3D Shape Search

8/3/2019 3D Shape Search

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3D Shape Search

Shiben Bhattacharjee

P.Dilip

S.Manikandan

Introduction:

Determining the similarity between 3D shapes is a fundamental task in

shape-based recognition, retrieval, clustering, and classification. Its main

applications have traditionally been in computer vision, mechanical

engineering, and molecular biology. Since most 3D file formats (VRML, 3D

Studio, etc.) have been designed for visualization, they contain only

geometric and appearance attributes, and usually lack semantic information

that would facilitate automatic matching. 3D models are most probablyacquired with the help of scanning devices or geometric manipulation tools

and thus they will have only geometric and appearance information and will

be completely devoid of semantic and structural information. Hence an

algorithm useful for recognition of 3D shapes will be very useful. The first

step would be to develop methods that can give us a measure of the

similarity/dissimilarity between any two shapes. This is what we have tried

to achieve in our project.

The challenging aspect of this problem is that it should be done quickly

while still being able to discriminate between similar and dissimilar shapes.

The key idea is to represent the signature of an object as a shape distributionsampled from a shape functionmeasuring global geometric properties of anobject. The primary motivation for this approach is to reduce the shape

matching problem to the comparison of probability distributions, which is

simpler than traditional shape matching methods that require pose

registration, feature correspondence, or model fitting.

Prior Methods:

Prior matching methods can be broadly classified based on theirrepresentation of the shape as:

2D Contours

3D Surfaces

3D Volumes

Structural models


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Statistics

Shape matching has been well-studied for 3D objects. For instance,

representations for registering and matching 3D surfaces include Extended

Gaussian Images, Spherical Attribute Images, Harmonic Shape Images, andSpin Images. Unfortunately, these previous methods usually assume that a

topologically valid surface mesh or an explicit volume is available for every

object. In addition, volumetric dissimilarity measures based on wavelets or

Earth Movers Distance usually rely upon a priori registration of objects

coordinate systems, which is difficult to achieve automatically and robustly.

Geometric hashing is a potential solution, but it requires a large amount of

storage for complex models. Model based approaches first decompose a 3D

object into a set of features and then compute a dissimilarity measure

between objects based on the differences between their features and their

spatial relationships. Examples representations of this type includegeneralized cylinders, superquadrics, geons, deformable regions, shock

graphs, medial axes and skeletons. These methods work best when 3D

models can be segmented into a canonical set of features naturally and

correspondences can be found between features robustly. Unfortunately,

these tasks are difficult and not always well-defined for arbitrary 3D

polygonal models. Moreover, feature detection and segmentation algorithms

tend to be sensitive to small perturbations to the model, placing undue

burden on subsequent feature correspondence and dissimilarity computation

steps. Also the combinatorial complexity of finding correspondences in largediscrete models usually leads to long computation times and/or large storage

requirements. Shapes have also been compared based on their statistical

properties. The simplest approach would be to compute the distances

between feature vectors of features that represent global geometric

properties such as circularity, eccentricity, algebraic moments, etc. Other

methods have compared discrete histograms of geometric statistics.

Overview of Approach:

The key idea is to represent the signature for a 3D model as a probabilitydistribution sampled from a shape function measuring geometric properties

of the 3D model. This is called a shape distribution. Once the shape

distributions has been computed for two 3D models then the

similarity/dissimilarity between them can be calculated using any metric that

measures distance between distributions. We transform any arbitrary 3D

model into a parameterized function that can be compared with others easily.


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The main advantage of this method is that the problem of shape comparison

is reduced to a problem of sampling, normalization and comparison of shape

distributions which are relatively easy compared to the prior methods which

require reconstructing a solid object or manifold surface from degenerate 3D

data, registering pose transformations, finding feature correspondences, or

fitting high-level models. This method is invariant to arbitrary rotations,

translations and mirroring. Invariance to scaling can be achieved by

normalizing the shape distributions before comparison. The method is also

robust since random sampling ensures that shape distributions are insensitive

to small perturbations like noise, blur, cracks, dust, etc in the input models.

