Exam 2021.02.15 PDF

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Chair of Graphics and Visualization Department of Informatics Technical University of MunichSPersonal stickerCompliance to the code of conduct I hereby assure that I solve and submit this exam myself under my own name by only using the allowed tools listed below.Signature or full name if no pen inpu...

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Chair of Graphics and Visualization Department of Informatics Technical University of Munich

Compliance to the code of conduct I hereby assure that I solve and submit this exam myself under my own name by only using the allowed tools listed below.

Personal sticker S5011

Signature or full name if no pen input available

Visual Data Analytics Exam: Examiner:

IN2026 / IN8019 / Endterm Prof. Dr. Rüdiger Westermann

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Monday 15th February, 2021 18:15 – 19:30

Date: Time:

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Working instructions • This exam consists of 9 pages with a total of 5 problems. Please make sure now that you received a complete copy of the exam. • The total amount of achievable credits in this exam is 80 credits. • Detaching pages from the exam is prohibited. • Allowed resources: – lecture scripts, books, dictionaries, own notes, exercise sheets – a non-programmable calculator • This PDF contains forms that you can directly fill out. It isnot necessary to print out the exam or to fill it out by hand and scan it. Please note that we will only grade the answers in the forms in the submitted PDF file. • Your answers need to be given in the format that is explained in the exercise text.We will not consider any additional information in the forms that is given by you. Only provide intermediate results when it is explicitly stated in the exercise text. • If you think there is a mistake in an assignment, note this in your solution and answer the question accordingly. • Do not write outside of the text box since this information is lost when TUMexam processes the page. • You are not allowed to have any contact or exchange with third parties during the examination, with the exception of the invigilator. • You are not allowed to communicate or discuss solutions, information, hints or assignments related to the examination during the examination period (including the upload time window). • Please note that sharing assignment sheets constitutes copyright infringement and legal action may be taken in such a case.

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Problem 1

General questions (5 credits)

For each of the following questions/statements,only one answer is correct. If any of the questions are unclear, you can state here the assumptions you made in your solution:

a) One of the stages in the visualization pipeline is filtering/data enhancement. Which of the following operations is NOT an example for this stage? Smoothing and noise suppression Resample data on a grid with different resolution Associate color and opacity to every voxel for rendering purposes Compute gradients or curvature b) Which of the following examples has a 1D domain (the independent variable) combined with 3D data values (the dependent variables)? A parametrized 3D curve (e.g., a stream line) A 3D vector field (e.g., atmospheric flow) A scalar volumetric data set (e.g., a CT scan) c) Two iso-contours for different iso-values in a 2D scalar field. . . . . . can meet at the non-zero divergence points (sources and sinks) of the scalar field’s gradient field. . . . are always orthogonal to the scalar field’s gradients. . . . can only be computed when the scalar field’s gradient field is divergence free. . . . can cross each other. d) Which of the following statements on data grids is true? An unstructured grid stores both vertex positions and neighborship information. A Cartesian grid must store the position of each vertex, but no neighborship information. A rectilinear grid contains only cells with equal length for each coordinate axis. e) A Delaunay triangulation of a set of points in 2D always yields a triangulation. . . . . . that maximizes the smallest angle between triangle edges. . . . that minimizes the length of all edges of the triangles. . . . with less triangles than any other triangulation.

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Problem 2

Interpolation (25 credits)

a) Given a triangle with verticesP1 = (0, 0), P2 = (1, 0), P3 = (0, 1) and corresponding scalar values f1 = 0, f2 = 1, f3 = 2. Compute the interpolated scalar values at points A = (0, 0.75) and B = (0.25, 0.5) using abarycentric interpolation of the scalar values given at the vertices. Use the functionf (X) = u · f1 + v · f2 + w · f3 to compute the interpolated scalar value at a point X. In your solution, specify the values of u, v, w together with the interpolated value for the points A and B, respectively.

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b) The mid-point decider can be used to decide ambigous cases in the marching squares algorithm. However, it cannot decide all cases correctly. Given a quadrilateral cell with scalar values f1 = 0, f2 = 21 , f3 = 2, f4 = 0 given at the vertices (see figure below). Specify for which iso-values c (with cmin ≤ c ≤ cmax) the mid-point decider would return the wrong connectivity information for an ambigous case. In your solution, what do cmin and cmax represent and what are their values?

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c) In each of the figures below, scalar values are given at the 16 vertices of a 2D Cartesian grid. The vertices are shown as black dots. Which of the four figures shows the filled iso-lines of a cell-wise bi-linear interpolation? Briefly explain your answer. Solutions without an explanation will not be graded.

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d) Given a 2D triangulation connecting a set of points labeled from A to K. Using theedge flip algorithm, which of the edges in the triangulation need to be changed in order to obtain a Delaunay triangulation?

Specify exactly how and in which order the edges need to be flipped (e.g., replace BC with AD, replace . . . ).

