6 7 8 9 10

Exercises - Part 21

For Chapters 06 to 10

See Exercise Sections inReinhard Klette: Concise Computer Vision

Springer-Verlag, London, 2014

1See last slide for copyright information.1 / 25

6 7 8 9 10

Agenda

1 Chapter 6

2 Chapter 7

3 Chapter 8

4 Chapter 9

5 Chapter 10

2 / 25

6 7 8 9 10

6.1–6.2

Exercise

Check lectures to Chapter 6 for equations given in inhomogeneouscoordinates. Express all those in homogeneous coordinates.

Exercise

Specify the point at infinity on the line 31x + 5y − 12 = 0.Determine the homogeneous equation of this line. What is theintersection point of this line with the line 31x + 5y − 14 = 0 atinfinity?Generalize by studying lines ax + by + c1 = 0 andax + by + c2 = 0, for c1 6= c2.

3 / 25

6 7 8 9 10

6.3

Exercise

Consider a camera defined by the following 3× 4 camera matrix:

C =

100

010

001

001

Compute the projections of the following 3D points (in worldcoordinates) with this camera:

P1 =

1111

, P2 =

11−11

, P3 =

3211

, and P4 =

0001

4 / 25

6 7 8 9 10

6.4

Exercise

Let pR = [x , y , 1]> and pL = [x ′, y ′, 1]>. Equation pTR · F · pL = 0

is equivalently expressed by

[xx ′ xy ′ x yx ′ yy ′ y x ′ y ′ 1

]

F11

F21

F31

F12

F22

F32

F13

F23

F33

= 0

where Fij are the elements of the fundamental matrix F. Nowassume that we have at least 8 pairs of corresponding pixels,defining the matrix equation

5 / 25

6 7 8 9 10

6.4 Continued

Exercise

x1x ′

1 x1y ′1 x1 y1x ′

1 y1y ′1 y1 x ′

1 y ′1 1

x2x ′2 x2y ′

2 x2 y2x ′2 y2y ′

2 y2 x ′2 y ′

2 1...

......

......

......

......

xnx ′n xny ′

n xn ynx ′n yny ′

n yn x ′n y ′

n 1

F11

F21

...F33

=

00...0

for n ≥ 8, expressed in short as

A · f = 0

Solve this equation for the unknowns Fij , considering noise orinaccuracies in pairs of corresponding pixels.

6 / 25

6 7 8 9 10

6.5

Exercise

Show that the following is true for any nonzero vector t ∈ R3:

1 [t]x · t = 0,

2 the rank of matrix [t]x is 2,

3 the rank of the essential matrix E = R[t]x is two,

4 the fundamental matrix F is derived from the essential matrixE by the formula F = K−TR EK−1

L ,

5 the rank of the fundamental matrix F is two.

7 / 25

6 7 8 9 10

Agenda

1 Chapter 6

2 Chapter 7

3 Chapter 8

4 Chapter 9

5 Chapter 10

8 / 25

6 7 8 9 10

7.1

Exercise

A smooth compact 3D set is compact (i.e., connected, boundedand topologically closed) and curvature is defined at any point ofits frontier (i.e., its surface is differentiable at any point).Prove (mathematically) that the similarity curvature measure

S(P) =

(κ3, 0) if signs of κ1 and κ2 are both positive

(−κ3, 0) if signs of κ1 and κ2 are both negative

(0, κ3) if signs of κ1 and κ2 differ, and |κ2| ≥ |κ1|(0,−κ3) if signs of κ1 and κ2 differ, and |κ1| > |κ2|

is scaling invariant, for any smooth compact 3D set.

9 / 25

6 7 8 9 10

7.2

Exercise

Instead of applying a trigonometric approach for structuredlighting, specify the required details for implementing alinear-algebra approach for structured lighting along the followingsteps:

1 From calibration we know the implicit equation for each lightplane expressed in world coordinates.

2 From calibration we also know in world coordinates theparametric equation for each ray from the camera’s projectioncenter to the center of a square pixel, for each pixel.

3 Image analysis gives us the ID of the light plane visible at agiven pixel location.

4 An intersection of the ray with the light plane gives us thesurface-point coordinates.

10 / 25

6 7 8 9 10

7.3–7.5

Exercise

Specify the fundamental matrix F for canonical stereo geometry.Consider also a pair of tilted cameras as shown in Figure ??, andspecify also the fundamental matrix F for such a pair of twocameras.

Exercise

Describe the Lambertian reflectance map in spherical coordinateson the Gaussian sphere (hint: isointensity curves are circles in thiscase). Use this model to answer the question why two light sourcesare not yet sufficient for identifying a surface normal uniquely.

Exercise

Why the statement “We also use Parseval’s theorem...” whenintroducing the gradient-field integration algorithms (with derivingsolutions in frequency space)?

11 / 25

6 7 8 9 10

Agenda

1 Chapter 6

2 Chapter 7

3 Chapter 8

4 Chapter 9

5 Chapter 10

12 / 25

6 7 8 9 10

8.1

Exercise

Assume that our stereo system needs to analyze objects being in adistance of at least a meters from our binocular stereo camerasystem. We have intrinsic and extrinsic camera parameterscalibrated. Determine value dmax based on those parameters andknown value a.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

13 / 25

6 7 8 9 10

8.2

Exercise

The figure on the page before shows two profiles. Are theseepipolar profiles which represent disparity vectors d? If so, whichdisparity vector? In reverse: Given are the disparity vectors

d1 = [f4, f5, . . . , f16]> = [1, 0, 1, 2, 3, 0, 1, 1, 0, 1, 2, 3, 2]>

d2 = [f4, f5, . . . , f16]> = [4, 3, 2, 1, 0, 1, 2, 4, 1, 2, 3, 2, 2]>

Draw the epipolar profiles defined by those two disparity vectors.Which profiles and which vectors satisfy the ordering constraint?