Another advantage of the above method is its efficiency. Construction and

comparison of shape distributions is very fast and efficient. Also this method

works regardless of they way in which the 3D model is represented. The

three main steps to be implemented are:

To select a discriminating shape function.

To construct shape functions for the given 3D models efficiently.

To compute a similarity/dissimilarity measure for a pair ofdistributions.

Below we have provided a block diagram of our approach.

Detailed Approach:

Selecting a shape function:

The most important issue is to select a shape function whose distribution

provides a good signature for the shape of a 3D polygonal model. An ideal

distribution would be invariant to tessellations and other transformations and

insensitive to noise, cracks, tessellation and insertion/removal of small

polygons. The shape function could also be domain specific, based on

visibility and surface attributes.


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Compute dissimilarity between the two shape distributions using measuressuch as

2, Bhattacharyya or LN Minkowski

Shapedistribution 1

Dissimilarity measure

Generate shape functions

Using measures such asD2,D3 or A3

Sample shape functionto get shape distribution

Shape function 2

Shapedistribution 2

Model 2

Generate shape functions

using measures such asD2,D3 or A3

Shape function 1

Sample shape functionto get shape distribution

Model 1


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The most common shape functions are:

A3:Measures the angle between three random points on the surfaceof a 3D model.

D1:

Measures the distance between a fixed point and one random point on the surface. The centroid of the boundary of the model is

generally used as the fixed point.

D2:Measures the distance between two random points on the surface.

D3:Measures the square root of the area of the triangle between threerandom points on the surface.

D4: Measures the cube root of the volume of the tetrahedron betweenfour random points on the surface.

The figure below shows a diagrammatic representation of the shape

functions A3, D1, D2, D3 and D4 respectively.

We have implemented the A3, D2 and D3 shape functions. They were

chosen because of the ease of implementation and their invarianceproperties. They are quick to compute, easy to understand and produce shape

distributions invariant to rigid motions (translations and rotations). They are

invariant to tessellation of the 3D polygonal model, since points are selected

randomly from the surface. They are insensitive to small perturbations due

to noise, cracks, and insertion/removal of polygons, since sampling is area

weighted. In addition, the A3 shape function is invariant to scale, while the

others have to be normalized to enable comparisons. Finally, the D2 and D3

shape functions provide a nice comparison of 1D and 2D geometric

measurements. The figure below shows the D2 shape distribution for somemodels. In each plot, the horizontal axis represents distance, and the vertical

axis represents the probability of that distance between two points on the

surface.


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Constructing shape distributions:

Once a shape function has been chosen the next issue is the construction and

storage of a representation of its distribution. For this we evaluate N samples

from the shape distribution and construct a histogram by counting how many

samples fall into each of B fixed size bins. From the histogram we

reconstruct a piecewise linear function with V (


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triangle, compute its area and store it in an array along with the cumulative

area of triangles visited so far. Next, select a triangle with probability

proportional to its area by generating a random number between 0 and the

total cumulative area and performing a binary search on the array of

cumulative areas. For each selected triangle with vertices (A; B; C),construct a point on its surface by generating two random numbers, r1 andr2, between 0 and 1, and evaluating the following equation:

P = (1 - r1)A + r1(1 - r2)B + r1r2C

Comparing shape distributions:

The next task is the comparison of two shape distributions to produce a

similarity/dissimilarity measure. There are many simple dissimilaritymeasures. Some of them are:

2 :D(f,g) = (f-g)2/(f+g)

Bhattacharyya :D(f,g) = 1 - fg

LN Minkowski norm of the pdf:

D(f,g) = (|f g|N)1/N

LN Minkowski norm of the cdf:D(f,g) = (|f* g*|N)1/N

x

Where f* = f-

Since each shape distribution is represented as a piecewise linear function,analytic computation of these norms can be done efficiently in time

proportional to the number of vertices used to store the distributions. For

certain shape functions, a normalization step has to be added to the

comparison process to account for differences in scale. The three most

common methods for normalization are:


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1) align the maximum sample values

2) align the mean sample values

3) search for the scale that produces the minimal dissimilarity measure

during each comparison.

