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Unformatted text preview: CS221 Exercise Set #6 1 CS 221 Exercise Set #6 1. Convolution (a) Suppose f,g are two functions with bounded support; i.e., there exist integer constants l f ,u f ,l g ,u g such that f [ i ] = 0 unless l f i u f and g [ i ] = 0 unless l g i u g . Give bounds on the support of h = f * g , that is, numbers l h ,u h such that h [ n ] = 0 unless l h n u h . (b) Let f [ i ] = 2 i for i { , 1 , 2 , 3 , 4 } ; f [ i ] = 0, otherwise. Let g [ i ] =  i  for i { 2 , 1 , , 1 , 2 } ; g [ i ] = 0, otherwise. What is the convolution h = f * g ? (c) Let k [ i ] be the kernel [0 . 5 , . 5]. Let f [ i ] = 2 i . What is the convolution k * f ? Show associativity of convolution explicitly in the case k * ( k * f ) = ( k * k ) * f . 2. Properties of the convolution operator In this problem, we will investigate some properties of the convolution operator. For simplicitly, we consider just the 1D convolution, defined for input image X and kernel K as: Y [ n ] = ( X * K )[ n ] = i X [ i ] K [ n i ]. (a) Prove that convolution operator is commutative: X * K = K * X . (b) The averaging (smoothing) filter that we studied in class can perform poorly when the image is very noisy. In this case, even a single bad pixel can add noise to the averageimage is very noisy....
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 '09
 KOLLER,NG

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