ANALYSIS TOOL AND COMPUTER SIMULATION OF A DOUBLE LOBED HYPERBOLIC OMNIDIRECTIONAL CATADIOPTRIC VISION SYSTEM

Copyight 04 y ABCM ANALYSIS TOOL AND COMPUTER SIMULATION OF A DOUBLE LOBED HYPERBOLIC OMNIDIRECTIONAL CATADIOPTRIC VISION SYSTEM Macello Mainho Rieio, macello@un. José Mauício S. T. da Motta, jmmotta@un. Depatamento de Engenhaia Mecânica, Faculdade de Tecnologia, Univesidade de Basília, Campus Univesitáio Dacy Rieio, Asa Note, 7090-900, Basília, Basil. Astact. An omnidiectional catadioptic vision system has the capacity to acquie images with a 360 field-of-view y using a conve mio. A hypeolic doule loed mio has the single view point popety, which allows the acquisition of two diffeent images of the same scene simultaneously, making possile steeoscopy. Coespondence of points etween the two images lies on a staight line, educing the computational compleity of the system, which is an advantage fo eal time steeo navigation systems. Using the single view point popety of hypeolic catadioptic systems, it is possile to pefom the ectification of cylindical panoamic images to omnidiectional images as a function of the mio design paametes. This wok pesents the constuction of a compute system to tansfom cylindical panoamic images to steeo omnidiectional catadioptic images y using a model of a doule loed hypeolic mio, allowing fee choice of the mio model geometic paametes to achieve omnidiectional steeoscopy. The system pefoms an assessment of the sensitivity of hypeolic doule loed mio paametes in the econstuction of 3D scenes. It is also discussed the influence the mio paametes has on the dimensional eos of the 3D scene econstuction pocess. Keywods: catadioptic systems, omnidiectional vision, hypeolic mio, steeoscopy. INTRODUCTION The development of systems that use panoamic images (images that captue 360 infomation fom scenes, also named omnidiectional images) has ecome a continuous eseach focus (Caon et al (0), Becke & Naya (998), Baeto & Aaujo (00)). Such vision systems could e used, fo instance, in ootic autonomous navigation.(coêa et al(006), Deccó (004)) By focusing on inceasing camea`s aility to captue infomation fom an envionment, the omnidiectional vision system aises a vaiety of techniques fo that pupose. One of those techniques is to uild an omnidiectional system that uses conve mios to impove the field-of-view of the camea. This type of system is called catadioptic omnidiectional vision system o catadioptic vision system. Among diffeent types of mios that can e comined with a camea, this wok has focus on doule loed hypeolic mios. Besides having the single view point popety when comined with the pespective camea, this mio pofile allows the acquisition of two diffeent images fom the same scene, enaling to pefom the omnidiectional steeoscopy, educing computational compleity to calculate distances etween ojects within the envionment (Souza & Motta 0). In this wok a compute system is poposed to simulate the effects of such mio paametes, aiming at studying the influence of doule loed hypeolic mio paametes in the 3D econstuction of simple scenes.. OMNIDIRECTIONAL CATADIOPTRIC VISION SYSTEM The catadioptic omnidiectional vision system is asically composed y a conve mio in font of a camea with the mio cente aligned with the optical ais of the camea. The system can e attached on the top of a oot fo ootic navigation. The envionment that suounds the mio is eflected to the optical cente of the camea and its image is pojected on a CCD optical chip. Some mio shapes when comined with cetain cameas povide the single view point popety and image unwaping without geometic non-linea distotions (Gassi, 006). Figue () shows the catadioptic omnidiectional vision system schema. 63