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Problem 3

Volume Rendering (14 credits)

A viewing ray intersects a volumetric data set during ray casting in direct volume rendering. The volume is thereby sampled at three locations with interpolated density values f1 = 75, f2 = 150, f3 = 125 (see left image). Given a transfer function (right image) which assigns RGB- andα-values to each sample during ray casting. The α-value represents the opacities, where 0 = completely transparent and 1 = completely opaque.

a) For each of the three sampling locations, what is the RGB-color andα-value assigned by the transfer function? Write your answer in the form: Sample 1: C1 = (R1, G1, B1), alpha1 = A1, Sample 2: . . .

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b) Determine the RGB-color andα-value seen along the viewing ray ifα-compositing and a front-to-back strategy are used. In your answer, state intermediate results for color Ci and αi after compositing the i-th sample point. Write your answer in the form: Sample 1: C1 = (R1, G1, B1), alpha1 = A1, Sample 2: . . .

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c) Given is a surface point P = (0, 0, 0) with normal N = ( √12 , √12 , 0). The position of a point light source is Lpos = (−1, 2, 0) and the viewer is positioned at Epos = (3, 0, 0). Compute the color of the specular reflection at the surface point using the Phong lighting model. The specular reflection coefficient isks = 1 and the specular exponent is n = 2. The color of both the light source and the object are white with RGB-values (1, 1, 1). The specular reflection is based on the angle between two vectors. In your solution, also name the two vectors and write each vector in the form V = (vx, vy, vz).

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Problem 4

Flow visualization (23 credits)

a) Two path lines are seeded at fixed locations in a 2D vector field. Neither of the seed points belongs also to the other path line. For which kind of flow can these path lines(self-)intersect? Only one answer is correct (1 credit). both for time-varying and steady flow only for steady but not time-varying flow only for time-varying but not steady flow neither for time-varying or steady flow b) Two streak lines are seeded at fixed locations in a 2D vector field. Neither of the seed points belongs also to the other streak line. For which kind of flow can these streak lines(self-)intersect? Only one answer is correct (1 credit). only for steady but not time-varying flow only for time-varying but not steady flow both for time-varying and steady flow neither for time-varying or steady flow 0 1 2 3 4 5

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c) Calculate the Jacobian matrix, divergence, and curl of the 3D vector field v(x , y , z) = (x − 2y, yz, y 2 − 1)T . In your solution, to specify a matrix that has multiple rows, separate the rows with semicolons, e.g.,   1

M = [1, 2, 3; 4, 5, 6; 7, 8, 9] =  4 7

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d) A critical point in a steady 2D vector fieldv(x, y) : R2 → R2 is a point (x, y ) where v(x, y) = (0, 0)T . How many critical points does the vector field v(x, y) = (2x − y 2 + 4, xy)T have, and where are these points located? In your solution, also specify the initial equation system that has to be solved.

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e) The figure below illustrates five vector fields which each have a critical point in (0, 0).

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For three of these vector fields, the eigenvaluesλ1 , λ2 of the respective Jacobian matrices have been computed (see the list below). Specify for each case, which type of critical point is represented and which figure illustrates the corresponding flow behaviour. 1. λ1 = 2,

λ2 = 1

2. λ1 = −1 + i, 3. λ1 = 3,

λ2 = −1 − i

λ2 = −1

f) Given a steady 2D vector field v(x, y) = (−y, x)T . Use the Midpoint integration method to compute the next two points x1 and x2 of a stream line with a seed point x0 = (0, 2)T and step size ∆t = 1. For each point of the streamline, specify the vector∆x resulting from the Euler step and the vectorvmid evaluated at the midpoint. Write your answer in the form: First point:delta_x = (ux, uy), v_mid = (vx, vx) , x_1 = (wx, wy) , Second point: . . .

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Problem 5

Visualization mapping (13 credits)

The figure below shows temperature values in Munich between January 31 and February 3 usingHorizon graphs.

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a) At what time and date is the maximum temperature and what is the approximate value?

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b) At what time and date is the minimum temperature and what is the approximate value?

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c) What is the approximate temperature value at the following points in time: a. Jan 31 at 15:00 b. Feb 1 at 6:00 c. Feb 2 at 12:00

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The following scatterplot matrix shows measurement data from penguins collected in the Antarctica. Properties such as culmen depth, flipper length, or body mass were collected per penguin. Points in the matrix are labeled with random numbers. In one of the scatterplots, points A, B, and C have been selected by a rectangular brush (gray background color). Which points in the other two scatterplots are consequently selected/highlighted when brushing and linking between the views is used?

If any of the questions below are unclear, what are the assumptions you made in your solution?

d) Which points in the matrix are selected/highlighted because of point A(2 credits) ? Only one answer is correct. points 6 and 13, but not point 9 points 6 and 9, but not point 13 points 6, 9, and 13 e) Which points in the matrix are selected/highlighted because of point B(2 credits) ? Only one answer is correct. points 5 and 19, but not points 2 or 16 points 2, 5, 16, and 19 points 2 and 16, but not points 5 or 19 f) Which points in the matrix are selected/highlighted because of point C(2 credits)? Only one answer is correct. points 1 and 12, but not points 8 or 17 points 8 and 17, but not points 1 or 12 points 1, 8, 12, and 17

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