midterm_fa11

# midterm_fa11 - (b What is the special name for this...

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CSE166 – Image Processing – Midterm Instructor: Prof. Serge Belongie http://www-cse.ucsd.edu/classes/fa11/cse166-a 12:00-12:50pm Friday Oct. 28, 2011. 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. (10 pts) In this problem, b N ( x ) denotes the 1D binomial kernel of length N . (a) Write down a one-line Matlab command to calculate b N ( x ) for arbitrary N . (b) What is b 5 ( x )? (c) Suppose you want to ﬁlter a 2D image with a 9 × 9 binomial kernel, but the only kernel you have available is b 5 ( x ). Explain how to do this. Use sketches and/or pseudocode to illustrate your answer. (d) What continuous function h σ ( x ) does b N ( x ) approach as N → ∞ ? Give the relationship between N and σ . 2. (4 pts) Consider the 3-tap ﬁltering operation g ( x ) = 1 4 [ f ( x - 1) - 2 f ( x ) + f ( x + 1)]. (a) Expressing this as a convolution g ( x ) = f ( x ) * h ( x ), what are the coeﬃcients of h ( x
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Unformatted text preview: )? (b) What is the special name for this ﬁltering operation? (c) You can achieve the same result via iterated convolution with a 2-tap ﬁlter. What is that ﬁlter? 3. (6 pts) Consider the M × N image f ( x,y ) = (-1) x for x = 0 ,...,N-1 and y = 0 ,...,M-1. (a) Describe what f ( x,y ) looks like as an image. (b) Put f ( x,y ) into the form e j 2 π u o · x , where x = ( x,y ) > and u o = ( u o ,v o ) > . (c) What is the Fourier transform F ( u,v )? Write it down for the case of M = N = 4. 4. (10 pts) Consider the 1D continuous kernel h ( x ) = e-x 2 / 2 σ 2 . (a) Give the name of this kernel and write the expression for its Fourier transform H ( u ). (b) Sketch h ( x ) and H ( u ) for two cases: large σ and small σ . (c) Calculate g ( x ) = dh ( x ) dx and its Fourier transform G ( u ). (d) Sketch g ( x ) and G ( u ) for two cases: large σ and small σ ....
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