Copyight 04 y ABCM Figue. Catadioptic omnidiectional vision system schema.. Single View Point The single view point (SVP) is a desied popety in catadioptic vision systems, also used to classify such systems (Yagi, 999). This popety allows the unwaping of omnidiectional catadioptic images to cylindical panoamic images, o even pespective images. The cylindical panoamic image can e unwaped into a cylindical plane aound the mio. Figue. Conve mios shapes: a) spheical mio. ) Hypeolical mio c) Paaolic mio (adapted fom Gassi J, 00). Figue (a) shows the association etween a spheical mio and a camea that does not povide the SVP popety. This configuation shows that the light eams oiginated fom envionment points P and P do not convege towads a common point (the mio pojection cente). This convegence chaacteizes the SVP. The spheical mio eflects the light eams to diffeent points geneating distoted and lued images (Souza & Motta, 0). The SVP is oseved in the othe two associations etween mios and cameas such as the hypeolical mio with pespective pojection cameas (Fig. 3) and the paaolic mio with othogaphic pojection cameas (Fig. 3c). The paaolic mio has the popety of eflecting the light eams paallel to the othogaphic camea s optical ais. In the hypeolical mio and pespective camea scheme (Fig. 3), the light eams oiginated fom the envionment convege towads the pojection cente F, and ae eflected y the mio suface to the camea focal point F. The association of hypeolical mio and pespective pojection camea is used in this wok with a doule loed hypeolical mio pofile. Such shape also has the SVP popety (Souza, 007). 64

Copyight 04 y ABCM. Hypeolic mio design A hypeole can e descied y two mioed cuves disconnected fomed y the intesection etween a conical suface and a plane. As a esult, a set of Cn(,y) points can e dawn, such that the diffeence etween the line segments d and d is constant, as shown in Fig. (3). The hypeole cuve is defined y. Figue 3. Hypeole (adapted fom Souza 007). ( y k) a ( j), () whee a and ae the hypeole semi-aes that have thei cente located in (j, k). If the hypeole cente is at the system oigin (j=0, k=0), Eq. () can e simplified to y a () If the oigin of the coodinate fame is moved fom the point (j, k) (Fig. 3) to the mio focus F (Fig. 4) and the camea focus coincides with focus F, then the geometic paametes of the association etween a hypeolic mio with a camea of pespective pojection can e found. Figue 4. Hypeolic omnidiectional system schema (Souza, 007). 65

Copyight 04 y ABCM In Fig. (4) f is the camea focal length, h is the distance etween the top of the mio suface and the camea focus, R is the coodinate of the mio suface at the mio top and α is the angle of vision. Equations (3), (4), and (5) topo elate these paametes. e y a a. e h e a.tan R topo (3) (4) (5) Fo the omnidiectional vision system to make use of all the camea CCD aea and to e compact, a good elation etween paametes h and R topo must e found empiically. Fom the elationship etween the simila tiangles F p p3 and F p p it is possile to define h as 4 f. R topo h T. R piel piel (6), whee R piel is the adius of the lagest cicumfeence that can e pojected on the camea CCD, in piels coodinates, and T piel is the dimension of each piel of the camea CCD in millimetes. If the piel cell is not squaed T piel must e consideed the shote dimension of the CCD. A elation c=a/ of the semi-aes must e detemined accoding to the mio pofile and angle of vision (Gassi J., 00), and must e chosen accoding to the designed system application. Once the c value is estalished its possile to calculate the semi-ais using Eq. (7). Equation (7) is deived fom Eq. (4) y using (, y) = ( R topo, y topo). h. c c. h R topo (7) Svooda et al (997) shows diffeent mios shapes y changing values of semi-ais a and. The mio shape defined y aove equations influence the unwaping pocess, as descied elow. 3. STEREO OMNIDIRECIONAL VISION Steeo vision pefoms 3D econstuction of ojects y using a pai of images acquied fom two diffeent positions. The spatial displacement etween the two camea positions poduces small dispaities on the image of the ojects. Using specific algoithms and knowing the camea paametes and the position of one camea elative to the othe it is possile to calculate the 3D coodinates of an oject using the dispaities in the images of the two cameas. Steeo algoithms cay out thee asic tasks: a) etaction of specific image featues, ) calculation of coespondences etween simila featues and c) tiangulation etween the coesponding featues. 