I am reading the paper

"A Method for Extraction of Bronchus Regions from 3D Chest X-ray CT Images by Analyzing Structural Features of the Bronchus" by Takayuki KITASAKA, Kensaku MORI, Jun-ichi HASEGAWA and Jun-ichiro TORIWAKI

and I run into a term I do not understand:


In equation (2), when we say "[] expresses the Gauss sign", what does it mean?

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    $\begingroup$ You may want to add the title of the paper, and its bibliographic information... $\endgroup$ – Arturo Magidin Feb 14 '12 at 5:12
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    $\begingroup$ Notation $\endgroup$ – Peđa Terzić Feb 14 '12 at 5:18
  • $\begingroup$ @pedja: seriously? Is that it? Gauss sign is just floor and ceiling functions? But then it must be either floor or ceiling, isn't it? Or is it a round function, where if the number is < 0.5, floor function is applied and vice versa? $\endgroup$ – Karl Feb 14 '12 at 5:25
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    $\begingroup$ @J.M. Well, voxel coordinates I imagine would need integer-valued arguments, so I think the last line would be evidence this is the NINT function... $\endgroup$ – anon Feb 14 '12 at 5:47
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    $\begingroup$ @J.M. I think pedja was just specifying what he meant. If there's good reason to believe this is a rounding function and these brackets are the rounding function attributed to Gauss with the same exact notation, then I think we have a match here. $\endgroup$ – anon Feb 14 '12 at 6:03

From the context (a change of scale using discrete units), this should certainly mean floor as on page 5 of Gauss's Werke 2

per signum $[x]$ exprimemus integrum ipsa $x$ proxime minorem, ita ut $x-[x]$ semper fiat quantitas positiva intra limites $0$ et $1$ sita

i.e. the next lower integer.

  • $\begingroup$ From Jeff Miller's Earliest Uses of Function Symbols: Greatest integer function (floor function). Until recently [x] has been the standard symbol for the greatest integer function. According to Grinstein (1970), "The use of the bracket notation, which has led some authors to term this the bracket function, stems back to the work of Gauss (1808) in number theory. The function is also referred to by Legendre who used the now obsolete notation E(x)." The Gauss reference is to Theorematis arithmetici demonstratio nova. Werke Volume: Bd. 2 p. 5. $\endgroup$ – Math Gems Feb 14 '12 at 15:37

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