# UNIVERSITY OF OSLO. Faculty of Mathematics and Natural Sciences

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2 Exercise 1.Using 1D projection histograms a) How are 1D projection histograms computed? Explain either using formulas or by words if you do not remember exact formulaes. 1D horizontal projection of the region: p h( x) f ( x, y) y 1D vertical projection of the region: p v ( y) f ( x, y) x b) You are given images like the music sheet below. How can you use 1D projection histograms to determine if the page is rotated? (Not to find the rotation angle, just IF it is rotated)? Discuss shortly how you could do this. Consider a projection onto the vertical axis. If the page is not rotated, you will see distict peaks in the histogram at the locations of the horisontal note lines. If the page is rotated, the peaks will not be distinct, more of a gradual increase/decrease. c) 1D projection histograms can also be useful for splitting connected objects. Consider the segmented symbols below. Describe briefly how you could use 1D projection histograms to split the two symbols that are merged in the image below. There are several ways to do this. One alternative is to use projection onto the horisontal axis. The gaps between the numbers will be mimimum values of the histograms. Minimum values are then good candidates for splitting locations, maube combined with other criteria like location of the split. 2

3 Exercise 2. Shape descriptors a) How can you compute the width and height of an object in a segmented binary image? We have presented several alternatives, list the alternatives shortly. Examples are bounding box (object-oriented would be better), major axis/minor axis, thickness from skeleton, dimensions of a fitted ellipse etc. Assume that we have a binary image f(x,y). A general discrete moment is defined as p q m pq x y f ( x, y) x y b) What is the definition of central moments? Central moments are defined as p q ( x x) ( y y) f ( x, y) pq x y c) What is the main advantage of using central moments? Position invariance. d) How could second order moments be useful for discriminating between the two symbols 0 and 1? For example using µ20 the spread along the x-axis. Exercise 3: Hough transform You are given two point is (x,y)-space: (1,1) and (0,2). a) Give the equations for the (a,b) representation of these two points. (1,1): b=1-a (0,2): b=2 b) Plot them in (a,b)-space. c) Explain the algorithm for finding lines in the polar representation ((,θ)- space). 3

4 Exercise 4. Texture a) Describe in short how to calculate a GLCM for an entire image. Explain the necessary concepts, using a sketch if you find it useful. Stikkord som må med for full score her er glidende vindu, inputparametre til algoritmen (retning og avstand), samt skisse av selve algoritmen. Kvantisering spørres om under, så det gir ikke poeng. Diskusjon om kanteffekter kan trekke opp. b) From an unknown texture sample you calculate a couple of GLCM features for three different step lengths and the four directions 0, 45, 90, 135. Looking at the output you notice that most features are nearly identical for all step lengths for the 0 direction, but wildly varying between step lengths for the other directions. What does this tell you about the texture? Teksturen er horisontal c) When analyzing texture it is often useful to quantize the graylevels in the image. Explain why. Comment on one problem with this approach. Kvantisering reduserer størrelsen på GLCM bl.a. Ved for grov kvantisering vasker en ut lavkontrast tekstur. d) For the following texture a coocurrence matrix is calculated (ignore problems with the edge pixels, i.e., do not evaluate coocurrences across the edges) j i The coocurrence matrix given h=(i,j) is 4

5 j i What is h? 1. (0,1) 2. (1,0) 3. (1,1) 4. (1,-1) 5. None of the above Exercise 5. Classification using Bayes rule pixel class channel 1 channel 2 channel 3 1 ω ω ω ω MEAN ω VAR ω ω ω ω ω MEAN ω VAR ω ω ω ω ω MEAN ω VAR ω ω 2 ω 3 ω 1 Figur 1: Training image For the multispectral image (3 channels) above, we have the training data picked from the indicated regions as given in the table Your task is to classify the pixels in the image below into class ω1 (background), ω2 (leaf), and ω3 (apple ) based on the information from the image in Figur 1. You are only allowed to use two channels. 5

6 a) Evaluate by visual inspection (using 2D feature spaces) which two channels are the most suited. a) 1 &2 eller 2&3 er begge brukende, men 2&3 er best for naturlig valg av klassevise avstandsmål data1 data2 data data1 data2 data data1 data2 data3 Feature Feature Feature Feature Feature Feature 2 b) b) Based on this you are designing a classifier using Bayes rule. The data is modeled with gaussian distributions having the same covariance matrix Σ 1 0. Write the 0 1 generic form of the discriminant function. 2 tilordnes til klasse j if, evt er direkte avstand til mean også akseptabelt 1 c) Classify the pixels below using the functions as in (b). Note that just giving the class for the pixels will not give any score even if correct. Whether you answer the classification problem graphically or numerically is equivalent. for alle tre kanaler pixel channel 1 channel 2 channel

7 ved bruk av 1 & : class 3 as : class 2 as : class 1 as : class 2 as ved bruk av 2 & : class 3 as : class 2 as : class 1 as : class 2 as Remember that, when assuming a 2D Gaussian distribution, the point probability of a point x=[x 1, x 2 ] T can be written on the form T x x1 1 x f exp 1/ 2 1/ 2 x2 (2 ) 2 x2 2 x2 2 where the probability distribution function takes as parameters the vector µ=[µ 1, µ 2 ] T and a matrix. Exercise 6. K-nearest neighbor classification In the following questions you will consider a k-nearest neighbor classifier using Euclidean distance metric on a binary classification task. We assign the class of the test point to be the class of the majority of the k nearest neighbors. Note that a point can be its own neighbor. 7

8 Figur 2: k-nn classification task, a plus indicates a positive example and a star a negative example. a) In the Figur 2, sketch the 1-nearest neighbor decision boundary for this dataset. b) How would the point (8,1) be classified using 1-nn? 8

9 a) b) Point (8,1) will be classified to class * 9

10 Candidate number: 10

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