Implementation Details:

We have assumed that the inputs are in .3ds file format in which the model

is described in terms of the triangles which make it up.

We have implemented the following shape functions:

D2

D3

A3

We have implemented the following dissimlarity measures:

2 Bhattacharyya LN Minkowski norm of the pdf (with N=2)

For our implementation we used N=1024*1024 samples, B=1024 bins and

V=64 vertices.

We used three dissimilarity measures:

2 Bhattacharyya LN Minkowski norm of the pdf (with N=2)


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Experimentation and Results:

We have used the following twenty models for our experiments:

Bed

Casio

Chair

Column

Guitar

Hand Set


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Helicopter

Lamp

Monitor

Car

All these models were downloaded from www.help3d.com
http://www.help3d.com/http://www.help3d.com/


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Dissimilarity matrix using D2 shape function:

Bed Car Casio Chair Column Hand Set Guitar Helicopter Lamp Monitor

B

e

d

0.0522843

0.0133494

0.0454714

0.114814

0.294435

0.0999145

0.319191

0.0893539

0.128534

0.739689

0.188293

0.13786

0.3022

0.0897885

0.106445

0.0515548

0.0131408

0.0431497

0.293866

0.0789532

0.115848

0.329812

0.0894503

0.116206

0.434307

0.118603

0.28882

0.0847076

0.0217109

0.0523883C

a

r

0.419527

0.116082

0.133715

0.208554

0.0556251

0.0985718

0.47835

0.332312

0.189802

0.345875

0.0953444

0.151279

0.33234

0.0942872

0.141081

0.329182

0.0997939

0.120671

0.431785

0.119739

0.109059

0.361702

0.149167

0.191049

0.498323

0.143756

0.143511

0.568582

0.159989

0.152268

Ca

s

i

o

0.147048

0.03959290.0695793

0.149304

0.04148340.0684619

0.007258

0.00608630

.0256732

0.134528

0.03717090.0601861

0.162574

0.04495130.0732475

0.133964

0.03821320.0645826

0.168176

0.04558080.0833615

0.232192

0.07027560.0845656

0.192067

0.05178470.0788495

0.242197

0.06630280.0882167

C

h

a

ir

0.253972

0.0725295

0.090112

0.0822553

0.0216659

0.055987

0.32746

0.0948865

0.136082

0.0602086

0.0282891

0.040196

0.358007

0.114853

0.104121

0.0918136

0.0280221

0.0598845

0.434149

0.120159

0.13538

0.24327

0.0639522

0.111686

0.150089

0.0442604

0.0561107

0.316772

0.0915136

0.100092

Co

l

um

n

0.212171

0.0579961

0.086671

0.326644

0.089046

0.108324

0.315641

0.0902578

0.118361

0.27633

0.0775159

0.0968909

0.0965449

0.0214857

0.030675

0.284597

0.0782868

0.10004

0.166407

0.0444839

0.073497

0.310268

0.0912713

0.0944473

0.345384

0.0987133

0.112072

0.328893

0.0947606

0.10915

H

a

nd

S

et

0.04367720.0257867

0.0740785

0.2828290.0729386

0.0324877

0.2318120.0646573

0.115926

0.06052640.157676

0.0879923

0.2528720.0708915

0.0916996

0.039174

0.0104488

0.0328211

0.3005060.08094

0.119391

0.2252310.0607063

0.0985716

0.1058260.0275538

0.0892811

0.213690.0563512

0.0886789

G

u

i

ta

r

0.200952

0.0523276

0.0923786

0.217088

0.0570994

0.0963588

0.170117

0.0465266

0.0986187

0.195456

0.0511755

0.0895305

0.188256

0.0552828

0.0836358

0.231626

0.0611373

0.0983544

0.0394368

0.00995864

0.0444481

0.135834

0.0382768

0.0577408

0.326717

0.0888849

0.112043

0.332712

0.0903135

0.114918

H

el

ico

p

t

e

r

0.277976

0.0757696

0.102148

0.08900440.0231368

0.0637402

0.1682560.0504726

0.106869

0.1625730.0423595

0.0777038

0.2056360.0582612

0.0852807

0.1536620.0402216

0.0764009

0.1872660.0490504

0.0849933

0.0366661

0.00944749

0.0372489

0.2982010.0808161

0.102827

0.4206830.116592

0.124092


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La

m

p

0.120502

0.0347483

0.0522851

0.392532

0.1112810.111408

0.593152

0.183220.153362

0.195299

0.05664540.0687402

0.503123

0.1718640.125172

0.229469

0.06525710.0823831

0.466972

0.1359940.133555

0.660728

0.2066580.149917

0.088332

0.0173754

0.0246162

0.1007

0.03202540.0470266

M

on

it

o

r

0.499516

0.172290.128299

0.108521

0.03019610.0557825

0.376139

0.1133050.139402

0.217875

0.07042790.0735501

0.411129

0.1243580.111261

0.508909

0.1694040.128134

0.552901

0.1634190.148006

0.292369

0.07806590.119659

0.155519

0.05190180.052943

0.0645298

0.0227079

0.0147232

Dissimilarity matrix using A3 shape function:


