Alberto Paoluzzi with Francesco Furiani and Claudio Paoluzzipaoluzzi/web/pao/doc/esa.pdf · Rapid...

Algebraic mining of solid models from images

Alberto Paoluzziwith Francesco Furiani and Claudio Paoluzzi

Dep. of Mathematics and Physics, University Roma Tre, Italy

March 13, 2014

A. Paoluzzi (Univ. Roma Tre) Mining of models from images March 13, 2014 1 / 38

Outline

Outline


Outline

1 Standard unification of topological structures

2 LAR: the Linear Algebraic Representation

LAR representations: CSR matrices

3 Proof of concept: multidimensional morphological operators


Standard unification of topological structures




Representation schememapping s : M → R from a space of math models M to computer representations R

M

DRV

s

1 The set M contains the mathematical models of the class of solid objectsthat the scheme aims to represent.

2 The set R contains symbolic representations, i.e. suitable data structures,built according to some appropriate computer grammar.



Representation scheme: Quad-Edge data structure(Guibas & Stolfi, ACM Transactions on Graphics, 1985)

(a) Edge record showing Next links.(b) A subdivision of the sphere.

(c) Data structure for the subdivision

Primitives for the Manipulationof General Subdivisions andthe Computation of VoronoiDiagrams

largely used in computationalgeometry algorithms and ingeometric libraries



Representation scheme: Partial-Entity data structure(Sang Hun Lee & Kunwoo Lee, ACM Solid Modeling, 2001)

Compact Non-Manifold Boundary Representation Based on PartialTopological Entities



Representation scheme: Coupling Entities data structure(Yamaguchi & Kimura, IEEE Computer Graphics and Applications, 1995)



Rethinking some foundationsdealing with Big Data and scalable architectures Google’s map-reduce

Emerging applications (e.g. space, nano& bio technology, medical 3D) requirethe convergence of shape synthesis andanalysis from:

computer imaging

computer graphics

computer-aided geometric design

discrete meshing of domains

physical simulations

The goals of unification,scalability, and massivelyparallel distributedcomputing

call for rethinking thefoundations of geometricand topological computing

GOAL: 103 times fasterand 104 times bigger



IEEE-SA P3332 WGSolid models from 3D medical images


http://standards.ieee.org/develop/project/3333.2.html

http://standards.ieee.org/develop/project/3333.2.html

LAR: the Linear Algebraic Representation




LAR: topology extraction from d-imagesFast algebraic extraction (GPGPU) of spongy bone’s exact topology



LARA standard for model topology through a general and simple repr scheme

Models: (co)chain complexes → Reprs: sparse binary matrices



Chain and cochain complexC 0,C 1,C 2,C 3 ≡ V, E ,F ,P ∆p ≡ Laplacian δ0, δ1, δ2 ≡ grad, curl, div



Chain complex (of chain spaces)

Sequence of linear spaces (over Z2) of d-cell subsets

Unit d-chains (single d-cell subsets), give the standard bases (Md rows) of d-chainspaces

M3

M2M1 M0

!3 !2 !1



Characteristic matrices of d-chain spacesMatrix representation of the basis — d-cells as subsets of vertices


LAR: the Linear Algebraic Representation LAR representations: CSR matrices

LAR representations: CSR matrices



Compressed Sparse Row (CSR) matrix storage

image from



Storage of LAR(X ) ≡ CSR(Md) matrixAmazingly compact storage of a solid model

REDUCED LAR

simplicial d-complexes: k = d + 1cuboidal d-complexes: k = 2d

Remark (Input and long-term storage)

space(LAR)= |FV | = 2|E | !!!

Remark (Full topology representation)

|VE |+ |VF | = 4|E |

Remark (Any topological queries)

single SpMV multiplication

Remark (Sparse Matrix-Vector Multiplication)

is one of the most important computationalkernels, for very effective iterative solutionmethods



Example: 3D simplicial gridBoundary computation via a single SpMV product: [c2] = [∂3] 1

matrix [∂3]



Image complex = implicit LAR of voxelscell description is made by 4 or 8 indices

cell[i,j,k] = [ (i , j , k), (i + 1, j , k), (i , j + 1, k), (i , j , k + 1), (i + 1, j + 1, k), (i +1, j , k + 1), (i , j + 1, k + 1), (i + 1, j + 1, k + 1) ]

the boundary matrix may be computed only one time (depending on the image size),and stored on disk



Paradigm: Divide et ImperaBottleneck of GPGPU: moving data from global to local memory

Solution: store the (sparse) [∂3] of n3 voxels in device Constant Memory, and move the(binary) coordinate vectors of chains in Private Memory



LAR competitive advantagesSeveral, and very important including mimetic and isogeometric methods

Unification

computer imagingcomputer graphicsCAD and CAGDdiscrete meshing of domainsphysical simulations

Compactness

amazing long-term storage ofsolid models

Dimensional independence

same algorithms on 2D, 3D,4D representation

Native parallelism

based on linear algebra andGPGPU implementation ofSpMV

Support of simulations via

∂3, ∂2, ∂1, δ0, δ1, δ2,∆p

Integrated manufacturing

e.g.: of composite materials

Rapid prototyping

tight integration of geometry& CAD/CAE/CAM


Proof of concept

Proof of concept: multidimensional morphologicaloperators


Proof of concept

Mathematical morphology (MM)From Wikipedia

Mathematical morphology (MM) is atheory and technique for the analysisand processing of geometricalstructures, based on set theory,lattice theory, topology, and randomfunctions.

MM is also the foundation ofmorphological image processing,which consists of a set of operatorsthat transform images according tothe above characterizations.

Figure : http://en.wikipedia.org/wiki/File:DilationErosion.png


Proof of concept

Generation of a random block imagewith random noise

Let us consider the chain γ ∈ C2 of white pixels

Figure : (a) PNG image; (b) exploded solid model of |γ| ⊂ E3


Proof of concept

Extraction of boundary of the white chainβ = ∂2(γ) ∈ C1


Proof of concept

Down(boundary)η = VE(∂2(γ)) ∈ C0


Proof of concept

Up(up(down(boundary)))β2 = FV(VE(∂2(γ))) ∈ C2 ≡ (FV ◦ VE ◦ ∂2)(γ)


Proof of concept

Boolean combination of chainsβ2 = (FV ◦ VE ◦ ∂2)(γ) Dilation = β2 − γ (yellow); Erosion = β2 ∩ γ (cyan)

Figure : (a) PNG image; (b) exploded solid model of γ ∈ E3


Proof of concept

64× 64 examplechain γ ∈ C2; chain β = ∂2(γ) ∈ C1


Proof of concept

64× 64 example(VE ◦ ∂2)(γ) ∈ C0 (EV ◦ VE ◦ ∂2)(γ) ∈ C1


Proof of concept

64× 64 exampleβ2 = (FE ◦ EV ◦ VE ◦ ∂2)(γ) ∈ C2 DIL(β2)(γ) = β2 − γ ERO(β2)(γ) = β2 ∩ γ


Proof of concept

A last word

Our morphological operators are multidimensional


Conclusion

Conclusion


Conclusion

LAR state of the art

A. DiCarlo, V. Shapiro, and A. Paoluzzi, Linear AlgebraicRepresentation for Topological Structures, Computer-Aided Design,Volume 46, Issue 1 , January 2014, Pages 269–274

Ongoing development (MIT licence) on GitHub, using OpenCL, Theopen standard for parallel programming of heterogeneous systems


Conclusion

TODOLooking for collaborations

and a grant in Space Horizon2020! ... :o)


Conclusion

Thank you for your interest!

QUESTIONS?

[email protected]


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