Expression 7: "G" left parenthesis, "x" , right parenthesis equals StartFraction, 1 Over StartRoot, left parenthesis, 2 pi , right parenthesis , EndRoot "s" , EndFraction exponent left parenthesis, negative 1 half left parenthesis, StartFraction, "x" Over "s" , EndFraction , right parenthesis squared , right parenthesisGx=12πsexp−12xs2
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Expression 8: "I" left parenthesis, "x" Subscript, 1 , Baseline , right parenthesis equals StartFraction, 1 Over 2 "r" , EndFraction Start integral from "x" Subscript, 1 , Baseline minus "r" to "x" Subscript, 1 , Baseline plus "r" , end integral, "G" left parenthesis, "x" , right parenthesis "d" "x"Ix1=12r∫x1+rx1−rGxdx
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opacity for a ray at x
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Expression 10: "O" Subscript, "g" , Baseline left parenthesis, "x" , right parenthesis equals 1. minus exponent left parenthesis, negative "k" "G" left parenthesis, "x" , right parenthesis , right parenthesisOgx=1.−exp−kGx
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opacity for a pixel of radius r
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Expression 12: "O" Subscript, "i" , Baseline left parenthesis, "x" Subscript, 1 , Baseline , right parenthesis equals StartFraction, 1 Over 2 "r" , EndFraction Start integral from "x" Subscript, 1 , Baseline minus "r" to "x" Subscript, 1 , Baseline plus "r" , end integral, "O" Subscript, "g" , Baseline left parenthesis, "x" , right parenthesis "d" "x"Oix1=12r∫x1+rx1−rOgxdx
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Expression 13: "y" equals .9 left brace, abs left parenthesis, "x" , right parenthesis less than or equal to "r" , right bracey=.9absx≤r