Sub-pixel precision using 1D parabola fitting

parabola.lua

-- Use three (consecutive) points to fit a 1D parabola
-- and return its maximum value. To achieve sub-pixel
-- precision for a 2D maxima (x,y), just fit the parabola over the
-- x and y coordinates separately with two neighboring points.
local function fitParabola(x1,x2,x3,y1,y2,y3)
  local x1_sqr = x1*x1
  local x2_sqr = x2*x2
  local x3_sqr = x3*x3
  
  local div = (x1_sqr-x1*(x2+x3)+x2*x3)*(x2-x3)
  local a = (x1*(y2-y3)-x2*(y1-y3)+x3*(y1-y2))/div
  local b = (x1_sqr*(y2-y3)-x2_sqr*(y1-y3)+x3_sqr*(y1-y2))/div
  
  return b/(2*a)
end