Partial Shape Matching
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Partial Shape MatchingPartial Shape Matching
Hu JianweiHu Jianwei2006-10-112006-10-11
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Shape RetrievalShape Retrieval
3D Query
3D Database
Best Match
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Global Shape MatchingGlobal Shape Matching
Skeleton Based SimilaritySkeleton Based Similarity Reeb Graph Based SimilarityReeb Graph Based Similarity
Shape HistogramsShape Histograms Light Field Descriptor Light Field Descriptor Extended Gaussian Image Extended Gaussian Image …………
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MergingMerging
Aligned ScansPartially Overlapping Scans
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The same class with different overall shapesThe same class with different overall shapes
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Partial Shape MatchingPartial Shape Matching
Partial matching is a much harder problem than global matching, since it needs to search for and define the subparts prior to measuring similarities.
Exhaustive Search?
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Partial Shape MatchingPartial Shape Matching
SegmentationSegmentation
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Salient Geometric Features for Partial Salient Geometric Features for Partial Shape Matching and SimilarityShape Matching and Similarity
ACM Transactions on Graphics, Vol. 25, No. 1, January 2006, Pages 130–ACM Transactions on Graphics, Vol. 25, No. 1, January 2006, Pages 130–150.150.
Ran Gal & Daniel Cohen-OrRan Gal & Daniel Cohen-Or
Tel-Aviv UniversityTel-Aviv University
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OutlineOutline
Local Surface DescriptorsLocal Surface Descriptors Salient Geometric FeaturesSalient Geometric Features Indexing and Geometric HashingIndexing and Geometric Hashing
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Quadric Fitting and Curvature EstimationQuadric Fitting and Curvature Estimation
Quadric FittingQuadric FittingDouros and Buxton [2002]Douros and Buxton [2002]
Curvature EstimationCurvature EstimationOhtake et al [2004]Ohtake et al [2004]
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Compact RepresentationCompact Representation
Defining the Local PatchesDefining the Local Patches Error Measuring: Squared Algebraic Error Measuring: Squared Algebraic
DistancesDistances Threshold: of the model bounding box Threshold: of the model bounding box
diagonal lengthdiagonal length
410
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Local Surface DescriptorsLocal Surface Descriptors
A Patch (Points)A Patch (Points) A PointA Point A CurvatureA Curvature
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Gaussian Curvature vs Max CurvatureGaussian Curvature vs Max Curvature
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Sample the Surface RandomlySample the Surface Randomly
Osada et al [2001]Osada et al [2001] Elad et al [2001]Elad et al [2001]
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OutlineOutline
Local Surface DescriptorsLocal Surface Descriptors Salient Geometric FeaturesSalient Geometric Features Indexing and Geometric HashingIndexing and Geometric Hashing
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Salient Geometric FeaturesSalient Geometric Features
A set of descriptors that have:A set of descriptors that have: A high curvature relative to their surroundingsA high curvature relative to their surroundings A high variance of curvature valuesA high variance of curvature values
A function of one free parameter: the scaleA function of one free parameter: the scale
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Saliency GradeSaliency Grade
31 2( ) ( ) ( ) ( )S Area d Curv d N F Var F The area of the patch associated with d relative to the sphere sizeThe curvature associated with dThe number of local minimums or
maximums curvatures in the cluster
The curvature variance in the cluster
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Saliency GradeSaliency Grade
31 2( ) ( ) ( ) ( )S Area d Curv d N F Var F
The saliency of the regionThe degree of interestingness of the cluster
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Saliency GradeSaliency Grade
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Saliency GradeSaliency Grade
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OutlineOutline
Local Surface DescriptorsLocal Surface Descriptors Salient Geometric FeaturesSalient Geometric Features Indexing and Geometric HashingIndexing and Geometric Hashing
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Indexing and Geometric HashingIndexing and Geometric Hashing
Each salient feature is associated with a Each salient feature is associated with a vector indexvector index
The vector index is defined by the four The vector index is defined by the four terms: terms:
Geometric hash table contains the vector Geometric hash table contains the vector indicesindices
( ), ( ), ( ), ( )Area d Curv d N F Var F
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Best TransformationBest Transformation
Compute the transformationCompute the transformation Triplet of pointsTriplet of points Voting system Voting system [Lamdan and Wolfson 1988][Lamdan and Wolfson 1988]
