Description

HANDWRITTEN TASKS.
∞
(1) Evaluate the sum: S X+ nxn−1,|x| < ,x ∈ R.
= 1 n

(2) Evaluate the sum: S X a|n| z−n,|a| < ,z ∈ C. Indicate the region of
= 1 n −∞
= convergence of the sum.
(3) Consider a cartesian coordinate system with orthogonal x and y axes. The axes are rotated counterclockwise by an angle of θ to obtain a transformed coordinate system. Derive a mapping between the points in the original coordinate system and those in the transformed coordinate system. Is the mapping linear? Is it invertible? Justify your answer. (4) Consider a unit cube in R3, subjected to a × transformation A. Compute the
3 3
volume of the transformed cube in each of the following cases:

1 a A 
( ) =  7
4 2
9
7 
3

9 
6 b
( ) 
1
A 
=  2
3 1
2
3 
1

2 
3

1 c A 
( ) =  4 2
5 
3
 6  d
( ) 
2
A 
=  1 2
3 − 
1
 0 
−
5 7 9
(5) Given n orthonormal column vectors ai,…,n, evaluate
n
S = Xai aTi ,
i=1
where T denotes the transpose operation.
1
PROGRAMMING TASKS.
(1) Write a Matlab/Python script to implement the following input-output operation on an input image (f denotes an 8-bit greyscale input, g denotes the output,
< τ ≤ τ < ):
0 1 2 255

 f(x,y), if 0 < τ1 ≤ f(x,y) ≤ τ2 < 255,
g(x,y) = τ1, if f(x,y) ,
 τ , if f x,y.
2 ( )
Sketch the input-output characteristics. Also display the input image as well as the output images on greyscale. The program should provide a slider control to select τ and τ respecting < τ ≤ τ < , and dynamically update the
1 2 0 1 2 255
output.
(2) Write a Matlab/Python script that accepts two greyscale images f and f2 as the inputs and gives the output g αf − α f , where. Display the
= 1 + (1 ) 2
input images as well as the output. The program should provide a slider control to select α on a scale of to and dynamically update the output.
0 1
(3) Write a Matlab/Python script to accept a greyscale image as input and generate a
3-D mesh plot with the pixel coordinates on the x and y axes and the greyscale intensity on the z axis.
(4) Write a Matlab/Python script to accept two greyscale images f1 and f2 as the input and generate the output g as follows:
(
, if f x,y < f x,y , g x,y 0 1( ) 2( )
( ) = , if f x,y ≥ f x,y .
255 1( ) 2( )
(5) Write a Matlab/Python script to generate and display a 2-D sinusoid f(m,n) = π . m π . n φ m,n , ,…, . The program should pro-
cos(2 0 2 + 2 0 3 + ); = 1 2 256 vide a slider control to select φ ∈ , π and dynamically update the display.
[0 2 )