With the increasing use of 3D objects and models, mining of 3D databases is becoming an important issue. However, 3D object recognition is very time consuming because of variations due to position, rotation, size and mesh resolution. A fast categorisation can be used to discard non-similar objects, such that only few objects need to be compared in full detail. We present a simple method for characterising 3D objects with the goal of performing a fast similarity search in a set of polygonal mesh models. The method constructs, for each object, two sets of multi-scale signatures: (a) the progression of deformation due to iterative mesh smoothing and, similarly, (b) the influence of mesh dilation and erosion using a sphere with increasing radius. The signatures are invariant to 3D translation, rotation and scaling, also to mesh resolution because of proper normalisation. The method was validated on a set of 31 complex objects, each object being represented with three mesh resolutions. The results were measured in terms of Euclidian distance for ranking all objects, with an overall average ranking rate of 1.29

du Buf, J. M. H.

Lam, Roberto

Sapientia

INVARIANT CATEGORISATION OF POLYGONAL OBJECTSUSING MULTI-RESOLUTION SIGNATURESRoberto LamInstituto Superior de Engenharia, Universidade do Algarve, Campus da Penha, Faro, Portugalrlam@ualg.ptJ.M.Hans du BufISR-Vision Laboratory, FCT, Universidade do Algarve, Campus de Gambelas, Faro, Portugaldubuf@ualg.ptKeywords: 3D shape matching - Volumetric models - Manifold Meshes.Abstract: With the increasing use of 3D objects and models, mining of 3D databases is becoming an important issue.However, 3D object recognition is very time consuming because of variations due to position, rotation, sizeand mesh resolution. A fast categorisation can be used to discard non-similar objects, such that only fewobjects need to be compared in full detail. We present a simple method for characterising 3D objects withthe goal of performing a fast similarity search in a set of polygonal mesh models. The method constructs,for each object, two sets of multi-scale signatures: (a) the progression of deformation due to iterative meshsmoothing and, similarly, (b) the influence of mesh dilation and erosion using a sphere with increasing radius.The signatures are invariant to 3D translation, rotation and scaling, also to mesh resolution because of propernormalisation. The method was validated on a set of 31 complex objects, each object being represented withthree mesh resolutions. The results were measured in terms of Euclidian distance for ranking all objects, withan overall average ranking rate of 1.29.1 INTRODUCTION ANDRELATED WORKOne might say that technological developments willlead us towards using increasingly complex illustra-tions, i.e., moving from 2D to 3D space. There aredigital scanners which produce 3D models of real ob-jects. CAD software can also produce 3D models,from complex pieces of machinery with lots of cor-ners and edges to smooth sculptures. Very complexprotein structures play an important role in pharma-cology and related medical areas. Many actors in theWorld Wide Web have started to incorporate 3D mod-els in sites and home pages. As a consequence ofthis trend, there is a strong interest in methods for 3Dsimilarity analysis (Bustos et al., 2005; Tangelder andVeltkamp, 2007). Similarity analysis is a fast way todiscard many irrelevant objects from a database, i.e.,before precise object recognition (matching) whichmay be very time consuming because of all variationsthat may occur: different position (object origin), ro-tation, size and also mesh resolution.Similarity analysis does not require precise shapecomparisions, global nor local. Instead, this approachis based on computing sets of features (FV or featurevector) of a query object and comparing its FV withall FVs of known objects in a database. The FVs canbe obtained by a variety of methods, from very sim-ple ones (bounding box, area-volume ratio, eccentric-ity) to very complex ones (curvature distribution ofsliced volume, spherical harmonics, 3D Fourier coef-ficients) (Saupe and Vranic, 2001; Pang et al., 2006;Sijbers and Dyck, 2002). We mention two approacheswhich are related to our own approach. Assfalg etal. (2006) projected a 3D object onto 2D curvaturemaps. This is preceded by smoothing and simplifica-tion of the polygonal mesh, and final retrieval is basedon comparing the 2D curvature maps. Chuang et al.