Prove that both 2-D continuous and discrete Fourier transforms are linear operations.
The linearity property of the 2-D continuous and discrete Fourier transforms can be proven through mathematical demonstrations.
1. 2-D Continuous Fourier Transform (CFT):
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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 7 × 7 arithmetic mean filter?
iii) A 9 × 9 arithmetic mean filter?
Consider a 3 × 3 spatial mask that averages the four closet neighbours of a point 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.
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 |
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.
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
Suppose that an image )y,x(f is convolved with a mask of size n × n (with cofficients 1/n2 ) to produce a smoothed image ).y,x(f (5) 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
and
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.
Apply Prewitt operators and Sobel operators for the image given below: