Prove that, for a zero-memory source with q symbols, the maximum value of the entropy is log q, which is achieved if and only if all source symbols are equiprobable. [Hint: Consider the quantity log q-H(z) and note the inequality In x ≤x-1]

Consider an 8-pixel line of intensity data, (108,139,135,244,172,173,56,99). If it is uniformly quantized with 4-bit accuracy, compute the rms error and rms signal-to- noise ratios for the quantized data.

The white bars in the test pattern shown are 7 pixels wide and 210 pixels high. The separation between bars is 17 pixels. What would this image look like after application of

i) A 3×3 arithmetic mean filter?

ii) A 7x7 arithmetic mean filter?

iii) A 9×9 arithmetic mean filter?

Consider a 3x3 spatial mask that averages the four closet neighbours of a point

(x, y), but excludes the point itself from the average.

i) Find the equivalent filter, H (u, v), in the frequency domain.

ii) Show that your result is a lowpass filter.

Prove that both 2-D continuous and discrete Fourier transforms are linear operations.

Consider the two image subsets, S₁ and S2, shown in the following figure. For V = {1}, determine whether these two subsets are (i) 4-adjacent, (ii) 8-adjacent, or

(iii) m-adjacent.

An automobile manufacturer is automating the placement of certain components on the bumpers of a limited-edition line of sports cars. The components are colour coordinated, so the robots need to know the colour of each car in order to select the appropriate bumper component. Models come in only four colours: blue, green, red, and white. Find a solution based on imaging and determine the colour of each car, keeping in mind that cost is the most important consideration.

Write an expression for 2-D continuous convolution.

Two images, f (x,y) and g (x,y), have histograms he and hg. Give the condition under which you can determine the histograms of

i) f(x,y)+g(x, y)

ii) f(x,y)-g(x,y)

iii) f(x, y)xg(x, y)

iv) f(x,y)+g(x,y)

Apply Prewitt operators and Sobel operators for the image given below:

What is Discrete Fourier Transform (DFT)? Find DFT of the function:

What do you understand by the term "Entropy" in context of any digital image? Calculate the entropy for the symbols, where probability distribution is given below:

Define the terms 'Sampling' and 'Quantization' in context of digital image processing. A medical image has size 8x8 inches, the sampling reduction is 5 cycles/mm, calculate the number of pixels required for the medical image.

Consider a linear, position-invariant image degradation system with impulse response

Supose that the input to the system is an image cosnsiting of a line of infinitesimal width located at x = a, and modeled by f(x, y) = 8(x-a), where 8 is an impulse. Assuming no noise, what is the output image g(x, y)?

Find an expression for the signature of each of the following boundaries, and plot the signatures.

i) An equilateral triangle

ii) A rectangle

iii) An ellipse

Explain how the MPP algorithm behaves under the following conditions: b

i) 1-pixel wide, 1-pixel deep indentations.

ii) 1-pixel wide, 2-or- more pixel deep indentations.

iii) 1-pixel wide, 1-pixel longprotrusions.

iv) 1-pixel wide, n-pixel long protrusions.

Suppose that an image f(x, y) is convolved with a mask of size n x n (with cofficients 1/n²) to produce a smoothed image .

i) Derive an expression for edge strength (edge magnitude) of the smoothed image as a function of mask size. Assume for simplicity that n is odd and that edges are obtained using the partial derivatives

ii) Show that the ratio of the maximum edge strength of the smoothed image to the maximum edge strength of the orginal is 1/n. In other words, edge strength is inversely proportional to the size of the smoothing mask.

A binary image contains straight lines oriented horizontally, vertically, at 45°, and at - 45°. Give a set of 3 x 3 masks that can be used to detect 1-pixel breaks in these lines. Assume that the intensities of the lines and background are 1 and 0, respectively.

The arithmetic decoding process is the reverse of the encoding procedure. Decode the message 0.23355 given the coding model.

Prove the following result:

Suppose A is a non-zero compact self-adjoint operator on a Hilbert space H over K. Prove that there exists a finite set {r₁,r₂,..., r_n} of a non-zero real numbers with  and an orthonormal set {w₁, w₂,..., w_n} in H such that

Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?