14 / 25

6 7 8 9 10

8.3

Exercise

In the lectures we discussed how 4-adjacency “grows” into theimage carrier by repeated creations of dependencies betweenadjacent pixels. At time t = 0 it is just the pixel itself (n0 = 1), attime t = 1 also the four four-adjacent pixels (n1 = n0 + 4 = 5), attime t = 2 also eight more pixels (n2 = n1 + 8 = 13).

How many pixels are in this growing set at time t ≥ 0 in general,assuming no limitation by image borders. At the time τ whenterminating the iteration, nτ defines the cardinality of the area ofinfluence.

Now replace 4-adjacency by 8-adjacency and do the samecalculations. As a third option, consider 4-adjacency but also aregular image pyramid “on top” of the given image.

15 / 25

6 7 8 9 10

8.4

Exercise

Stereo matchers have to work on any input pair? Fine, here is one- see the figure above.

Assume the simple AD data cost function and discuss (as a“Gedanken experiment”) outcomes of “The winner takes all”,DPM with ordering constraint, multi-scanline DPM withsmoothness constraint, and of BPM for this stereo pair.

16 / 25

6 7 8 9 10

Agenda

1 Chapter 6

2 Chapter 7

3 Chapter 8

4 Chapter 9

5 Chapter 10

17 / 25

6 7 8 9 10

9.1–9.2

Exercise

In the lecture we discussed a proposal how to detect keypoints atsubpixel accuracy. Assume values aN , aE , aS , aW (for the4-adjacent pixel locations) and ap (at the detected keypoint pixel)for function g and provide a general solution for subpixel accuracy.

Exercise

The algorithm (provided in the lectures) for plane fitting to asparse set of 3D points lists two procedures ransacFitPlane andransacRefinePlaneModel. Specify such two procedures for initialand refined plane fitting following the general RANSAC idea.

18 / 25

6 7 8 9 10

9.3–9.5

Exercise

For lane-border detection it was suggested in the lectures togenerate a bird’s-eye view by using a homography, defined by fourmarked points being corners of a trapezoid in the image, butactually corners of a rectangular region on the road. Specify thishomography.

Exercise

Explain the motivations behind the definitions of particle weightsgiven in the lectures for the presented lane-border detectionapproach.

Exercise

Show that F∆t = I + ∆tA + ∆t2

2 A2 for matrix A as defined in thefirst example in the Kalman filter section.

19 / 25

6 7 8 9 10

Agenda

1 Chapter 6

2 Chapter 7

3 Chapter 8

4 Chapter 9

5 Chapter 10

20 / 25

6 7 8 9 10

10.1–10.2

Exercise

Continue the calculations (at least one more iteration step) for thegiven AdaBoost example.

Exercise

Do manually AdaBoost iterations for six descriptors x1 to x6 whenhaving three weak classifiers (i.e. w = 3), denoted by h1, h2, andh3, where h1 assigns class number “+1” to any of the sixdescriptors, classifier h2 assigns class number “-1” to any of the sixdescriptors, and classifier h3 assigns class number “+1” to x1 tox3, and class number “-1” to x4 to x6.

21 / 25

6 7 8 9 10

10.3

Exercise

Let S = {1, 2, 3, 4, 5, 6}, X and Y are random variables defined onS, with X = 1 if the number is even, and Y = 1 if the number isprime (i.e. 2, 3, or 5). Let the values of joint or conditionalprobabilities p(x , y) and p(y |x) be defined as follows:

p(x , y) = P(X = x ,Y = y)

p(y |x) = P(X = x |Y = y)

Give values for all possible combinations, such as for p(0, 0) orp(0|1).

22 / 25

6 7 8 9 10

10.4

Exercise

Consider a finite alphabet S = {a1, . . . , am} and two differentrandom variables X and Y taking values from S, withpj = P(X = aj) and qj = P(Y = aj). The relative entropy ofdiscrete probability p with respect to discrete probability q is thendefined as

H(p|q) = −m∑j=1

pj · log2

pj

qj

Show that

1 H(p|q) ≥ 0

2 There are cases where H(p|q) 6= H(q|p.

3 H(p|q) = 0 iff p = q (i.e. pj = qj , for j = 1, . . . ,m).

23 / 25

6 7 8 9 10

10.5–10.6

Exercise

Let X be a random variable that takes values in the alphabetS = {A,B,C ,D,E}.Calculate the Huffman codes (not explained in this book; checkother sources if needed) for the following two probabilitydistributions:

1 Let X takes values A, B, C , D, or E with uniform probability1/5.

2 Now consider probabilities P(X = A) = 1/2,P(X = B) = 1/4, P(X = C ) = 1/8, P(X = D) = 1/16, andP(X = E ) = 1/16.

Exercise

Verify that H(Y |c) = 0 in the given example for illustratingconditional entropy.

24 / 25

6 7 8 9 10

Copyright Information

This slide show was prepared by Reinhard Klettewith kind permission from Springer Science+Business Media B.V.

The slide show can be used freely for presentations.However, all the material is copyrighted.

R. Klette. Concise Computer Vision.c©Springer-Verlag, London, 2014.

In case of citation: just cite the book, that’s fine.

25 / 25