(1991) and Suzuki (2007) used the fractal dimensionfor characterising 3D objects.The intrinsic nature of the objects may posesome constraints, and some methods may be moresuitable—and faster—for the extraction of FVs thanothers. For example, methods based on spherical har-monics and 3D Fourier coefficients are not suitablefor concave objects (non-star-shaped), whereas othermethods have problems with open (non-closed) ob-jects. Some limitations can be solved by combin-ing two or more methods. However, since many ob-jects can yield very similar FVs by applying only onemethod, i.e., mathematically possibly an infinite num-ber of objects, normally several methods are com-bined to achieve the best results.2 OVERVIEW OF OURAPPROACHWe use a set of 31 models, each one represented withfour different mesh resolutions. The models were se-lected from the (AIM@SHAPE, 2008) database. Thisdatabase has high-definition objects which can beconverted to other mesh resolutions by means of oneparameter between 9.9 (max mesh size) and 5.5 (minmesh size). The models were downloaded in PLYformat and only “watertight” ones—closed, withoutgaps and with regular meshes—were selected. Fig-ure 1 shows a few examples, and Table 1 a list of allobjects with their mesh resolutions: the first three res-olutions are used for creating the training set FVs, thefourth one as test object for similarity search.In order to obtain invariance to translation andscale (size), the models were normalised to the uni-tary sphere (radius 1.0) after the origin of the modelswas translated to the center of the sphere. Rotation in-variance is achieved by the fact that our FV is globalto the model as proven in (Vranic´, 2004). Invarianceto mesh resolution is obtained by proper feature nor-malisations, which will be explained below. We ap-ply two different but complementary methods in orderto generate two kinds of features for object retrieval.These are based on mesh smoothing (Section 2.1) andon dilation and erosion (Section 2.2).2.1 Mesh smoothingMesh smoothing serves to reduce noise, for examplefor decreasing the mesh size by re-triangulation ofplanar areas. (Glendinning and Herbert, 2003) usedsmoothing of principal components for shape classi-fication in 2D. Here, the idea is related to iterativeand adaptive (nonlinear) mesh smoothing in 3D, i.e.,smoothing in quasi-planar regions but not at sharpedges (Lam et al., 2001). However, here we simplyapply the linear version which will smooth the meshat all vertices: it starts by eliminating very sharp ob-ject details like in- and protruding dents and bumps,and then, after more iterations, less sharp details. Thesum of the displacements of all vertices, in combi-nation with the contraction ratio of the surface area,Table 1: All 31 models with their mesh resolutions, the lastresolution was used in similarity search.N Model Resolutions1 Blade 6.5; 7.5; 9.9; 8.02 Bimba 6.0; 8.5; 9.5; 8.03 Block 5.0; 6.5; 8.0; 8.54 Bunny 6.5; 7.5; 9.9; 8.05 Cow 6.0; 6.4; 9.9; 7.16 Cow2 6.0; 7.5; 9.9; 8.97 DancingChildren 6.0; 7.5; 9.9; 6.88 Dragon 6.0; 8.0; 9.5; 7.79 Duck 6.0; 7.5; 9.9; 6.710 Eros 6.0; 7.5; 9.9; 6.511 Fish 6.0; 7.5; 8.0; 8.012 FishA 6.0; 7.5; 9.9; 7.013 GreekSculpture 6.5; 7.0; 7.7; 8.514 IsidoreHorse 6.0; 7.5; 9.9; 7.015 Mouse 6.0; 7.5; 9.9; 7.816 Pulley 6.0; 7.5; 9.9; 7.017 Torso 6.0; 7.5; 9.9; 7.718 CamelA 6.0; 7.5; 9.9; 7.819 Carter 6.0; 8.5; 9.5; 7.320 Chair 6.5; 7.5; 9.9; 6.921 Dancer 6.0; 7.5; 99; 7.722 Dente 6.0; 7.5; 9.9; 7.023 Elk 6.0; 7.5; 9.9; 7.924 Grayloc 6.0; 7.5; 9.9; 7.825 Horse 6.0; 7.5; 9.9; 8.026 Kitten 6.0; 7.5; 9.9; 7.327 Lion-dog 6.0; 7.5; 9.9; 8.028 Neptune 6.0; 8.0; 9.5; 7.629 Ramesses 6.0; 7.5; 9.9; 8.030 Rocker 6.0; 7.5; 9.9; 7.131 Squirrel 6.0; 7.5; 9.9; 7.2generates a quadratic function which can characterisethe model quite well.If Vi, i = 1,N, is the object’s vertex list with as-sociated coordinates (xi,yi,zi), the triangle list T (V )can be used to determine the vertices at a distance ofone, i.e., all direct neighbour vertices connected to Viby only one triangle edge. If all neighbour verticesof Vi are Vi, j, j = 1,n, the centroid of the neighbour-hood is obtained by V¯i = 1/n∑nj=1 Vi, j. Each vertexVi is moved to V¯i, with displacement D¯i = ||Vi− V¯i||.The total displacement is D = ∑Ni=1 D¯i. The