Journal of Advanced Mechanical Design, Systems, and Manufacturing

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1 Journal of Advanced Mechancal Desgn, Systes, and Manufacturng Vol.4, No., Robust Method for Poston Measureent of Verte of Polyhedron Usng Shape fro Focus * akash HARADA ** **Departent of Mechancal Engneerng, Faculty of Scence and Engneerng, Knk Unversty, 3-4-, Kowakae, Hgashosaka, Osaka , Japan E-al: harada@ech.knda.ac.jp Abstract A robust easureent ethod for easurng the verte poston of a sall polyhedron usng 3D age processng s proposed. he shape fro focus ethod s appled for obtanng 3D postons of surfaces of the polyhedron. he n-focus functon s usually averaged for reducng nose, but the resultant functon lacks sharp edge nforaton of the target. he poston of the verte s ndrectly calculated fro 3D data usng the geoetrcal odel of the polyhedron. In ths study, relatonshps between easureent nose and (a) the approprate nuber of data values requred for averagng of focal easureent and (b) the area of data ftted by the Gaussan functon are nvestgated. hen, error propagaton analyss s appled to the equatons of the ndrect easureent of the verte poston. By usng the geoetrcal characterstcs of the polyhedron, the verte poston easureent s ade robust aganst easureent errors. he effectveness of the proposed ethod s confred by nuercal sulatons and eperents. Key words: Shape Measureent, Iage Processng, Abrasve Gran, Robust Measureent, Error Analyss, Shape fro Focus. Introducton *Receved 8 Sep., 9 (No. -8-7) Japanese Orgnal : rans. Jpn. Soc. Mech. Eng., Vol.74, No.743, C (8), pp (Receved 5 Jan., 8) [DOI:.99/jads.4.49] Copyrght by JSME A robust easureent ethod for easurng the verte poston of a sall polyhedron usng 3D age processng s proposed. hs ethod can be used for easurng the protruson heght of daond abrasve grans for precse grndng work. he shape fro focus (SFF) ethod s appled to obtan 3D postons on surfaces of the polyhedron. he SFF ethod can be used to detect 3D postons by usng sple and nepensve hardware. In the real world, there est several types of nose such as dark current nose, and quantzaton of the sgnal of CCDs and averagng of the n-focus functon are usually appled to reduce such nose () (4). However, the averaged functon lacks sharp edge nforaton of the easureent target. Hence, we easure the poston of a verte fro ndrect calculaton usng 3D poston data and the geoetrcal odel of the polyhedron. he easureent error propagates to the poston error of the verte through the ndrect calculaton. However, the error analyss of the SFF ethod has not been carred out yet. In ths study, the error propagaton analyss s appled to the poston easureent of the verte, and a easureent ethod that s robust aganst nose s proposed. Frst, relatonshps between age nose and the n-focus functon are nvestgated. An approprate data range for fttng the Gaussan functon and the data area for averagng are deterned. hen, two ndrect calculaton ethods for calculatng the verte poston on surfaces of the polyhedron 3D poston data are proposed. By applyng the error propagaton analyss to the ndrect calculaton, the contrbuton of the easureent error of 49

