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Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud

Jaya Sreevalsan-Nair Beena Kumari

Graphics-Visualization-Computing Lab (GVCL), International Institute of Information Technology, Bangalore (IIITB), India.

Dagstuhl Seminar 16142: Multidisciplinary Approaches to Multivalued Data: Modeling, Visualization, Analysis April 3-8, 2016.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Agenda

Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Agenda

Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

3D Urban LiDAR Point Cloud

Data acquisition using airborne LiDAR, where point cloud (3D points) describes topography; Depending on acquisition settings, dataset could contain additional information such as multiple returns, intensity, color, etc. Region of our interest: urban settings, usually consisting of buildings, asphalt & natural ground, vegetation, etc.

Image courtesy: Keil Schmid, Kirk Waters, Lindy Dingerson, Brian Hadley, Rebecca Mataosky, Jamie Carter, and Jennifer

• Dare. Lidar 101: An introduction to lidar technology, data, and applications. NOAA Coastal Services Center, 2012.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

What is Augmented Classification ?

A combination of structural and contextual classifications∗ – explicitly building class tuples in order to enable structural operations on points in specific object classes, such as curve extractions in buildings, asphalt ground (road), etc.

* B. Kumari and J. Sreevalsan-Nair. 2015. An interactive visual analytic tool for semantic classification of 3D urban LiDAR point cloud. In Proceedings of the 23rd SIGSPATIAL International Conference on Advances in Geographic Information Systems (GIS ’15). ACM, New York, NY, USA, Article 73.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Structural Classification

[Keller11] proposed structural classification using principal component analysis of covariance matrix in local (spherical) neighborhood;

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Structural Classification

[Keller11] proposed structural classification using principal component analysis of covariance matrix in local (spherical) neighborhood; Structural classes, given λ2 ≥ λ1 ≥ λ0 :

◮ (L) disc-shaped neighborhood = planar type, i.e.

Pd = {p ∈ P|λ0/λ2 < ǫ};

◮ (M) cylindrical-shaped neighborhood = line/curve type, i.e.

Pc = {p ∈ P|λ1/λ2 < ǫ};

◮ (R) spherical neighborhood = point type, i.e.

Ps = {p ∈ P|λ0/λ2 ≥ ǫ}.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Contextual Classification

We proposed hierarchical clustering, using visualization in choosing clustering parameters (3D color maps/heatmaps of parameters) – in our implementation:

◮ we classify into 4 object classes: buildings, asphalt ground, natural

ground, vegetation;

◮ we use hierarchical expectation-maximization (HEM); ◮ we use one parameter for clustering at a time, from a n-dimensional

parameter space;

◮ we use a tree-visualizer which enables user to decide the parameter based

• n heatmaps of parameters;

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Contextual Classification

We proposed hierarchical clustering, using visualization in choosing clustering parameters (3D color maps/heatmaps of parameters) – in our implementation:

◮ we classify into 4 object classes: buildings, asphalt ground, natural

ground, vegetation;

◮ we use hierarchical expectation-maximization (HEM); ◮ we use one parameter for clustering at a time, from a n-dimensional

parameter space;

◮ we use a tree-visualizer which enables user to decide the parameter based

• n heatmaps of parameters;

Clustering parameters used: normalized height, height-difference, intensity, properties derived from local geometry (linearity, planarity, curvature, difference of normals, surface residual, etc.)

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Contextual Classification

We proposed hierarchical clustering, using visualization in choosing clustering parameters (3D color maps/heatmaps of parameters) – in our implementation:

◮ we classify into 4 object classes: buildings, asphalt ground, natural

ground, vegetation;

◮ we use hierarchical expectation-maximization (HEM); ◮ we use one parameter for clustering at a time, from a n-dimensional

parameter space;

◮ we use a tree-visualizer which enables user to decide the parameter based

• n heatmaps of parameters;

Clustering parameters used: normalized height, height-difference, intensity, properties derived from local geometry (linearity, planarity, curvature, difference of normals, surface residual, etc.) Preserved spatial locality using region-growing, with post-processing of merging clusters in the leaf nodes of our tree visualizer, to form 4 classes.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Contextual Classification

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Contextual Classification

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Contextual Classification

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Contextual Classification

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Results: Contextual Classification

Area 1 of the Vaihingen dataset (1,79,997 points): (L) Orthoimage, (R) our contextual classification.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Results: Augmented Classification

Structural classification + contextual classification = augmented classification (using class tuples)

Area 1 of the Vaihingen dataset (1,79,997 points)

Expected benefits: preserving structural classification explicitly for curve extraction in object classes.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Motivation for Ongoing Work

Structural classification is important for the classification itself, as well as its use in contextual classification.

