Write an algorithm to find the first two largest numbers in an in array of integers. For example, given the input ,2 ,3 ,7 − ,4 ,1,5 the algorithm should output ,7 .5 State precisely a loop invariant for you algorithm. Prove that your loop invariant holds and hence conclude that your algorithm works.

) Given an image f(x,y) with Fourier transform F(u,v) obtain the fourier transform of .
c) Given that the 2-D Fourier Tranform is real and even, obtain the constraints on the image characteristics.

Define the following with suitable examples
i) Unsharp marking

ii) High boost filtering

iii) High frequency filterin

 Filter the given  gray level image with

i)  mean filter using zero padding

ii) 3×3 weighted mean filter using zero padding with mask

iii) 3×3 median filter processing only such pixels that have all the needed neighbours.
iv) Laplacian filter with the given mask and reflecting the border pixels.

 Describe the homographic filtering. Explain why the filtering scheme is effective for the applications it is used.

Given that

where f and g are real images and h is a spatial filter.

Obtain G(u,v), in terms of F(u,v), and H(u,v), the 2–D Fourier transform of g(x,y).

 A bullet is 2.5 cm long, 1cm wide and its range of speed is . The bullet is flight is captured by a camera that exposes the scene for k sec and the bullet occupies 10% of the horizontal resolution of  frames. Find
i) Automatic segmentation of the bullet.

ii) Automatic determination of speed of the bullet.

 Perform histogram equalization for the following histogram.

 Apply the perception algorithm to the following pattern classes:
,

.
Let  and 
Also, Sketch the decision surface.

Suppose a low-pass spatial filter is formed by averaging the four immediate neighbours of a point (x, y) but excluding the point itself. Find the equivalent filter H(u,v) in the frequency domain.

 In image restoration, how are the noise parameters estimated?
b) Assume that the noise is estimated as exponential, with mean  Variance . How will you estimate the parameter 'a' of pdf of exponential Noise?

Explain in detail the adaptive mean and median filters.
b) Obtain mean and variance of the following noise pdfs:
i) 
ii) 
iii) 

Given an image with uniform histogram. Explain the effect of applying following compression techniques:
i) Huffman,

ii) Golomb,

iii) LZW,

iv) Prediction coding, and

v) Optimal Quantization

Explain the functioning of an adaptive, local noise reduction filter.

) Propose a gray level slicing algorithm capable of producing the 4-bit plane of an 8-bit monochrome image.

 Explain the Hough transform for edge linking with suitable example.

Show that Sobel masks can be implemented by one pass of differencing mask of the form [−1 0 1](or its vertical counterpart) followed by a smoothing mask of the form [1 2 1] (or its vertical counterpart).

 Verify Lagrange's mean value theorem for the function f defined by  over 

If , show that:

Hence find the value of .

Solve the equation:

given that its roots are in G.P.
b) Evaluate: