1

Digital Image Processing

Midterm Exam

November 9, (Tue.), 2010

Name: _______________________________________

Student ID: ___________________________________

Email Address: ________________________________

Notes:

1. Exam duration: 150 minutes (from 2:20pm – 4:50pm.)

2. Open-book exam (books, lecture notes, graded homework,

etc.)

3. One-line calculator.

4. Show all answers on sheets.

Problem Weight Score

1 7

2 15

3 17

4 13

5 20

6 14

7 14

Total 100

2

Problem1 (7 points)

(True or false) You will get -1 points for each wrong answer as penalty

(a) We can reconstruct the original image from its skeletonizing image since the

skeleton is preserved.

(b) The kernel [-1 2 -1] is meant to approximate first order derivative.

(c) Rotate P(x,y) by an angle θ clockwise with respect to the reference point R(u,v).

The resulting position is:

P’( cosθ(x-u)-sinθ(y-v)+u, sinθ(x-u)+cosθ(y-v)+v ).

(e) There are 4 connected components in the figure below. (4-connected rule for

object and 8-connected rule for background)

(f) Mean-square-error is a good criterion for measuring the quality of halftoned image.

(g) K-means algorithm is a supervised classification method.

(h) More features will lead to a better classification result.

3

Problem2 (15 points)

Suppose that the gray scale is of range [0, 9] instead of [0, 255].

(a) (2 point) Please plot the histogram of the following 8x8 image.

0 5 7 7 5 8 7 8

7 2 6 2 6 5 6 8

6 9 7 7 0 7 2 7

6 6 1 7 6 7 7 5

9 6 0 7 8 2 6 7

2 8 8 2 7 6 7 8

7 3 2 6 1 7 5 8

9 9 5 6 7 7 7 7

(b) (5 points) Please perform histogram equalization of the above image according to

the formula:

T�x� � round � cdf�x� � cdf���N � N � cdf���� � �L � 1��, � x: pixel value N: image size L: max gray scale value* Please output the resultant image and its corresponding histogram.

(c) (2 points) What will happen if we apply histogram equalization to the result of (b)

again?

(d) (2 points) Can histogram equalization always provide a better result? State your

reasons.

(e) (4 points) “Histogram matching” is a useful contrast manipulation technique

which transforms an image’s histogram to match the one of another image.

Please describe clearly how you achieve it.

4

(Answer)

5

Problem3 (17 points)

Fig. 3-1 Fig3-2

(a) (12 points) Please write down how you wrap Fig. 3-1 to Fig. 3-2 explicitly.

The flower shape can be described by the formula

r � 16 , 240 cos�4θ� , 0 r � 1x2 , y2 θ � tan45�y/x�* (b) (3 point) While finding the corresponding coordinates, we may choose forward

treatment or backward treatment. Is forward treatment better than backward

one? Explain your reasons.

(c) (2 point) Can we perfectly wrap Fig. 3-2 back to Fig. 3-1 since the exact wrapping

function is given? Why or why not?

6

(Answer)

7

Problem4 (13 points)

(a) (2 points) Plot the gradient curve of the following 1-D signal.

(b) (2 points) Explain the meanings of gradient obtained from a 2D image in terms of

magnitude and orientation.

(c) (2 points) Convolve the following 4x4 array with the Sobel kernel that detects

horizontal edges. (Use even boundary extension)

33 62 55 73

21 77 65 150

10 29 212 198

17 34 83 142

(d) (3 points) Compare the first order and second order edge detection methods.

(e) (2 points) For median filter, what’s the difference between large kernel size and

small one? Which one outperforms the other?

(f) (2 points) Given an image with impulse noise, Alice applies MINMAX after

MAXMIN while Bob adopts MAXMIN after MINMAX. Would they get the same

results? Why or why not?

8

(Answer)

(a)

1-D signal

Your

answer

Problem 5 (20 points)

(a) (10 points) Please apply three morphological operation

skeletonizing, to the following images

state and plot the results

(b) (5 points) Apply mask A to the

show the result after one iteration.

(a)

(c) (5 points) Suppose B is a binary image and J, K

as follows. Please explain

where 7 is the erosion operator and

lease apply three morphological operations, shrinking, thinning, and

, to the following images (Fig. 5-1) until reaching the convergent

and plot the results in the provided answer sheet.

Fig. 5-1

Apply mask A to the following image to implement dilation filter and

show the result after one iteration.

(a) Input binary image (b) Mask A

Fig. 5-2

B is a binary image and J, K are two different kernels specified

Please explain the purpose of this operation: �B 7is the erosion operator and B9 is the complement of B.

J

K

9

, shrinking, thinning, and

) until reaching the convergent

image to implement dilation filter and

Mask A

nt kernels specified

7 J�;�B9 < K�

is the complement of B.

(Answer)

(a)

Shrinking

Thinning

Skeletonizing

10

(Answer)

(b)

(c)

11

12

Problem 6 (14 points)

(a) (4 points) Given the following data points, please perform k-means algorithm to

classify these data points to two clusters with two initial centroids, (-0.5,-0.5) and

(0,2). To simplify the computation, please use 1-norm (>x>5 � ∑ |x�|��A5 ) instead

of usually use 2-norm to compute distance between points.

A B C D E F G H I J

(0,0) (0,1) (1,0) (-1,0) (0,-1) (2,1) (2,2) (3,1) (3,2) (5,5)

(b) (4 points) Will the classification result be different if we randomly choose these

two initial centroids?

(c) (3 points) Design and explain a method to choose initial centroids in order to get

more reasonable result efficiently.

(d) (3 points) What will happen if k becomes 3 in this case?

13

(Answer)

Problem 7 (14 points)

(a) (2 points) The Discrete Cosine Transform (DCT) is a widely used transform. The

definition of DCT is

2),(

CvuF =

where i, u = 0, 1, …, M

are determined by

What is the value of F(0, 0) if the image f(

10 20

90 100

160 150

80 70

10 20

90 100

160 150

80 70

(b) (4 points) F(0,0) is also known a

from Part (a), please explain what the meaning of DC coefficient

called a “DC value”.

(c) (8 points) Fig. 7-1 shows 8x8

transform, please show

the pattern in the position (0,2) below.

The Discrete Cosine Transform (DCT) is a widely used transform. The

2

)12(cos

2

)12(cos

)()( 1

0

1

0

fN

vj

M

ui

MN

vCuC M

i

N

j

ππ ++∑∑

−

=

−

=

, M-1 and j, v = 0, 1, …, N-1, and the constants C(u) and C(v)

==

otherwiseC

,1

0,2

2)( ξξ .

hat is the value of F(0, 0) if the image f(i, j) is given as:

20 30 40 50 60 70 80

100 110 120 130 140 150 160

150 140 130 120 110 100 90

70 60 50 40 30 20 10

20 30 40 50 60 70 80

100 110 120 130 140 150 160

150 140 130 120 110 100 90

70 60 50 40 30 20 10

F(0,0) is also known as a DC coefficient. By the computation experience

(a), please explain what the meaning of DC coefficient

shows 8x8 2-D DCT basis functions. By definition of DCT

transform, please show in a mathematical way that F(0,2) is actually related to

the pattern in the position (0,2) below.

Fig. 7-1: 2-D DCT basis functions

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The Discrete Cosine Transform (DCT) is a widely used transform. The

),( jif

1, and the constants C(u) and C(v)

By the computation experience

is and why it is

. By definition of DCT

actually related to

15

(Answer)