Bed

0.004216440.00381456

0.0135032

0.145050.050939200.0983142

0.1250110.03936020.0781532

0.0441840.01386860.0422063

0.1144950.03165160.0814197

0.02482480.0114010.0310763

0.114630.03171860.0713441

0.03470730.01101070.0375937

0.0592390.01856730.0434032

0.07544250.02256150.0529078

C

a

r

0.0101219

0.00343161

0.0224428

0.00765467

0.00148054

0.0108831

0.170982

0.0498428

0.0949803

0.0272805

0.0074783

0.0306139

0.166476

0.0442097

0.0986837

0.0139099

0.0067783

0.0222298

0.156052

0.0413385

0.0879157

0.028849

0.00779045

0.0363666

0.0401417

0.0112206

0.0309834

0.0623095

0.0167953

0.0457645

Ca

si

o

0.09550480.035414

0.0939397

0.09100890.043675

0.0490551

0.0638363

0.0301901

0.0272665

0.1753770.0591867

0.114298

0.1164230.039981

0.103827

0.08508440.0340585

0.0843634

0.09315680.033176

0.0787549

0.1157650.0419848

0.102549

0.1924280.0647543

0.116499

0.2249190.074361

0.123626

C

h

air

0.0142615

0.00404615

0.0211118

0.206531

0.0677262

0.119322

0.207326

0.0602538

0.102107

0.0112393

0.00287439

0.0198576

0.175851

0.0460025

0.0969335

0.0266154

0.0097598

0.0319454

0.17599

0.0464942

0.0894268

0.032551

0.00825472

0.0355719

0.0198752

0.0052228

0.0208734

0.0406968

0.0104195

0.0369811

C

ol

u

m

n

0.170562

0.04476760.0916795

0.151657

0.05028140.112895

0.047175

0.01669650.0605939

0.318812

0.08593470.123911

0.0291969

0.0074451

0.0302281

0.17623

0.04871340.0975809

0.0389598

0.009879810.0476944

0.127296

0.03272850.0789242

0.350787

0.09768410.125701

0.395886

0.1094960.137036

H

an

dS

et

0.013613

0.004286010.0226769

0.191794

0.06342940.115694

0.181753

0.05357150.096742

0.0195414

0.005466320.0278876

0.157582

0.04173030.0944764

0.019639

0.0022811

0.0269308

0.15834

0.04204890.0865032

0.027248

0.007349820.0326801

0.0287238

0.007958750.0273789

0.0498595

0.0132240.0426767

G

u

it

a

r

0.230938

0.06179910.112425

0.0976072

0.03587340.0915725

0.0507674

0.01766420.0534917

0.42825

0.1224340.149948

0.12182

0.03164350.0805519

0.223792

0.06310610.114053

0.0348139

0.00889461

0.0472251

0.208574

0.0565640.112671

0.449195

0.1327330.150144

0.498997

0.1460070.160049


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He

l

i

co

pte

r

0.04113310.0108325

0.0407805

0.1116380.0416252

0.105959

0.08607870.0272823

0.0723593

0.1210550.0312614

0.0688789

0.07919260.0202234

0.0641514

0.03316360.011328

0.0418439

0.0528050.0136017

0.0560766

0.00856322

0.00215824

0.017457

0.1405380.0378528

0.0694883

0.1877640.0494744

0.0856219

L

a

m

p

0.040641

0.0109277

0.0378313

0.286167

0.0925995

0.135436

0.289402

0.0846529

0.120719

0.0208332

0.0055996

0.0845546

0.233085

0.0619826

0.111442

0.0755015

0.0223731

0.0541963

0.262274

0.0708704

0.109038

0.0789199

0.0200798

0.0562777

0.0128054

0.00333187

0.0163738

0.0860541

0.0227068

0.0661458

Mo

n

it

or

0.0751856

0.02054670.048842

0.342664

0.1060130.136499

0.338834

0.09862680.124487

0.0229878

0.006431170.0274046

0.275089

0.07550310.117653

0.122618

0.03519740.0661826

0.324674

0.08989140.117111

0.136426

0.03618540.0700221

0.0237703

0.006533340.0293461

0.010221

0.0032190

0.0185209

Dissimilarity matrix using D3 shape function:


