Solve tasks 1.1.2, 1.1.3, 1.2.2, 1.2.3 and 2.2.2. Do not use any built-in functions in Matlab. Use only addition, subtraction, multiplication, division, square root, power, matrix inverse, matrix transpose, summation, mean, min, max, integral2, eig, load(), size(). For simplicity, we start with simple quadratical surfaces of the general formulation: z= f(x,y) = Ax2 + By2 + Cxy + Dx + Ey+ F. (1) The objective is to estimate the unknown parameters A, B, C, D, E and F using the observed points in the provided file 'quadratic surface.mat'. The fitting error is (2) which is minimized when a = 0, δε = 0, ΔΕ = 0, 0 = 0, δ£ = 0 and of = 0. Refer to the lecture notes', use linear regression to estimate the unknown parameters A, B, C, D, E and F. N Σ(Ax + By? + Cxy + Dx + Ey + F –z)2 t= Task 1.1.1 Derive six linear quati involving the unknown parameters A, B, C, D, E and F based on the error minimization condition. Then, reformulate these linear equations into the form of Eq.(3). Namely, derive the matrix XEROX6 and vector Y € Ro in Eq.(3). Completed task 1.1.1 Σε Σαν xy Σαν Σε Σχένι Σε xí ΓΧου Χοι ΧΟΖ Χος Χος Χος Χιο Χης Χης Χ13 Χ14 Χις 1X20 Χ21 Χ22 Χ23 Χ24 Χης 1X30 Χ31Χ32 X33 X34 Xas D ΙΧΑΟ ΧΑΙ ΧΑΖ Χ43 ΧΑΑ Χός E X50 X51 X52 X53 X54 X55] F Task 1.1.2 Estimate the unknown parameters A, B, C, D, E and F by solving Eq.(3). Task 1.1.3 Report the volume bounded by the estimated surface and the ground plane (z = 0) by evaluating the integral Josxeseisygy f(x,y)dxdy. Σχέν Σαν Exy yi C Σε λα λα λα λα λα Σαν Σαν xy x² Σχινι Σ Σ Σανι yi xiyi Yi y χένι Σκιν xi yi xi yi x²yi x? Xiyi Συ Σαν Στ y Σ» Σαν Σε Xiyi Xi XiVi y τω Yi y Xiyi Xi Yi N tooomi (3) II x? Zi y? Zi XiYiZi XiZi YiZi MI Zi The quadratic surface model assumes smooth and simple shapes. A bivariate cubic function can model more complex shapes of a bulk. In this context, the sensed surface is approximated by the following cubic function, z = f(x,y) = Ax3 + By + Cxy + Dxy2 + Ex? + Fy? + Gxy +Hx+ Iy + 1 The objective is to estimate the unknown parameters A, B, C, D, E,F,G,H, and J with the observed points in the provided file 'cubic surface.mat". The fitting error is (4) (5) which is minimized when of = 0, 0 = 0, 8€ = 0, 0 = 0,0€ = 0,8F = 0, 86 = 0, 95 = 0, f = 0 and af = 0. Task 1.2.1 Derive ten linear equations involving the unknown parameters A, B,C,D,E,F,G,H, I and J based on the error minimization condition. Then, reformulate these linear equations into the form of Eq.(6). Namely, derive the matrix XER 10x10 and vector Y€ Rio in Eq.(6). N (Ax3 + By + Cxy; + Dxy? + Ex? + Fy? + Gxy; + Hx; + lyi+1-z)2 ΓΧου Χοι Χοζ Χος Χου Χος Χος Χοτ Χοβ Χορζ ΓΑ Χ1 Χ12 Χ14 Χης Χίο Χιτ Χ18 Χιο B Χιο Χ21 Χ22 Χ24 Χης Χζο Χατ Χ28 Χ 20 Χ13 Χ23 X33 ΧΑ3 ΧΑΑ Χ31Χ32 X34 Χος ΧΑΙ ΧΑΖ ΧΑς Χ30 Χ3τ Χ38 Χ 30 ΧΑΟ Χιτ Χ18 Χιο Xss Xso Χιτ Χ18 Χιο Χος Χου Χοτ Χος Χο Xso Xs1 Xs2 X53 Xst | Χου Χοι Χοζ Χος Χολ ΙΧΤο Χτζ Χτζ ΧΤΑ Χτς Χτο Χτ ΧτΒ Χτο Χτι X80 X1 X2 X3 X4 X5 X86 X87 XB8 X89 LXgo Χοι Χοζ Χοζ Χολ Χος Χος Χοτ Χοβ Χορζ xy; 1X20 | X30 ΧΑΟ Σ Xiy Xiyi αν xfyi xy Σαν Σε Σαν Σαν xy _x?y? κέν Σαν Σαν Σ Σαν xy y xy x{y} xy xyi Xiyi x¡y? y² x Σχν xyi xy Σε κένι Σαν Σαν Σαν x Σκv? Σαν Σε Σχνι xy xy xiyi και Συ xy Σαν Σαν Σ Σαν Ση Xiyi y Σαν Σαν xy; x{y} x{y} yi Σαν Σ xy xy Σαν Σως Σαν Σαν Σαν Σα xty? x¡y{ x?y? xy x{y} Xiy Σε x xy xyi xy xfyi x?y? yi xy MI y D ULOI- E x} yi H | 11 Yg xyi xy xyi x Σ Σαν Σ [x²y? [x²y. Σxy? x} yi Σ xy Σ Xi Yi xyi xy Σε Συ Σαν Στέν Σαν Σχνι Σαν Σε x² yi Task 1.2.2 Estimate the unknown parameters A, B, C, D, E,F,G,H,I and J by solving Eq.