Relative Positioning from Model Indexing

Transcription

1 Reative Positioning from Mode Indexing Stefan Carsson Computationa Vision and Active Perception Laboratory (CVAP)* Roya Institute of Technoogy (KTH), Stockhom, Sweden Abstract We show how to determine the position of a camera reative to a mode of the environment using features extracted from one image. The method is based on view variation of cross-ratios and is independent of camera caibration. Mode Indexing From Images Recognition and pose estimation of objects in images is in genera based on extracting features such as points and ines from the image and finding corresponding features in mode objects. The combinatoria compexity of matching image descriptors to modes can be considerabe. A way to reduce this is to use index tabes where image descriptors are used as indexes to a tabe containing corresponding mode descriptors. These tabes can be constructed off-ine which means that space is traded off for time [] [] [] []. Another factor affecting the compexity of recognition is the fact that the image of an object, depends on the viewpoint of the camera. This has initiated work in finding feature descriptors invariant to viewpoint [] []. For configurations of points and ines invariants can in genera be found ony when they are contained in a panar surface. For genera configurations of points and ines, any descriptor computed from image data wi vary with viewpoint. [] [5]. Since camera viewpoint in D is parameterized by parameters, variation of image descriptors wi in genera be -dimensiona. It is however possibe to choose descriptors that have far ess dimensionaity of variation than. In the case when imaging is approximated by an affine mapping, it has been shown [] [] that it is possibe to compute image descriptors from points that exhibit ony -dimensiona variation of viewpoint. For a specific configuration of points, the -D image descriptor wi be contained in a -D surface. In this work we wi consider the more genera case of imaging by perspective mapping. Since one of the main appications we have in mind is moving patform navigation, perspective effects can be substantia. In projective and perspective transformations the cross ratio is a fundamenta invariant of point and ine sets. We wi show that for points in an image, cross ratios can be computed. These cross ratios wi exhibit -dimensiona variation with viewpoint. This is a very simpe resut due to the fact that the cross ratios are invariant to rotations of the camera around the point of projection. The variation is therefore due entirey to camera transation. In the imit of arge reative viewing distances the image mapping wi become parae and the dependence of image data on camera position *Adcress: NADA, KTH, S- Stockhom, Sweden Emai: stefanc@bion.kth.se BMVC doi:.5/c..

2 wi reduce to that of the direction to the camera described by parameters, The cross ratios wi therefore asymptoticay be contained in a -D surface. An interesting aspect of using cross ratios for indexing is the fact that they are independent of camera caibration. This is assuming that the imaging is inear. Cross ratios wi be reated to camera position in a non inear way. When imagemode feature correspondence has been estabished, this reation can be inverted and the camera position reative to the mode can be found in a caibration independent way. Due to the compexity of non inear inversion this is impemented as a tabe ookup where cross ratios are used to index directy into a tabe of camera positions. In an appication we wi consider the probem of finding the position of a camera reative to a mode of the environment. By considering camera motion restricted to a panar ground, the variation of cross ratios on viewpoint wi be - dimensiona even in the perspective mapping case, and -dimensiona in the affine imit. By using ony the horizonta coordinates of vertica edge segments, cross ratios can be computed from points. This fact has previousy been expoited for navigation. [] In our case, we can use cross ratios computed from points as entries to the index tabe. This wi reduce the compexity of the index tabes consideraby. View Variation of Cross Ratios Since a genera viewpoint in R has degrees of freedom, the In image coordinates of a set of n points wi be confined to fi-dimensiona surface in R n. Having a -parameter variation is definitey too arge to be manageabe for recognition and pose estimation using indexing. We therefore consider the cross ratio of a group of 5 points arbitrariy paced in R. A cross-ratio of a set of 5 points in R with image coordinates (xi, j/i)... (xx,, j/s) can be defined as: / yi / / ys /5 y / y «/ ys y5 If a points are copanar in R a change of viewpoint wi correspond to a projective transform of the image coordinates. The projective transform is inear in the homogeneous coordinates (a:,-, j/,-, ) and the cross ratio wi therefore be invariant. If the points are not a copanar, a change of viewpoint wi no onger correspond to a projective transformation. The cross ratio wi therefore vary with the viewpoint. The image coordinates are reated to the position of a point in a camera centered coordinate system as: () x,- = Z? Y () The coordinates of the points in a word centered reference frame are denoted as (X\,Y\,Z\)... (XSJYS, Z5) and the coordinates of the projection point of the camera is (X C,Y C,Z C ) This point is taken as the origin of the camera centered

3 system. The coordinates in the camera centered system are reated to those of the word centered system by a inear transformation: ^ () This transformation can be factored into camera rotation R and projection matrix P The projection matrix contains scae parameters for image coordinates and other parameters that can in principe be obtained by caibration. By using cross ratios as image descriptors however, both the rotation and projection matrix wi cance out and there wi be no need for camera caibration. Using eq. and in the expression for the cross ratio we get: () a = X\ X c X X c X$ X c Yi Y - Y c n - Y c Z\ Z c Z Z c Zx, Z c X\ X c X X c X5 X c Y x Y - Y c y 5 - Y c Z\ Z c Z Z c Zz Z c X X c X X c X5 X c Y y - Y c Y 5 - Y c Z Z c Z^ Zc Z5 Z c X X c X X c X5 X c Y - y c y - Y C y 5 - Y C Z Z c Z^ Z c Z$ Z c where the factors /Zf and determinant of the inear transformation matrix Pi? are canceed out. We see that the cross ratio computed from the image coordinates of 5 points in genera position in R is the ratio of two poynomias in the camera coordinates (X C,Y C, Z c ) and totay independent of camera rotation. The set of cross ratios <i... <Tjt computed from a set of points are therefore in genera contained in a -dimensiona surface in fc-space. This -dimensiona surface is a characteristic of the point set, independent of viewpoint. Using cross ratios we have therefore reduced the parametric variation on viewpoint from to. In the imit of parae projection this variation is reduced even further, from to. The determinants in eq. 5 can be expanded as: (5) X\ X c X X c Y Y - Y c Z\ Z c Z Z c Yh - Yc /S5 ZJC X\ X Vi Y Z\ Z X c X c Yx y Y c Z\ x Y c x 5 Y c z z 5 V V V Y V V Z c Z c Z c etc. A determinants in eq. 5 can therefore be expressed in Xc,Y c,zc as:... = A n X c + B n Y c + C n Z c + D n n=... () and the cross ratio: _ (A t X c + BjY e + C X Z C + A)(^ X C + g y c + C Z C + >a) (A X C + B Y C + C Z C + D ){A A X C + B Y C + C Z C + D A ) ()

4 8 If wefix the coordinates of the points in the word centered system and introduce spherica coordinates for the camera position (X C,Y C, Z c ), we get: X c = r Y c = Z c = r f z (,<f>) (8) Inserting this into eq. and taking the imit of arge observation distance r we get: r-&* = (*/!(»;*) C f z {,<j>)) The variation of the cross ratio with viewpoint is therefore reduced to variation with respect to the two parameters and <f>, i.e. the anguar position of the camera reative to the word reference frame. This resut shoud be compared with that of [] where it is shown that assuming para perspective projection, it is possibe to choose image descriptors from points that wi have -dimensiona variation with viewpoint. The price for generaizing to perspective projection is that we need points instead of. Feature Indexing and Pose Estimation for a Camera on Panar Ground. Construction of Index Tabes and Feature Matching The cross ratios computed from image data can be used as indexes to a tabe containing mode features. In order to be efficient these tabes must not be too arge. The tabes are quantized versions of the index space with each axis representing an index feature. A critica factor is of course the dimensionaity of the tabe. For points in genera position in R and perspective projection we woud in genera need a -D index space with each axis representing a cross ratio. Various point configurations wi be represented as -D surfaces in this -D index space. In the imit when parae projection is vaid we can consider -D surfaces in the -D index space. The dimensionaity of the index space can aso be reduced by restricting the camera positions. This wi be the case e.g. when a camera is constrained to move on a panar surface. Our probem is to find the position reative to a mode of the environment, of a camera moving on a panar ground foor. In this case perspective effects cannot in genera be negected, since the extent of the object, in this case the spatia environment, wi be of the same order of magnitude as the distance to the object features. The fact that we are moving on a panar ground foor means however that ony two position parameters X C,Y C need to be considered. Features groups in the form of cross ratios can therefore be represented as a -D surface and the dimensionaity of the index space can be taken to be. As features we wi consider horizonta coordinates V{ of vertica ines in the image. Cross ratios can then be computed from groups of coordinates xi... x and from eq. 5 we see that they wi vary with camera position as: X x x x X x X A Vi *i Vi -y e -n x Y x Y x Y x Y x n XA Y

5 MO I ^ Jk > f> p \ - - Camera position ima e ^^ Came a position image I 5 cm D D Figure : Map of the mode environment with mode features. The area and samping grid for computing tripe cross ratio indexes is indicated and the two camera positions for testing position estimation are marked by the shaded circes The mode in our case is a ist of X, Y coordinates of features in the environment that coud give rise to vertica ines in the image. The environment is part of a room with a door opening into a corridoor. A map of the room with mode features is shown in fig The index space was constructed in the foowing way:. For X c, Y c camera positions chosen on a grid with cm spacing within the rectange in the map fig. and a combinations of features in the mode, ordered cockwise as seen from the camera position, cross ratios were computed by seection od sets of points.. The cross ratios were used as indexes to a -dimensiona index tabe. The tabe was divided into bins chosen as cubes with sides.. Whenever a cross ratio tripet indexed a bin of the tabe, the mode features ji... je together with the position X c, Y c were recorded in that bin. The indexing for feature matching works as foows:

6 . A combinations of horizonta coordinates of extracted vertica ines i'i...ie are used to compute cross ratio tripets.. The cross ratiotripets are used to index into the tabe. For each mode group of features ji...je found in the bin, an association tabe of image and mode features is incremented one step at positions (n, ji)...(i,j ). Position Estimation Pose estimation in the genera -D case invoves determining the parameters of rotation and transation describing camera position. This requires at east image points and their corresponding mode points. Methods have been presented based on seecting point tripets from image and mode and for each tripet pair compute a tentative pose. Correct tripet pairs wi then give custers in pose parameter space. [] Since this method is based on seecting groups of features from both image and mode the compexity wi in genera be higher than using indexing where seection is ony of features in the image. In the case where a priori information about possibe image mode matchings is avaiabe the compexity of this method can be reduced. Given matched image and mode features from indexing, pose estimation is in genera a non inear over determined probem provided more than features have been matched. With a caibrated camera both rotation and transation parameters of the camera can be obtained. In the case of no caibration information we can sti obtain camera position parameters X c, Y c, Z c by using the reation between cross ratios and camera position of eq. 5 From eq. we see that given cross ratios and mode coordinates, the position parameters are reated by a second order poynomia equation. Soving these equations is a non-trivia matter. In order to avoid compicated non-inear equation soving the index tabes were equipped with the information about position corresponding to each tripe cross ratio entry. Every tripet cross ratio therefore indexes a number of positions corresponding to those from which this tripet was computed from mode data. Using ony index tabe positions corresponding to correcty matched features the indexed positions can be accumuated in a matrix and arge vaues of this matrix shoud correspond to probabe camera positions. Note that, camera rotation is not obtained by this method but requires some form of caibration of the camera. Experimenta Resuts In order to test mode indexing and pose estimation the camera was paced in two different positions indicated in fig.. For each position vertica edge segments were extracted by a very simpe and robust procedure based on averaging image intensity in the vertica direction for each horizonta image coordinate. In order to cope with sight, variations from ideay vertica edges, the image was divided into horizonta stripes. In each stripe vertica edges were extracted independenty and subsequenty inked across stripes based on horizonta proximity. Figs. and show extracted edge segments and resuting number of associations between image and mode features using the indexing method described earier. The size of the number at position i,j of this tabe can be seen as a measure of support for the hypothesis that image feature i shoud be matched to

