to transformations• Insensitive to noise Insensitive to topology• Robust to degeneracies
Images courtesy of Viewpoint & Stanford
Different Tessellations
Different Genus
Challenge
Need shape descriptor that is:• Concise to store• Quick to compute• Efficient to match• Discriminating• Invariant to transformations• Insensitive to noise• Insensitive to topologyRobust to degeneracies
Images courtesy of Utah & De Espona
No Bottom!
&*Q?@#A%!
Taxonomy of Shape Descriptors
Structural representations• Skeletons• Part-based methods• Feature-based methods
Statistical representations• Voxels, moments, wavelets, …• Attributes, histograms, ...• Point descriptors
Taxonomy of Shape Descriptors
Structural representations• Skeletons• Part-based methods• Feature-based methods
Statistical representations• Voxels, moments, wavelets, …• Attributes, histograms, ...• Point descriptors
Images courtesy of Amenta & Osada
Taxonomy of Shape Descriptors
Structural representations• Skeletons• Part-based methods• Feature-based methods
Statistical representations• Voxels, moments, wavelets, …• Attributes, histograms, ...• Point descriptors
Image courtesy of De Espona
?
Taxonomy of Shape Descriptors
Structural representations• Skeletons• Part-based methods• Feature-based methods
Statistical representations• Voxels, moments, wavelets, …• Attributes, histograms, ...• Point descriptors
?
Statistical Shape Descriptors
Alignment-dependent• Voxels• Wavelets• Moments• Extended Gaussian
Image• Spherical Extent
Function• Spherical Attribute
Image
Alignment-independent• Shape histograms• Harmonic descriptor• Shape distributions
Feature Vectors
Map shape onto point in multi-dimensional space• Similarity measure is distance in feature space
Feature 2
Fea
ture
1
File cabinets
Tables
Desks
Image courtesy ofMao Chen
Feature Vectors
Cluster, classify, recognize, and retrieve similarfeature vectors using standard methods
Feature 2
Fea
ture
1
File cabinets
Tables
Desks
Image courtesy ofMao Chen
What feature vectors?
Voxels
Use voxel values as feature vector (shape descriptor)• Feature space has N3 dimensions
(one dimension for each voxel)
• d(A,B) = ||A-B||N
Example:
( )d =,
NA B A-B
Voxels
Can store distance transform (DT) in voxels
• ||A-DT(B)||1 represents sum of distances from every point on surface of A to closest point on surface of B
Distance TransformSurface
Image courtesy ofMisha Kazhdan
Voxels
Can store distance transform (DT) in voxels
• ||A-DT(B)||1 represents sum of distances from every point on surface of A to closest point on surface of B
Distance TransformSurface
Image courtesy ofMisha Kazhdan
Voxels
Can build hierarchical search structure• e.g., interior nodes store MIV and MSV
Image courtesy ofDaniel Keim, SIGMOD 1999
Voxel Retrieval Experiment
Test database is Viewpoint household collection1,890 models, 85 classes
153 dining chairs 25 livingroom chairs 16 beds 12 dining tables
8 chests 28 bottles 39 vases 36 end tables
Evaluation Metric
Precision-recall curves• Precision = retrieved_in_class / total_retrieved• Recall = retrieved_in_class / total_in_class
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
Recall
Pre
cisi
on
1
Evaluation Metric
Precision-recall curves• Precision = 0 / 0• Recall = 0 / 5
44 55 66
77
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
Recall
Pre
cisi
on
1
11 22 33
9988
Ranked Matches
Query
Evaluation Metric
Precision-recall curves• Precision = 1 / 1• Recall = 1 / 5
44 55 66
77
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
Recall
Pre
cisi
on
1
11 22 33
9988
Ranked Matches
Query
Evaluation Metric
Precision-recall curves• Precision = 2 / 3• Recall = 2 / 5
44 55 66
77
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
Recall
Pre
cisi
on
1
11 22 33
9988
Ranked Matches
Query
Evaluation Metric
Precision-recall curves• Precision = 3 / 5• Recall = 3 / 5
44 55 66
77
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
Recall
Pre
cisi
on
1
11 22 33
9988
Ranked Matches
Query
Evaluation Metric
Precision-recall curves• Precision = 4 / 7• Recall = 4 / 5
44 55 66
77
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
Recall
Pre
cisi
on
1
11 22 33
9988
Ranked Matches
Query
Evaluation Metric
Precision-recall curves• Precision = 5 / 9• Recall = 5 / 5
44 55 66
77
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
Recall
Pre
cisi
on
1
11 22 33
9988
Ranked Matches
Query
Voxel Retrieval Experiment
Test database is Viewpoint household collection1,890 models, 85 classes
153 dining chairs 25 livingroom chairs 16 beds 12 dining tables
8 chests 28 bottles 39 vases 36 end tables
Voxel Retrieval Results
0 0.2 0.4 0.6 0.8
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0.6
0.8
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Recall
Pre
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Voxels
Random
Voxels
PropertiesDiscriminating Insensitive to noise Insensitive to topologyRobust to degeneraciesQuick to compute• Efficient to match?X Concise to storeX Invariant to transforms
Wavelets
Define shape with wavelet coefficients
16,000 coefficients 400 coefficients 100 coefficients 20 coefficients
Image courtesy ofJacobs, Finkelstein, & Salesin
Wavelets
Descriptor 1:• Given an NxNxN grid, generate an NxNxN array of
the wavelet coefficients for the standard Haar basis functions
Jacobs, Finkelstein, & SalesinSIGGRAPH 95
Wavelets
Descriptor 1:• Given an NxNxN grid, generate an NxNxN array of
the wavelet coefficients for the standard Haar basis functions
Descriptor 2:• Truncate: Find the m largest coefficients and set
all others equal to zero• Quantize: Set the non-zero coefficients to +1 or –1
depending on their sign
Jacobs, Finkelstein, & SalesinSIGGRAPH 95
Jackie Chan Example
Original Image (256x256)
Truncated And Quantized to 5000
Truncated And Quantized to 1000
Truncated And Quantized to 500
Truncated 100
Truncated 50
Truncated 10
Torus Example
Torus Truncated to 5000
Torus Truncated to 1000
Torus Truncated to 500
Torus Truncated to 100
Torus Truncated to 50
Wavelets
Distance Function 1:• The query metric is defined by:
where A[i,j,k] and B[i,j,k] are the truncated and quantized coefficients and wi,j,k are weights, fine tuned to the database.
kji
kji kjiBkjiAwBAd,,
,, ,,,,),(
Jacobs, Finkelstein, & SalesinSIGGRAPH 95
Wavelets
Distance Function 2:• The query metric can be approximated by:
to enable efficient indexing and search.
0),,(:,,
,, ),,,,(),(kjiAkji
kji kjiBkjiAwBAd
Jacobs, Finkelstein, & SalesinSIGGRAPH 95
Wavelets
Properties Insensitive to noise Insensitive to topologyRobust to degeneraciesQuick to computeEfficient to matchConcise to store• Discriminating?X Invariant to transforms
Jacobs, Finkelstein, & SalesinSIGGRAPH 95
Moments
Define shape by moments of inertia:
surface
rqppqr dxdydzzyxm
Moments Retrieval Experiment
Test database is Viewpoint household collection1,890 models, 85 classes
153 dining chairs 25 livingroom chairs 16 beds 12 dining tables
8 chests 28 bottles 39 vases 36 end tables
Moments Retrieval Results
0 0.2 0.4 0.6 0.8
0
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0.6
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Recall
Pre
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1
Voxels
Moments [Elad et al.]
Random
Moments Retrieval Results
0 0.2 0.4 0.6 0.8
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Recall
Pre
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Voxels
Moments [Elad et al.]
Random
Moments
Properties Insensitive to topologyRobust to degeneraciesQuick to computeEfficient to matchConcise to storeX Insensitive to noiseX Invariant to transformsX Discriminating
Extended Gaussian Image
Define shape with histogram of normal directions• Invertible for convex objects• Spherical function
3D Model EGI
EGI Retrieval Experiment
Test database is Viewpoint household collection1,890 models, 85 classes
153 dining chairs 25 livingroom chairs 16 beds 12 dining tables
8 chests 28 bottles 39 vases 36 end tables
EGI Retrieval Results
0 0.2 0.4 0.6 0.8
0
0.2
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0.6
0.8
1
Recall
Pre
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Voxels
Moments [Elad et al.]
EGI [Horn 84]
Random
Extended Gaussian Images
Properties Insensitive to topologyQuick to computeEfficient to matchConcise to storeX Insensitve to noiseX Robust to degeneraciesX Invariant to transformsX Discriminating
Other Rotation-Dependent Descriptors
Spherical Extent Functions(Vranic & Saupe, 2000)
Shape Histograms (sectors)(Ankherst, 1999)
Shape Descriptors IIShape Descriptors II
Thomas Funkhouser
CS597D, Fall 2003Princeton University
Thomas Funkhouser
CS597D, Fall 2003Princeton University
Taxonomy of Shape Descriptors
Structural representations• Skeletons• Part-based methods• Feature-based methods
Statistical representations• Voxels, moments, wavelets, …• Attributes, histograms, ...• Point descriptors
Statistical Shape Descriptors
Alignment-dependent• Voxels• Wavelets• Moments• Extended Gaussian
Image• Spherical Extent
Function• Spherical Attribute
Image
Alignment-independent• Shape histograms• Harmonic descriptor• Shape distributions
Statistical Shape Descriptors
Alignment-dependent• Voxels• Wavelets• Moments• Extended Gaussian
Image• Spherical Extent
Function• Spherical Attribute
Image
Alignment-independent• Shape histograms• Harmonic descriptor• Shape distributions
Alignment
Translation (Center of Mass)
Scale (Radial Deviation)
n
iip
nc
1
1
n
iip
ns
1
21
Alignment
Rotation (PCA)• Principal axes are eigenvectors associated with
largest eigenvalues of 2nd order moments covariance matrix
PCAComputation
Principal Axis Alignment
Alignment
Rotation (PCA)• Principal axes are eigenvectors associated with
largest eigenvalues of 2nd order moments covariance matrix
Not very robust!
Alignment
Mirror• PCA does not give directions for principal axes
Need heuristics to determine positive axes!
Alignment-Independent Descriptors
Observation: it is difficult to normalize for differences in rotation and mirroring
Motivation: build a shape descriptor that is invariant to rotations and mirrors and as discriminating as possible
Three mugs aligned automatically with PCA
Shape Histograms
Shape descriptor stores histogram of how much surface resides at different radii from center of mass
Image courtesy of Ankerst et al, 1999
Shape Histograms (shells)(Ankherst, 1999)
Radius
Shape Histograms
Shape descriptor stores histogram of how much surface resides at different radii from center of mass
Image courtesy of Misha Kazhdan
ShapeDescriptor
3D Model SphericalDecomposition
0.7
0.3
0.1
Shape Histogram Experiment
Test database is Viewpoint household collection1,890 models, 85 classes
153 dining chairs 25 livingroom chairs 16 beds 12 dining tables
8 chests 28 bottles 39 vases 36 end tables
Shape Histogram Retrieval Results
Precision-recall curves (mean for all queries)
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
Recall
Pre
cisi
onShape Histogram [Ankerst et al.]
EGI [Horn]
Moments [Elad et al.]
Random
1
Shape Histograms
Properties Insensitive to noise Insensitive to topologyRobust to degeneraciesQuick to computeEfficient to matchConcise to store Invariant to rotations• Discriminating?
Harmonic Shape Descriptor
Key idea:• Decompose each sphere into irreducible
set of rotation independent components• Store “how much” of the model resides
in each component
3D Model ShapeDescriptor
HarmonicDecompositions
Step 1: Normalization
Normalize for translation and scale
3D Model
Step 2: Voxelization
Rasterize polygon surfaces into 3D voxel grid
3D Voxel Grid
Step 3: Spherical Decomposition
Intersect with concentric spheres
Spherical Functions
Step 4: Frequency Decomposition
Represent each spherical function as a sum of harmonic frequencies (orders)
Spherical Functions
Represent each spherical function as a sum of harmonic frequencies (orders)
Step 4: Frequency Decomposition
SphericalFunctionSphericalFunction
Spherical Functions
Represent each spherical function as a sum of harmonic frequencies (orders)
Step 4: Frequency Decomposition
+ + += …SphericalFunction
Harmonic Decomposition
Represent each spherical function as a sum of harmonic frequencies (orders)
Step 4: Frequency Decomposition
=
+ + +
+ + +
Constant 1st Order 2nd Order
= …
…
SphericalFunction
Represent each spherical function as a sum of harmonic frequencies (orders)
Step 4: Frequency Decomposition
=
+ + +
+ + +
Frequency Decomposition
= …
…
SphericalFunction
Amplitudes are invariant to rotation
Step 5: Amplitude Computation
Store “how much” (L2-norm) of the shape resides in each harmonic frequency of each sphere
Frequency Radius
Harmonic Shape Descriptor
Matching Harmonic Descriptors
Define similarity as L2-distance between descriptors• Enables nearest neighbor indexing and fast search
• Provides lower bound for L2-distance between models
, = -
-
-
-
Sim
Harmonic Shape Descriptor
PropertiesConcise to store?• Quick to compute?• Insensitive to noise?• Insensitive to topology?• Robust to degeneracies?• Invariant to transforms?• Efficient to match?• Discriminating?
Frequency Radius
2048 bytes per model(16 frequencies x 32 radii x 4 bytes)
Harmonic Shape Descriptor
PropertiesConcise to storeQuick to compute?• Insensitive to noise?• Insensitive to topology?• Robust to degeneracies?• Invariant to transforms?• Efficient to match?• Discriminating?
1.6
seco
nd
s (o
n
avera
ge)
Polygons
Voxels
SphericalDecomposition
FrequencyDecomposition
HarmonicShapeDescriptorfrequency radius
Harmonic Shape Descriptor
PropertiesConcise to storeQuick to compute?• Insensitive to noise?• Insensitive to topology?• Robust to degeneracies?• Invariant to transforms?• Efficient to match?• Discriminating?
1.6
seco
nd
s (o
n
avera
ge)
Polygons
Voxels
SphericalDecomposition
FrequencyDecomposition
HarmonicShapeDescriptorfrequency radius
Harmonic Shape Descriptor
PropertiesConcise to storeQuick to compute Insensitive to noise Insensitive to topologyRobust to degeneracies• Invariant to transforms?• Efficient to match?• Discriminating?
Rasterize polygon surfaces(no solid reconstruction)
Harmonic Shape Descriptor
PropertiesConcise to storeQuick to compute Insensitive to noise Insensitive to topologyRobust to degeneracies Invariant to transforms• Efficient to match?• Discriminating?
RotationMirrorTranslation (w/ normalization)Scale (w/ normalization){
Harmonic Shape Descriptor
PropertiesConcise to storeQuick to compute Insensitive to noise Insensitive to topologyRobust to degeneracies Invariant to transformsEfficient to match?• Discriminating? 0.0
0.5
1.0
1.5
2.0
0 5000 10000 15000 20000
Database size (models)
Se
arc
h t
ime
(s
ec
s)
IndexedNot In
dexed
0.23 secondsto search
17,500 models
Harmonic Shape Descriptor
PropertiesConcise to storeQuick to compute Insensitive to noise Insensitive to topologyRobust to degeneracies Invariant to transformsEfficient to match?Discriminating?
Harmonic Matching Results
Test database is Viewpoint household collection1,890 models, 85 classes
153 dining chairs 25 livingroom chairs 16 beds 12 dining tables
8 chests 28 bottles 39 vases 36 end tables
Harmonic Retrieval Results
Precision-recall curves (mean for all queries)
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
Recall
Pre
cisi
onHarmonic Shape Descriptor
Shape Histogram [Ankerst et al.]
EGI [Horn]
Moments [Elad et al.]
Random
1
Statistical Shape Descriptors
Alignment-dependent• Voxels• Wavelets• Moments• Extended Gaussian
Image• Spherical Extent
Function• Spherical Attribute
Image
Alignment-independent• Shape histograms• Harmonic descriptorShape distributions
Shape Distributions
Motivation: general approach to finding a common parameterization for matching
3D SurfaceAudio
2D Contour 3D Volume
Shape Distributions
Key idea: map 3D surfaces to common parameterization
by randomly sampling shape function
3D Models D2 Shape Distributions
Randomlysampleshape
function
SimilarityMeasure
Distance
Distance
Pro
babili
tyPro
babili
ty
Which Shape Function?
Implementation: simple shape functions based on
angles, distances, areas, and volumes
A3(angle)
D1(distance)
[Ankerst 99]
D2(distance)
D3(area)
D4(volume)
D2 Shape Distribution
Properties• Concise to store?• Quick to compute?• Invariant to transforms?• Efficient to match?• Insensitive to noise?• Insensitive to topology?• Robust to degeneracies?• Discriminating?