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Time and StorageTime and Storage
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Self-SimilaritySelf-Similarity
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Shape AlignmentShape Alignment
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Partial Shape RetrievalPartial Shape Retrieval
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Retrieval ResultRetrieval Result
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Spin Image vs SGFsSpin Image vs SGFs
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Registration of Synthetic ScansRegistration of Synthetic Scans
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ReconstructionReconstruction
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Local Feature Extraction and Local Feature Extraction and Matching Partial ObjectsMatching Partial Objects
Computer-Aided Design 38 (2006) 1020–1037Computer-Aided Design 38 (2006) 1020–1037
Dmitriy Bespalov, William C. Regli & Ali ShokoufandehDmitriy Bespalov, William C. Regli & Ali Shokoufandeh
Drexel UniversityDrexel University
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Traditional CAD RepresentationTraditional CAD Representation
Watertight boundary-Watertight boundary-representation solidrepresentation solid Implicit surfacesImplicit surfaces Analytic surfacesAnalytic surfaces NURBS, etcNURBS, etc
Topologically and Topologically and geometrically consistentgeometrically consistent
Produced by kernel Produced by kernel modelers and CAD modelers and CAD systemssystems
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Traditional CAD RepresentationTraditional CAD Representation
Usually a mesh or point Usually a mesh or point cloudcloud
Usually an approximate Usually an approximate representationrepresentation
Sometimes error proneSometimes error prone Produced by CAD Produced by CAD
systems, animation tools, systems, animation tools, laser scanners, etclaser scanners, etc
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What is a Scale-Space What is a Scale-Space Representation?Representation?
Commonly used for Coarse-to-Commonly used for Coarse-to-Fine representations of an objectFine representations of an object
Very popular in Computer VisionVery popular in Computer Vision Basic Idea:Basic Idea:
At each At each scalescale, topologically relevant , topologically relevant components will components will decomposedecompose the the object into so called object into so called salient partssalient parts
RecursiveRecursive application of this application of this paradigm will create the object’s paradigm will create the object’s scale space hierarchyscale space hierarchy
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Why Scale-Space Representation?Why Scale-Space Representation?
A unified framework for matchingA unified framework for matching Different features can be parameterized as Different features can be parameterized as
different scale-space decompositionsdifferent scale-space decompositions Robust & consistent across noisy and diverse Robust & consistent across noisy and diverse
data setsdata sets
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MethodMethod
Start with CAD modelStart with CAD model Perform geometry-based decompositionPerform geometry-based decomposition Construct hierarchical “feature” graphConstruct hierarchical “feature” graph Use hierarchical matching to compare graphsUse hierarchical matching to compare graphs
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Algorithm Overview (I)Algorithm Overview (I)
1. Given model P, compute mesh representation M
2. Define measurement function: Our d is the maximum angle on an angular shortest path distance function between every two faces on M will be captured in a pair-wise distance matrix D.
d(t1,t2)
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Algorithm Overview (II)Algorithm Overview (II)
3. 3. DecomposeDecompose MM into components relevant using a into components relevant using a sisingular value decompositionngular value decomposition of distance matrix of distance matrix DD
Compute the SVD decompositionCompute the SVD decomposition
with with
Compute the order-k compression matrixCompute the order-k compression matrix
Let denote the jLet denote the jthth column of , column of ,
Form sub-feature as the union of facesForm sub-feature as the union of faces
withwith
TD U V
1 2( , , , )nDiag
( )1( , , ,0, ,0)k T
kD UDiag V
jC ( )kD
1{ , , }kS c c
jM
it M ( , ) ( , )i i jd t S d t c
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Algorithm Overview (III)Algorithm Overview (III)
4. Recursive 4. Recursive feature decompositionfeature decomposition using two using two principle components creates binary principle components creates binary feature feature treestrees
feature tree for swivelfeature tree for simple_bracket
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Algorithm Overview (IV)Algorithm Overview (IV)
5. Compare feature trees (bottom up dynamic programming) using The Largest Common Subgraph Algorithm [Ullmann JR. 1976]
swivelsimple_bracket
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When to Stop?When to Stop?
2 3,i jt M t M m l i jt t t t
. .s t
2 3 ( , ) ( , )m l m l i jt M t M t t D t t
The feature is decomposed into sub-features and if the angular distance between components of and is large.
