this is assignment 3 of computer vision which is been given to batch of 2023, Assignments of Computer Science

Download this is assignment 3 of computer vision which is been given to batch of 2023 and more Assignments Computer Science in PDF only on Docsity! CAP5416: Assignment #3, Due Date:Nov. 7th 2023 Make sure that you writing is legible, or else, type your answers using your favorite text formatter. 1. Show that the curve f(x) that minimizes the integral∫ 2 1 √ 1 + f ′2 x dx, with f(1) = 0, and f(2) = 1. is a circle. What is its radius and what are the coordinates of its center? 2. Given n data points {xi, yi}i=1,..,n such that f(xi) = yi, how many unknowns are there if a quadratic spline is to be fitted to this data? Deduce the conditions required to solve the unknowns. Do you have enough conditions? If not, what conditions can you impose? 3. Given a set of sample measurements of a one dimensional curve in the image plane, f(x), what is the purpose of minimizing the following functional: E(S) = ∫ {λ(S′′(x))2 + (f(x)− S(x))2 ∑ k δ(x− xk)}dx Describe the significance of each term on the right hand side of the above equation. Assume that λ is a constant regularization parameter. Then, write down the Euler-Lagrange equation for this minimization problem. 4. We know that the length element of a curve f(x) is given by ds = √ 1 + (f ′(x))2dx. Show that the area element of the image surface I(x, y) is given by dA = √ 1 + I2 x + I2 ydxdy. Next, find the Euler-Lagrange equations for the functional ∫ dA. 5. Image restoration is a very important problem in computer vision and image processing. Let Ω ⊂ R2 denote the image domain and f(x, y) represent the given observed image data. The restored image u(x, y) can be computed via the following minimization. min u E(u) = ∫ ∫ Ω ( |∇u|+ λ 2 ‖u− f‖22 ) dxdy . Write down the Euler-Lagrange equation for this minimization and simplify the expressions. 1