(6). Task 1.2.3 Report the volume bounded by the estimated surface and the ground plane (z = 0) by evaluating the integral Josxetine fraisysyf(x,y)dxdy. XiVi Xi Xiyi y τω XiYi Yi xzi yi zi x²yizi x₁y} Zi xzi y} Zi N XiYiZi Xi Zi Σχετι τω Ζή The explicit representation z=f(x,y) introduced in Topic 1 has a significant limitation in efficiently representing arbitrary surfaces, particularly when a location (x,y) corresponds to multiple height z. In contrast, implicit represen- tation can easily represent arbitrary topological structures. In this method, the surface S is implicitly defined by the zero-level set of a function f(x.y.z), denoted as $ = {(x,y,z)|f(x,y,z) = 0}. For simplicity, here we constrain the form of f(x, y, z) to a quadratic function: f(x, y, z)= Ax² +By²+Cz²+Dxy+Eyz+Fxz+Gx+Hy+lz+J. The error on fitting an implicit representation is N c = = E(Ax} + By? + Cz? +Dxy + Ey,z, + Fxz, + Gx + Hy, + Iz + 1 – 02 i-1 which is minimized when f = 0, 0 = 0, 0 = 0,86 = 0, 8£ = 0,8£ = 0, 8c = 0, @f = 0, f = 0 and of = 0. Task 2.1 Derive ten linear equations involving the unknown parameters A,B,C,D,E,F,G,H, I and J based on the error minimization condition. Then, reformulate these linear equations into the form of Eq.(8). Namely, derive the matrix X € 110x10 in Eq.(8). Σχετι xzi Σχ xt x?y? xz x²³ yi ΓΧου Χοι Χοζ Χοζ Χος Χος Χος Χοτ Χος Χορτ [Α] 1Χ10 Χ1 Χ12 Χ13 Χ14 Χης Χιό Χιτ Χ18 Χιο B Χης Χης Χατ Χ28 Χο 1Χ20 Χ21 Χ22 Χ 23 C Χ24 X31 X32 X33 X34 X30 Χ35 Χ30 Χ3 X38 Χ30 D Χ40 ΧΑΙ ΧΑΖ ΧΑ3 ΧΑΑ ΧΑς ΧΑς Χιτ Χ18 Χιο | Xso Xει Χεζ Χεζ Xs4 Xss Xso Χιτ Χ18 Χιο ΙΧσο Χοι Χοζ Χοζ Χός Χος Χος Χοτ Χός Χερ ΙΧτο Χτι Χτζ Χτζ ΧΤΑ Χτς Χτο Χτ Χτ8 Χτο Xgo Χοι Χgz XB3 X84 Χος Χου Χοτ Xes Xeo LXgo Χοι Χοζ Χοζ Χολ Χος Χος Χοτ Χος Χαολ yz x Συνz? x? Yizi yizi αν Viz Σχετι Σχετινες Σχ x;Z Σχ Σαν Σε xiz x² yi Task 2.2 For the observed points in 'implicit_l.mat', estimate their corresponding parameters A, B, C, D, E, F,G,H,1 and J. Then, based on the value and signs of these parameters, classify the type of underlying surface. • Hint: directly solving Eq. (8) will result in a solution whose elements are all zero. To address this prob- lem, we can use the eigenvector of X corresponding to the smallest eigenvalue of X as an estimate of [A, B, C, D,E,F,G,H,1,JJT. The relationship between this eigenvector v and the corresponding ground truth parameters θ = [A,B,C,D,E,F,G,H,1,JJT is v = αθ + n where a is a scale factor and n denotes the model noise. Σ» Σαν Σνική xy y? Zi • For the observed points in file 'implicit_2.mati, estimate their corresponding parameters A, B, C, D, E, F,G,H,I and J. Then, based on the value and signs of these parameters, classify the type of underlying surface. • Note: Refer to Fig.2, recognize the surface type by analyzing the values and signs of surface parameters. You must not draw a conclusion via eyeballing. Στο x Σxy Ex?z Σχένι Σχέντι Σχει Στ yt yz x¡y? Σνακ Σαν Σ Στο Xiyiz? Viz Σ xiz xi yi Σχν? x?y? Σκινέζι Zi Zi Σνικ κινέζι χένιτι Σκινιζέ Σχινιτι x?yiZi yizi Σχετιν Σχένι Σχνι x² yi ουυοΙ· H yz || XiYiZi Στο Σχινι Συ XiVi - xri νέζι Σχινιτι yzi XiYiZi yız? xiz ....... xiz Σχένιαι Σκιντέ Σ Σχέτ xz XiZi Xiyiz? XiYiZi Σχέτι Σ. Σχινι Xi Yi Σ XiZi Xi WW x? Yi xiy y? Zi Σχινιτι XiYiZi Viz Σ z² xv? Σχινίζε Σχινι Xi Yi y? YiZi xzi Yi y? Zi Viz Σχει Στ XiZi Σ YiZi Στ Xiyi YiZi Zi XiZi 0 Xi Yi Zi Ellipsoid Eliptic parabolaid Non-degenerate real quadric surfaces Hyperbolic paraboloid Hyperboloid of one sheet or Hyperbolic hyperboloid Hyperboloid of two sheets or Eliptic hyperboloid 2² + + 12 # T