en-tire procedure is repeated 10 times, because we aremainly interested in the deformation of the object atthe start, when there still are many object details,and more iterations do not add useful informationanymore. Hence, displacements are accumulated byAl = ∑lm=1 Dm with m = 1...10. In order to obtaininvariance to mesh size, in each iteration m the dis-Figure 1: Examples of models: Squirrel (top), IsidoreHorse(middle) and GreekSculpture (bottom). Low resolutions atleft and high ones at right.placement Dm is corrected usingDm := Dm · NPm ·NA10 ·Sm , (1)with N the total number of vertices, NPm the numberof participating vertices (in non-planar regions whichcontributed to the displacement), Sm the surface of theobject (sum of all triangles) after the smoothing step,and A10 the final, maximum accumulated displace-ment after all 10 iterations. Then the curve of eachobject and each mesh resolution is further normalisedby the total contraction ratio defined by S10/S0 (finalsurface and original surface), and the three curves (10data points) are averaged over the three mesh resolu-tions. In the last step, the averaged Al is least-squaresapproximated by a quadratic polynomial in order toreduce 10 parameters to 3. Figure 3 shows represen-tative examples of curves Al . It should be stressedthat, in contrast to the second method as described be-low, no re-triangulation of the mesh of the object aftereach iteration is done, i.e., the number of vertices—and triangles—remains the same. Figure 2 shows amodel and the influence of mesh smoothing.Figure 2: Mesh smoothing applied to IsidoreHorse. Fromtop to bottom: original and smoothed meshes after 3, 6 and10 iterations.Figure 3: Characteristic curves from mesh smoothing of theBimba and IsidoreHorse models.2.2 Mesh dilation and erosionThe second method is based on the estimation of the3D fractal dimension by applying a sphere as struc-tural element with increasing radius. A sphere is ap-plied to the model, its origin being placed at eachvertex. This yields two surfaces: the dilated surfacegrows and the eroded one shrinks as a function ofsphere radius, both showing less object detail. In-stead of computing the fractal dimension, we com-pute the volume between the two surfaces as a func-tion of sphere radius in order to obtain characteristiccurves, like the ones described in the previous sec-tion, for characterising the objects. The growth (in-creasing radius) of the sphere is related to the meshresolution of the model. Therefore, before comput-ing the volumes, we eliminate vertices which are in-side the neighbourhood defined by the sphere with theradius used in the dilation-erosion process. If ∆L isthe difference between the maximum and minimumedge lengths of a model, i.e., ∆L = Lmax−Lmin, then∆R = 0.05∆L. Hence, the radius at iteration m isRm = m∆R, which results in volumes V0 (the volumeof the original model) and Vm (the volume between di-lated and eroded models at iteration m). For obtaininginvariance to mesh size, we applyVm :=Vm · V0Rm ; m = 1,2, ... (2)As can be seen in Fig. 4, the dilation-erosion curvesare quite similar, for different mesh resolutions, at thestart of the process, but then start to diverge when theradius becomes too big and noise is introduced to themodel. Therefore we averaged the Vm of the threemesh resolutions and only included in the FV two pa-rameters: V0 and V2. Figure 5 shows the Bimba modelwith erosion and dilation.Figure 4: Characteristic curves from mesh dilation and ero-sion of the Mouse and Squirrel models.3 RESULTS AND DISCUSSIONThe 31 models listed in Table 1 were used, eachwith four mesh resolutions. As explained before,the first three mesh resolutions were used for con-structing the FV of the model, and the last one wasused for testing. Each model was characterised by7 parameters, 5 from the method described in Sec-tion 2.1 (surface of original model after normalisationto unit sphere; contraction ratio after 10 iterations;3 coefficients of the quadratic approximation of thesmoothing curves), and 2 from Section 2.2 (volumeof original model after normalisation to unit sphere;volume between dilated-eroded surfaces after 2 itera-tions). The FVs of the objects’ test resolutions werecompared with all FVs of the database, and the ob-jects were sorted by using the Euclidean distance be-tween the FVs. Table 2 lists the results, starting withthe object