The vertex-covering number of an odd cycle is 1 more than its independence number.

K₄, as a plane graph, is self-dual.

Gr¨otzsch graph is Eulerian.

For some nonnegative integers d₁, d₂, . . . , d_n the sequence (max{d₁, n}, d_2, d₃, . . . , d_n) is graphic.

Every 2-connected bipartite graph is Hamiltonian.

Every tree has a perfect matching.

Construct a 4-directional and 8-directional chain code for the shape of the alphabet ‘C’. Hence, obtain its difference code and shape number.

Two images g (x, y) 1 and g (x, y) 2 have histograms 1 hg and hg , 2 respectively. Give the condition under which the histograms of the following can be determined:

i) g₁ (x, y) g₂ (x, y)
 ii) g₁ (x, y) g₂ (x, y) 
 iii) g₁ (x, y) g₂ (x, y) 
 iv) g₁ (x, y) g₂ (x, y)

Hence, obtain the histogram in each case in terms of hg1 and hg . 2

Find the entropy of the image

Compute the degree of compression that can be achieved using (i) Huffman coding, and (ii) Run length coding, assuming two-bits to represent the pixel value and twobits to represent the run length.

What is Histogram Equalization? Why is it performed? Perform histogram equalization of the following image:

Define the following intensity transformation functions, with suitable examples: 
 i) Log Transformations
 ii) Power-Law (Gamma) Transformations. 

Image segmentation.

MPP Algorithm

Digital Watermarking and its Applications

Principal Component Analysis

Let the salt and pepper noise have the following pdf:

Obtain the mean and variance of this distribution.

Given a four-symbol source {a, b, c, } with source probabilities {0.1, 0.4, 0.3, 0.2}, arithmetically encode the sequence b b a d c.

Calculate the following for the data given below:

i) Entropy

ii) Coding redundancy of Binary code

iii) Coding redundancy of Huffman code

[Given, log 0.05= -4.32] 

The following pattern classes have Gaussian probability functions:

w :{(1, 1)^T , (3, 1)^T , (3, 3)^T , (1,3)^T}  and

w :{(5, 5)^T , (7, 5)^T , (7, 7)^T , (5, 7)^T }

Assuming that ,P(w₁ ) P(w₂) = obtain the equation of the Bayes’ decision boundary between these two classes.