29th August 1999

Maximum Marks 30 ............................... 2 hours

* Solve the following problems:

1. When are the D4 and D8 distances equal. Draw the trace of the pixels having equal D4 and D8 distances from a point.

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   Figure 1
   

2. Fig. 1 shows an image with two different kinds of objects in it, object A and object B. Describe a method, that can be automated, for (a) identifying these in the image and (b) counting how many object A's and how many object B's are there in the image. (The background has gray level value 255 and the object pixels have gray level value 0.)

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3. Given that any two tristimulus coordinate systems based on different sets of primary sources are linearly related give a method for obtaining the transformation matrix to convert from one system to the other. More specifically, let X_k and Y_k, k=1, 2, 3, be two sets of primary sources, in the X coordinate system w_k denotes the amount of X_k^th primary required to match a given colour and in the Y coordinate system w'_k denotes the amount of Y_k^th primary required to match the same. Give a procedure for getting the transformation matrix A to get the values of w'_k given w_k.

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4. The eigenvalues of an eight-channel multispectral image are [6.1 168 0.08 13 64 214 1.2 0.2]. What will be the RMS error if you use K-L Transform for 2:1 data compression?

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5. Show that the following property of separable sequences is valid

   x(n₁,n₂)=x(n₁)x(n₂) w X(k₁,k₂)=X(k₁)X(k₂)

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6. For a uniform picture, the entire (D)FT is concentrated into a single point, the (0,0) location in the frequency plane.

   (a) What would be the effect of applying a low pass filter to such an image? Why?

   (b) What is the physical significance of this in terms of the Fourier domain representation of low frequency spatial domain variations?

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7. Consider the vector x and an orthogonal transform A
   Let a₀ and a₁ denote the columns of A^T (that is, the basis vectors of A). The transformation y=Ax can be written as y₀=a₀^Tx, y₁=a₁^Tx. Represent the vector x in Cartesian coordinates on a plane. Show that the transform A is a rotation of the coordinates by theta degrees and y₀ and y₁ are the projections of x in the new coordinate system (see Fig. 2).
   Figure 2
   

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8. Figure 3 shows a scene with background gray-level of 0 and contains 3 rectangles, A, B, C with gray-levels 64, 64, and 192, respectively. A and C are moving in x-direction with velocity v. The output of the camera is connected to an RGB monitor. Design a system so that vertical edges of rectangles A and B are displayed in red and blue respectively. Horizontal edges of C should be displayed in green. You may assume that A and C move by one pixel between successive frames. The rest of the scene should be displayed as dark. Specify all parameters used in the system. You may assume that both rectangles are completely contained in the field-of-view of the camera, and A and B do not overlap as A moves. You may neglect the corner effects.
   Figure 3

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9. Figure 4 shows a vertical edge at x=0. Intensities on two sides of the edge are 0 and 100. Plot the response r of x-directional Sobel operator around x=0. Specify the values of the response at x=-3,-2,-1,0,1,2,3. If you apply the x-directional Sobel operator to the response image r, what is the final response R? Specify a mask that generates R directly.
   Figure 4

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10. Specify n and eta for a Butterworth Low-pass filter so that |Hlow(u,v)|=0.25 for D/D0=1, and |Hlow(u,v)|=0.9287 for D/D0=0.4. Using this low-pass filter, adder(s) and multiplier(s), build a high-pass filter so that |Hhigh(u,v)|=0.5 at D/D0=1. At D/D0=0.4, what is the gain of the resulting high-pass filter?

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11. Figure 5 shows the histogram of an image. Plot the histogram of the negative of the image, given by 255-gray. What is the probability of (a) gray-level >=192, (b) gray-level<64, and (c) 64<= gray-level <128 occurring in the original image? Design a transformation function so that P(gray=0) = 1/16, P(gray=255) = 1/16, and P(gray=128) = 7/8 in the transformed image?
    Figure 5
    

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