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CG UFRGS

On the Generalization of Subspace Detection in Unordered Multidimensional Data

Leandro Augusto Frata Fernandes

laffernandes@inf.ufrgs.br

Manuel Menezes de Oliveira Neto

• liveira@inf.ufrgs.br

Advisor

Data Alignments

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Data Alignments are Everywhere

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EBSD image for Particle's Crystalline Phase Identification Robot Navigation Counting Process in Clonogenic Assays

Points in the Plane as Input

Conventional Techniques

⚫ Designed to detect specific types of data alignments ⚫ Designed to a given type of input data

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ρ = x cos θ + y sin θ

Straight-Line Detection

Normal equation of the line as model

( x – xc )2 + ( y – yc )2 – r2 = 0

Circle Detection

Center-radius parameterization as model

Drawbacks due to Specialization

• It requires different formulations
• It requires different implementations
• It prevents the development of generally

applicable optimizations

• Concurrent detection may be a challenging task

Key Observation

⚫ Linear subspaces can be

interpreted as some types of data alignments

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3-D Representational Space 2-D Base Space

Vectors can be interpreted as points or directions within the Homogeneous Model

3-D Representational Space 2-D Base Space

2-D subspaces can be interpreted as straight lines within the Homogeneous Model 3-D subspaces can be interpreted as circles within the Conformal Model

4-D Representational Space 2-D Base Space

Thesis Statement

⚫ It is possible to define a parameterization for linear

subspaces which is independent of:

▪ The dimensionality of the intended subspace ▪ The model of geometry ▪ The input data type

⚫ It is possible to develop a generalized approach for

the automatic detection of data alignments

▪ Closed-form solution ▪ It can be applied to any alignments that can be characterized by a linear subspace

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The proposed subspace detector framework was formulated with Geometric Algebra

Contributions

⚫ A general framework for subspace detection in

unordered multidimensional datasets

⚫ A new rotation-based parameterization scheme

for subspaces

⚫ Two mapping procedures for input subspaces

▪ One for (exact) input subspaces ▪ One for input subspaces with Gaussian distributed uncertainty

⚫ An algorithm that identifies local maxima in a

multidimensional histogram

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Detecting Different Types of Alignments

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Conformal Model Input : tangent directions (2-D subspaces) Output : circles (3-D subspaces) Homogeneous Model Input : points (1-D subspaces) Output : straight lines (2-D subspaces)

Detection on Heterogeneous Datasets

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Conformal Model Input : points (1-D subspaces) circles (3-D subspaces) line (3-D subspace) Output : spheres (4-D subspaces) plane (4-D subspace) Homogeneous Model Input : points (1-D subspaces) plane (3-D subspaces) Output : straight lines (2-D subspaces)

Outline

⚫ Overview ⚫ Parameterization of Subspaces ⚫ Voting Process for Input Subspaces ⚫ Voting Process for Input Subspaces with Uncertainty ⚫ Conclusions and Future Work

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Proposed Subspace Detection Framework

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Initialization Voting Process Peak Detection Input Subspaces Detected Subspaces

Parameter Space for p = 3, n = 4

Proposed Subspace Detection Framework

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• Setup the model function for p-D subspaces

in n-D representational space

• Create an accumulator array representing the

parameter space for p-D subspaces Initialization Voting Process Peak Detection Input Subspaces Detected Subspaces

Accumulator Array for p = 3, n = 4 Model function for p-D subspaces Model function for 3-D subspaces in 4-D representational space

Proposed Subspace Detection Framework

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• Input data must be encoded into the same model

as intended subspaces

• Map each input subspace to the parameter space

and increment related bins of the accumulator array Initialization Voting Process Peak Detection Input Subspaces Detected Subspaces

q1 q2 q3

Input subspaces interpreted as points

Input Subspaces Voting Process

Accumulator Array for p = 3, n = 4

Proposed Subspace Detection Framework

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Initialization Voting Process Peak Detection Input Subspaces Detected Subspaces

