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Problems, INF 269, Chapter 2 This set of problems is intended for Wednesday September 4, 2002. It starts with a set of theoretical problems, with the possibility to discuss the various consequences of the “theorems” shown, and to perform some interactive demonstrations. Then comes a programming problem, also with possibilities for discussions and demo. • Theoretical problems 1) Show that if f (x, y) has a Fourier transform F (u, v), then |ab|1/2 f (ax, by) has a Fourier transform |ab|−1/2 F ua , vb . Discuss the consequences of this. 2) Show that if f (x, y) has a Fourier transform F (u, v), then f (x − a, y − b) has a Fourier transform F (u, v)e−2πi(au+bv) . Discuss the consequences of this. 3) Use the convolution theorem to obtain an expression for the convolution of two 1D zero-mean Gaussians √ (x)2 1 − (x)22 1 e− 2a2 ∗ √ e 2b 2πa 2πb and similarly for the 2D circular symmetric case 1 − x2 +y2 2 1 − x2 +y2 2 2a e ∗ e 2b 2πa2 2πb2 • Practical programming exercise: (this will not count as a mandatory exercise) 4) ∗ Write a program that creates an N ×N image where the pixel value increases linearly from f (x, y) = ∆ in the lower left hand corner to f (x, y) = 255 − ∆ in the upper right hand corner. ∗ Implement Algorithm 2.3 on page 36 of the textbook to add zero-mean Gaussian noise to the image f , thus creating a new image g. Do this for a couple of values of the standard deviation σ. What limitations does ∆ put on σ ? ∗ Plot the distribution (histogram) of ε = f (x, y) − g(x, y) for the chosen values of σ. ∗ Compute the actual mean and standard deviation of these distributions/histograms, and explain if these eventually deviate from a “perfect” zero-mean Gaussian. ∗ How can you use local first order statistics to find whether the noise in an image is additive or multiplicative? See you on September 4! 1