Exam text content

Week exam 1

1. General questions

(a) What is a Gaussian filter and where it can be applied?

(b) What is the benefit of using homogenous coordinates in the case of pinhole camera
model?

(c) How Fourier transform could be used in calculating the linear filtering result?

(d) What is a Gaussian image pyramid?

2. Transformations

(a) A perspective camera has the following camera matrix:

toid
P=/|0 11 2
0011

Determine the image points corresponding to 3D point X = (6,2,2). Report your answer in
non-homogenous coordinates.

(b) Write the matrix equations for 3D similarity, affine, and perspective transformations. Use
homogenous coordinates. How many degrees of freedom each transform has and how
many point correspondences are needed to estimate them?

3. Homogenous coordinates

(a) Convert the following (in-homogenous) points into homogenous coordinates (1, 5), (100,
500), and (4, 4, 1). Similarly, convert the following homogenous points into corresponding
in-homogenous form (i.e. to normal coordinates) (1,5,1), (7,1,3), (24,12,6) and (8,6,1,2).
What does a homogenous point (1, 1, 0) correspond to?

(b) A line ax + by + c = 0 can be presented in a vector form as |=(a,b,c)" and, using '
homogenous coordinates, a point x is on the line | if x™l = 0. The intersection of two lines | i
and I’ is given by the vector cross product between | and |’. Similarly, the line | passing
though points x and x’ is given by the vector cross product between x and x’. Use
homogenous coordinates and above formulas to determine the intersection of lines 11 and
12. The |1 runs through points (2,4) and (8,8), and I2 runs through points (14,10) and (18,6).

Hint: The 3D vector cross product is calculated as:

a,) (| | a,b,—b,4, ‘
a, |x) 5, |=| 4,2, — 8,4, i

a,) \b,) \a,b, -b,a,

 

Week exam 2
1. General questions

(a) What is the main goal in image retrieval task?

(b) What do hyperparemeters mean in image classification (give one example)?

(c) What is k-nearest neighbour classifier? What are its pros and cons?

(d) Give one example how pertained classification network can be used in image retrieval?
2. Neural networks

(a) What is a Perceptron? Explain the construction (hint: use picture) and how it can be trained to
perform classification task (assume you have training samples with input feature vector x and
class label 1 or -1).

(b) In Figure 1 below you see a very small neural network, which has one input unit, one hidden
unit (logistic), and one output unit (linear). The nonlinear function o in the logistic unit is defined
by the formula o(z) = 1/(1 + e-2). Let’s consider one training case. For that training case, the
input value is 1 (as shown in the figure) and the target output value t is 2. We are using the
standard squared loss function: E = (t - y)#/2, where y is the output of the network. The values
of the weights and biases are shown in the figure and they have been constructed in such a
way that you don’t need a calculator. Hint: the derivative of logistic function is defined as d/dx
6(x) = 6(x)(1-0(x)). Answer the following questions:

i. What is the output of the hidden unit and the output unit, for this training case?
ii. What is the loss, for this training case?
iii. What is the derivative of the loss with respect to w2, for this training case?

|
|
|
bias= Linear output unit
w= +4
bias= +2-—-»| Logistic hidden unit
wi=-2
Figure 1: A simple neural network -
Input unit

3. Image retrieval

| (a) Describe the bag-of-visual-words image representation technique. How it can be utilised in
image retrieval?

(b) Figure 3 (below) illustrates a database of four images and corresponding visual words for each
image (W1, W2, ...). Construct an inverted index for this example dataset.

(c) We have a database of 10 images. Our retrieval algorithm has ranked them in the following
order with respect to a given query (see query and ranked database in Figure 2 below). Based
on the manual annotations, we know that the images with green box are relevant to the
current query. Draw a precision-recall curve for the retrieval result (use the axis given in Fig 2).

 

   
   
   
   
     

Database images Corresponding
visual words

=> wi wi wa

Dataset size: 10 images
Relevant (total): 5 images

    

Precision = #relevant / #returned
Recall = #relevant / #total relevant

 

 

Figure 2

 

 

Results (ordered)

 

=| a> ws wi we

 

 

 

Figure 3
Week exam 3

1. General questions

(a) What is the goal in object category detection and how it differs from image classification
and object segmentation?

(b) Name the main components in the sliding window based object detector.

(c) What is bootstrapping and how it can be used in training detectors?

(d) What is the difference between one and two stage CNN object detectors?

2. Classical object detectors

(a) Describe different phases in extracting Histogram of Oriented Gradients (HoG). Use picture.

(b) The following image (Figure 1) depicts an example detection result. The blue boxes are the
know ground truth locations of the objects and the red boxes are the obtained detection. The
number next to each detection denotes the corresponding ranking (i.e. detection 1 has the
highest classification score, detection 2 next highest, and so on.). The corresponding
intersection over union values are: 1) 0.9, 2) 0.57, 3) 0, and 4) 0.49 (i.e. the loU measure for
each detection with respect to the highest overlapping ground truth). Draw the corresponding
precision-recall curve using 0.5 loU value as a detection threshold.

Hint:
Precision = #returned correct detections / #returned detections
Recall = #detected objects / #total number of objects

 

 

 

Figure 1
3. CNN based detectors

(a) Explain the main phases in the “CornerNet” object detection approach.

(b) The following image (Figure 2) depicts the Faster-RCNN object detector. Shortly describe the
objective of each component in the system (i.e. what it takes in and what it aims to produce as

  

an output).
ROI classifier and regressor
Proposals
Feature map
Base network
Figure 2
Week exam 4

1. General questions

(a) What are the main stages in Canny edge detector?