2 Journal of Advanced Mechancal Desgn, Vol. 4, No., Systes, and Manufacturng the ponts on the surface to the calculaton error of the verte usng the geoetrcal characterstcs of the polyhedron s reduced. he relatonshps between data area and (a) the nuber of poston data values and (b) the easureent error of the verte poston are also nvestgated. he effectveness of the proposed easureent ethod s confred by nuercal sulatons and eperents.. Error Propagaton Analyss of In-focus Measureent. Deternaton of Shape fro SFF Method () (4) he dstance d between a caera and a easureent target s changed as shown n Fg.. he relaton between a focus easure FM(,y) at the poston (,y) of a CCD and d s approately epressed by a Gaussan functon as FM F F ep( ( d ) / σ ) () p d p he peak value F p and the standard devaton σ F of the Gaussan functon are deterned by the paraeter of the caera lens and the teture frequency of the target surface. FM(,y) s calculated usng the su odfed Laplacan (SML) and the pel ntensty of the CCD as follows: ML(, y) I(, y) I( p, y) I( p, y) I(, y) I(, y p) I(, y p) () M N FM (, y) ML( n, y ) (N )(M ) M n N (3) In Eq. (), p denotes dstance for the dervaton of the SML. Equaton (3) ples that FM s averaged usng ( N ) (M ) data. FM s calculated for dfferent values of d usng Eq. (3). hen, the n-focus poston d p, the peak of the focus easure, s obtaned by fttng FM data wth the Gaussan functon. In ths paper, ths fttng calculaton s called as n-focus easureent. r Sensor Plane Lens d Focal Plane Object Plane Fg. Defocused age FM Fp d p σ F FM FM Fg. d - d d d Estated n-focus poston d 493

3 Journal of Advanced Mechancal Desgn, Vol. 4, No., Systes, and Manufacturng. Error n In-focus Measureent FM also ncludes nose such as dark current and quantzaton of the sgnal of CCD. Such nose propagates to the n-focus poston error. Several ages of 5 lnes/ Ronch rulngs (Fg. 3) were obtaned by changng d n steps of 5-µ ptches. A telecentrc lens wth a agnfcaton of, an effectve F nuber of., and a workng dstance of 65 and a CCD caera wth 39 4 pels n / nch, and a pel sze of 4.65 µ 4.65 µ were used as the sensng devce. Fgure 4 llustrates the result of the n-focus easureent. FM along a lne perpendcular to the rulng s calculated for the condtons p and (N ). d p s calculated usng the nonlnear least squares ethod fro Eq. () and the easured FMs. However, each FM does not precsely ft to the Gaussan functon because of the nose n FM data. hs leads to the ntroducton of an error n d p. As observed n Fg. 4, dffracton rng patterns appear near both ends of the bell-shaped Gaussan functon. he data correspondng to ths part are obtaned fro fttng calculatons. Fro the eperent, the standard devaton of the Gaussan functon, σ F, s found to be µ. Because the ages are obtaned by changng d n steps of 5 µ ptches, ( µ)/(5 µ) 9 data values are ncluded n ± σf area. Fgure 5 (a) shows the relatonshp between the standard devaton of the n-focus poston, σ dp, and the data area used for the n-focus easureent. he black crcles n Fg. 5 (a) ndcate the eperental results. Each σ dp s calculated along a lne of length 3 µ n the age. In Fg. 5 (a), 3 data values fro the peak and both sdes of the Gaussan functon are used for the n-focus easureent. Fg. 3 Ronch rulngs 4 3 FM d d Fg. 4 Gaussan nterpolaton eperent 8 analyss Data Area ( σ 4 F ) (a) data area-error σ dp µ Fg. 5 σ dp µ eperent 8 analyss Data Nuber (b) data nuber-error Fttng error of Gaussan functon 494

