Colorado School of Mines. Computer Vision. Professor William Hoff Dept of Electrical Engineering &Computer Science.

Transcription

1 Proeor William Ho Dept o Electrical Engineering &Computer Science

2 Uncertaint

3 Uncertaint Let a that we have computed a reult (uch a poe o an object), rom image data How do we etimate the uncertaint in our anwer? For reerence, ee the paper Ho, W. A. and T. Vincent, Anali o Head Poe Accurac in Augmented Realit, IEEE Tran. Viualization and Computer Graphic, Vol 6., o. 4, pp ,. Link at 3

4 Review - Mean and variance Aume we have a random variable (e.g., a meaurement) We have a ample o value,,, We etimate it mean (epected value) b We etimate it variance b m E( ) i i i i m The tandard deviation i a good meaure o variabilit o I probabilit denit o i Gauian, then a ample i will all within m± 68% o the time 4

5 Review - Covariance Covariance (o two variable, and ) i a matri C Or, i we have a vector = ( ) T C i i i i i i i m m m m T E C μ μ 5 E E E E E C m m m m m m m m m m

6 Eample C = C = ote O diagonal value are mall i variable are independent O diagonal value are large i variable are correlated (the var together) Matlab cov unction 6

7 Function o random variable Let a we have random variable and, and a unction z = (,) I we know the uncertaint in and, what i the uncertaint in z? Simple cae z = a, where a i a contant m z z m z i z z i 7

8 Function o random variable Sum o two variable z = + m z z m z i z z i 8

9 Eample - Uncertaint in tereo diparit Diparit i the dierence in poition d = R L Aume Then Meaurement uncertaint i R =, L = piel That R, L are independent (i.e., RL = ) d R L RL 9

10 General cae Sa we have z = (), where z and are vector We can alo write a Or T z i z i z C z μ z μ T C E zμ z μ z z z T Cz E z z It can be diicult to compute the uncertaint analticall But ou can alwa get a numerical etimate o C z, near the current olution or z=()

11 Computing Covariance Doing a Talor erie epanion o z=() and ignoring higher order derivative So where C i the covariance matri o the vector T T z T T T T T E E E E C z z C z M M M j i Recall (AB) T = B T A T

12 Eample etimating tereo Z error We want to ind C Z, i we know Z = b/d Let Z=(), where = ( b d) T. The acobian o Z=() i ow ind C = E( T ), auming error in,b,d are independent Then C Z = C T where i the acobian evaluated at the current value o / / / d b d d b d b d b d b E

13 Probabilit Denit Let aume that error have a Gauian probabilit denit p The probabilit denit or an - dimenional vector i ai ai 3 T C p ep C 3

14 Interpreting Probabilit Denit Look at where the probabilit i a contant. Thi i where the eponent i a contant: T C z Thi i the equation o an ellipe. For eample, with uncorrelated error thi reduce to z Can chooe z to get deired probabilit. For z=3, the cumulative probabilit i about 97%. 4

15 - ai - ai Plotting Contour o contant probabilit ai ai 5

16 % Show covariance o two variable clear all cloe all Matlab code randn('tate',); p = randn(4,); p =.5 * randn(4,); % p = randn(4,); % p = p +.5*randn(4,); plot(p,p, '+'), ai equal; ai([ ]); C = cov(p,p) Cinv = inv(c); detcqrt = qrt(det(c)); % Plot the probabilit denit, % p(,) = (/(pi det(c)^.5))ep(- Cinv /) L = 3.; delta =.; [,] = mehgrid(-l:delta:l,-l:delta:l); or i=:ize(,) or j=:ize(,) = [(i,j); (i,j)]; X(i,j) = (/(*pi*detcqrt)) * ep( -.5*'*Cinv* ); end end hold on % mehc(,,x); % thi doe a urace plot contour(,,x); % thi doe a contour plot label(' - ai'); label(' - ai'); 6

17 Etimation o Poe Covariance Recall how we olved or the poe o an object = (a,a,az,t,t,tz) uing iterative leat quare Given n meaured D image point {p i }, and the correponding 3D point on the object {P i } We know how to predict the image point, given an etimate o the poe, p = (P, ) i the unction that return the predicted image point p = (p,p, ) T given the object point P = (P,P, ) T To etimate the poe, we took the derivative o p = (P, ) to get T p where i the acobian o, evaluated at (P, ) Then we olved or the correction: T T p and added the correction to, and repeated the tep until convergence 7

