midterm_fa06 - f linear Is it shift invariant(b Writing...

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CSE166 – Image Processing – Midterm Instructor: Prof. Serge Belongie 11:00am-12:20pm Tuesday Oct. 31, 2006. On this exam you are allowed to use a calculator and one 8.5” by 11” sheet of notes. The total number of points possible is 30. In order to get full credit you must show all your work . Good luck! 1. (7 pts) Consider the 1D continuous kernel h ( x ) = e - x 2 / 2 σ 2 . (a) Give the name of this kernel. (b) What type of filter is h ( x ): lowpass, bandpass, or highpass? (c) Sketch h ( x ) and its Fourier transform H ( u ) for σ = 0 . 5. Repeat for σ = 2. Label the axes in each of your four plots. (d) Suppose we want to convert this filter into a real, even-symmetric bandpass filter with a passband centered around u = ± u o . What would we multiply h ( x ) by in the spatial domain to achieve this effect? What is the name of the resulting filter? 2. (7 pts) Suppose you are given an image f ( x, y ) and you produce a new image g ( x, y ) by subtracting f ( x, y ) from a copy of itself shifted two pixels to the right. (a) Is this operation on
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Unformatted text preview: f linear? Is it shift invariant? (b) Writing this operation in the form g = f * h , what choice of h will produce the desired g ? (c) Let f represent an image of a white disk on a black background. Sketch f and k g k . 3. (8 pts) Let W=dftmtx(4) . (a) Explain in words what each column of W represents. (b) Write down the result of the operation (1/4)*W*W’ . What property of W does this illus-trate? 4. (8 pts) Recall that the chi-squared distance between a pair of K-bin histograms h i ( k ) and h j ( k ) is given by: χ 2 ( i, j ) = 1 2 K X k =1 [ h i ( k )-h j ( k )] 2 h i ( k ) + h j ( k ) (a) For the use of this distance to be valid, what two conditions must each histogram satisfy? (b) What is the chi-squared distance between two identical histograms? Show that χ 2 ( i, j ) cannot be smaller than this value. (c) What is the largest possible chi-squared distance between two histograms? Show that χ 2 ( i, j ) cannot exceed this value. 1...
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