4 Journal of Advanced Mechancal Desgn, Vol. 4, No., Systes, and Manufacturng σ dp decreases when the data area s larger than ± σ F /. Fgure 5 (b) shows σ dp for dfferent nuber of data values,.e., 3, 5, and 9 n ± σf data area. σ dp s ndependent of the nuber of data values. Data area s assued to be equal to ± σf and nuber of data values s assued to be equal to 3 for robust easureent for practcal cases..3 Error Analyss of In-focus Measureent he error propagaton analyss of ult-nputs and ult-outputs (5) s appled to Eq. () after takng the logarth of each sde. he error propagaton equaton becoes F F p [ ln( FM ),..., ln( FM )] [ ln( Fp ) d p ( σf )] p d d σf d K d σ F p p K ( d d p ) ( σf ) ( d ) K d p ( σf ) (4) where the superscrpt denotes the pseudo-nverse of a atr, and K s the nuber of data values used for the n-focus calculaton of regresson. In a prevous study, σ FM, the standard devaton of ln(fm), was found to be equal to.4. he standard devatons of the n-focus poston wth the change n data area and data nubers are calculated to be σ FM.4. he results are shown n Fg. 5, ndcated by whte squares. he results of the error analyss roughly concde wth the eperental results. hs shows that the propagaton of the n-focus easureent error s can be epressed by Eq. (4)..4 Relatonshp between Error Propagaton and Averagng Iages of the Ronch rulngs of 3, 5, and 7 lnes/ are obtaned by changng d n steps of 5 µ ptches. Settng p n Eq. (), the standard devaton of the n-focus poston s derved fro the age data by changng the averagng area N fro to 4. Eperental results and results of the error propagaton analyss are shown n Fg. 6 by black crcles and whte squares, respectvely. he results of the error propagaton analyss also roughly concde wth the eperental results. he data range for the n-focus calculaton of regresson s ± σf for each case. σ dp of the n-focus poston s suffcently sall when the averagng sze N s larger than. F p of the Gaussan functon ncreases wth the spatal frequency of the Ronch rulngs. On the other hand, for a gven spatal frequency of the Ronch rulngs, F p s reans constant wth the changed n the averagng sze. Because F p s not ncluded n the error propagaton atr gven n Eq. (4), the easureent error s not nfluenced by the value of F p. σ dp µ Fg. 6 eperent analyss 3lnes/ 5lnes/ 7lnes/ 3 4 N Error of n-focus poston 495

5 Journal of Advanced Mechancal Desgn, Systes, and Manufacturng 5 Vol. 4, No., σ FM (.) 4 3 Fg. 7 3lnes/ 5lnes/ 7lnes/ 3 4 N Standard devaton of focus easure z Σ θ easureet data z Σ p j Σ θz z y y π y Fg. 8 Geoetrcal odel of polyhedron he eperental relatonshp between σ FM and the averagng sze wth a change n the spatal frequency of the rulngs s shown n Fg. 7. σ FM decreases wth ncreasng averagng sze. Consequently, σ dp of the n-focus poston decreases wth ncreasng averagng sze, as shown n Fg Measureent of Verte Poston of Polyhedron 3. Measureent Methods of Edge Shape Abrasve grans of daond, the easureent target of our research, have the shape of a polyhedron. he edges and peaks of the polyhedron functon as cuttng edges durng grndng. he SSF ethod can be used to detect ponts on a flat surface of a polyhedron, but shape around edge s obtunded by the averagng of the prevous secton. Hence, by usng the data of ponts on surfaces ecluded around the edge of a polyhedron, the poston of a peak can be ndrectly easured fro the geoetrcal characterstcs of the polyhedron. 3. Geoetrcal Model of Polyhedron he coordnate syste Σ of a polyhedron s shown n Fg. 8. he pont p, the orgn of Σ, concdes wth a peak of the polyhedron, the poston of whch s to be easured. Let π (,...,N) be a surface of the polyhedron ncludng the pont p. On each surface, the orgn of the coordnate syste Σ concdes wth p and z as s noral to π. he coordnate transforaton fro Σ to Σ s epressed by hoogeneous transforaton as follows: A Rot( z, θ ) Rot(, θ ) (5) z where Rot(k,θ) s a 4 4 rotatonal atr around k (k, y, z) as wth angle θ. he coordnate syste for the easureent of ponts on surfaces s defned as Σ, as shown n 496