• Search for peaks of votes in the accumulator
• The coordinates of the peaks correspond to

the most significant p-D subspaces

q1 q2 q3

Detected Subspaces Peak Detection

Accumulator Array for p = 3, n = 4 Input subspaces interpreted as points

Parameterization of Subspaces

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Vectors with the same attitude Attitude

Non-Metric Properties of Subspaces

Attitude The equivalence class, for any Weight The value of in , where is a reference with the same attitude as Orientation The sign of the weight relative to

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Reference Weighted vector Reference Positive orientation

+

Negative orientation

Parameterization of Vectors

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Reference Vector

here, n = 3

Arbitrary Vector

Parameterization of Vectors

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describes the weight and orientation of a vector where n – 1 rotation angles describe the attitude of a vector in n-D here, n = 3

Parameterization of Pseudovectors

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Pseudovector Vector

The parameterization of vectors naturally extends to pseudovectors through duality

here, n = 3 where

Parameterization of Arbitrary Subspaces

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(n – 2)-D subspace (n – 1)-D subspace (pseudovector) n-D subspace (pseudoscalar) 0-D subspace (scalar) 1-D subspace (vector) 2-D subspace

… …

Avoiding Ambiguous Representations

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Parameterized 2-D Subspace (interpreted here as straight line) 4-D Representational Space

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Proposed Parameterization

Reference subspace m = p (n – p) rotation operations m = p (n – p) rotation angles for attitude Scalar for weight and orientation

Parameter Space for p-Dimensional Subspaces

+ orientation – orientation

Parameter Space

Proposed Subspace Detection Framework

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Initialization Voting Process Peak Detection Input Subspaces Detected Subspaces

Model function for p-D subspaces in n-D Parameter space

Properties of the Parameterization

⚫ It is independent of the type of input data ⚫ It is independent of the geometric interpretation of

parameterized p-D linear subspaces

⚫ It uses the smallest set of parameters in the

representation of p-D linear subspaces

⚫ It defines a coordinate chart for the Grassmannian ⚫ It can properly represent all p-D linear subspaces

▪ The open affine covering cannot do it

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Proposed Subspace Detection Framework

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Initialization Voting Process Peak Detection Input Subspaces Detected Subspaces

q1 q2 q3

Input Data

Input Subspaces Voting Process

Accumulator Array Mapping and Voting

Voting Process for Input Subspaces

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The Mapping Procedure

⚫ It identifies in parameter space

▪ All p-D subspaces that contain a given r-D subspace ( r ≤ p ) ▪ All p-D subspaces contained in a given r-D subspace ( r ≥ p )

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e.g., straight line detection from points e.g., straight line detection from planes

Venn Diagram

• f Dimensions

Input Intended Total space Equivalent by Duality Venn Diagrams

• f Dimensions

We want the parameter vectors (sets of rotation angles) for the sequence of m rotation operations related to the input

The Mapping Procedure

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the procedure identifies all p-D subspaces For a given input subspace Starting from the last to the first one

A given parameter can assume a single value

• r all values in

A vector (r = 1) mapped to a discrete parameter space, for p = 3 and n = 4

Results

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22 most relevant detected lines from points

Conformal Model r = 2, p = 3, n = 2+2

166 most relevant detected circles from tangent directions

Homogeneous Model r = 1, p = 2, n = 2+1

70 most relevant detected lines from points

Results

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Conformal Model r = 1 and 3, p = 4, n = 3+2

3 most relevant detected spheres/planes from points, lines and circles

Homogeneous Model r = 1 and 3, p = 2, n = 3+1

2 most relevant detected lines from points and plane 3 most relevant detected planes from points

Homogeneous Model r = 1, p = 3, n = 3+1

Voting Process for Input Subspaces with Uncertainty

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Input Uncertain Data

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Exact Point Uncertain Point Exact Straight Line Uncertain Straight Line

Expectation Distribution’s Envelope (Three Standard Deviations) Distribution’s Envelope (Three Standard Deviations)

Sampling-Based Approach

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Parameter Space Input Straight Line with Uncertainty Straight Line Samples Mapping Procedure Mapped Samples

