## Digital Signal Processing - Dec 2012

### Computer Engineering (Semester 7)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1(a)** Prove that 2D Fourier Transform Matrix is Unitary matrix.(5 marks)
**1(b)** Derive 8D Laplacian filter Mask (3 X 3)(5 marks)
**1(c)** Derive matrix representation of 1D Walsh transform for N=4 from forward Walsh transform function.(5 marks)
**1(d)** State fidelity objective and subjective criteria of Image evaluation.(5 marks)
**2(a)** Let X(k)={ 1, -2, 1-j, 2j, 0, _, _, _}is the 8 point DFT of the real valued sequence x(n)

i) What is 8 point DFT P(k) such that p(n)=(-1)^{n} x(n)?

ii) What is 8 point DFT Q(k) such that q(n)=(-1)^{n-4} x(n-4)?(6 marks)
**2(b)** Derive the equation of contrast stretching transformation function as given in figure below. Apply the contrast stretching transformation function on the input image F and obtain the output image.

(6 marks)
**2(c)** Given

(i) Find 3 bit IGS coded image and calculate compression factor and bits per pixel BPP.

(ii) Find decoded image and calculate MSE and PSNR.(8 marks)
**3(a)** Using FFT, find the response of the system to the input x(n) when

(6 marks)
**3(b)** Given

Determine the output image using power law transformation s=(r)^{2}(6 marks)
**3(c)** Segment the following given image such that the difference between maximum intensity value and minimum intensity value in segmented region is less than 18 using split and merge technique.

(8 marks)
**4(a)** Let x(n) be four point sequence with X(k)={1,2,3,4}. Find the DFT of the following sequence using X(k)

(6 marks)
**4(b)** F is given as follows:

i) If the gray level intensity changes are to be made as shown in the figure below, derive the necessary expression for obtaining the new pixel value using slope.

ii) Obtain the new image by applying the above mentioned transformation function.

iii) Plot the histogram of input and output image.

iv) Compare the histogram of input and output image.(8 marks)
**4(c)** Given

Apply the following filter mask W1,W2,W3 on the input image F and obtain the output image.(6 marks)
**5(a)** Find the response of the system to the input x(n) given h(n) using Z-transform

(6 marks)
**5(b)** Explain Trimmed average filter, find the trimmed average value of the input image F at the centre position for R=2 and S=1 where R is the number of consecutive pixels to be trimmed from the min extreme and S is the number of consecutive pixels to be trimmed from max extreme.

(6 marks)
**5(c)** Given

(i) Find the Huffman coded image of the following encoder:

(ii) Calculate Bits per pixel (BPP) and percentage of compression of compressed image. Do not consider the payload of Huffman table.(8 marks)
**6(a)** Given

(i) What are the frequencies in radians in the resulting DT signal x(n).

ii) If x(n) is passed through an ideal interpolator then what is the reconstructed signal.(6 marks)
**6(b)** Applying Horizontal and Vertical line detection mask on the following image F. Use appropriate threshold value. Assume virtual Rows and Column by repeating border pixel values.

(6 marks)
**6(c)** Assume that edge in the gray level image starts in the first row and end in the last row. Find the cost of all possible edges using the following cost function.Cost (p,q)=Imax-|f(p)-f(q)|Where Imax is the maximum intensity value in the image and f(p) and f(q)are pixel value at point p and q resp. Find the edge with minimum value of cost.Plot the graph

(8 marks)
**7(a)** How to find the inverse one dimensional DFT using forward DITFFT flowgarph?(5 marks)
**7(b)** Derive High Boost filter mask (3x3)(5 marks)
**7(c)** Bit Reversal technique in FFT.(5 marks)
**7(d)** Image Enhancement using LOG Transformation and power law transformation.(5 marks)