6 Journal of Advanced Mechancal Desgn, Systes, and Manufacturng Fg. 8. Postons of easured ponts on each surface are epressed as Vol. 4, No., j [, y, z, ] (j,..., M j ) (6) j j j he superscrpt on the left ndcates the coordnate syste n whch the varable s defned. A coordnate transforaton atr, whch epresses the orgn (p,q,r) and drectons (λ,ω,θ) of Σ referred n Σ, s defned as A rans( p, q, r) Rot( z, θ) Rot( y, ω) Rot(, ) (7) λ he orgn p (p,q,r) s the poston of the verte to be easured. wo type of ethods for easurng the verte poston, () wthout and () wth usng the geoetrcal characterstcs of the polyhedron, are nvestgated. 3.3 Measureent Method wthout Usng Geoetrcal Characterstcs of Polyhedron Each surface π (,..., N) at the easureent coordnate Σ can be epressed as a b y z d (8) When ponts on a surface are gven by Eq. (6), the paraeters a, b, and d (,..., N) of the surface can be obtaned by solvng the followng equaton: a b y z d j,..., M ) (9) j j j ( j hen, by usng the solutons a, b. and d, the poston of the verte s obtaned by solvng the followng equaton: a p b q r d (,..., N) () he solutons of Eqs. (9) and () are gven as a b d p a q r an M b b N y y M d d N z z M () hs ethod requres only the nforaton regardng the ntersecton pont of the verte p and the surface π. Geoetrcal characterstcs of the polyhedron, such as those gven by Eq. (5), are not necessary for ths ethod. Hence, ths ethod s useful for easurng the shape of abrasve grans, whch have uneven geoetrcal characterstcs such as shape and sze. 3.4 Measureent Method of Geoetrcal Characterstcs of Polyhedron he daond grans selected for precse grndng have shape of unfor polyhedron. Ape angles or angles between surfaces of the polyhedron are consdered as paraeters n the geoetrcal odel gven by Eq. (5). he easured data are ftted to the odel surface usng the coordnate transforaton atr gven by Eq. (7) by the nonlnear least squares ethod. 497

7 Journal of Advanced Mechancal Desgn, Systes, and Manufacturng he surface π n the reference frae Σ can be epressed as n n y y n z [ n n n ] A [ ] z n y z Vol. 4, No., () where n s unt noral vector of π defned on Σ. he dstance d j between the easureent pont j and the odel surface s derved fro Eq. (7) as follows: d n A ( c) n A ( c) (,..., N, j,..., M ) (3) j j j j For splcty, the paraeters of the coordnate transforaton are grouped as a vector c p, q, r, λ, ω, θ. c s obtaned by teratve calculaton usng the Jacoban atr J cd as follows: [ ] c c J J cd cd d c d c NMN [ d,..., d d ] M,..., d c d c 6 NMN 6 NMN (4) he poston of the verte s gven by the convergng values of p, q, r of c n Eq. (4). If the easureent data j ncludes errors, the errors propagate to the easureent of the verte poston va Eqs. () and (4). In the followng secton, a easureent ethod that s robust aganst these errors s derved by applyng the error propagaton analyss. 4. Robust Measureent of Verte Poston 4. Error Propagaton Analyss Frst, the error propagaton equaton for easureent wthout usng the geoetrcal characterstcs s derved. he easureent odel gven by Eq. (9) s totally dfferentated wth respect to j and the paraeters of the surface, a, b, and d. he result s epressed by the followng atr forula: a a a y a [ a b d ] M [, y, z,..., z ] M a, b, a, b, M (5) where a s an error propagaton atr whose sze s ( 3 3M ). In the slar anner, the easureent odel gven by Eq. () s totally dfferentated wth respect to the verte poston (p, q, r) and the paraeters of the surface, a, b, and d. he result s epressed by the followng atr forula: 498