18 Etimation o Poe Covariance ow we want to ind the 66 covariance matri o the poe error C = E ( T ) From beore, we had where + i the puedoinvere So C i Simpliing 8 T E p p C p p T T T T E p p C T T E p p T C p ow, C p i the covariance o the meaured image point I independent, we have C p

19 Eample Model baed poe etimation, rom 6 target iducial Poe: X = Covariance: C =

20 Etimating Meaurement Uncertaintie How to etimate the uncertaint in locating image point? Could etimate accurac analticall, rom knowledge o the image noie and the perormance o the eature operator in noie Another wa i to look at the image reidual error I we ue image point to determine the poe, and get a um-o-quared reidual o r i Then an etimate o the image error i p p r 6 i,p i becaue we are umming a total o value, and there are 6 parameter to be it

21 z Drawing in 3D Tranorm wirerame model uing H Draw egment uing line([pstart(,i) pend(,i)],... - [pstart(,i) pend(,i)],... [pstart(3,i) pend(3,i); Tranorm target point uing H Draw target uing plot3(p_c(,:), P_c(,:), P_c(3,:), '+'); Create a bo camera model a a et o 5 polgon (peci the vertice) Draw bo uing ill3(xvertice,yvertice,zvertice,'c');

22 Covariance a Ellipoid The joint probabilit denit or n-dimenional vector i T C ep C p A urace o contant probabilit i T C z Thi i the equation o an ellipoid in n dimenion For a 6-DOF poe, a 66 covariance matri i diicult to viualize We etract jut the 33 ubmatri correponding to the tranlation

23 Matlab Ellipoid unction [,,z] = ellipoid(c,c,zc,rad,rad,zrad) generate (,,z) vertice or an ellipoid centered at c,c,zc radii are rad,rad,zrad ellipoid i aligned with,,z ae We need to determine the radii rotate the vertice appropriatel 3

24 Rotating and caling We have the equation o the ellipoid T C z Let = R, where R i the rotation matri that align the ellipoid with the,,z ae Then T D = z, where D i a diagonal matri Or, T R T D R = T (R T D R) = z Thu, C - = R T D R We can get R b taking the SVD o C - The length o ai i i /qrt( i /z^) Uing SVD, C - = U D V T 4

25 % We will draw the ellipoid deined b the urace ' Cinv = z^, % where Cinv = Cz^-. For z=3, thi i the 97% probabilit urace. % Firt, center the ellipoid at the location o the model. c = X(4); c = X(5); zc = X(6); % Let = R, where R i the rotation matri that align the ae. % Then '*D* = z^, where D i a diagonal matri. % Or, 'R'*D*R = z^. '(R'DR) = z^. % So Cinv = R'DR. Thi i jut taking the SVD o Cinv. [U,S,V] = vd(inv(cz)); R = V; % The length o the ellipoid ae are len=/qrt(si/z^) % where Si i the ith eigenvalue rad = qrt( 9/S(,) ); rad = qrt( 9/S(,) ); zrad = qrt( 9/S(3,3) ); [,,z] = ellipoid(,,,rad,rad,zrad); % Rotate the ellipoid or i=:ize(,) or j=:ize(,) Y = R' * [(i,j); (i,j); z(i,j)]; r(i,j) = Y()+c; r(i,j) = Y()+c; zr(i,j) = Y(3)+zc; end end ur(r,r,zr); 5

26 z Eample 6 -

27 More Eample (Ho & Vincent paper) Uncertaint ellipoid The rotational uncertaint i depicted a elongated cone about each ai. A enor etimate the poe o the target on the top o the head. It covariance i hown a a mall ellipoid, barel viible. Uing the known poe o the glae with repect to the target, the poe o the glae i then etimated, along with it covariance matri (larger ellipoid). 7

28 Combining poe etimate The ellipoid rom two enor are nearl orthogonal. The ellipoid correponding to the combined etimate i much maller and i contained in the volume o interection. 8