3. Doule loed hypeolic mios steoscopy The techniques used to econstuct a scene fom a pai of omnidiectional steeo images estict the effectiveness of the system. One of these limitations is the eistence of an invisile space egion, in the motion diection (Coêa, 006). The availaility of only one camea and the need to captue at least one pai of steeo images equies the location of the system at two places to captue the two images. As the map of the envionment is still not availale, the motion to the second location may cause cashes with eisting ojects in the envionment. To educe such limitations of the omnidiectional steeo vision system, a mio with two hypeolic loes can e used to acquie simultaneously oth images of a pai of steeo images, using a camea of pespective pojection. 66

Copyight 04 y ABCM The doule loed hypeolic mio has alignment among thei loes focal points and the camea optical ais. (Fig. 5). The eflected image fom mio s loes has a small diffeence povoked y spatial displacements of thei loes. This featue allow to pefom the omnidiectional steeoscopy. Figue 5 display the omnidiectional steeoscopy ased in doule loed hypeolic mio in association with pespective pojection camea. Figue 5. Omnidiectional steeoscopy ased in doule loed hypeolic mio and a camea with pespective pojection. Fo each mio loe, it s necessay to compute the paametes using Eqs. (3), (5), (6) and (7). In ode to pefom this task, it is equied to assign values fo R topo (intenal loe) and R topo (etenal loe). The camea paametes (f, R piel, R piel and T piel ) ae known. Net one can calculate h, a, e fo the intenal loe and h, a, e fo the etenal loe. As an eample, some paametes ae calculated in Tale (Eq. 8), defining the mio pofile. Tale - Omnidiectional vision system paametes. Constant paametes Calculated paametes R topo f T Loe pie R piel c h a (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (degee) Intenal (0) 3 307,5 33,9 56,05 37,37 86,5 6 0,009 Etenal (0) 60 504 0,9,6 78,07 86,75 70,04 y( ) 78,07. 86,75 56.05. 37,37 6,7 67,37 p/ 3 60 e -60 p/ - 3 3-3 (8) With the mio equation defined, it is possile to simulate the pofile and imaging of a doule loed hypeolic omnidiectional mio. Stating fom the cylindical panoamic image it is possile to pefom the invese unwaping to geneate catadioptic images, using the mio paametes. With the compute simulation model and a testing methodology the influence of the paametes on the mio imaging pefomance can e validated. This methodology can e used specially to minimize eos and educe unepected imaging in an actual mio design. 4. THE PROPOSED COMPUTER SYSTEM The poposed compute system uses panoamic images which ae unwaped into omnidiectional catadioptic images in ode to simulate the doule loed hypeolical mio imaging pocess. The omnidiectional catadioptic image is pocessed fo the 3D oject econstuction fom the suounding envionment. The unwaped steeo images ae concentic, enaling the etaction of geometic infomation fom the 3D scene. Howeve, the following estictions ae posed: i) eos caused y the camea intinsic paametes o hadwae failues ae not consideed and, ii) panoamic images may have vaious hoizontal and vetical dimensions. With these estictions the system was conceived to allow 67