B

ed

0.0327302

0.00849057

0.0284896

0.379821

0.1095510.120999

0.0417333

0.01079920.0327831

0.170054

0.05317460.0615466

0.124462

0.03736430.0501542

0.0133044

0.00359880.018848

0.248694

0.07055370.0865092

0.481247

0.1524510.127637

0.0694547

0.01912470.0440143

0.175601

0.05670660.0634833

Ca

r

0.04438910.0112317

0.0427367

0.02937580.00810733

0.0135278

0.07705820.0206918

0.0540544

0.09998840.0265844

0.0534111

0.07748010.0200781

0.0488325

0.04849340.0135166

0.0347547

0.2416050.063568

0.100368

0.4663920.132629

0.138444

0.1078380.0294916

0.0478917

0.02651920.0071350

0.0197639

Ca

s

i

o

0.243787

0.0700586

0.108643

0.0377096

0.00970812

0.0384095

0.0147549

0.0040314

0.021809

0.323228

0.100256

0.124915

0.287808

0.0906141

0.117904

0.256429

0.0702284

0.101876

0.220266

0.0679777

0.108243

0.310446

0.0984646

0.123222

0.49245

0.150667

0.138529

0.430892

0.134614

0.137111

C

h

a

i

r

0.0167279

0.00436188

0.0156326

0.31148

0.0934991

0.124627

0.0510386

0.016441

0.0385243

0.00701696

0.00185362

0.0086747

0.0127953

0.0032865

0.0153713

0.0731016

0.0217155

0.0412352

0.105575

0.0271053

0.063799

0.270645

0.0752666

0.101281

0.177155

0.0481545

0.0684482

0.0445461

0.011476

0.0376032

Co

l

um

n

0.0751491

0.02289260.0404894

0.384661

0.1139360.139444

0.0925858

0.02708530.0549862

0.0539122

0.01413020.0383307

0.0308669

0.0078131

0.0322462

0.0849065

0.02596010.0446783

0.180522

0.04675170.0881209

0.281535

0.07485820.114232

0.495764

0.17440.122265

0.113251

0.02893230.165221


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Ha

n

d

Se

t

0.0581505

0.0147192

0.0465353

0.483987

0.141694

0.142768

0.0961692

0.0259387

0.0574079

0.0323132

0.00846672

0.0276565

0.128734

0.0354246

0.0582171

0.0092468

0.0049978

0.0120316

0.307161

0.0827662

0.106454

0.56737

0.169545

0.147115

0.0392941

0.010309

0.0305147

0.0472766

0.0134368

0.0301559

G

ui

t

a

r

0.2179010.061245

0.0825684

0.2365210.071607

0.116855

0.2073480.0598802

0.0832519

0.1401270.0360641

0.0798013

0.1699520.0443815

0.0812911

0.3232080.0928635

0.102795

0.0265747

0.0069416

0.0266502

0.02702870.00696932

0.0301038

0.5427090.16125

0.135479

0.2399790.063332

0.101997

H

el

i

co

pt

er

0.2440830.0701102

0.0867554

0.3304530.101431

0.136274

0.2466480.0724132

0.0917394

0.1627390.0426508

0.0840356

0.1670870.0445949

0.0804238

0.3848720.113887

0.114443

0.0379780.00980866

0.0322755

0.0291623

0.00737316

0.0379532

0.6115670.187487

0.146975

