text: We consider [1] the pixels as small rectangles. This supposition is adequate if the light sensitive elementary devices (the CCD or CMOS cells) of the digitizing device, (a scanner or a digital camera) have rectangular apertures and the spaces between the apertures are negligible. Then the digitizing device assigns to a pixel a grey value proportional to the portion of the area of the corresponding rectangle covered by the light area.
  The areas of the intersection SA (small area) and GA (big area) are defined in each of the cases by different functions of t and alpha.
  The boundary of a binarized image (the red solid line) with the sub-pixel points SP labeled by blue crosses. The threshold of the binarization lies in the middle between the maximum and minimum gray value.
  Definition of the sub-pixel boundary SPB: The sub-pixel boundary of a region R in a gray value image (with no shading) is a polygon whose vertices are the sub-pixel points. An edge of the polygon connects each two sub-pixel points corresponding to subsequent boundary cracks.
  3. Compute the curvature at each sub-pixel point with a great chord (about 1/10 of the number of cracks in a closed curve) and store the results.
  4. Read the points anew, estimate the second derivative of the curvature with respect to the arc length (as an estimate of F4 in (3) of the lecture "Curvature of curves"), choose for each point the optimal chord length: optim Δx = (48*ε/F4)1/4 and recompute the curvature by means of the optimal chord.
  3. In a digital image the uncertainty of the coordinates is about 1 pixel and this amount strongly influences the derivatives.
  To measure the curvature of the boundary of an approximately homogeneous subset S of a 2D image we suggest to use the gray values at the boundary of S to improve the precision of estimating the coordinates of the points of the boundary. We suggest a method of estimating the coordinates with a precision of about 1/100 of a pixel. This gives us the possibility to estimate the curvature with a higher precision.