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US20070106719(Google Patent) (USPTO: 20070106719)
"The present invention relates generally to digital image processing. More specifically, the present invention relates to an algorithm for determining a precise integer square root applicable in image processing functions. The square root is a mathematical operation that is useful in color science and image manipulation algorithms. For instance, the square root may be useful in calculating the Euclidean distance between any two points in a two or three dimensional space. Distance calculations between pixels may be used in a variety of image processing functions, including for example, red eye reduction, resizing, noise suppression, and other filtering operations. As an example, for a two dimensional image represented by pixels having a range of intensities and unique coordinates, the distance between the pixels may be determined as the square root of the sum of the squared differences between coordinate values. In equation form, distance D may be represented by: 
D=√{square root over ((x ₂ −x ₁)²+(y ₂ −y ₁)²)}　　(1) 
where the x values are respective x coordinates of two pixels and the y values are respective y coordinates of the same two pixels."

US6963895(Google Patent)
"When X_i is assembled in this manner, the corresponding values of R_S are as listed in Table 1 below.

 

TABLE 1
Eodd	Cout	R	RS	Xi
1	0	(2√2 − 1, 2.0)	(2√2 − 1)/8, 0.25	2.0, √2 + 0.5
0	1 or 0	[1.0, 2.0)	0.25, 0.5	√2 + 0.5, √2
1	1	[1.0, 2√2 − 1]	0.5, (2√2 − 1)/2	√2, 1.0

 

For each row, the (R_S, X_i) pairs define a three slope, piece-wise linear approximation to 1/√{square root over (R_S)}. The R_S values are derived directly from R by right shifts of 3, 2 and 1 respectively."