shape matching for model alignment 3d scan matching and registration, part i iccv 2005 short course...
TRANSCRIPT
![Page 1: Shape Matching for Model Alignment 3D Scan Matching and Registration, Part I ICCV 2005 Short Course Michael Kazhdan Johns Hopkins University](https://reader035.vdocuments.site/reader035/viewer/2022081515/56649e795503460f94b798dd/html5/thumbnails/1.jpg)
Shape Matching for Model Alignment3D Scan Matching and Registration, Part I
ICCV 2005 Short Course
Michael Kazhdan
Johns Hopkins University
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Goal
Given two partially overlapping scans, compute the transformation that merges the two.
Partially Overlapping Scans Aligned Scans
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Approach
1. (Find feature points on the two scans)
Partially Overlapping Scans
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Approach
1. (Find feature points on the two scans)
2. Establish correspondences
Partially Overlapping Scans
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Approach
1. (Find feature points on the two scans)
2. Establish correspondences
3. Compute the aligning transformation
Aligned ScansPartially Overlapping Scans
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Outline
Global-Shape Correspondence– Shape Descriptors– Alignment
Partial-Shape/Point Correspondence– From Global to Local– Pose Normalization– Partial Shape Descriptors
Registration– Closed Form Solutions– Branch and Bound– Random Sample Consensus
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Correspondence
Goal:
Identify when two points on different scans represent the same feature
?
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Correspondence
Goal:
Identify when two points on different scans represent the same feature:
Are the surrounding regions similar?
?
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Global Correspondence
More Generally:
Given two models, determine if they represent the same/similar shapes.
≈?
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Global Correspondence
More Generally:
Given two models, determine if they represent the same/similar shapes.
≈?
Models can have different: representations, tessellations, topologies, etc.
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Global Correspondence
Approach:
1. Represent each model by a shape descriptor:
– A structured abstraction of a 3D model– That captures salient shape information
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Global Correspondence
Approach:
1. Represent each model by a shape descriptor.
2. Compare shapesby comparing theirshape descriptors.
≈?
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Shape Descriptors: Examples
Shape HistogramsShape descriptor stores a histogram of how
much surface area resides within different concentric shells in space.
ModelShape Histogram (Sectors + Shells)
Represents a 3D model by a 1D (radial) array of values
[Ankerst et al. 1999]
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Shape Descriptors: Examples
Shape HistogramsShape descriptor stores a histogram of how
much surface area resides within different sectors in space.
ModelShape Histogram (Sectors + Shells)
[Ankerst et al. 1999]
Represents a 3D model by a 2D (spherical) array of values
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Shape Descriptors: Examples
Shape HistogramsShape descriptor stores a histogram of how
much surface area resides within different shells and sectors in space.
ModelShape Histogram (Sectors + Shells)
[Ankerst et al. 1999]
Represents a 3D model by a 3D (spherical x radial) array of values
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Shape Descriptors: Challenge
The shape of a model does not change when a rigid body transformation is applied to the model.
Translation
Rotation
≈
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Shape Descriptors: Challenge
In order to compare two models,we need to compare themat their optimal alignment.
-
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Shape Descriptors: Challenge
In order to compare two models,we need to compare themat their optimal alignment.
-
-
-
-
min- =
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Shape Descriptors: Alignment
Three general methods:– Exhaustive Search– Normalization– Invariance
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Shape Descriptors: Alignment
Exhaustive Search:– Compare at all alignments
Exhaustive search for optimal rotation
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Shape Descriptors: Alignment
Exhaustive Search:– Compare at all alignments– Correspondence is determined by the
alignment at which models are closest
Exhaustive search for optimal rotation
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Shape Descriptors: Alignment
Exhaustive Search:– Compare at all alignments– Correspondence is determined by the
alignment at which models are closest
Properties:– Gives the correct answer– Even with fast signal processing tools, it is
hard to do efficiently
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Shape Descriptors: Alignment
Normalization:Put each model into a canonical frame– Translation:– Rotation:
Translation
Rotation
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Shape Descriptors: Alignment
Normalization:Put each model into a canonical frame– Translation: Center of Mass– Rotation:
Initial Models Translation-Aligned Models
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Shape Descriptors: Alignment
Normalization:Put each model into a canonical frame– Translation: Center of Mass– Rotation: PCA Alignment
Translation-Aligned Models Fully Aligned Models
PCA Alignment
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Shape Descriptors: Alignment
Normalization:Put each model into a canonical frame– Translation: Center of Mass– Rotation: PCA Alignment
Properties:– Efficient– Not always robust– Cannot be used for feature matching
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Shape Descriptors: Alignment
Invariance:Represent a model by a shape descriptor that is independent of the pose.
-
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Shape Descriptors: Alignment
Example: Ankerst’s ShellsA histogram of the radial distribution of surface area.