◮ Is there a better model that can yield structural classification ? ◮ Using its second order tensor definition, tensor visualization can be applied to

understand the covariance matrix.

(Left) Saliency map where R, G, B channels correspond to line-, surface- and point-type feature points, and (right) superquadric glyph visualization of orientation of structure tensor for Area 1 of the Vaihingen dataset (1,79,997 points). The cyan box shows where tensors representing tree foliage show less random orientation than expected. The red boxes show where intersection line of planes in the building roof is faint, which is not correct, as there is a sharp change in surface normals along this intersection line.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Agenda

Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Covariance Matrix & Structure Tensor

Disambiguation of terminologies:

◮ Covariance matrix [Point sampled geometry][Hoppe92] and Structure

tensor [LiDAR data analysis][Gross06]: S =

• y∈N(x)

(y − ¯ x)(y − ¯ x)T

◮ Structure tensor [Image processing/computer vision]:

S = Gσ∇f (x)∇f (x)T where Gρ is the Gaussian with zero mean and standard deviation σ Both are symmetric second-order tensors.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Covariance Matrix & Structure Tensor

Some more:

◮ A tensor for representing local structure [Knutsson89] – mapping a vector

y onto the tensor S: S =

1 y(yyT )

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Covariance Matrix & Structure Tensor

Some more:

◮ A tensor for representing local structure [Knutsson89] – mapping a vector

y onto the tensor S: S =

1 y(yyT )

◮ Covariance matrix [Statistics & Probability] uses statistical description.

S =

1 N(x)

• y∈N(x)

(y − ¯ x)(y − ¯ x)T where ¯ x is the same as average/mean, which comes from Cov(X, Y ) = E[(X − E[X])(Y − E[Y ])] – here, the position coordinates are random variables.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Classification Using Covariance Matrix [PSG]

Algorithm used in [Keller11]:

• 1. Preprocessing – Octree construction, outlier removal.
• 2. Stochastic point classification using multiscale approach –

2.1 Find which class a point belong to in each class – collect a “vote” 2.2 The average number of votes across scales, for a point belonging to each class = likelihood of a point to fall into one of the three classes, given by: {Lc, Ld, Ls}

We can say that these three values, {Lc, Ld, Ls} give Di Zenzo multi-value geometry –

◮ as the structure tensor [Image processing/computer vision] is a good

estimator of local multi-valued geometry, where values come from eigen-analysis of structure tensor (also called Di Zenzo matrix). [Tschumperlé06] .

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Brief Introduction to Tensor Voting

Unified framework that uses perceptual organization based on Gestalt psychology

◮ Gestalt principles: proximity, similarity, good continuation, closure.

Salient features:

◮ Data model: second order symmetric, non-negative definite tensors ◮ Information propagation using voting – contributions are ball, stick, or

plate tensors.

◮ Shape of tensors gives the orientation of structure.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Voting Tensor Types

Image courtesy: slides by Mordohai http://cs.unc.edu/ mordohai

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Alternative Symmetric Second-order Tensors

Showing similar structural classes – Image courtesy: [Wang13]

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Alternative Symmetric Second-order Tensors

Formulation of ball tensor (for unoriented points): N =

• y∈N(x)

nynT

y =

• y∈N(x)

exp y−x2

σ2

• .(Id −

(y−x)(y−x)T (y−x)T (y−x))

where ny is the normal vector at y, σ is scale factor (neighborhood size, here). Rewriting, upon performing PCA, we get N =

3

• j=1

λj.ej.eT

j .

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Alternative Symmetric Second-order Tensors

Formulation of ball tensor (for unoriented points): N =

• y∈N(x)

nynT

y =

• y∈N(x)

exp y−x2

σ2

• .(Id −

(y−x)(y−x)T (y−x)T (y−x))

where ny is the normal vector at y, σ is scale factor (neighborhood size, here). Rewriting, upon performing PCA, we get N =

3

• j=1

λj.ej.eT

j .

[Wang13] apply diffusion to alleviate:

◮ fast diffusion across edges, and slow diffusion along edges.