Error-Propagation-Based Approach

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Parameter Space Input Random Multivector Variable (Straight Line with Uncertainty) Extended Mapping Procedure Analytically Defined Envelope

Error-Propagation-Based Approach

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Input Random Multivector Variable (Straight Line with Uncertainty) Parameter Space Auxiliary Space Orthonormal Basis

• f Eigenvectors

Eigenvectors-Aligned Bounding Box Mean Parameter Vector Gaussian Distribution

Error Propagation vs. Sampling

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First-Order Error Propagation Sampling Smoother Distributions of Votes

Conclusions and Future Work

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Summary

⚫ A framework for detecting emerging data alignments

in unordered noisy multidimensional data

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Initialization Voting Process Peak Detection Input Subspaces Detected Subspaces

• A parameter space for

p-D linear subspaces

• A mapping and a voting

procedure for (exact) input subspaces

• A mapping and a voting

procedure for input subspaces with uncertainty

• The dissertation also

describes a new peak detection scheme

Usability

⚫ The parameter space definition requires

▪ The dimensionality p of the intended subspace ▪ The dimensionality n of the representational space

⚫ The accumulator array definition requires

▪ Some discretization criteria

⚫ The application of the approach is straightforward

▪ Perform one of the generalized voting procedures ▪ Perform the peak detection scheme

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Euclidean Homogeneous Conformal Conic n → d d+1 d+2 6 Frees 1-D direction 1 1 2 2-D direction 2 2 3 3-D direction 3 3 4 Flats Flat point 1 2 Straight line 2 3 Plane 3 4 Rounds Point pair 2 Circle 3 Sphere 4 Tangents Point 1 1-D tangent direction 2 2-D tangent direction 3 Point Sets Point 1 Point pair 2 Point triplet 3 Point Quadruplet 4 Conics Circle, Ellipse, Straight line, Hyperbola, Parallel line pair, Intersecting line pair 5 n = 2+2 p = 3 p = 2 n = 2+1

Generalization of the Hough Transforms

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Parabolas from Points

Sklansky (1978)

2-D Space

Circles from Points

Duda and Hart (1972)

2-D Space

Ellipses from Points with Normal Direction

Bennet et al. (1989)

2-D Space

Straight Lines from Points

Hough (1959) Duda and Hart (1972)

2-D Space

Straight Lines from Points with Normal Direction

O’Gorman and Clowes (1973)

2-D Space

Circles from Points with Normal Direction

Kimme et al. (1975) 2-D Space

Oriented Flat Spaces from Points

Achtert et al. (2008)

n-D Space

Non-Analytical Shapes in Volumetric Images

Wang and Reeves (1990) 3-D Space

Non-Analytical Shapes in Images

Ballard (1981) 2-D Space Proposed Approach

p-D Subspaces from Any Combination of Subspaces

n-D Space

Proposed Approach

p-D Subspaces from Any Combination of Subspaces with Uncertainty

n-D Space

Analytical Shapes Representable by Linear Subspaces Specific Geometric Shapes Non-Analytical Shapes

Properties of the Proposed Approach

⚫ Desirable properties

▪ Closed-form solution ▪ Generally applicable ▪ It represents the subspaces in the most compact way ▪ Robust to the presence of noise and discontinuities ▪ Suitable for massively parallel architectures

⚫ Drawbacks

▪ Large memory requirements ▪ Computational cost

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Future Work

⚫ The development of generally-applicable optimizations

A next step is the generalization of

• L. A. F. Fernandes, M. M. Oliveira. Real-time line detection through an

improved Hough transform voting scheme. Pattern Recognit., 41(1), 2008.

⚫ Detection of manifolds with boundary

A next step is the generalization of

• M. Atiquzzaman, M. W. Akhtar. A robust Hough transform technique for

complete line segment description. Real-Time Imaging, 1(6), 1995.

⚫ Detection of arbitrary shapes

▪ Radial basis functions (RBF)

⚫ Data mining

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Questions?

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Closed-Form Framework for Subspace Detection Concurrent Detection and Heterogeneous Input Datasets Error-Propagation-Based Voting Scheme for Input Uncertain Data