(b) Outline the cost function that is minimised when fitting a line with least squares method.
(no need to solve it)

(c) Why it is usually beneficial to sample minimal subset of data points in RANSAC instead of
using more data points?

(d) What is the main motivation in using “robust cost functions” in model fitting instead of
normal quadratic function used in vanilla Least Square fitting?

2. Local features

(a) Figure 1 illustrates three different kinds of local image areas (the box). For each case, explain if
it makes a good local keypoint or not. Justify your answer. (local key point = an image point
that can be accurately and reliably detected from multiple images from the same scene).

 

 

 

Figure 1

(b) Describe how scale normalised Laplacian of Gaussian function (see figure 2) can be used in
scale covariant blob detection.

og &
Hs ,Og
ox” oy

2 _ 2
Vom =O

 

Boe ee ee

 

Figure 2. Scale normalised Laplacian of Gaussian
3. Robust model fitting
(a) Describe the main stages of the RANSAC algorithm in the general case.

(b) What is the idea in Hough transform and how it can be used in model fitting? Give one
example.

Week exam 5

1. General questions

(a) What is the brightness constraint in optical flow estimation?

(b) What is so called aperture problem?

(c) What are motion field and optical flow? What is the main difference?
(d) What kind of features are good for tracking and why?

2. 2D transformations

(a) Figure 1 depicts two images of database objects and a scene where they need to be
detected. Describe the main steps how this kind of object instance recognition task can be
solved using local features and image alignment. For each step, explain the main goal and
name at least one method to implement this.

Aaya

i

 

Figure 1
 

(b) Figure 2 depicts two images taken from the same scene. Describe the main steps how these
images can be aligned to form a panorama image shown in Figure 3. For each step, explain
the main goal and name at least one method to implement this.

  

Figure 1: image pair from the same scene Figure 2: panorama image

3. Optical flow and tracking
(a) Assume we have two frames obtained at time instants (t-1) and t as shown in Figure 3. In

optical flow, our target is to estimate the motion (u,v) of a pixel at position (x,y). Starting from
the brightness constraint, derive the optical flow equation:

VI -(u,v) +I, =0

How many unknown this equation has per pixel? Hint: I(x +u(x, y), V+V(x, y),t)
= I(x, y,t)+1,u(x, y) +I, v(x, y)

  
 

(ey)
\ displacement = (u,v)

°
(e+u,y+v)

Ia)

Figure 3. 1(x,y,t) denotes the brightness of a pixel at position (x,y) at time instant 1.

 

(b) Explain the multi-resolution approach for optical flow estimation. What are the main
advantages of the approach?

Week exam 6

1. General questions

(a) Why the recovery of the scene structure from a single image is an ill-posed problem?
(b) What does auto calibration mean in the context of camera calibration?

(c) What is the relation between depth and disparity in stereo vision?

(d) What is the main difference between essential and fundamental matrices?
2. Camera calibration and single view metrology

(a) Briefly explain the “linear method” for camera calibration? What are the pros and cons of this
approach?

(b) Figure 2 illustrates a scenario, where we are trying to estimate the height H (distance between
top T and bottom B) from a single image using the known reference height R. We have
detected the image points t, r, and b that correspond to 3D points T, R, B, respectively. The
image point vz is the vanishing point in the vertical direction. Show how the height H can be
obtained using points t, r, b, and . Hint: use the cross ration of four points defined as

1 P;—Pil [P, — Pal
ha -P,| IP, ~ P\|

    
 

T (top of object)

R (reference point)

Figure 1: Estimating the height H from a single image using reference height R.

Vue

3. Epipolar geometry and stereo

(a) What is epipolar line and how it relates observations in two cameras? How Essential
and Fundamental matrices are related to these?

(b) Figure 2 presents a stereo system with two parallel pinhole cameras separated by a
baseline b so that the centers of the cameras are cj = (0,0,0) and c; =(b,0,0). Both
cameras have the same focal length f. The point P is located in front of the cameras
and its disparity d is the distance between corresponding image points, i.e., d=|xi - xr.
Assume that d = 4 cm, b = 12 cm, and f = 2 cm. Compute Zp.

P= (Xp,Yp,Zp)

Figure 2: Top view of a stereo pair where
two pinhole cameras are placed side by side.

Week exam 7

 

1. General questions

(a) What is the projective ambiguity in the context of Structure from Motion.

(b) What are the main differences between multi-view stereo and Structure from Motion?
(c) What is bundle adjustment and why it is important in Structure from Motion?

(d) What is inverse depth and why it is used in some multi-view stereo applications?
2. Structure from Motion

(a) We know that all images used in Structure from Motion (SfM) are captured by a single moving
camera. How this information can be used to “upgrade” projective SfM solution? Give rough
idea how can be done (no need to solve).

(b) You are given m images of n fixed 3D points

l.e. you have m cameras, n 3D points, and each point is detected in every camera (see
illustration in Figure 1). The task is to estimate m projection matrices Pj and n 3D points Xj
from mn correspondences xj (up to projective transformation). Explain the main steps in
solving this problem in sequential manner.

Aux, = PIX, L=1,..,m, fal ...n

 

 

 

 

 

Figure 1: Illustration of the setup in Structure from Motion problem in the case of 3 cameras.

3. Multi-view geometry

(a) Explain the main principle in epipolar geometry based multi-view stereo reconstruction
approach.

(b) What is space carving method and how it works in multi-view stereo reconstruction?