Hence, with diffusion parameter δD, N can be modified to: S =

3

• j=1

exp − λj

δD

• .ejeT

j .

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Results

(a) (b) (c)

Saliency map where R, G, B channels correspond to line-, surface- and point-type feature points, for Area 1 of the Vaihingen dataset (1,79,997 points), using (a) covariance matrix, (b) normal voting tensor, (c) diffused normal voting tensor.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Results

Superquadric glyph visualization of orientation of (top) covariance matrix and (bottom) diffused normal voting tensor, respectively, for Area 1 of the Vaihingen dataset (1,79,997 points). The cyan boxes highlight the comparison of randomness

• f orientation in the tree foliage. The red boxes highlight the sharpness of intersection line of planes in the gabled roof.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Results

(a) (b) (c)

Saliency map where R, G, B channels correspond to line-, surface- and point-type feature points, for blade model (8,82,954 points), using (a) covariance matrix, (b) normal voting tensor, (c) diffused normal voting tensor.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Agenda

Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Interpretation of Results

◮ Order of eigenvalues: matters for classification. ◮ In normal voting tensor, the eigenvalues are reversed in order compared to that

• f covariance matrix – diffusion flips it back to the right order. Thus,

covariance matrix and diffused normal voting tensor become comparable.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Interpretation of Results

◮ Order of eigenvalues: matters for classification. ◮ In normal voting tensor, the eigenvalues are reversed in order compared to that

• f covariance matrix – diffusion flips it back to the right order. Thus,

covariance matrix and diffused normal voting tensor become comparable.

◮ Why did the results get better?

– [Tombari10] makes a distinction between the surface reconstruction/fitting and local reference frame computation –

◮ which means substituting the centroid with the point itself, adding a

distance-based weight and using unit vectors in the construction of covariance matrix.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Interpretation of Results

◮ Order of eigenvalues: matters for classification. ◮ In normal voting tensor, the eigenvalues are reversed in order compared to that

• f covariance matrix – diffusion flips it back to the right order. Thus,

covariance matrix and diffused normal voting tensor become comparable.

◮ Why did the results get better?

– [Tombari10] makes a distinction between the surface reconstruction/fitting and local reference frame computation –

◮ which means substituting the centroid with the point itself, adding a

distance-based weight and using unit vectors in the construction of covariance matrix. Thus, changing from: S =

y∈N(x)

(y − x)(y − x)T to: S =

y∈N(x)

y − x. (y−x)(y−x)T

(y−x)T (y−x)

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Correction to Covariance Matrix

(a) (b) (c)

Saliency map where R, G, B channels correspond to line-, surface- and point-type feature points, for Area 1 of the Vaihingen dataset (1,79,997 points), using (a) covariance matrix, (b) modified covariance matrix, (c) diffused normal voting tensor.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Correction to Covariance Matrix

(a) (b) (c)

Saliency map where R, G, B channels correspond to line-, surface- and point-type feature points, for blade model (8,82,954 points), using (a) covariance matrix, (b) modified covariance matrix, (c) diffused normal voting tensor.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Some more analysis

Disambiguation of terminology:

◮ Covariance matrix [Point sampled geometry] (S): outer product of

tangent vector of locally fitted plane.

◮ Covariance matrix [Tensor voting] (N): outer product of normal vector of

locally fitted plane.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Some more analysis

Disambiguation of terminology:

◮ Covariance matrix [Point sampled geometry] (S): outer product of

tangent vector of locally fitted plane.

◮ Covariance matrix [Tensor voting] (N): outer product of normal vector of

locally fitted plane. Observation: the tensor types of covariance matrix and diffused normal voting tensor are different.

◮ Covariance matrix [Hoppe92, Keller11]: contravariant tensor – (2,0)

tensor

◮ Normal voting tensor/structure tensor: covariant tensor – (0,2) tensor

Then how were the tensor fields substitutable ?

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Some more analysis

Disambiguation of terminology:

◮ Covariance matrix [Point sampled geometry] (S): outer product of

tangent vector of locally fitted plane.

◮ Covariance matrix [Tensor voting] (N): outer product of normal vector of

locally fitted plane. Observation: the tensor types of covariance matrix and diffused normal voting tensor are different.

◮ Covariance matrix [Hoppe92, Keller11]: contravariant tensor – (2,0)

tensor

◮ Normal voting tensor/structure tensor: covariant tensor – (0,2) tensor

Then how were the tensor fields substitutable ? It turns out the local reference frame is unchanged when doing PCA of covariant matrix and normal voting tensor.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Some more analysis

Disambiguation of terminology:

◮ Covariance matrix [Point sampled geometry] (S): outer product of

tangent vector of locally fitted plane.