0.2111380.0557542

0.0956866

L

a

mp

0.045404

0.01203

0.0336688

0.470789

0.149609

0.145583

0.108127

0.0333027

0.0570018

0.0600541

0.0158268

0.0363025

0.0532586

0.01376

0.0362319

0.0707109

0.0224154

0.0401059

0.223492

0.0588989

0.0913808

0.454051

0.130162

0.132049

0.0620269

0.00827331

0.0162766

0.039308

0.0109072

0.0278697

M

o

n

i

to

r

0.128709

0.0331342

0.0671721

0.571569

0.171318

0.149556

0.185845

0.0494486

0.0764842

0.296337

0.0881494

0.0942851

0.263499

0.0764136

0.0880259

0.0829025

0.0219923

0.0520367

0.44566

0.12708

0.125146

0.715772

0.228828

0.16361

0.143613

0.0391429

0.1459

0.0292735

0.0090351

0.0094933

Conclusion:

The above method is very simple to implement. It is also very fast it takes

around fifteen seconds to compute all three dissimilarity measures between

two 3D models given in the .3ds format. Invariance and robustnessproperties can be satisfied using shape functions and norms with the desired

properties.

The above tables give dissimilarity measures. They absolute values have no

particular meaning. What we must observe is the relative values. So we have

to see how a model compares with respect to the others and with respect to

another model of the same thing. For example we can see that the


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dissimilarity measure for bed against bed is almost half of all the other

values except for bed against handset. This is because both bed and handset

have the same basic structure and the comparison is invariant to scale.

We can observe that the A3 shape function is the best as it is able to

distinguish better. Also the LN

Minkowski norm with N=2 is providing the

best dissimilarity measure.

Improvements:

Development of benchmark databases containing degenerate 3D polygonal models so that different shape analysis methods can be

compared.

More sophisticated shape functions based on domain specificinformation or local geometric properties.

More efficient shape distribution sampling and reconstructionmethods, possibly based on adaptive strategies.

Combining shape distributions with other attributes (e.g. surfacecolors, moments, etc.) for improved discriminability.

References:

Shape DistributionsRobert Osada, Thomas Funkhouser, Bernard Chazelle, and David Dobkin

Princeton University, Princeton, NJ 08540, USA

www.help3d.com for the 3D models.

www.google.com
http://www.help3d.com/http://www.google.com/http://www.google.com/http://www.help3d.com/

3D Shape Search

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