Shells Histogram
Radius
Are
a
[Ankerst et al. 1999]
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Shape Descriptors: Alignment
Invariance:Power spectrum representation– Fourier transform for translations– Spherical harmonic transform for rotations
Frequency
En
erg
y
Frequency
En
erg
y
Circular Power Spectrum Spherical Power Spectrum
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Translation Invariance
1D Function
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Translation Invariance
+ + += + …
Cosine/Sine Decomposition
1D Function
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Translation Invariance
=
+ + +
Constant
= + …
Frequency Decomposition
1D Function
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Translation Invariance
=
+ + +
+
Constant 1st Order
= + …+
Frequency Decomposition
1D Function
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Translation Invariance
=
+ + +
+ +
Constant 1st Order 2nd Order
= + …+
Frequency Decomposition
1D Function
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Translation Invariance
=
+ + +
+ + +
Constant 1st Order 2nd Order 3rd Order
= + …
+ …
+
Frequency Decomposition
1D Function
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Translation Invariance
Frequency subspaces are fixed by rotations:
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Translation Invariance
Frequency subspaces are fixed by rotations:
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Translation Invariance
Frequency subspaces are fixed by rotations:
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Translation Invariance
Frequency subspaces are fixed by rotations:
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+ + + + …+
Translation Invariance
= + + +
Constant 1st Order 2nd Order 3rd Order
+ …
Frequency Decomposition
=Amplitudes invariant
to translation1D Function
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Rotation Invariance
Represent each spherical function as a sum of harmonic frequencies (orders)
+ += +
+ + +
Constant 1st Order 2nd Order 3rd Order
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Rotation Invariance
Frequency subspaces are fixed by rotations:
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Rotation Invariance
Frequency subspaces are fixed by rotations:
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Rotation Invariance
Frequency subspaces are fixed by rotations:
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Rotation Invariance
Frequency subspaces are fixed by rotations:
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+ ++
Rotation Invariance
Store “how much” (L2-norm) of the shape resides in each frequency to get a rotation invariant representation
= + + +
Constant 1st Order 2nd Order 3rd Order
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Shape Descriptors: Alignment
Invariance:Represent a model by a shape descriptor that is independent of the pose.
Properties:– Compact representation
– Not always discriminating
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Outline
Global-Shape Correspondence– Shape Descriptors– Alignment
Partial-Shape/Point Correspondence– From Global to Local– Pose Normalization– Partial Shape Descriptors
Registration– Closed Form Solutions– Branch and Bound– Random Sample Consensus
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From Global to Local
To characterize the surface about a point p, take a global descriptor and:
– center it about p (instead of the COM), and– restrict the extent to a small region about p.
Shape histograms as local shape descriptors
p
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From Global to Local
Given scans of a model:
Scan 1 Scan 2
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From Global to Local
• Identify the features
Scan 1 Scan 2
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From Global to Local
• Identify the features• Compute a local descriptor for each feature
Scan 1 Scan 2
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From Global to Local
• Identify the features• Compute a local descriptor for each feature• Features correspond → descriptors are similar
Scan 1 Scan 2
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Pose Normalization
From Global to Local Translation: Accounted for by centering the
descriptor at the point of interest. Rotation: We still need to be able to match
descriptors across different rotations.
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Pose Normalization
Challenge– Since only parts of the models are given, we
cannot use global normalization to align the local descriptors
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Pose Normalization
Challenge– Since only parts of the models are given, we
cannot use global normalization to align the local descriptors
Solutions– Normalize using local information
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Local Descriptors: Examples
Variations of Shape Histograms:For each feature, represent its local geometry in cylindrical coordinates about the normal
nn
nn
height
radius
angle
Feature point
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Local Descriptors: Examples
Variations of Shape Histograms:For each feature, represent its local geometry in cylindrical coordinates about the normal– Spin Images: Store energy
in each normal ringn
nn
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Local Descriptors: Examples
Variations of Shape Histograms:For each feature, represent its local geometry in cylindrical coordinates about the normal– Spin Images: Store energy
in each normal ring– Harmonic Shape Contexts:
Store power spectrum ofeach normal ring
n
nn
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Local Descriptors: Examples
Variations of Shape Histograms:For each feature, represent its local geometry in cylindrical coordinates about the normal– Spin Images: Store energy
in each normal ring– Harmonic Shape Contexts:
Store power spectrum ofeach normal ring
– 3D Shape Contexts: Searchover all rotations about thenormal for best match
n
nn
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Outline
Global-Shape Correspondence– Shape Descriptors– Alignment
Partial-Shape/Point Correspondence– From Global to Local– Limitations of Global Alignment– Partial Shape Descriptors
Registration– Closed Form Solutions– Branch and Bound– Random Sample Consensus
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Registration
Ideal Case:– Every feature point on one scan has a
(single) corresponding feature on the other.
pi
qi
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Registration
Ideal Case:– Every feature point on one scan has a
(single) corresponding feature on the other.– Solve for the optimal transformation T:
n
iii qTp
1
2)(
pi
qi
[McLachlan, 1979][Arun et al., 1987][Horn, 1987][Horn et al., 1988]
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Registration
Challenge:– Even with good descriptors, symmetries in the
model and the locality of descriptors can result in multiple and incorrect correspondences.