◮ Covariance matrix [Tensor voting] (N): outer product of normal vector of

locally fitted plane. Observation: the tensor types of covariance matrix and diffused normal voting tensor are different.

◮ Covariance matrix [Hoppe92, Keller11]: contravariant tensor – (2,0)

tensor

◮ Normal voting tensor/structure tensor: covariant tensor – (0,2) tensor

Then how were the tensor fields substitutable ? It turns out the local reference frame is unchanged when doing PCA of covariant matrix and normal voting tensor. It works for the classification task.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Where can these fields not be substituted ?

For 3D reconstruction, [Viksten07] have constructed a local descriptor:

◮ for range data, ◮ to describe one or more planes or lines in a local region, ◮ using a fourth order tensor from tensor product of contravariant and

covariant second order tensors, which are covariant matrices of tangent and normal vectors, respectively. S22(p) = S20(p) ⊗ S02(p) = (wx(i).xxT ) ⊗ (wn(i).nnT )

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Where can these fields not be substituted ?

For 3D reconstruction, [Viksten07] have constructed a local descriptor:

◮ for range data, ◮ to describe one or more planes or lines in a local region, ◮ using a fourth order tensor from tensor product of contravariant and

covariant second order tensors, which are covariant matrices of tangent and normal vectors, respectively. S22(p) = S20(p) ⊗ S02(p) = (wx(i).xxT ) ⊗ (wn(i).nnT ) Observations:

◮ For covariance matrix [Keller11] is to be the S20 tensor, using the diffused

normal voting tensor, which is a (0,2) tensor as a substitute, will not work.

◮ This is useful in our case to extract planes in gabled roofs of building.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Open Questions

How important is to understand the construction of the tensor ?

◮ Can we really substitute any second order tensor field with another

second order tensor field that fulfil the same purpose ?

◮ Will a formulation of (1,1) tensor work in the discussed case ? ◮ What are the semantics of these different tensor types ? ◮ What are good measures for the “substitutability” of tensor fields ?

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

Summary

◮ We substituted covariance matrix with diffused normal voting tensor to

improve structural classification in augmented classification.

◮ Yet to do – quantify “effectiveness” of the substitution. ◮ We analyzed why covariance matrix can be substitued with diffused

normal voting tensor for structural classification.

◮ We found an application where these fields cannot be substituted.

Substitutability of Symmetric Second-order Tensor Fields: An Application in Urban LiDAR 3D Point Cloud Jaya Sreevalsan-Nair Beena Kumari Motivation: Augmented Classification Covariance Matrix: an Alternative Discussions

References

[Keller11] Patric Keller, Oliver Kreylos, Marek Vanco, Martin Hering-Bertram, Eric S Cowgill, Louise H Kellogg, Bernd Hamann, and Hans Hagen. Extracting and visualizing structural features in environmental point cloud LiDaR data sets. In Topological Methods in Data Analysis and Visualization, pages 179–192. Springer, 2011. [Hoppe92] Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Surface reconstruction from unorganized points. In: SIGGRAPH, pp. 71Ű78 (1992) [Knutsson89] Knutsson, H. (1989). Representing local structure using tensors. In 6th Scandinavian Conference on Image Analysis, Oulu, Finland (pp. 244-251). Linköping University Electronic Press. [Tombari10] Tombari, F., Salti, S., Di Stefano, L. (2010). Unique signatures of histograms for local surface description. In Computer Vision-ECCV 2010 (pp. 356-369). Springer Berlin Heidelberg. [Wang13] Wang, S., Hou, T., Li, S., Su, Z., Qin, H. (2013). Anisotropic elliptic pdes for feature classification. Visualization and Computer Graphics, IEEE Transactions on, 19(10), 1606-1618. [Tschumperlé06] Tschumperlé, D. (2006). Fast anisotropic smoothing of multi-valued images using curvature-preserving PDE’s. International Journal of Computer Vision, 68(1), 65-82. [Viksten07] Viksten, F., Nordberg, K. (2007). A geometry-based local descriptor for range data. In Digital Image Computing Techniques and Applications, 9th Biennial Conference of the Australian Pattern Recognition Society on (pp. 210-217). IEEE.