Prove that both 2-D continuous and discrete Fourier transforms are linear operations.
The Fourier transform, both in its continuous and discrete forms, is a linear operation, which means it satisfies two properties: additivity and homogeneity (scalar multiplication).
Additivity:
Let \( f_1(x) \) and \( f_2(x) \) be two functions, and let their Fourier transforms be \( F_1(k) \) and \( F_2(k) \), respectively. The Fourier _________ __________ ___ ____ _________ __________.
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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)?
Write an expression for 2-D continuous convolution.
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.
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.
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.
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]
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:
| Symbol | Probability |
| 1 | 0.4 |
| 2 | 0.3 |
| 3 | 0.1 |
| 4 | 0.1 |
| 5 | 0.1 |