with the smallest distance, then the objectwith the next smallest distance, and so forth, until thefifth object. Table 2 shows that in 26 of 31 casesthe correct object was ranked first. Similar objects(IsidoreHorse, 14; CamelA, 18; Horse, 25) were al-ways ranked at position 1 to 3. The average rankingrate R = (1/31)∑31i=1 Pi, where Pi is the ranked posi-tion of object i, is 1.29. This means that the majorityof objects is ranked at position 1 or 2, at least withinthe first 3 or 5 positions for narrowing a full objectcomparison in a big and complex database. It shouldbe stressed that, although 26 objects were ranked first,Figure 5: Mesh dilation and erosion applied to the Bimbamodel. From top to bottom: original plus eroded and dilatedmodels after 4 iterations.this does not mean that the correct object has beenidentified. The most similar object may have been de-tected, but in real conditions, i.e., with big and com-plex databases like protein structures, the search hasbeen narrowed in order to save time for detailed objectcomparisons. Future work involves improving furtherthe methods described in this paper, but in relation todetailed object comparisons which may provide addi-tional parameter models for a similarity search. Mostimportant is to improve the erosion-dilation methodsuch that identical curves are obtained for differentobject resolutions at larger sphere radii, after whichthe relative importance of the individual features andfeature sets can be studied.Table 2: Results.N Testing Model Ordered output1 Blade 1-6-30-22-122 Bimba 2-26-31-22-153 Block 3-27-9-15-314 Bunny 4-7-23-14-255 Cow 12-5-6-17-116 Cow2 6-12-30-22-17 DancingChildren 7-4-23-14-108 Dragon 8-31-23-9-29 Duck 9-31-15-8-210 Eros 18-25-10-14-2611 Fish 11-6-12-1-3012 FishA 12-6-1-5-1113 GreekSculpture 13-28-20-17-514 IsidoreHorse 14-25-18-10-2615 Mouse 15-31-9-22-216 Pulley 16-24-19-3-917 Torso 17-5-12-20-618 CamelA 25-18-14-10-2619 Carter 19-24-16-8-920 Chair 20-17-5-12-121 Dancer 11-12-5-6-2122 Dente 22-26-30-2-623 Elk 23-7-4-14-824 Grayloc 24-19-16-8-925 Horse 25-14-18-10-2626 Kitten 26-2-22-10-3027 Lion-dog 27-3-9-15-1728 Neptune 28-21-5-20-1729 Ramesses 29-10-14-4-730 Rocker 22-30-26-2-631 Squirrel 31-15-9-2-22ACKNOWLEDGEMENTSWe would like to thank the author of Binvox software,Patrick Min. Binvox was used in order to compute thevolume of the models.Research supported by the Portuguese Foundationfor Science and Technology (FCT), through the pluri-annual funding of the Inst. for Systems and Roboticsthrough the POS Conhecimento Program (includesFEDER funds), and by the FCT project SmartVision:active vision for the blind (PTDC/EIA/73633/2006).REFERENCESAIM@SHAPE. (2008). http://www.aimatshape.netAssfalg, J., Del Bimbo, A. and Pala, P. (2006). Content-based retrieval of 3D models through curvature maps:a CBR approach exploiting media conversion. Multi-media Tools Appl., 31(1):29–50.Bustos, B., Keim, D. A., Saupe, D., Schreck, T. and Vranic´,D.V. (2005). Feature-based similarity search in 3Dobject databases. ACM Comput. Surv., 37(4):345–387.Chuang, K., Valentino, D. and Huang, H. (1991). Measure-ment of fractal dimension using 3-D technique. Proc.SPIE, Vol 1445,:341-34.Glendinning, R.H. and Herbert, R.A. (2003). Shape classi-fication using smooth principal components. PatternRecognition Letters, 24(12):2021-2030.Lam, R., Loke, R. and du Buf, H. (2001). Smoothing andreduction of triangle meshes. Proc. 10th PortugueseComputer Graphics Meeting, 97–105.Pang, M., Wenjun, D., Gangshan, W. and Fuyan, Z. (2006).On volume distribution features based 3D model re-trieval. Proc. Advances in Artificial Reality and Tele-Existence ICAT, LNCS Vol. 4282, 928–937.Saupe, D. and Vranic´, D.V. (2001). 3D model retrieval withspherical harmonics and moments. Proc. DAGM Sym-posium, 392–397.Sijbers, J. and Dyck, D.V. (2002). Efficient algorithm forthe computation of 3D Fourier descriptors. Proc. Int.Symp. 3D Data Processing Visualization and Trans-mission, 640–643.Suzuki, M. (2007). A three dimensional box countingmethod for measuring fractal dimension of 3D mod-els. Proc. 11th IASTED Int. Conf. on Internet andMultimedia Systems and Applications, PID:577-082.Tangelder, J. and Veltkamp, R. (2007). A survey of contentbased 3D shape retrieval methods. Multimedia Toolsand Applications, 441–471.Vranic´ D.V. (2004). 3D Model Retrieval. Ph.D. ThesisUniversity of Leipzig.