D2 Shape Distribution
PropertiesConcise to store?Quick to compute?• Invariant to transforms?• Efficient to match?• Insensitive to noise?• Insensitive to topology?• Robust to degeneracies?• Discriminating? 512 bytes (64 values)
0.5 seconds (106 samples)
Distance
Pro
babili
ty
Skateboard
D2 Shape Distribution
PropertiesConcise to storeQuick to compute Invariant to transforms?• Efficient to match?• Insensitive to noise?• Insensitive to topology?• Robust to degeneracies?• Discriminating?
TranslationRotationMirror{
Normalized Means
Scale (w/ normalization)
Skateboard Porsche
Distance
Pro
babili
ty
Skateboard
D2 Shape Distribution
PropertiesConcise to storeQuick to compute Invariant to transformsEfficient to match?• Insensitive to noise?• Insensitive to topology?• Robust to degeneracies?• Discriminating?
Porsche
D2 Shape Distribution
PropertiesConcise to storeQuick to compute Invariant to transformsEfficient to match Insensitive to noise? Insensitive to topology?Robust to degeneracies?• Discriminating?
1% Noise
D2 Shape Distribution
PropertiesConcise to storeQuick to compute Invariant to transformsEfficient to match Insensitive to noise Insensitive to topologyRobust to degeneraciesDiscriminating?
D2 Shape Distribution Results
Question• How discriminating are
D2 shape distributions?
Test database• 133 polygonal models• 25 classes
4 Mugs
6 Cars
3 Boats
D2 Shape Distribution Results
D2 distributions are different across classes
D2 shape distributions for 15 classes of objects
D2 Shape Distribution Results
D2 distributions for 5 tanks (gray) and 6 cars (black)
Distance
Pro
babili
ty
D2 Shape Distribution Results
Similarity Matrix• Darkness
representssimilarity
Blocks• Tanks, cars• Airplanes• Humans• Helicopters
al bl btbp bt cr cr cw hr hn lp lg me mg ok pn pe pe re sd sa sp sb te tk
animal
ball
beltblimp
boat
car
chair
claw
helicopter
human
lamp
lightning
missle
mug
openbook
pen
phone
plane
rifle
skateboard
sofa
spaceship
sub
table
tank
al bl btbp bt cr cr cw hr hn lp lg me mg ok pn pe pe re sd sa sp sb te tk
animal
ball
beltblimp
boat
car
chair
claw
helicopter
human
lamp
lightning
missle
mug
openbook
pen
phone
plane
rifle
skateboard
sofa
spaceship
sub
table
tank
D2 Retrieval Experiment
Test database is Viewpoint household collection1,890 models, 85 classes
153 dining chairs 25 livingroom chairs 16 beds 12 dining tables
8 chests 28 bottles 39 vases 36 end tables
D2 Retrieval Results
Precision-recall curves (mean for all queries)
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
Recall
Pre
cisi
onHarmonic Shape Descriptor
D2 Shape Distribution [Osada et al.]
Shape Histogram [Ankerst et al.]
EGI [Horn]
Moments [Elad et al.]
Random
1
Shape Distributions
Next steps:• Better shape functions• Better comparsion methods• Analysis apps
D2 Shape Distribution Results
D2 shape distributions for 15 classes of objectsLine Segment
Recognizing gross shapes with D2 distributions
D2 Shape Distribution Results
Recognizing gross shapes with D2 distributions
D2 shape distributions for 15 classes of objects
Circle
D2 Shape Distribution Results
Recognizing gross shapes with D2 distributions
D2 shape distributions for 15 classes of objectsCylinder
D2 Shape Distribution Results
Recognizing gross shapes with D2 distributions
D2 shape distributions for 15 classes of objects
Sphere
D2 Shape Distribution Results
Recognizing gross shapes with D2 distributions
D2 shape distributions for 15 classes of objectsTwo Spheres
Taxonomy of Shape Descriptors
Structural representations• Skeletons• Part-based methods• Feature-based methods
Statistical representations• Voxels, moments, wavelets, …• Attributes, histograms, ...• Point descriptors
Taxonomy of Shape Descriptors
Structural representations• Skeletons• Part-based methods• Feature-based methods
Statistical representations• Voxels, moments, wavelets, …• Attributes, histograms, ...Point descriptors Next Time!