3D Description of the Human Body Shape Using Karhunen-Loève Expansion

Transcription

1 Internatonal Journal of Informaton Technology Vol. 8, No. 2 September D Descrpton of the Human Body Shape Usng Karhunen-Loève Expanson Zouhour Ben Azouz (1, 2), Marc Roux (2), Rchard Lepage (1) (1): Laboratore d magere de vson et d ntellgence artfclle École de Technologe supéreure de Montréal ( 2 ): Vsual Informaton Technology Insttute for Informaton Technology Natonal Research Councl Abstract The Karhunen-Loève expanson s used for a compact 3D descrpton of the human body shape. In ths paper we show that a compact 3D descrpton can be acheved usng a small set of egenvectors. Expermental results usng a database of three-dmensonal body scans are presented. Applcatons are also dscussed. 1.Introducton The measurement of the human body (Anthropometry) s an essental part of the engneerng desgn of cars, arcraft, workspaces, and clothng, to name a few. Tradtonally such measurements nvolve the lnear dstances between anatomc landmarks and the crcumferental values at predefned locatons. Practcally, such measurements are lmted to a set of about 100 values. There are, however, problems nherent wth the use of tradtonal anthropometry as outlned n Robnette et. al [1]. Frst, the set of collected values s so small that the only vald attempt to reconstruct the orgnal shape s lmted to very smple geometres. Moreover, n most cases the set of unconnected (unregstered) values does not provde an accurate reconstructon of the orgnal subject. Reconstructon ambgutes are due to the lack of a coordnate reference system. Furthermore, there can be large dfferences n measurement values between observers who collect the data even f they use the same measurement protocol. Full body 3D dgtzng, a recently developed technology as outlned n Robnette et. al [1], allows one to ncrease the number of measurements from a hundred to a mllon, n ten s of seconds and thus provdes a better descrpton of the human body surface. Furthermore each value n the data set s related to a common coordnate system, allowng an accurate and detaled 3D reconstructon.

2 Int. J. of Informaton Technology Vol 8, No. 2 CAESAR project (Cvlan Amercan and European Surface Anthropometry Resource) as outlned n Robnette et al [2], s the frst three-dmensonal surface anthropometry survey performed n both U.S. and Europe. Durng ths project, body measurements have been taken for people between the ages of 18 and 65 n three countres: U.S, Netherlands, and Italy. Besdes tradtonal measurement, full body 3-D scans have been recorded n three postures as shown n fgure 1. Two scanners were used to record full body 3-D scans: a Cyberware WB4 scanner[3] n the Unted States and Italy, and a Vtronc scanner [4] n the Netherlands. Fgure 1. The three CAESAR postures The CAESAR project s a collaboratve effort wth partners from several countres such as the Ar Force Research Laboratory (AFRL) n Oho, the Socety of Automotve Engneers (SAE), the Netherlands Organzaton of the Appled Scentfc Research (TNO) and the Natonal Research Councl of Canada (NRC). After collectng a large number a body scans, the challenge s to analyze the human body shape varablty usng the provded 3-D database n order to mprove the szng strategy n the desgn of many products. To acheve ths goal, a compact descrpton of the human body shape s requred. Even though a 3D scan provdes a good descrpton of the body surface, t represents a large amount of data (thousands of polygons), whch s not convenent to study the varablty of the human body shape. The descrpton should also provde a fathful reconstructon of the orgnal shape; otherwse t wll not be more valuable than tradtonal measurements. Ths paper proposes a technque to acheve ths goal. Among the methods proposed n the lterature of 3-D objects modelng, three-dmensonal mplct functons such as algebrac functons, superquadrcss and hyperquadrcs provde a compact descrpton. Algebrac functons used by Sullvan et. al [5] and Keren et. al [6] requre hgh order terms and several constrants to represent sgnfcant shapes. Superquadrcs ntroduced by Pentland [7] and modfed by Barr [8] and Solna et. al [9] are a famly of parametrc shapes whch are extensons of ellpsods. Hyperquadrcs proposed by Hanson [10], are volumetrc shape representaton. Even though they have more representaton power than superquadrcs, hyperquadrcs have a dffculty n representng objects wth concavtes. Ohush et.al [11] developed the extended hyperquadrcs to overcome ths problem. However t has been demonstrated that the presented mplct functons are not adequate to represent complex objects such as the human body. 27