Invariant Categorisation of Polygonal Objects using Multi-resolution Signatures

Universidade do Algarve

INVARIANT CATEGORISATION OF POLYGONAL OBJECTS
USING MULTI-RESOLUTION SIGNATURES
Roberto Lam
Instituto Superior de Engenharia, Universidade do Algarve, Campus da Penha, Faro, Portugal
rlam@ualg.pt
J.M.Hans du Buf
ISR-Vision Laboratory, FCT, Universidade do Algarve, Campus de Gambelas, Faro, Portugal
dubuf@ualg.pt
Keywords: 3D shape matching - Volumetric models - Manifold Meshes.
Abstract: With the increasing use of 3D objects and models, mining of 3D databases is becoming an important issue.
However, 3D object recognition is very time consuming because of variations due to position, rotation, size
and mesh resolution. A fast categorisation can be used to discard non-similar objects, such that only few
objects need to be compared in full detail. We present a simple method for characterising 3D objects with
the goal of performing a fast similarity search in a set of polygonal mesh models. The method constructs,
for each object, two sets of multi-scale signatures: (a) the progression of deformation due to iterative mesh
smoothing and, similarly, (b) the influence of mesh dilation and erosion using a sphere with increasing radius.
The signatures are invariant to 3D translation, rotation and scaling, also to mesh resolution because of proper
normalisation. The method was validated on a set of 31 complex objects, each object being represented with
three mesh resolutions. The results were measured in terms of Euclidian distance for ranking all objects, with
an overall average ranking rate of 1.29.
1 INTRODUCTION AND
RELATED WORK
One might say that technological developments will
lead us towards using increasingly complex illustra-
tions, i.e., moving from 2D to 3D space. There are
digital scanners which produce 3D models of real ob-
jects. CAD software can also produce 3D models,
from complex pieces of machinery with lots of cor-
ners and edges to smooth sculptures. Very complex
protein structures play an important role in pharma-
cology and related medical areas. Many actors in the
World Wide Web have started to incorporate 3D mod-
els in sites and home pages. As a consequence of
this trend, there is a strong interest in methods for 3D
similarity analysis (Bustos et al., 2005; Tangelder and
Veltkamp, 2007). Similarity analysis is a fast way to
discard many irrelevant objects from a database, i.e.,
before precise object recognition (matching) which
may be very time consuming because of all variations
that may occur: different position (object origin), ro-
tation, size and also mesh resolution.
Similarity analysis does not require precise shape
comparisions, global nor local. Instead, this approach
is based on computing sets of features (FV or feature
vector) of a query object and comparing its FV with
all FVs of known objects in a database. The FVs can
be obtained by a variety of methods, from very sim-
ple ones (bounding box, area-volume ratio, eccentric-
ity) to very complex ones (curvature distribution of
sliced volume, spherical harmonics, 3D Fourier coef-
ficients) (Saupe and Vranic, 2001; Pang et al., 2006;
Sijbers and Dyck, 2002). We mention two approaches
which are related to our own approach. Assfalg et
al. (2006) projected a 3D object onto 2D curvature
maps. This is preceded by smoothing and simplifica-
tion of the polygonal mesh, and final retrieval is based
on comparing the 2D curvature maps. Chuang et al.
