On the Generalization of Subspace Detection in Unordered Multidimensional Data Leandro Augusto Frata Fernandes laffernandes@inf.ufrgs.br Manuel Menezes de Oliveira Neto oliveira@inf.ufrgs.br Advisor CG UFRGS
Data Alignments 2
Data Alignments are Everywhere Counting Process in Clonogenic Assays EBSD image for Particle's Crystalline Phase Identification Robot Navigation 3
Conventional Techniques ⚫ Designed to detect specific types of data alignments ⚫ Designed to a given type of input data Straight-Line Detection Circle Detection Drawbacks due to Specialization ( x – x c ) 2 + ( y – y c ) 2 – r 2 = 0 ρ = x cos θ + y sin θ • It requires different formulations Center-radius parameterization as model Normal equation of the line as model • It requires different implementations • It prevents the development of generally Points in the Plane as Input applicable optimizations • Concurrent detection may be a challenging task 4
2 -D Base Space 2 -D Base Space 2 -D Base Space Key Observation ⚫ Linear subspaces can be interpreted as some types of data alignments Vectors can be interpreted 2 -D subspaces can be 3 -D subspaces can be interpreted as straight lines as points or directions interpreted as circles within the Homogeneous Model within the Conformal Model within the Homogeneous Model 3 -D Representational Space 4 -D Representational Space 3 -D Representational Space 5
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 The proposed subspace ▪ Closed-form solution detector framework was formulated ▪ It can be applied to any alignments that can be with Geometric Algebra characterized by a linear subspace 6
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 7
Detecting Different Types of Alignments Homogeneous Model Conformal Model Input : points ( 1 -D subspaces) Input : tangent directions ( 2 -D subspaces) Output : straight lines ( 2 -D subspaces) Output : circles ( 3 -D subspaces) 8
Detection on Heterogeneous Datasets Homogeneous Model Conformal Model Input : points ( 1 -D subspaces) Input : points ( 1 -D subspaces) plane ( 3 -D subspaces) circles ( 3 -D subspaces) line ( 3 -D subspace) Output : straight lines ( 2 -D subspaces) Output : spheres ( 4 -D subspaces) plane ( 4 -D subspace) 9
Outline ⚫ Overview ⚫ Parameterization of Subspaces ⚫ Voting Process for Input Subspaces ⚫ Voting Process for Input Subspaces with Uncertainty ⚫ Conclusions and Future Work 10
Proposed Subspace Detection Framework Input Detected Subspaces Subspaces Initialization Voting Process Peak Detection 11
Proposed Subspace Detection Framework Input Detected Subspaces Subspaces Initialization Voting Process Peak Detection Accumulator Array Parameter Space for p = 3 , n = 4 for p = 3 , n = 4 • 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 Model function for 3 -D subspaces in 4 -D representational space Model function for p -D subspaces 12
Proposed Subspace Detection Framework Input Input Detected Subspaces Subspaces Subspaces Initialization Voting Process Voting Process Peak Detection Accumulator Array for p = 3 , n = 4 • 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 q 2 q 1 q 3 Input subspaces interpreted as points 13
Proposed Subspace Detection Framework Input Detected Detected Subspaces Subspaces Subspaces Initialization Voting Process Peak Detection Peak Detection Accumulator Array for p = 3 , n = 4 • Search for peaks of votes in the accumulator • The coordinates of the peaks correspond to the most significant p -D subspaces q 2 q 1 q 3 Input subspaces interpreted as points 14
Parameterization of Subspaces 15
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 Vectors with the same attitude Attitude Reference Positive orientation + Reference Weighted vector Negative orientation - 16
Parameterization of Vectors Reference Vector Arbitrary Vector here, n = 3 17
Parameterization of Vectors describes the weight and orientation of a vector n – 1 rotation angles describe the attitude of a vector in n -D where here, n = 3 18
Parameterization of Pseudovectors The parameterization of vectors naturally extends to pseudovectors Vector through duality Pseudovector where here, n = 3 19
Parameterization of Arbitrary Subspaces 0 -D subspace (scalar) 1 - D subspace (vector) 2 - D subspace … … ( n – 2) - D subspace ( n – 1) - D subspace (pseudovector) n - D subspace (pseudoscalar) 20
Avoiding Ambiguous Representations Parameterized 2 -D Subspace (interpreted here as straight line) 4 -D Representational Space 21
Proposed Parameterization Reference subspace Parameter Space for + orientation p -Dimensional Subspaces m = p ( n – p ) rotation operations – orientation m = p ( n – p ) rotation angles for attitude Scalar for weight and orientation 22
Proposed Subspace Detection Framework Input Detected Subspaces Subspaces Initialization Voting Process Peak Detection Parameter Space Model function for p -D subspaces in n -D Parameter space 23
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 24
Proposed Subspace Detection Framework Input Input Detected Subspaces Subspaces Subspaces Initialization Voting Process Voting Process Peak Detection Accumulator Array q 2 Mapping q 3 q 1 and Voting Input Data 25
Voting Process for Input Subspaces 26
The Mapping Procedure ⚫ It identifies in parameter space Venn Diagrams Venn Diagram of Dimensions of Dimensions ▪ All p -D subspaces that contain a given r -D subspace ( r ≤ p ) e.g., straight line detection from points Equivalent ▪ All p -D subspaces contained by Duality in a given r -D subspace ( r ≥ p ) e.g., straight line detection from planes Total space Intended Input 27
The Mapping Procedure For a given input subspace A vector ( r = 1 ) mapped to a discrete parameter space, for p = 3 and n = 4 the procedure identifies all p -D subspaces A given parameter can assume a single value or all values in related to the input We want the parameter vectors (sets of rotation angles) for the sequence of m rotation operations Starting from the last to the first one 28
Homogeneous Model r = 1 , p = 2 , n = 2+1 Results 22 most relevant detected lines from points Conformal Model r = 2 , p = 3 , n = 2+2 166 most relevant detected circles from tangent directions 70 most relevant detected lines from points 29
Homogeneous Model r = 1 and 3 , p = 2 , n = 3+1 Results 2 most relevant detected lines Conformal Model from points and plane r = 1 and 3 , p = 4 , n = 3+2 Homogeneous Model r = 1 , p = 3 , n = 3+1 3 most relevant 3 most relevant detected spheres/planes detected planes from points, lines and circles from points 30
Voting Process for Input Subspaces with Uncertainty 31
Input Uncertain Data Expectation Distribution’s Envelope Distribution’s Envelope ( Three Standard Deviations ) ( Three Standard Deviations ) Uncertain Point Exact Point Uncertain Straight Line Exact Straight Line 32
Sampling-Based Approach Mapped Samples Straight Line Samples Mapping Procedure Input Straight Line Parameter Space with Uncertainty 33
Error-Propagation-Based Approach Analytically Defined Envelope Extended Mapping Procedure Input Random Parameter Space Multivector Variable (Straight Line with Uncertainty) 34
Error-Propagation-Based Approach Parameter Space Mean Parameter Vector Input Random Multivector Variable (Straight Line with Uncertainty) Eigenvectors-Aligned Bounding Box Gaussian Distribution Auxiliary Space Orthonormal Basis of Eigenvectors 35
Smoother Distributions of Votes Error Propagation vs. Sampling First-Order Error Propagation Sampling 36
Conclusions and Future Work 37
Recommend
More recommend