Copyight 04 y ABCM paamete choices y the use such as f and T piel (camea paametes), R topo and R piel (to e used in unwap pocess) and c= a/ fo each loe. The system input is a cylindical panoamic image. To unwap the input image the system offes two options to geneate the catadioptic image as defined in section 3, using the paametes of the doule loed hypeolic mio. The featues used y the system ae image cones and edges. These featues ae simple to detect and pesent in most of simple ojects. The algoithm to etact image featues applies a nonlinea median filte on the catadioptic images to attenuate the effects of quantization geneated fo the unwaping pocess. Net, edge detection is pefomed (Ballad & Bown, 99), followed y the cone detection y using the Hais algoithm (Hais & Stephens, 998). Figue 6 shows, as an eample, a panoamic image with 60000 piels. Figue (7a) displays the esulting unwapped image y using the mio pofile defined with the paamete values shown in Ta (). Figue (7) displays the image filteing esult, Fig. (7c) shows the edges detected, wheeas Fig. (7d) shows the cones detected. Figue 6. Cylindical 60000 piels panoamic image geneated y PovRay softwae. Figue 7. a) Rectified image. ) filteed y a Median non-linea filte. c) edge detection. d) cone detection. The coespondence etween simila featues (supposed to e the images of the same oject point on the two loes) on a steeo catadioptic omnidiectional system can e esticted to the seach of a point in a adial line. In the poposed system, this is caied out y finding a point c (an oject cone in the etenal loe) that lies on the polonged line that passes on the image cente and on the point which is the simila cone on the intenal loe (c ). This staight line has the same length of the adius of the lagest cicle that can e pojected on the camea CCD. The seach windows used to pefom the coespondence ae the cicula windows poposed y Souza (007). The cicula windows W, assigned to point c, and W to point c, ae used to calculate the sum of squaed diffeences (SSD) in the neighohood of the points. The pai of points that esults in the lowest SSD is chosen as coespondent points. Figue 8 shows a pai of coespondent points found in the catadioptic omnidiectional image (maked in ed), on the adial staight line. Net to finding all possile pais of coespondent points, the steeo algoithm is to pefom the tiangulation task. 68

Copyight 04 y ABCM Figue 8 Pai of coespondent points (maked in ed). Since the coespondent point pais ae known, pai of points c and c can e used to calculate the coodinates of point P(X, Y), as showed in Fig. (9). Figue 9. Geometic scheme to map the envionment: a) tiangulation, ) camea CCD (adapted fom Souza, 0). Figue 9 pesents the geometic scheme to etact the coodinates of P. Such coodinates can e calculated y using tigonometic equations to find the diection of the light ays that geneate the image. Fig (9) shows the geomety fom which the P point coodinates can e calculated. The tigonometic equations used ae: Oc Oc height widith i j height widith i j (9) (0) In Eq. (9) and Eq. (0) height is the Camea CCD vetical dimension and width is its hoizontal dimension. The simila tiangles F Oc and F O P can e used to elate e z in Eq. (). In the same way, the simila tiangles F Oc and F O P elate and z 3 in Eq. (). z f () z f () Meging Eq.(4) with Eq. (9) the intenal loe pofile equations ae otained in Eq. (3) and Eq. (4). Equation (5) and Eq. (6) can e elated y meging Eq. (4) and Eq.(0), desciing the etenal lo pofile. Equation (3) and Eq. (4) calculates the coodinates of point P, ), while Eq.(5) and Eq.(6) calculates the coodinates of point P, ). ( z ( z 69