(1991) and Suzuki (2007) used the fractal dimension
for characterising 3D objects.
The intrinsic nature of the objects may pose
some constraints, and some methods may be more
suitable—and faster—for the extraction of FVs than
others. For example, methods based on spherical har-
monics and 3D Fourier coefficients are not suitable
for concave objects (non-star-shaped), whereas other
methods have problems with open (non-closed) ob-
jects. Some limitations can be solved by combin-
ing two or more methods. However, since many ob-
jects can yield very similar FVs by applying only one
method, i.e., mathematically possibly an infinite num-
ber of objects, normally several methods are com-
bined to achieve the best results.
2 OVERVIEW OF OUR
APPROACH
We use a set of 31 models, each one represented with
four different mesh resolutions. The models were se-
lected from the (AIM@SHAPE, 2008) database. This
database has high-definition objects which can be
converted to other mesh resolutions by means of one
parameter between 9.9 (max mesh size) and 5.5 (min
mesh size). The models were downloaded in PLY
format and only “watertight” ones—closed, without
gaps and with regular meshes—were selected. Fig-
ure 1 shows a few examples, and Table 1 a list of all
objects with their mesh resolutions: the first three res-
olutions are used for creating the training set FVs, the
fourth one as test object for similarity search.
In order to obtain invariance to translation and
scale (size), the models were normalised to the uni-
tary sphere (radius 1.0) after the origin of the models
was translated to the center of the sphere. Rotation in-
variance is achieved by the fact that our FV is global
to the model as proven in (Vranic´, 2004). Invariance
to mesh resolution is obtained by proper feature nor-
malisations, which will be explained below. We ap-
ply two different but complementary methods in order
to generate two kinds of features for object retrieval.
These are based on mesh smoothing (Section 2.1) and
on dilation and erosion (Section 2.2).
2.1 Mesh smoothing
Mesh smoothing serves to reduce noise, for example
for decreasing the mesh size by re-triangulation of
planar areas. (Glendinning and Herbert, 2003) used
smoothing of principal components for shape classi-
fication in 2D. Here, the idea is related to iterative
and adaptive (nonlinear) mesh smoothing in 3D, i.e.,
smoothing in quasi-planar regions but not at sharp
edges (Lam et al., 2001). However, here we simply
apply the linear version which will smooth the mesh
at all vertices: it starts by eliminating very sharp ob-
ject details like in- and protruding dents and bumps,
and then, after more iterations, less sharp details. The
sum of the displacements of all vertices, in combi-
nation with the contraction ratio of the surface area,
Table 1: All 31 models with their mesh resolutions, the last
resolution was used in similarity search.
N Model Resolutions
1 Blade 6.5; 7.5; 9.9; 8.0
2 Bimba 6.0; 8.5; 9.5; 8.0
3 Block 5.0; 6.5; 8.0; 8.5
4 Bunny 6.5; 7.5; 9.9; 8.0
5 Cow 6.0; 6.4; 9.9; 7.1
6 Cow2 6.0; 7.5; 9.9; 8.9
7 DancingChildren 6.0; 7.5; 9.9; 6.8
8 Dragon 6.0; 8.0; 9.5; 7.7
9 Duck 6.0; 7.5; 9.9; 6.7
10 Eros 6.0; 7.5; 9.9; 6.5
11 Fish 6.0; 7.5; 8.0; 8.0
12 FishA 6.0; 7.5; 9.9; 7.0
13 GreekSculpture 6.5; 7.0; 7.7; 8.5
14 IsidoreHorse 6.0; 7.5; 9.9; 7.0
15 Mouse 6.0; 7.5; 9.9; 7.8
16 Pulley 6.0; 7.5; 9.9; 7.0
17 Torso 6.0; 7.5; 9.9; 7.7
18 CamelA 6.0; 7.5; 9.9; 7.8
19 Carter 6.0; 8.5; 9.5; 7.3
20 Chair 6.5; 7.5; 9.9; 6.9
21 Dancer 6.0; 7.5; 99; 7.7
22 Dente 6.0; 7.5; 9.9; 7.0
23 Elk 6.0; 7.5; 9.9; 7.9
24 Grayloc 6.0; 7.5; 9.9; 7.8
25 Horse 6.0; 7.5; 9.9; 8.0
26 Kitten 6.0; 7.5; 9.9; 7.3
27 Lion-dog 6.0; 7.5; 9.9; 8.0
28 Neptune 6.0; 8.0; 9.5; 7.6
29 Ramesses 6.0; 7.5; 9.9; 8.0
30 Rocker 6.0; 7.5; 9.9; 7.1
31 Squirrel 6.0; 7.5; 9.9; 7.2
generates a quadratic function which can characterise
the model quite well.
