GLOBAL-LOCAL APPROXIMATIONS

Transcription

1 GLOBAL-LOCAL APPROXIMATIONS Puneet Singla MAE 2 Department of Mechanical & Aerospace Engineering University at Buffalo psingla/mae2/ October 31, 27 PUNEET SINGLA 1 / 1

2 MOVING LEAST SQUARES Let f (x) : U R n R be an unknown continuous function and ˆf (x) = m j=1 a j(x)p j (x)) be the approximation of f (x) at the point x U. Then given n distinct data points x i and m basis vectors p j (usually m < n) which span the approximation space U, the moving least square approximation of f (x) is defined as: ˆf (x) = m a j(x)p j (x)) j=1 which minimizes the following weighted least square error function, J(x): J(x) = n i=1 [ ˆf (x i ) f (x i )] 2 w i (x) (1) where w i represent some non-negative weight functions which are a function of the evaluation point x. Further a = {a 1,...,a m } is given by following expression: a = (P T WP) 1 P T Wf (2) PUNEET SINGLA 2 / 1

3 where P = W = p 1 (x 1 ) p m (x 1 )... p 1 (x n ) p m (x n ) w 1 (x)... w n (x) (3) (4) f = [ f (x 1 ),..., f (x n )] T () PUNEET SINGLA 3 / 1

4 MLS The main difference between the conventional Gaussian Least Squares procedure and moving least squares is the choice of the weight function. In the conventional least squares procedure, the weight matrix is chosen to be the inverse of the measurement error covariance matrix and is assumed to be constant. In the moving least squares procedure the weight matrix is chosen such that the observations near to the evaluation point get more weightage than the observations that are far away from the evaluation point. PUNEET SINGLA 4 / 1

5 MLS The main drawback of the moving least squares approximation is that it is valid only at one evaluation point, x and therefore a new linear system dictated by equation 2, needs to be solved when the evaluation point is changed. However, if the weight functions have local support (domain) then we need to solve the linear system in the neighborhood of the evaluation point x thus bringing down the computational cost. Further, Levin and Wendland [?,?] have shown that if we have polynomial basis functions and the support of the weight function is local and proportional to some mesh size ρ then we have approximation error of order O(ρ d + 1), where d is the degree of polynomial basis functions. Also, the support or domain of the weight functions dictate the local approximation space and according to the Stone-Weierstrass theorem we require this support to be compact. PUNEET SINGLA / 1

6 MLS: WEIGHT FUNCTIONS WEIGHT FUNCTIONS Hence, we are interested in non-negative continuous weight functions defined on a locally compact subset of approximation space. To be more precise, we can define weight functions as follows: w : Ω R is a weight function if it satisfies the following conditions: 1 Ω is a compact space. 2 w(r) >, r [,1), where r = y x. 3 w(r) =, r 1. 4 w(r) is a monotonically decreasing function. w(r) δ(r) as Ω shrinks to point x. PUNEET SINGLA 6 / 1

7 MLS: WEIGHT FUNCTIONS Three commonly used weight functions are the exponential, the cubic spline and the quintic spline [?]. w i (x) = { e ( x x i /c)2 e (r i /c)2 1 e (r i /c)2, x x i r i, x x i > r i (6) w i (x) = { ) 2 ( ) 3 ( ) 4 1 6( di r i + 8 di r i 3 di r i, di = x x i r i, d i > r i (7) PUNEET SINGLA 7 / 1

8 MLS: ALGORITHM The local approximation procedure can be outlined as follows: 1 Given the evaluation point, (x, f (x)), construct the following compact support, Ω for weight function w: Ω = {y : y U, y x ρ} (8) 2 Find the moving least squares approximation by solving the normal equations (2) using the points in set Ω. The necessary condition for the MLS solution to exist is that the rank of the matrix P T WP should be at least m. As a consequence of this the domain of definition Ω should consist of at least m nodal points. 3 The accuracy of this approximation will depend upon the value of ρ and the degree of the polynomial, d. If the desired accuracy is not met with the initial value of ρ and d then either reduce ρ or increase d and repeat from step 2. PUNEET SINGLA 8 / 1

9 MLS: ALGORITHM It should be noticed that one can avoid solving the system of linear equations in step 3 by choosing an orthonormal basis with respect to the weight function, w. PUNEET SINGLA 9 / 1

10 NUMERICAL SIMULATION: VISION SENSOR CALIBRATION Vision based sensors have found immense applications not only in aerospace industry but manufacturing inspection and assembly. However, no sensor is perfect!. In order to achieve high precision information from these sensors, those systematic effects which tend to introduce error in the information must be accounted for. These effects can include lens distortion and instrument aging. PUNEET SINGLA 1 / 1

11 CALIBRATION MODEL COLLINEARITY EQUATIONS x i = f C 11r xi +C 12 r yi +C 13 r zi C 31 r xi +C 32 r yi +C 33 r zi + x, i = 1,2,,N (9) y i = f C 21r xi +C 22 r yi +C 23 r zi C 31 r xi +C 32 r yi +C 33 r zi + y, i = 1,2,,N (1) C i j are the unknown elements of rotation matrix C associated to the orientation of the image plane with respect to some reference plane. f is known focal length. (x i,y i ) are the known image space measurements for the i th line of sight, (r xi,r yi,r zi ) are the known object space direction components of the i th line of sight and N is the total number of measurements. x and y refers to the principal point offset. PUNEET SINGLA 11 / 1

12 CALIBRATION PROCEDURE Generally, the focal plane calibration process is divided into two major parts: 1 Calibration of principal point offset (x,y ) and focal length ( f ). 2 Calibration of the non-ideal focal plane image distortions due to all other effects (lens distortions, misalignment, detector alignment, etc.). The implicit pin-hole camera model is not exact so we need to find the best effective estimates of principal point offset (x,y ) and focal length ( f ). PUNEET SINGLA 12 / 1

13 SIMULATION PARAMETERS Field of View (FOV): 8 8. Principal point offset of x =.7mm and y =.2mm. The focal length of the camera is assumed to be mm. The true lens distortion is assumed to be given by following models [?] Φ = { r r 2 r 3 r 4 } ; δx = xφ T a & δy = yφ T b(11) where, r = x 2 + y 2 (12) PUNEET SINGLA 13 / 1

14 TRUE DISTORTION 8 6 x 1 3 True Distortion y x FIGURE: True distortion map. We seek to reduce these errors to the order of 1µ radians PUNEET SINGLA 14 / 1

15 RESULTS Approximated Distortion 1 x 1 3 y (a) Approximated Distortion Surface x Approximated Error Distortion 1 1 x 1 y (b) Net approximation error surface FIGURE: MLS approximation results. x PUNEET SINGLA 1 / 1