Copyight 04 y ABCM fe a f e a f a (3) f z (4) fe a f e a f a (5) f z (6) Similaity of tiangles also can e used to elate the coodinates of point P(X, Y) with the coodinates of the points P (, z) and P (, z). Using the simila tiangles F PO and F P O one can fomulate Eq. (7) that epess the elationship etween P(X, Z) and P (, z). In the same way, the similaity of tiangles F PO and F PO is fomulated y Eq. (8), that elates P(X, Z) and P, ). ( z X e Z (7) e z X e Z (8) e z Meging Eq. (7) and Eq. (8) yields Eq. (9) and Eq.(0), the point P coodinates. ( e e ) X ( e z ) (e z ) (9) X (e z ) Z e (0) Fo each pai of coespondent image points, the system calculates the coodinates (X, Z) of selected points of an oject within the envionment with oigin at the camea focus point, making possile the constuction of an envionment map. By using such coodinates and the angle θ associated to each point (Fig. 9), the distance etween the mio focus and an oject point P(X,) in pola coodinates can e calculated, and then making possile to constuct a map of points P n (X n, n ) (Fig. 0) with n <. 5. METHODOLOGY TO EVALUATE RESULTS The calculated oject coodinate points P n (X n, n ) (Fig. 0) in a pola coodinate system can e tansfomed into a ectangula coodinates system, p n( n, yn), y using Eq. () and Eq. (), in such a way that they can e used to compae the econstucted oject coodinates with the oject known wold coodinates to evaluate the system accuacy. Figue 0. Tansfomation etween pola and ectangula coodinates. 70

Copyight 04 y ABCM y cos( n ) () n X n n X sen( n ) () n Since P w ( w, y w, z w ) and p, y ) ae known, the econstuction eo can e calculated y using the Euclidean distance d etween n n( n n p and P w (point coodinate in the envionment) as d ( p, ) ( ) ( ) n P y n P z y n Pn (3) Equation (3) shows the econstuction eos and can e used to estimate the measuement accuacy of the poposed system. In this wok, the Z coodinate (vetical distance etween the system focal point and the envionment point) was not used fo eo calculations since no vetical displacement was pogammed in the epeiments. To calculate the eo esults, a simulated envionment with a ay tacing tool PovRay was ceated. Such envionment was designed to evaluate the system pefomance and to calculate the distance etween scene points and the system focal point, accoding to the mio paametes. The envionment is composed y two cues with diffeent sizes. The lage cue (edge = 50 cm) is displayed in gay colo and the smallest cue (edge = 30 cm) is displayed yellow colo. The scale of the envionment model used was :0, i.e. an oject vete located on coodinate point p pov = (5cm 8,5cm 8,5cm) in the PovRay tool has its actual wold location at p a =(50 cm, 85cm 85cm). Tale shows the position of the vetices of two cues. The coodinate point Vi (i, yi, zi) epesents the closest cue vete to the system focal point (maked as ed in Fig. ) wheeas Vf (f, yf, zf) epesents the fathe vete. Tale. Position of the ojects inside designed scene. Oject Edge size PovRay vete Envionment vete Cue (yellow) Cue (gay) System focal point (ed) 30 cm 50 cm Vi=(5 8,5 8,5) Vi=(50 85 85) Vf=(8,5,5) Vf=(80 5 5) Vi=(5 7,5 7,5) Vi=(50 85 85) Vf=(8,5,5) Vf=(80 5 5) n/a P=(0 0 0) P=(00 00 00) Figue. Designed epeimental envionment. The ed point epesents the system focal point. The simulation pocess implemented makes use of diffeent distances etween the system focal point and the cues (anging fom 50 cm to 00 cm). In each of the fou epeimental uns, such distance is added y 50 cm. In each un a new cylindical panoamic image is simulated, and the steps descied so fa in this aticle ae pefomed. The fou epeiment uns ae pefomed with diffeent mio paametes, that ange accoding to Ta. (3). In this wok the eos associated with econstucted edges ae named as shape eo, wheeas the eo associated to the vetices coodinates ae named as position eo. 7