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Registration
Exhaustive Search:– Compute alignment error at each permutation
of correspondences and use the optimal one.
n
iii
ET
pTp1
2
3
))((minargminargError
encecorrespond possible of Settionstransformabody rigid of Group3E
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Registration
Exhaustive Search:– Compute alignment error at each permutation
of correspondences and use the optimal one.
Given points {p1,…,pn} on the query, if pi matches mi different target points:
n
iii
ET
pTp1
2
3
))((minargminargError
i
n
i
m
1
||=44=256 possible permutations
encecorrespond possible of Settionstransformabody rigid of Group3E
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Branch and Bound
Key Idea:– Try all permutations but terminate early if the
alignment can be predicted to be bad.
By performing two comparisons, it was possible to eliminate 16
different possibilities
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Branch and Bound
Goal:– Need to be able to determine if the alignment
will be a good one without knowing all of the correspondences.
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Branch and Bound
Goal:– Need to be able to determine if the alignment
will be a good one without knowing all of the correspondences.
Observation:– Alignment needs to
preserve the lengthsbetween points in a single scan.
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Branch and Bound
Goal:– Need to be able to determine if the alignment
will be a good one without knowing all of the correspondences.
Observation:– Alignment needs to
preserve the lengthsbetween points in a single scan.
d1 d2
d1d2
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Random Sample Consensus
Observation:– In 3D only three pairs of corresponding points
are needed to define a transformation.
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Random Sample Consensus
Algorithm:– Randomly choose three points on source– For all possible correspondences on target:
• Compute the aligning transformation T • For every other source point p :
– Find correspondingpoint closest to T(p)
• Compute alignment error
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Random Sample Consensus
Algorithm:– Randomly choose three points on source– For all possible correspondences on target:
• Compute the aligning transformation T • For every other source point p :
– Find correspondingpoint closest to T(p)
• Compute alignment error
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Random Sample Consensus
Algorithm:– Randomly choose three points on source– For all possible correspondences on target:
• Compute the aligning transformation T • For every other source point p :
– Find correspondingpoint closest to T(p)
• Compute alignment error
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Random Sample Consensus
Algorithm:– Randomly choose three points on source– For all possible correspondences on target:
• Compute the aligning transformation T • For every other source point p :
– Find correspondingpoint closest to T(p)
• Compute alignment error
p T(p)
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Random Sample Consensus
Algorithm:– Randomly choose three points on source– For all possible correspondences on target:
• Compute the aligning transformation T • For every other source point p :
– Find correspondingpoint closest to T(p)
• Compute alignment error
Error = 0
p T(p)
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Random Sample Consensus
Algorithm:– Randomly choose three points on source– For all possible correspondences on target:
• Compute the aligning transformation T • For every other source point p :
– Find correspondingpoint closest to T(p)
• Compute alignment error
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Random Sample Consensus
Algorithm:– Randomly choose three points on source– For all possible correspondences on target:
• Compute the aligning transformation T • For every other source point p :
– Find correspondingpoint closest to T(p)
• Compute alignment error
pT(p)
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Random Sample Consensus
Algorithm:– Randomly choose three points on source– For all possible correspondences on target:
• Compute the aligning transformation T • For every other source point p :
– Find correspondingpoint closest to T(p)
• Compute alignment error
![Page 80: Shape Matching for Model Alignment 3D Scan Matching and Registration, Part I ICCV 2005 Short Course Michael Kazhdan Johns Hopkins University](https://reader035.vdocuments.site/reader035/viewer/2022081515/56649e795503460f94b798dd/html5/thumbnails/80.jpg)
Random Sample Consensus
Algorithm:– Randomly choose three points on source– For all possible correspondences on target:
• Compute the aligning transformation T • For every other source point p :
– Find correspondingpoint closest to T(p)
• Compute alignment error
Error > 0
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Summary
Global-Shape Correspondence– Shape Descriptors
• Shells (1D)• Sectors (2D)• Sectors & Shells (3D)
– Alignment• Exhaustive Search• Normalization• Invariance
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Summary
Partial-Shape/Point Correspondence– From Global to Local
• Center at feature• Restrict extent
– Pose Normalization• Normal-based alignment
– Partial Shape Descriptors• Normalization/invariance• Normalization/exhaustive-search
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Summary
Registration– Closed Form Solutions
• Global Symmetry• Local self similarity
– Branch and Bound• Inter-feature distances for early termination
– Random Sample Consensus• Efficient transformation computation