8 Journal of Advanced Mechancal Desgn, Systes, and Manufacturng p a pa Vol. 4, No., pa p a a a N b b N [ p q r] [ a a ] N p, q, p, q, (6) where pa s a ( 3 3N) error propagaton atr. Fro Eqs. (5) and (6), the propagaton of the easureent error of the poston data to the calculaton error of the verte poston p s epressed as p p p p a [ ] an N (7) where p s a ( 3 3N M ) error propagaton atr. Eq. (7) s the error propagaton equaton for easureent wthout usng geoetrcal characterstcs. Net, we derve the error propagaton equaton for easureent usng the geoetrcal characterstcs of the polyhedron. he easureent odel gven by Eq. (3) s totally dfferentated by easureent data and the paraeter of the coordnate transforaton c. he result s epressed by a atr foraton as follows: c J c dj c c J cd d j J j d d y j j [ p q r λ ω θ] j J dnmn d z j (8) where p s a ( 6 3N M ) error propagaton atr. By etractng p fro Eq. (8), the equaton for the error propagaton fro to p n ths ethod can be derved as follows: p (9) c where c s a ( 3 3N M ) atr whch s the upper part of the error propagaton atr c. 4. Evaluaton of Error Propagaton For splcty, Eqs. (7) and (9) are rewrtten as follows: p () he range of the poston easureent errors s gven as ± elj (l, y, z). Sngular 499

9 Journal of Advanced Mechancal Desgn, Systes, and Manufacturng value decoposton s appled after noralzng the error propagaton atr as follows: Vol. 4, No., e I e S dag( s, s, s ) 3 NMNz U [ S ] V () where U and V are noral unt atrces and S s a dagonal atr of the sngular values s, s, and s 3. he set of noralzed errors n the drect easureent data s consdered as a unt hypersphere n a 3 N M j densonal space. hs sphere of the drect easureent error s projected onto a hyperellpsod of the ndrect easureent paraeter p va the error propagaton equaton. Lengths and drectons of the ellpsod are defned by the sngular values s and the colun vector of the atr U, respectvely. he saller the sngular values of Eq. (), the lower s the nfluence of the drect easureent error on the ndrect easureent,.e., robust easureent s acheved. he sngular values of the error propagaton atr are used as a easure of robustness. 4.3 Robust Measureent of Verte Poston of Polyhedron he daond tp of a cro Vckers hardness testng achne s used as an eaple of the easureent target. he tp s shaped lke a pyrad wth a face angle of 36. Net, we carry out nuercal sulatons to test the robust easureent ethod. A data set of ponts on each surface s selected, as shown n Fg. 9. he area of easureent s descrbed wth offset and square sze, as shown n the fgure. Data around the edges are ecluded by the offset because shape around the edge obtunded by the averagng. easureent data easureent data j π π π π p. π 3 offset square sze y top vew Fg. 9 Paraeters of easureent z offset square sze y µ µ z µ y µ µ z µ µ µ (a) ethod (b) ethod Fg. Error ellpsod 5

10 Journal of Advanced Mechancal Desgn, Vol. 4, No., Systes, and Manufacturng he value of the coordnate transforaton c, the poston and orentaton of the polyhedron n the easureent coordnate frae Σ, does not nfluence the error propagaton. Hence, c [,,,,, ] s gven n ths secton. Frst, a data set wth an offset of.6, a square sze of., and 49 data values n the square are selected on each surface. Wth addng flat dstrbuted error of ± µ range on each data, the poston of the verte s calculated by ethod, wthout usng the geoetrcal characterstcs of the polyhedron and ethod, usng the geoetrcal characterstcs. he dots n Fg. ndcate easureent errors n the verte poston obtaned by carryng out calculatons tes. Error ellpsods of ± 3σ are derved fro Eq. () and are superposed on Fg.. As shown n Fg., scope of the easureent error concdes wth volue of the error ellpsod. hus, the valdty of the analyss of the error ellpsod has been confred. Net, the relatonshps between the robustness of the easureents and (a) offset, (b) square sze, and (c) the nuber of data values are eaned by usng the sngular values of the error propagaton atr as a easure. As shown n Fg. (a) and (b), the easureent of the verte poston becoes ore robust when the offset s reduced and the square sze s ncreased. In all cases, ethod, usng the geoetrcal characterstcs of the polyhedron, s ore robust than ethod, wthout usng the. Postonal data of the z as drecton are nether nfluenced by values of the offset nor the square sze. he easureents becoe ore robust when nuber of data values s ncreased, as shown n Fg. (c). However, the robustness s not strongly affected by ecessvely large nuber of data values. he apltude of the sngular value ndcates the denson of error propagaton. hs ples that f the sngular value equals and the standard devaton of each data equals µ, the standard devaton of the easureent of the verte poston equals µ. 4.4 Measureent of Actual arget he verte poston of an actual daond tp was easured by the proposed ethod. A telecentrc lens wth a agnfcaton of and NA.3 and a CCD caera wth sze of / nch and 39 4 pels were used as the sensng devce. Several ages were obtaned by changng d n steps of µ ptches. Each age was copressed to sze for reducng coputatonal burden. sngular value () () (3) (4)..6. offset (a) sngular value () () (3) (4).6..4 square sze (b) () () ethod,,y drectons () () ethod,,y drectons (3) (3) ethod, z drecton (4) (4) ethod, z drecton data nuber (c) Fg. Error analyss of easureent of verte poston sngular value 5