Copyight 04 y ABCM Tale 3. Mio ange paametes. Paamete Mín value Aveage Ma value Incement R topo (mm) 5 37,5 60 5 R topo (mm) 45 67,5 90 5 R piel (piels) 30 80 0 0 R piel 300 45 550 5 (piels) c a / 0,5,4,3 0. c a / 0,5,4,3 0. When a paamete is to e vaied in the simulation analysis, the aveage value of the emaining paametes shown in Ta. 3 ae taken as constants in a un. Fo eample, when c is in focus, the othes paametes ae taken as thei aveage values displayed in Ta. (3). The set of camea s intinsic paametes wee fied in f= 6 mm and T 0,009 mm. piel 6. ANALYSIS OF THE RESULTS The esults otained fom the epeiments to analyze the system accuacy y vaying the catadioptic doule loed mio paametes show that the calculated eos have an epected ehavio, eflecting the aility of the poposed system to maintain the geometic shapes of the envionment s ojects duing the econstuction task. Howeve, vaiations of the oject distance within the scene appea to e a estiction fo pope applications of the omnidiectional system. It was oseved that a facto that can educe the system accuacy y inceasing the quantization effects (Fig. 8a) is the atio etween the vetical esolution of the cylindical panoamic and the R piel paamete that defines the catadioptic image size. Equation 4 poposes a atio that geneates satisfactoy esults, minimizing the quantization effects. R piel V 3 pn (4) By using a value fo R piel that satisfies Eq. (3), the image pocess that applies a median filte to the image can e skipped, educing the computational cost of the system. Figue () shows two gaphical esults of the eos associated to specific mio geometic paametes of the epeiment uns. Figue. Aveage eo: a) Position eo e ) Shape eo. 7

Copyight 04 y ABCM It can e oseved that R topo, R piel and c have moe effect on the position eos than R topo, R piel and c. That means, thei vaiation (R topo, R piel and c ) poduce lage vaiations in the position eos of the vetices. This effect can also e oseved when one looks to the shape eos (edge eos), in special R piel. The vaiation of R topo and R topo shows that y inceasing the hypeole ase, so vaying the mio loe pofile, the eos ae educed. This happens ecause data otained fom the scene y using lage mios ings aout moe infomation to the camea CCD. It can e seen that thee is a lage eo vaiation as a consequence of vaiations of R topo than with R topo in all epeiment uns. Regading the vaiation of the paametes R piel and R piel, the esults show that the eos ae educed y inceasing the value of R piel, ut R piel has smalle effect on eos when its value ae median (anging fom 60 to 90 piels). It can e oseved that when the aea of intenal loe of the catadioptic image eceeds 50% of the whole cicumfeence adius, the efficiency of the system deceases. Concening the vaiation of the paametes c and c, it is shown that thee is a tendency to eduction in eos when thei values anges aound thei median values and wosen eos when thei values ae in the etemes of thei ange. In othe wods, the est esults ae achieved when thei values anges fom 0.9 to.7. Such paametes have influence on the mio cuvatue and highlight that loes with accentuated cuvatues, in geneal, incease the eos. Howeve, the esults show that the paamete c must e lage than c. The constaint c, 5c has shown to poduce satisfactoy esults. The aveage eos ae shown in Fig. (3). Figue 3a depicts the aveage position eos of the fou uns. Figue 3 shows the aveage shape eo. Tale (4) displays the atio etween position and shape eos and the actual vete distance used in the epeimental uns. Figue 3. Aveage eo in 4 epeiment un: a) aveage position eo and ) aveage shape eo. Tale. 4. Ratio etween distances of epeimental uns and the aveage of oth position and shape eos. Eo/Distance (mm) 500 mm 000 mm 500 mm 000 (mm) Position eo 80:500 (0,6) 90:000 (0,09) 0:500 (0,073) 0:000(0,06) Shape eo 0:500 (0,0) 4:000(0,04) 6:500(0,0) 9:000(0,0095) The aveage position eos shown in Fig. (3) allow one to oseve that the poposed system has limited accuacy. Howeve, the oseved aveage shape eos allows to one to conclude that the oject econstuction pefomed y the system makes it feasile to e used in some applications. The elatively lage position eos can e associated to the compession of data that occus when using hypeolic mios. Such compession implies in loss of infomation aout the scene, inceasing the aveage position eo. An altenative to educe the eos