If Vi, i = 1,N, is the object’s vertex list with as-
sociated coordinates (xi,yi,zi), the triangle list T (V )
can be used to determine the vertices at a distance of
one, i.e., all direct neighbour vertices connected to Vi
by only one triangle edge. If all neighbour vertices
of Vi are Vi, j, j = 1,n, the centroid of the neighbour-
hood is obtained by V¯i = 1/n∑nj=1 Vi, j. Each vertex
Vi is moved to V¯i, with displacement D¯i = ||Vi− V¯i||.
The total displacement is D = ∑Ni=1 D¯i. The en-
tire procedure is repeated 10 times, because we are
mainly interested in the deformation of the object at
the start, when there still are many object details,
and more iterations do not add useful information
anymore. Hence, displacements are accumulated by
Al = ∑lm=1 Dm with m = 1...10. In order to obtain
invariance to mesh size, in each iteration m the dis-
Figure 1: Examples of models: Squirrel (top), IsidoreHorse
(middle) and GreekSculpture (bottom). Low resolutions at
left and high ones at right.
placement Dm is corrected using
Dm := Dm · NPm ·NA10 ·Sm , (1)
with N the total number of vertices, NPm the number
of participating vertices (in non-planar regions which
contributed to the displacement), Sm the surface of the
object (sum of all triangles) after the smoothing step,
and A10 the final, maximum accumulated displace-
ment after all 10 iterations. Then the curve of each
object and each mesh resolution is further normalised
by the total contraction ratio defined by S10/S0 (final
surface and original surface), and the three curves (10
data points) are averaged over the three mesh resolu-
tions. In the last step, the averaged Al is least-squares
approximated by a quadratic polynomial in order to
reduce 10 parameters to 3. Figure 3 shows represen-
tative examples of curves Al . It should be stressed
that, in contrast to the second method as described be-
low, no re-triangulation of the mesh of the object after
each iteration is done, i.e., the number of vertices—
and triangles—remains the same. Figure 2 shows a
model and the influence of mesh smoothing.
Figure 2: Mesh smoothing applied to IsidoreHorse. From
top to bottom: original and smoothed meshes after 3, 6 and
10 iterations.
Figure 3: Characteristic curves from mesh smoothing of the
Bimba and IsidoreHorse models.
2.2 Mesh dilation and erosion
The second method is based on the estimation of the
3D fractal dimension by applying a sphere as struc-
tural element with increasing radius. A sphere is ap-
plied to the model, its origin being placed at each
vertex. This yields two surfaces: the dilated surface
grows and the eroded one shrinks as a function of
sphere radius, both showing less object detail. In-
stead of computing the fractal dimension, we com-
pute the volume between the two surfaces as a func-
tion of sphere radius in order to obtain characteristic
curves, like the ones described in the previous sec-
tion, for characterising the objects. The growth (in-
creasing radius) of the sphere is related to the mesh
resolution of the model. Therefore, before comput-
ing the volumes, we eliminate vertices which are in-
side the neighbourhood defined by the sphere with the
radius used in the dilation-erosion process. If ∆L is
the difference between the maximum and minimum
edge lengths of a model, i.e., ∆L = Lmax−Lmin, then
∆R = 0.05∆L. Hence, the radius at iteration m is
Rm = m∆R, which results in volumes V0 (the volume
of the original model) and Vm (the volume between di-
lated and eroded models at iteration m). For obtaining
invariance to mesh size, we apply
Vm :=Vm · V0Rm ; m = 1,2, ... (2)
As can be seen in Fig. 4, the dilation-erosion curves
are quite similar, for different mesh resolutions, at the
start of the process, but then start to diverge when the
radius becomes too big and noise is introduced to the
model. Therefore we averaged the Vm of the three
mesh resolutions and only included in the FV two pa-
rameters: V0 and V2. Figure 5 shows the Bimba model
with erosion and dilation.