11 Journal of Advanced Mechancal Desgn, Systes, and Manufacturng Vol. 4, No., easureent data estated verte.4 z y z.4.35 ethod ethod Fg. Eperental result of easureent he result of the easureent s shown n Fg.. A data set wth an offset of approately.6, the square sze of., and 49 data values n the area were easured on each surface. he poston of the verte was obtaned usng the geoetrcal characterstcs of the polyhedron (ethod ). he result was obtaned fro the nuercal sulatons descrbed n the prevous secton showed that robust easureent was acheved. A coparson wth the true value wll be carred out n future studes. Fro Fg., a flat surface of the polyhedron was easured as a step pyrad by usng the SFF ethod. A systeatc error due to dffracton by the lens and quantzaton of the age data s ntroduced n the easureent. In ths study, the systeatc error has not been dealt wth, only the rando error s consdered. he error analyss of the systeatc error wll also be consdered n future studes. 5. Concluson A robust easureent ethod for the easureent of the poston of a verte of a polyhedron has been proposed. he shape fro focus (SFF) ethod s appled to obtan 3D postons of surfaces of the polyhedron. he poston of the verte s ndrectly calculated fro the 3D data. Robust easureent s acheved as follows: () At the pont easureent on the surface by the SFF ethod, data area for the n-focus easureent s larger than ± σf of the Gaussan functon. () In the case of the calculaton of the verte poston, robust easureent s acheved by usng the geoetrcal characterstcs of the polyhedron. (3) For the data acquston of the surface, the easureent of the verte poston becoe ore robust when the offset s decreased and the square sze s ncreased. he effectveness of the proposed easureent ethod has been confred by carryng out nuercal sulatons and eperents. References () Nayer, S.K. and Nakagawa Y., Shape fro Focus: An effectve Approach for Rough Surface, Proceedngs of IEEE Internatonal Conference on Robotcs and Autoaton, 5

12 Journal of Advanced Mechancal Desgn, Vol. 4, No., Systes, and Manufacturng (99), pp () Nayer, S.K., Shape fro Focus Syste, Proceedngs of IEEE Internatonal Conference on Coputer Vson and Pattern Recognton, (99), pp (3) Nar, H. N. and Stewart, C.V., Robust Focus Syste, Proceedngs of IEEE Internatonal Conference on Coputer Vson and Pattern Recognton, (99), pp (4) Subbarao, M. and Surya, G., Depth fro Defocus: A Spatal Doan Approach, Internatonal Journal of Coputer Vson, 3, 3, (994), pp (5) Harada,., Kotan, K. and Myosh,., In-Process Error Copensaton for ooth For Grndng Works, JSME Internatonal Journal, Seres C, Vol. 44, No. (), pp (6) Paul, R.P., Robot Manpulators, (98), pp. 9-9, he MI Press. 53