oseved in the epeiments can e the use of smalle piel sizes (e.g. T piel =0.005mm) and lage esolutions. Such choices would incease the amount of image data collected y the CCD. 7. CONCLUSIONS In this wok, it was developed a compute system to simulate the geneation of omnidiectional images fom cylindical panoamic images with cadadioptic systems and also to check the influence of the geometic paametes involved on the imaging pocess. The esults shows that the system can e used as an aid fo the design and constuction of omnidiectional catadioptic systems with doule loe hypeolic mios aiming at steeoscopy. The poposed compute system allows the manipulation of these paametes in ode to evaluate thei influence on the accuacy of envionment 3D econstuction and also to simulate the metological pecision of a doule loed hypeolic mio aleady designed. 73

Copyight 04 y ABCM Fou uns of epeiments wee poposed to validate the use of the developed system. This validation involved the design of a second envionment using the endeing tool PovRay. In each un of epeiments vaiations wee pefomed in the omnidiectional system paametes. At each change of paametes, position and shape eos at diffeent visile vete o edge y the system wee calculated. Along the epeiments, the ojects within the envionment wee displaced and paametes ecalculated. It was oseved that R topo, R piel and c have moe influence in the scene econstuction than R topo, R piel and c. It can e also oseved that the effect of the scene oject distance fom the mio system has shown to e an impotant estiction to an omnidiectional vision system ased on hypeolic doule loed mios. 8. REFERENCES Ballad, D. H., & Bown, C. M., 98. "Compute Vision". Englewood Cliffs, New Jesey, USA: Pantice-Hall. Hais, C., & Stephens, M., 988. "A Comined Cone and Edge Detecto". Poceedings of the 4th Alvey Vision Confeence, p. 47-5. Baeto, J. P., & Aaujo, H., 00. "Issue on the Geomety of Cental Catadioptic Image Fomation". Compute Vision and Patten Recognition (CVPR 00) Vol., (pp. II - 4-47). Kawai, Havai. Beke, S., & Naya, S. K.., 998. "A Theoy of Catadioptic Image Fomation". Intenational Confeence on Compute Vision, p. 3-0. Bomay, India. Caon, G., Mouaddi, E. M., and Machand, E., 0, "3D Model Based Tacking Fo Omnidiectional Vision: A New Spheical Appoach". Rootics and Autonomous Systems, 60(8), 056-068. Coêa, F. R., Guizilini, V. C., & Okmoto-J., J. (006). "Omndicetional Steeovision system with tow-loe Hypeolic Mio fo oot navegation". ABCM Symposium Seies in Mechatonics, Vol., pp. 653-660. Deccó, C. C. (004). "Constuçã de Mapas de Amiente paa Navegação de Roôs Móveis com Visão Omnidiecional Estéeo". Tese (Doutoado). São Paulo, SP, Basil. Gassi J, V., 00. "Sistema de Visão Omnidiecional Aplicado no Contole de Roôs Móveis". Dissetação ( Mestado). São Paulo, SP. Gassi J, V., and Okamoto J, J. 006. "Development of an Omnidietional Vision System". Jounal of the Bazilian Society of Mechanical Sciences and Engineeing Vol. 8, No., 658-68. Souza, G.G., 007. "Simulation and Design of an Omnidiectional Catadioptic Steeo Vision System With a Hypeolic Doule Loe Mio Fo Envionment Mapping Fom Moile Roots". 9th Intenational Congess of Mechanical Engineeing (COBEM). Basilia. Souza, G.G., and Motta, J.M.S.T., 0 Simulation and Design of an omnidiectional catadioptic steeo vision system with a hypeolic doule loed mio fo envionment mapping fom moile oots. In ABCM Symposium Seies in Mechatonics, Vol. 5, p. 9-8. Svooda, T., Pajdla, T., and Hilavac, V., 997. "Cental Panoamic Cameas: Geomety and Design". Czech Technical Univesity, Faculty of Eletical Engineeing, Cente fo Machine Peception, Paga, Czech Technical Univesity. Yagi, Y., 999. "Omnidiectional Sensing and Its Applications". IEICE Tasnections on Infomation and Systems, p. 568-579. VE8-D, Nº 3. 9. RESPONSIBILITY NOTICE The authos ae the only esponsile fo the pinted mateial included in this pape. 74