Tetra

The questions below are due on Thursday May 07, 2026; 02:00:00 PM.
 
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Let f[r,c] represent the following image as an array of pixel brightnesses in the range [-{1\over2},{1\over2}] where black corresponds to -1/2, white corresponds to +1/2, and the background grey corresponds to 0. The pixels are indexed by their row number r (with -{1\over2}R\le r\lt{1\over2}R) and column number c (with -{1\over2}C\le c\lt{1\over2}C), and R=C=200.

Consider eight images that are derived from f[r,c] -- four of which are defined by spatial functions h_1[r,c] and h_2[r,c]:

h_1[r,c]=\sin\left({40\pi r\over R}\right)
h_2[r,c]=\cases{1&if $r$ is even\cr0&otherwise\cr}
and four of which are defined by frequency functions H_3[k_r,k_c] and H_4[k_r,k_c]:
H_3[k_r, k_c] = j\sin\left({40\pi k_r\over R}\right)
H_4[k_r, k_c] =\cases{1&if $k_r$ is even\cr0&otherwise\cr}

Determine the images that are reprsented by each of the following expressions (1-8):

Expression 1:  (f\times h_1)[r,c]

Expression 2:  (f h_1)[r,c]

Expression 3:  (f\times h_2)[r,c]

Expression 4:  (f h_2)[r,c]

Expression 5:  (f\times h_3)[r,c]

Expression 6:  (f h_3)[r,c]

Expression 7:  (f\times h_4)[r,c]

Expression 8:  (f h_4)[r,c]

Locate the corresponding image from the panels below (A-L). Briefly describe your reasoning for each part.

    A     B     C
    D     E     F
    G     H     I
    J     K     L

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