Computer Graphics

Questions & Answers

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QUESTION 1 Which of the following is the focal length of the pinhole camera?D QUESTION 2 Which of the following are True Statements (Select all that apply) QUESTION 3 In which of the following scenarios can you use a weak perspective camera model for the target object? QUESTION 4 Which of the following could affect the intrinsic parameters of a camera? QUESTION 5 Which of the following is an Applications of Computer Vision? QUESTION 6 For 2D coordinate transformations, what are the DoF (Degrees of Freedom) for rigid transformation, affine transformation and perspective transformation? QUESTION 7 Which of the following is(are) correct? QUESTION 8 Which of the following 2D planer transformations has the largest degree of freedom? QUESTION 9 Which of the following are True Statements (Select all that apply) QUESTION 10 Take a picture with your camera that shows at least one set of parallel lines in perspective such that the vanishing point appears in the field of view. Mark the parallel lines and the vanishing point and upload the file. Extra Challenge (not required): Have the picture contain two different pairs of parallel lines that define two vanishing points, and therefore define a vanishing line. Mark the parallel lines, the vanishing points, and the vanishing line and upload the file.

Q1- If you have the following DOG images in one octave of the SIFT detector,imagine they are stacked over each other, determine which of the shaded points is a potential interest point? (2 marks)

Consider a ray with direction vector de R" being reflected from a mirroring surface resulting in reflection vectorr. At the point of intersection, the surface has the normal vector n. See Figure 1 for the setup. (a) Given d and r, determine the normal vector n, of the surface at the point of reflection! (2P) (b) Given d and n, determine the direction vector r for the reflection of the ray! (3P)

Let foo, fio, foi, and fi1 be four coefficients at the corners of a uniform square and let f(x, y) be the function that performs a bilinear interpolation of these coefficients. Bilinear interpolation can be expressed as the consecutive application of linear interpolation along each dimension. Show that the result of bilinear interpolation is independent of the order of linear interpolations, i.e., that linear interpolation can be performed in x-direction first followed by the y-direction or vice versa.

Given are the Bézier points of a cubic Bézier curvex(t) with \mathbf{b}_{0}=\left(\begin{array}{l} 0 \\ 0 \end{array}\right), \quad \mathbf{b}_{1}=\left(\begin{array}{c} 0 \\ 27 \end{array}\right), \quad \mathbf{b}_{2}=\left(\begin{array}{l} 27 \\ 27 \end{array}\right), \quad \mathbf{b}_{3}=\left(\begin{array}{c} 27 \\ 0 \end{array}\right) Use the de Casteljau algorithm to determine the value x(3) as well as all intermediate points!

Q2- Briefly describe, how does SIFT satisfy the rotation invariant property? (1mark)

Given are 3 points po = (0,0), P₁ = (1,0), P2 = (0, 1) equipped with the scalar values so = 1,81 = 2, 82 = 3. (a) Determine the interpolating function s(x, y) (i.e., write down the formula) for linear interpolation. (b) Compute the values of s(1/2, 0), s(0, 1/2), s(1/2,1/2), s(2/3, 2/3).

\text { (a) Find a cubic Bézier curve } \mathbf{x}(t), \mathbf{x}:[0,1] \rightarrow \mathbf{R}^{2} \text { with } \mathbf{x}(0)=\left(\begin{array}{l} 0 \\ 0 \end{array}\right) \quad \text { and } \quad \mathbf{x}(1)=\left(\begin{array}{l} 9 \\ 0 \end{array}\right) \text { which intersects itself orthogonally at } x\left(\frac{1}{4}\right) \text { and } x\left(\frac{3}{4}\right) \text {. } \text { (b) Construct a non-trivial Bézier curve } \mathbf{x}(t), \mathbf{x}:[0,1] \rightarrow \mathbb{R}^{2} \text { of degree } 4 \text { with } \mathbf{b}_{2}=\mathbf{x}\left(\frac{1}{2}\right) \text {. }

Figure 2 shows six curves and their supposed control polygons. Two of the control polygons are the Bézier control polygons for the Bézier curve drawn with it; the other four are not. For every case: Does the control polygon correspond to the curve? If not, check all violated properties of Bézier curves!You may assume that none of the control points overlap or are repeated You may assume that none of the control points over lapor are repeated.

Given is a 2D triangle with data values fo, fi, and f2 at the vertices x0, X₁, and x2 that are interpolated linearly on a triangle by f(x, y)=\alpha_{0}(x, y) f_{0}+\alpha_{1}(x, y) f_{1}+\alpha_{2}(x, y) f_{2} using linear barycentric coordinates a; as basis functions. The constant gradient of this function \nabla f=\nabla \alpha_{0} f_{0}+\nabla \alpha_{1} f_{1}+\nabla \alpha_{2} f_{2} is represented as the linear combination of constant basis function gradient vectors Vai E R². Determine closed form expressions of all Vai with respect to the vertex coordinates xi. Is there a geometric interpretation of these basis function gradients?

Given is a rectilinear grid with scalar data values at its vertices as shown in Figure 1. We assume bilinearinterpolation inside the grid cells which gives us the scalar field f(x,y). (a) Compute the following values: ƒ(2, 1.5) • ƒ(2,3) f(5,3) b) Compute the following values: f(1.5, 1.75) • f(3.25, 3.5) f(3.25, 4.25) (c) Compute the following values: f(1, 1) f(4,3) •f(7,5) ) Determine the formula for the gradient of f. (e) Compute the gradient at (2,3).

Q3- You come up with a CNN classifier. For each layer, calculate the number of weights, the size of the associated feature maps. The notation follows the convention: (2 marks) CONV-K-Nof them of size K × K, Padding P and stride S respectively.– P – S denotes a convolutional layer with N filters, each POOLK indicates a K × K pooling layer with stride K and padding 0. FC-N stands for a fully connected layer with N neurons.

The cells of a structured grid are usually saved linearly in memory. For example, in 2D the cells (i, j) are saved to the linear memory space at indices according to the following scheme: \text { Let a structured grid in } 3 \mathrm{D} \text { consist of }\left(n_{x}, n_{y}, n_{z}\right) \text { grid points in each dimension. } (a) Find the map (i, j, k) →l that maps 3D cell indices to linear cell indices. Assume that the first dimension(i) varies fastest in the linear memory space. (b) Find the map l → (i, j, k) that maps linear cell indices to 3D cell indices. Assume again that the first dimension (i) varies fastest in the linear memory space.

The color value of an individual pixel in Phong illumination is determined by different parameters: ●light colors (c, for specular and diffuse light, ca for ambient light) high-light color cp and surface color crspecular coefficient n .specular coefficient p. We consider a sphere with constant surface color cr=0.5 and otherwise varying lighting properties. For simplicity,all colors are in grayscale. Match each of the spheres (a)-(d) in Figure 2 with one the settings (1)-(6) below. (2) Ca = 0.0,C₁ = 0.5,Cp = 0.5,p=1.5 (1) Ca = 0.5,C₁ = 0.0,Cp = 0.0,p = 1.5 (3) Ca=0.5,CI= 0.0,=Cp = 0.5,P = 8 (4) Ca = 0.5,C₁ = =0.5,Cp = 0.0,P = 32 (5) Ca = 0.5,Cl = 0.5,Cp = 0.5,p = 8 (6) ca = 0.5,CĮ = 0.25,Cp = 0.25,P = 32

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