3 Zouhour Ben Azouz, Marc Roux, Rchard Lepage, 3D Descrpton of the Human Body Shape Usng Karhunen-Loève Expanson In the approach presented n ths paper we consder the fact that the analyzed 3D objects belongs to the same famly. Intutvely we can thnk that the famly s low dmensonal. It follows that any member mght be represented by a small number of parameters. The Karhunen-Loève expanson, as outlned n Fukanaga [12], s used to extract a small subset of shapes, whch allow the reconstructon of any of the shapes contaned n the space spanned by ths subset. The basc approach s smlar to the one used n face recognton as outlned n Krby et. al[13]. The data set consdered here s three-dmensonal body scans and the extracted shapes wll be named egenpersons equvalently to the egenfaces defned n face recognton experments. 2.Method 2.1 Data pre-processng For the purpose of ths work we assume that the heght of the human body s uncorrelated wth ts shape. In other words a short person and a tall person could have the same shape even though they don t have the same heght. Although wthn the tradtonal anthro-pometrc studes, the human heght s a measure that has been largely nvestgated, the descrpton of the human shape s stll a challenge. That s why n ths work we elmnated the varablty related to the heght by normalzng the heght of all the persons n the database. Before the normalzaton though, t s mportant to algn all the persons n order to have a common center of gravty. After algnment and normalzaton, the varablty n the 3D data corresponds only to the varablty of shape. We adopted a voxel descrpton of the 3D data. In ths case a cube of 1m of dmensons has been sampled to n voxels, where n = (200) 3. A functon descrbng the occupancy of those voxels s equal to one f the voxel contans at least one pont, and zero otherwse Karhunen-Loève expanson We consder the use of the Karhunen-Loève expanson, also known as Prncpal Component Analyss (PCA) as outlned n Lebart et al [14]. The man dea of the approach s to fnd a set of vectors whch best account for the dstrbuton of the body shape wthn the entre shape space. Each 3D person s converted nto a vector form mean person over a set of N persons s gven by: Ψ descrbng the occupancy of the n voxels. The Ψ = N 1 Ψ N = 1 (2.1) Devaton vectors Φ are generated and arranged n a data set matrx A as follows: A = Φ = Ψ Ψ Φ1 Φ 2... Φ [ ] N (2.2) (2.3) 28

4 Int. J. of Informaton Technology Vol 8, No. 2 The egenvectors of the covarance matrx C defned as : T C = A A (2.4) form the orthonormal bass that optmally spans the subspace of the human shape. The egenvectors, and ther correspondng egenvalues, of ths n x n symmetrc matrx C are ranked such that λ > for ( < j). The magntude of egenvector ( ) λ j λ j u j and s gven by: It then follows that any vector Φ Φ s equal to the varance n the data set spanned by ts correspondng λ = σ = 1 N 2 j j Φ. u j (2.5) N = 1 n the data set, A, can be optmally approxmated by: M j= 1 c j u j 0 M n Thus, the n-dmensonal body shape devaton vector can be re-defned as a lnear combnaton of egenvectors determned by M coeffcents denoted by c j gven by: c = Φ. u (2.7) j Computng the egenvectors and egenvalues of C s however mpractcal. Determnng the egenvectors u ' and egenvalues λ of the N x N matrx gven by ' j j (2.6) ' T C = AA (2.8) represents a computatonally feasble alternatve to resolve the problem. In fact, snce the number of persons s less than the number of voxels ( N < n ), there wll be only N meanngful egenvectors rather than n. Moreover, the frst N egenvalues and egenvectors of C are drectly computed as follows: u [ 0, N -1] ' λ = λ (2.9) 1 ' = A u [ 0, N -1] (2.10) λ The qualty of the reconstructon s dependant on the fracton of the total varance contaned n the M egenvectors used n the reconstructon. Ths fracton s gven by: M = 1 N = 1 q M q = λ λ (2.11) M Thus each 3-D body scan wll be characterzed by a set of M coeffcents, whch represent a compact and relable descrpton. 29