7 mode feature j. Finding the best association for each image feature wi in genera be combinatoriay very compex. By simpe threshoding severa hypothetica matches can be discarded at the outset however. From figs. and it can be seen that for most image features the correct mode feature dominates the row in the association tabe. Another factor reducing the compexity of finding associations is the fact that feature ordering from eft to right wi in genera be invariant over a positions. This imposes constraints on possibe pairwise matchings of image and mode features. The main factors determining the compexity of finding correct associations wi be the presence of cutter ines that are not modeed and the disappearance of modeed ines from the image. Work is presenty going on in testing the robustness of the method against extra cutter and disappearing mode features. Every cross ratio tripet associated with correcty matched features can be used to index the ook up tabe containing the position from which this tripet was computed. These positions are accumuated and figs. and 5 shows the argest accumuated positions obtained using correcty matched features together with correct camera positions. Note that the grid spacing is cm so the accuracy averaged argest accumuated positions is around - cm. From the spread of the accumuated positions we see that position uncertainty is argest in the direction to the features and substantiay ess in the orthogona direction. This effect is more pronounced in position. This can be expected since especiay in position the image mapping shoud be cose to parae and the features are concentrated in a rather sma anguar sector, which means that perspective wi be very sma, i.e. cross ratios wi not vary substantiay in the direction to the imaged features. References [] Y. Lamdan, J.T. Schwartz, and H. J. Wofson, Object recognition by affine invariant matching. In: Proc. CVPR-88, pp. 5-, (88) [] C.A. Rothwe, A.P. Zisserman, J.L. Mundy, D.A. Forsyth, Efficient Mode Library Access by Projectivey Invariant Indexing Functions, In: Proc. CVPR-, pp. -, () [] D.T. Cemens and D.W. Jacobs, Space and time bounds on indexing -D modes from -D images, IEEE Trans, on Pattern Anaysis and Machine Inteigence,, pp -. () [] D. Jacobs, Space efficient D mode indexing, IU-workshop pp. -5 () [5] J. B. Burns, R. S. Weiss and E.M. Riseman, View variation of point-set and ine-segment features, IEEE Trans, on Pattern Anaysis and Machine Inteigence, 5, pp 5-8, () [] K. Astrom, A Correspondence Probem in Laser Guided Navigation, Proc. Symposium on Image Anaysis, Uppsaa, Sweden,pp. -, - March () [] G. Stockman, Object recognition and ocaization via pose custering, CVGIP, -8, (8)

8 Vertica ines extracted: Image 5 8 IS <O> <> <5> <8> <> 8 5 <> <> <888> <> <> <85> <> < outside picture ii ode f eat u r e Figure : Number of associations between image feature (i) and mode feature (j) from indexing. Correct associations are marked with < >. Image feature which is a shadow edge and feature do not have any corresponding mode features.

9 Vertica ines extracted: Image : ) <> 8 8 <> <> <> 8 <EO> 8 8 <> 8 8 <> 5 <> <> 8 5 <E> outside picture M O de f e a t u r e - outside picture Figure : Number of associations between image feature (i) and mode feature (j) from indexing. Correct associations are marked with < >. Image features and and features and are the same edge spit into two. Feature is an extra edge not present in the mode.

10 I!!!!!! I I I I I I I I I I I I I I I I I I I I I I I I I I I I I cm I!... _ _ I I I I I *8 -I I I 8MMI 5 L:LI_^-_-_-_-_-_-_-_-_-_-_:_:_I^^^.;LL_-_-_J -c- >. 5 cm Figure : argest numbers accumuated positions from the index tabe representing reations between cross ratio tripets and x-y position. The grey circe is the correct camera position for image. * I II 8 I I I too 8 J cm I I 8 O ' II 8 -I I v L-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_-_H 5 cm Figure 5: argest numbers accumuated positions from the index tabe representing reations between cross ratio tripets and x-y position. The grey circe is the correct camera position for image.