CIS 663 Biometrics

Assignment 2

**This assignment is due by the week 7 live session. If you make any assumptions, clearly state them in your answer. **

- The following represents a 10 x 10-pixel grayscale. 0 represents black and 255 represents white.

0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |

0 | 0 | 3 | 3 | 3 | 3 | 3 | 2 | 1 | 1 |

0 | 0 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 |

0 | 0 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 |

0 | 0 | 0 | 1 | 1 | 3 | 4 | 4 | 4 | 4 |

0 | 0 | 0 | 0 | 0 | 4 | 4 | 4 | 1 | 0 |

5 | 5 | 0 | 0 | 0 | 4 | 4 | 4 | 0 | 0 |

5 | 5 | 0 | 0 | 0 | 4 | 4 | 4 | 0 | 0 |

5 | 5 | 0 | 0 | 0 | 0 | 5 | 5 | 0 | 0 |

5 | 5 | 0 | 0 | 0 | 0 | 5 | 5 | 0 | 0 |

- Convert the image to an integral image. (10pt)
- Using the integral image, compute the sum of area from (2,2) to (5,7), shaded red above. Show your steps. (10pt)
- Using the grayscale image from Question 1, apply the following Haar filter to all positions that are feasible. (20pts)

- In Viola-Jones face detection algorithm, explain what cascading is and why it is important. (20pt)
- (20pts) Consider the following labeled data (x, y) ∈ R2 (i is the example index):

i | x | y | Label |

1 | 11 | 3 | – |

2 | 10 | 1 | – |

3 | 4 | 4 | – |

4 | 12 | 10 | + |

5 | 2 | 4 | – |

6 | 10 | 5 | + |

7 | 8 | 8 | – |

8 | 6 | 5 | + |

9 | 7 | 7 | + |

10 | 7 | 8 | + |

In this problem, you will use Adaboost to learn a hidden function from this set of training examples. We will use two rounds of AdaBoost to learn a hypothesis for this data set. In round number t, AdaBoost chooses a weak learner that minimizes the weighted error(t). As weak learners, you will use axis parallel lines of the form

(a) Label + if x > a, else – or

(b) Label + if y > b, else -, for some integers a, b (either one of the two forms, not a disjunction of the two).

- a) The first step of AdaBoost is to create an initial data training data weight distribution D1. What are the initial weights given to data points with index 4 and 7 by the AdaBoost algorithm,

respectively?

- b) Which is the hypothesis h1 that minimizes the weighted error in the first round of AdaBoost, using the distribution D1 computed in the above question?
- c) What is the weighted error of h1 computed above?
- d) After computing h1 in the previous questions, we proceed to round 2 of AdaBoost. We begin by recomputing data weights depending on the error of h1 and whether a point was

(mis)classified by h1. What are the weights given to data points with index 4 and 7 according to the distribution after round 1, D2, respectively?

- e) Which is the hypothesis h2 that minimizes the weighted error in the second round of AdaBoost, using the distribution D2

computed in the above question?

- f) What is the weight assigned to the hypothesis of round 2, h2
- g) Now that we have completed two rounds of AdaBoost, it is time to create the final output hypothesis. What is the final weighted hypothesis after two rounds of AdaBoost?

**Formulas: **

** **

** **** **Where ei = 0 if input *i *is classified correctly and 1 if classified incorrectly.

- What is Principle Component Analysis and how does it relate to face recognition? (20pts)