Figure 4: Characteristic curves from mesh dilation and ero-
sion of the Mouse and Squirrel models.
3 RESULTS AND DISCUSSION
The 31 models listed in Table 1 were used, each
with four mesh resolutions. As explained before,
the first three mesh resolutions were used for con-
structing the FV of the model, and the last one was
used for testing. Each model was characterised by
7 parameters, 5 from the method described in Sec-
tion 2.1 (surface of original model after normalisation
to unit sphere; contraction ratio after 10 iterations;
3 coefficients of the quadratic approximation of the
smoothing curves), and 2 from Section 2.2 (volume
of original model after normalisation to unit sphere;
volume between dilated-eroded surfaces after 2 itera-
tions). The FVs of the objects’ test resolutions were
compared with all FVs of the database, and the ob-
jects were sorted by using the Euclidean distance be-
tween the FVs. Table 2 lists the results, starting with
the object with the smallest distance, then the object
with the next smallest distance, and so forth, until the
fifth object. Table 2 shows that in 26 of 31 cases
the correct object was ranked first. Similar objects
(IsidoreHorse, 14; CamelA, 18; Horse, 25) were al-
ways ranked at position 1 to 3. The average ranking
rate R = (1/31)∑31i=1 Pi, where Pi is the ranked posi-
tion of object i, is 1.29. This means that the majority
of objects is ranked at position 1 or 2, at least within
the first 3 or 5 positions for narrowing a full object
comparison in a big and complex database. It should
be stressed that, although 26 objects were ranked first,
Figure 5: Mesh dilation and erosion applied to the Bimba
model. From top to bottom: original plus eroded and dilated
models after 4 iterations.
this does not mean that the correct object has been
identified. The most similar object may have been de-
tected, but in real conditions, i.e., with big and com-
plex databases like protein structures, the search has
been narrowed in order to save time for detailed object
comparisons. Future work involves improving further
the methods described in this paper, but in relation to
detailed object comparisons which may provide addi-
tional parameter models for a similarity search. Most
important is to improve the erosion-dilation method
such that identical curves are obtained for different
object resolutions at larger sphere radii, after which
the relative importance of the individual features and
feature sets can be studied.
Table 2: Results.
N Testing Model Ordered output
1 Blade 1-6-30-22-12
2 Bimba 2-26-31-22-15
3 Block 3-27-9-15-31
4 Bunny 4-7-23-14-25
5 Cow 12-5-6-17-11
6 Cow2 6-12-30-22-1
7 DancingChildren 7-4-23-14-10
8 Dragon 8-31-23-9-2
9 Duck 9-31-15-8-2
10 Eros 18-25-10-14-26
11 Fish 11-6-12-1-30
12 FishA 12-6-1-5-11
13 GreekSculpture 13-28-20-17-5
14 IsidoreHorse 14-25-18-10-26
15 Mouse 15-31-9-22-2
16 Pulley 16-24-19-3-9
17 Torso 17-5-12-20-6
18 CamelA 25-18-14-10-26
19 Carter 19-24-16-8-9
20 Chair 20-17-5-12-1
21 Dancer 11-12-5-6-21
22 Dente 22-26-30-2-6
23 Elk 23-7-4-14-8
24 Grayloc 24-19-16-8-9
25 Horse 25-14-18-10-26
26 Kitten 26-2-22-10-30
27 Lion-dog 27-3-9-15-17
28 Neptune 28-21-5-20-17
29 Ramesses 29-10-14-4-7
30 Rocker 22-30-26-2-6
31 Squirrel 31-15-9-2-22
ACKNOWLEDGEMENTS
We would like to thank the author of Binvox software,
Patrick Min. Binvox was used in order to compute the
volume of the models.
Research supported by the Portuguese Foundation
for Science and Technology (FCT), through the pluri-
annual funding of the Inst. for Systems and Robotics
through the POS Conhecimento Program (includes
FEDER funds), and by the FCT project SmartVision:
active vision for the blind (PTDC/EIA/73633/2006).
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Invariant Categorisation of Polygonal Objects using Multi-resolution Signatures

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