1M
2M 3M
2M3M
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Example DecompositionExample Decomposition
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Feature Decomposition on Noisy DataFeature Decomposition on Noisy Data
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Partial Shape MatchingPartial Shape Matching
Retrieved Relevant models
Relevant models
andrecall
Retrieved Relevant models
Retrieved models
andprecision
A precision–recall graph for retrieval experiment using: 1. A Reeb Graph technique; 2. a Scale–Space technique with the max-angle distance function and simple sub-graph isomorphism for matching; 3. the original Scale–Space technique with a geodesic distance function; 4. a random retrieval technique.
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Fidelity ExperimentsFidelity Experiments
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Retrieval experimentRetrieval experiment
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Partial Shape Matching of 3D Partial Shape Matching of 3D Shapes with Priority-Driven SearchShapes with Priority-Driven Search
Eurographics Symposium on Geometry Processing (2006)Eurographics Symposium on Geometry Processing (2006)
T.Funkhouser & P.ShilaneT.Funkhouser & P.Shilane
Princeton UniversityPrinceton University
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System ExecutionSystem Execution
Preprocessing phasePreprocessing phase Constructing RegionsConstructing Regions Computing Shape DescriptorsComputing Shape Descriptors Selecting Distinctive FeaturesSelecting Distinctive Features
Query phaseQuery phase Creating Pairwise Feature CorrespondencesCreating Pairwise Feature Correspondences Searching for the Optimal Multi-Feature MatchSearching for the Optimal Multi-Feature Match
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Constructing RegionsConstructing Regions
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Computing Shape DescriptorsComputing Shape Descriptors
Generate and store a representation of the
shape for each spherical region Spin image Spin image [Johnson A. & Hebert M. 1999][Johnson A. & Hebert M. 1999]
Extended Gaussian Image Extended Gaussian Image [Horn, 1984][Horn, 1984]
Spherical Harmonics Spherical Harmonics [Frome A. et al. 2004][Frome A. et al. 2004]
…………
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Selecting Distinctive FeaturesSelecting Distinctive Features
Compute the difference between every two Compute the difference between every two descriptors in the data basedescriptors in the data base
Build a rank-to-difference mapping (RTD)Build a rank-to-difference mapping (RTD) Distinctive Feature: both similar to features in Distinctive Feature: both similar to features in
few other objects (ones in the same class) and few other objects (ones in the same class) and different from the rest (objects in other classes)different from the rest (objects in other classes)
2L
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Discounted Cumulative GainDiscounted Cumulative Gain
1
12
, 1
,log ( )
i ii
G i
DCG GDCG otherwise
i
22
11
log ( )
M
C
j
DCGDCG
j
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Selecting Distinctive FeaturesSelecting Distinctive Features
The selection algorithm iteratively chooses The selection algorithm iteratively chooses the feature with highest DCG whose positithe feature with highest DCG whose position is not closer than a Euclidean distance on is not closer than a Euclidean distance threshold to the position of any previously threshold to the position of any previously selected featureselected feature
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Result of Result of Preprocessing PhasePreprocessing Phase
A small set of featuresA small set of features Each feature associated with Each feature associated with
A positionA position A normalA normal A radiusA radius A shape descriptorA shape descriptor A rank-to-difference mapping (RTD)A rank-to-difference mapping (RTD) A retrieval performance score (DCG)A retrieval performance score (DCG)
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The Cost FunctionThe Cost Function
1 2( , ) ( , )rankC Rank RTD D Rank RTD D
1 2
1 2radius
R RC
RAVG RAVG
1 1 2 2normalC r n r n ��������������������������������������������������������
rank radius normalcorrespondence rank rank radius radius normal normalC C C C
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The Cost FunctionThe Cost Function1 2
1 2
1 2
1 2
max( , )length
L LLAVG LAVG
CL L
LAVG LAVG
1 1 2 2 1 1 2 2 1 1 2 2orient a b a b a a b bC n n n n v n v n v n v n ������������������������������������������������������������������������������������������������������������������������������������������������������������������������
length orientchord length length orient orientC C C
, ,
( ) ( , )match correspondence chordi k i j k i j
C C i C i j
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Query PhaseQuery Phase
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Comparison to Previous MethodsComparison to Previous Methods