5 Zouhour Ben Azouz, Marc Roux, Rchard Lepage, 3D Descrpton of the Human Body Shape Usng Karhunen-Loève Expanson 3. RESULTS The above method s appled to a subset of the CAESAR database. Three hundred scans of male subjects n the standng posture have been used to extract the egenpersons. The frst 9 egenpersons are shown n Fgure 2. Only the voxels correspondng to postve values are vsualzed. Fgure 2. The frst 9 egenpersons. The vsualzed ponts correspond to voxels havng postve values. Fgure 3 shows the varaton of the percentages of the varablty as a functon of the egenpersons s number. We notce that 80% of the varablty nduced by the tranng set s spanned by 185 egenpersons percentage of varablty number of egenpersons Fgure 3 Varaton of the percentage of varablty wth the number of egenpersons 30

6 Int. J. of Informaton Technology Vol 8, No. 2 The orgnal scans of two subjects (Fgure 4) ncluded n the tranng set have been sampled. By usng 185 egenpersons, 100% of ponts enclosed n the sampled models are reconstructed n both cases. Fgure 4. Orgnal scans of subjects ncluded n the tranng set (4a and 4d), correspondng sampled data (4b and 4e) and reconstructed shapes ( 4c and 4f). Fgure 5 shows an example of body scan ncluded n the tranng set. Its reconstructon usng 185 egenpersons generates a model, contanng % of the total ponts of the sampled model. A trangulated model(fgure 5d) generated from the reconstructed cloud of pont, usng nnovmetrc software [15], shows that the body shape s reconstructed even though around 40% of the ponts are mssng. The dstance between the ponts of the orgnal model and the surface generated from the reconstructed cloud of ponts are characterzed by a dstrbuton havng a null mean and a standard devaton equal to 1.2 mm whch s less than the samplng nterval equal to 5mm. Usng 260 egenpersons ncreases the percentage of reconstructed ponts to 99.73%. The fracton of varablty spanned by those egenpersons s 95% of the total varablty. 31

7 Zouhour Ben Azouz, Marc Roux, Rchard Lepage, 3D Descrpton of the Human Body Shape Usng Karhunen-Loève Expanson Fgure 5. Reconstructon of a subject ncluded n the tranng set usng dfferent numbers of egenpersons. Orgnal scan (5a), sampled data (5b), reconstructon usng 185 egenpersons (5c), trangulated model generated from the reconstructed cloud of ponts (5d) and reconstructon wth 260 egenpersons (5e). Fgures 6 and 7 show the reconstructons of two bodes scans not ncluded n the tranng set used to extract the egenpersons. In the frst case 47% of the ponts are reconstructed. In the second case the percentage of reconstructed ponts s 58%. The dstrbuton of the dstance between the orgnal model and the trangulated model generated from the reconstructed cloud of ponts (fgure 6d and 7d) have respectvely an average of 1.7mm and 0.5 mm. The standard devaton s respectvely of 5mm and 3.99 mm. Agan, the qualty of the reconstructon s quet good even though many ponts are mssng. 32

8 Int. J. of Informaton Technology Vol 8, No. 2 Fgure 6: Orgnal scans of a subject non-ncluded n the tranng set (6a), correspondng sampled data ( 6b), reconstructon usng 185 egenpersons (6c) and trangulated model generated from the reconstructed cloud of ponts (6d). The reconstructon could be mproved wth an approprate choce of the tranng set. It s worth to menton that hands poston represents a nosy source of varablty wthn the data set. 33

9 Zouhour Ben Azouz, Marc Roux, Rchard Lepage, 3D Descrpton of the Human Body Shape Usng Karhunen-Loève Expanson Fgure 7: Orgnal scans of a subject non-ncluded n the tranng set (7a), correspondng sampled data ( 7b), reconstructon usng 185 egenpersons (7c) and trangulated model generated from the reconstructed cloud of ponts (7d). 4.Applcatons One of the applcatons of the proposed method s n the understandng of the human shape varablty and ts dstrbuton for a gven populaton n the desgn of products such as seats, workstatons and clothng. In clothng applcatons for example, a better understandng of the human shape varablty would lead to an mproved szng strategy thereby reducng nventory and unsold tems. In computer smulaton for the desgn of automobles, the method should allow the valdaton of human models and help n the selecton of cases representatve of the target populaton for whom such car models are desgned. 5.Concluson Ths paper presents an approach to compactly represent the human body shape and whch allows 3D reconstructon for vsual evaluaton of human models, ether real or computer generated. It also provdes a new tool for the statstcal analyss of 3D anthropometrc data such as the ones collected n the CAESAR project. Ths prelmnary study shows that 185 egenpersons represents 80% of the varablty n a set of 300 male subjects. References [1], K.M. Robnette, M.W. Vanner, M. Roux, and P.R.M. Jones, 3-D Surface Anthropometry: Revew of Technologes (L'Anthropométre de surface en tros dmensons: examen des technologes). Neully-sur-Sene: North Atlantc Treaty Organzaton Advsory Group for Aerospace Research & Development, Aerospace Medcal Panel, [2] K. M. Robnette, H.Daanen, E.Paquet The Caesar Project: A 3-D Surface Anthropometry Survey. Second Internatonal Conference on 3-D Dgtal Imagng and Modelng, pp ,1999. [3] Cyberware Inc [4] Vtronc Inc [5] S. Sullvan, L. Sandford, and J. Ponce. Usng geometrc dstance fts for 3-D objects modelng and recognton. IEEE Trans. Pattern analyses and Machne Intellgence, vol. 16, no. 13, pp , [6] D. Keren, D. Cooper, and J. Subrahmona. Descrbng complcated objects by mplct Polynomals. IEEE Trans. Pattern analyses and Machne Intellgence, vol. 16, no. 1, pp ,

10 Int. J. of Informaton Technology Vol 8, No. 2 [7] A.P. Pentland. Perceptual Organzaton and the representaton of natural form. Artfcal Intellgence, vol.28, no.2, pp , [8] A.H. Barr. Global and local deformaton of sold prmtves. Computer Graphcs vol.18, no. 3, pp , [9] F. Solna and R. Bajcsy. Recovery of parametrc models from range mages : The case for superquadrcs wth global deformatons.. IEEE Trans. Pattern analyses and Machne Intellgence, vol. 12, no. 2, pp , [10] A. Hanson. Hyperquadrcs: smoothly deformable shapes wth convex polyhedral bounds. CVGIP, vol.44, pp , [11] M.Ohuch, T.Sato. Three-dmensonal shape modelng wth extended hyperquadrcs. Thrd Internatonal Conference on 3-D Dgtal Imagng and Modelng, pp , [12] K.Fukunaga, Introducton to Statstcal Pattern Recognton. New York Academc, [13] M.Krby, L.Srovch. Applcaton of the Karhunen-Loève Procedure for the Charactersaton of Human Faces. IEEE Transactons on Pattern Analyss and Machne Intellgence, (1): , [14] L.Lebart, A. Morneau, M.Pron.Statstque exploratore multdmensonnelle. Dunod [15] InnovMetrc Software Inc 35